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MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS TENTH EDITION

Walter Nicholson Amherst College

Christopher Snyder Dartmouth College

Microeconomic Theory Basic Principles and Extensions Tenth Edition Walter Nicholson Christopher Snyder VP/Editorial Director: Jack W. Calhoun Editor-in-Chief: Alex von Rosenberg Executive Editor: Mike Roche Sr. Developmental Editor: Susan Smart Sr. Content Project Manager: Cliff Kallemeyn

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To Beth, Sarah, David, Sophia, and Abby

To Maura

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About the Authors

Walter Nicholson is the Ward H. Patton Professor of Economics at Amherst College. He received his B.A. in mathematics from Williams College and his Ph.D. in economics from MIT. Professor Nicholson’s principal research interests are in the econometric analyses of labor market problems including unemployment, job training, and the impact of international trade. He is also the co-author (with Chris Snyder) of Intermediate Microeconomics and Its Application, Tenth Edition (Thomson/South-Western, 2007). Professor Nicholson and his wife, Susan, live in Amherst, Massachusetts, and Naples, Florida. What was previously a very busy household, with four children everywhere, is now rather empty. But an ever-increasing number of grandchildren breathe some life into these places whenever they visit, which seems far too seldom. Christopher M. Snyder is a Professor of Economics at Dartmouth College. He received his B.A. in economics and mathematics from Fordham University and his Ph.D. in economics from MIT. Before coming to Dartmouth in 2005, he taught at George Washington University for over a decade, and he has been a visiting professor at the University of Chicago and MIT. He is currently President of the Industrial Organization Society and Associate Editor of the International Journal of Industrial Organization and Review of Industrial Organization. His research covers various theoretical and empirical topics in industrial organization, contract theory, and law and economics. Professor Snyder and his wife Maura Doyle (who also teaches economics at Dartmouth) live within walking distance of campus in Hanover, New Hampshire, with their three elementary-school-aged daughters.

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Brief Contents

Part 1

Introduction Chapter 1: Economic Models Chapter 2: Mathematics for Microeconomics

Part 2

Choice and Demand Chapter Chapter Chapter Chapter Chapter Chapter

Part 3

3: 4: 5: 6: 7: 8:

Preferences and Utility Utility Maximization and Choice Income and Subtitution Effects Demand Relationships among Goods Uncertainty and Information Strategy and Game Theory

491 521

Pricing in Input Markets Chapter 16: Labor Markets Chapter 17: Capital and Time

Part 7

391 441

Market Power Chapter 14: Monopoly Chapter 15: Imperfect Competition

Part 6

295 323 358

Competitive Markets Chapter 12: The Partial Equilibrium Competitive Model Chapter 13: General Equilibrium and Welfare

Part 5

87 113 141 182 202 236

Production and Supply Chapter 9: Production Functions Chapter 10: Cost Functions Chapter 11: Proﬁt Maximization

Part 4

3 19

573 595

Market Failure Chapter 18: Asymmetric Information Chapter 19: Externalities and Public Goods

627 670

ix

x

Brief Contents

Brief Answers to Queries

701

Solutions to Odd-Numbered Problems

711

Glossary of Frequently Used Terms

721

Index

727

Contents

Preface

xix

PART 1

INTRODUCTION

CHAPTER 1 Economic Models

1

3

Theoretical Models 3 Veriﬁcation of Economic Models

3

General Features of Economic Models

5

Development of the Economic Theory of Value Modern Developments 16 Summary 17 Suggestions for Further Reading

18

CHAPTER 2 Mathematics for Microeconomics

19

Maximization of a Function of One Variable Functions of Several Variables

8

19

23

Maximization of Functions of Several Variables Implicit Functions 31

28

The Envelope Theorem 32 Constrained Maximization 36 Envelope Theorem in Constrained Maximization Problems

42

Inequality Constraints 43 Second-Order Conditions 45 Homogeneous Functions Integration 56

53

Dynamic Optimization

60

Mathematical Statistics Summary 74

64

Problems 75 Suggestions for Further Readings

79

Extensions: Second-Order Conditions and Matrix Algebra

81 xi

xii

Contents

PART 2

CHOICE AND DEMAND

CHAPTER 3 Preferences and Utility

87

Axioms of Rational Choice Utility 88 Trades and Substitution

87 91

A Mathematical Derivation 97 Utility Functions for Speciﬁc Preferences The Many-Good Case Summary Problems

85

100

104

105 106

Suggestions for Further Reading 109 Extensions: Special Preferences 110

CHAPTER 4 Utility Maximization and Choice An Initial Survey

113

114

The Two-Good Case: A Graphical Analysis The n-Good Case 118 Indirect Utility Function

114

124

The Lump Sum Principle 125 Expenditure Minimization 127 Properties of Expenditure Functions Summary 132 Problems

130

132

Suggestions for Further Reading Extensions: Budget Shares 137

136

CHAPTER 5 Income and Substitution Effects Demand Functions

141

141

Changes in Income 143 Changes in a Good’s Price

144

The Individual’s Demand Curve

148

Compensated Demand Curves 151 A Mathematical Development of Response to Price Changes Demand Elasticities 158 Consumer Surplus 165 Revealed Preference and the Substitution Effect Summary Problems

172 173

169

155

Contents

Suggestions for Further Reading

176

Extensions: Demand Concepts and the Evaluation of Price Indices

CHAPTER 6 Demand Relationships Among Goods The Two-Good Case

182

182

Substitutes and Complements 184 Net Substitutes and Complements 186 Substitutability with Many Goods Composite Commodities 188

188

Home Production, Attributes of Goods, and Implicit Prices Summary Problems

191

195 195

Suggestions for Further Reading 199 Extensions: Simplifying Demand and Two-Stage Budgeting

CHAPTER 7 Uncertainty and Information Mathematical Statistics

202

202

Fair Games and the Expected Utility Hypothesis The von Neumann–Morgenstern Theorem 205 Risk Aversion 207 Measuring Risk Aversion The Portfolio Problem

203

209 214

The State-Preference Approach to Choice under Uncertainty The Economics of Information 221 Properties of Information The Value of Information Flexibility and Option Value Asymmetry of Information Summary 226

221 222 224 225

Problems 226 Suggestions for Further Reading

231

Extensions: Portfolios of Many Risk Assets

CHAPTER 8 Strategy and Game Theory Basic Concepts

200

236

Prisoners’ Dilemma 237 Nash Equilibrium 240

236

232

216

178

xiii

xiv

Contents

Mixed Strategies Existence

247

251

Continuum of Actions 252 Sequential Games 255 Repeated Games 259 Incomplete Information

268

Simultaneous Bayesian Games

268

Signaling Games 273 Experimental Games 281 Evolutionary Games and Learning Summary 283 Problems

282

284

Suggestions for Further Reading 287 Extensions: Existence of Nash Equilibrium

PART 3

288

PRODUCTION AND SUPPLY

CHAPTER 9 Production Functions

295

Marginal Productivity 295 Isoquant Maps and the Rate of Technical Substitution Returns to Scale

293

298

302

The Elasticity of Substitution 305 Four Simple Production Functions 306 Technical Progress Summary 315 Problems

311

315

Suggestions for Further Reading 319 Extensions: Many-Input Production Functions

CHAPTER 10 Cost Functions Deﬁnitions of Cost

320

323 323

Cost-Minimizing Input Choices Cost Functions 330

325

Cost Functions and Shifts in Cost Curves 334 Shephard’s Lemma and the Elasticity of Substitution Short-Run, Long-Run Distinction Summary Problems

350 351

344

344

Contents

Suggestions for Further Reading

354

Extensions: The Translog Cost Function

CHAPTER 11 Proﬁt Maximization

358

The Nature and Behavior of Firms Proﬁt Maximization 359 Marginal Revenue

355

358

361

Short-Run Supply by a Price-Taking Firm Proﬁt Functions 369 Proﬁt Maximization and Input Demand Summary 380 Problems

365 374

381

Suggestions for Further Reading 385 Extensions: Applications of the Proﬁt Function

PART 4

386

COMPETITIVE MARKETS

389

CHAPTER 12 The Partial Equilibrium Competitive Model Market Demand

391

391

Timing of the Supply Response 395 Pricing in the Very Short Run 395 Short-Run Price Determination

396

Shifts in Supply and Demand Curves: A Graphical Analysis Mathematical Model of Market Equilibrium 403 Long-Run Analysis 406 Long-Run Equilibrium: Constant Cost Case Shape of the Long-Run Supply Curve

407

410

Long-Run Elasticity of Supply 412 Comparative Statics Analysis of Long-Run Equilibrium Producer Surplus in the Long Run 416 Economic Efﬁciency and Welfare Analysis Price Controls and Shortages

419

422

Tax Incidence Analysis 423 Trade Restrictions 427 Summary Problems

431 432

Suggestions for Further Reading

401

436

Extensions: Demand Aggregation and Estimation

438

413

xv

xvi

Contents

CHAPTER 13 General Equilibrium and Welfare Perfectly Competitive Price System

441

441

A Simple Graphical Model of General Equilibrium with Two Goods Comparative Statics Analysis 451 General Equilibrium Modeling and Factor Prices Existence of General Equilibrium Prices 455 General Equilibrium Models

442

453

462

Welfare Economics 466 Efﬁciency in Output Mix 469 Competitive Prices and Efﬁciency: The First Theorem of Welfare Economics Departing from the Competitive Assumptions 475 Distribution and the Second Theorem of Welfare Economics Summary Problems

Suggestions for Further Reading 486 Extensions: Computable General Equilibrium Models

PART 5

476

481 482

MARKET POWER

487

489

CHAPTER 14 Monopoly 491 Barriers to Entry

491

Proﬁt Maximization and Output Choice

493

Monopoly and Resource Allocation 497 Monopoly, Product Quality, and Durability

501

Price Discrimination 503 Second-Degree Price Discrimination through Price Schedules Regulation of Monopoly

510

Dynamic Views of Monopoly Summary 513

513

Problems 514 Suggestions for Further Reading

518

Extensions: Optimal Linear Two-part Tariffs

CHAPTER 15 Imperfect Competition

521

Short-Run Decisions: Pricing and Output Bertrand Model

523

519

521

508

471

Contents

Cournot Model

524

Capacity Constraints

531

Product Differentiation Tacit Collusion 537

531

Longer-Run Decisions: Investment, Entry, and Exit Strategic Entry Deterrence 547 Signaling

551

How Many Firms Enter? Innovation 558 Summary Problems

554

560 561

Suggestions for Further Reading

565

Extensions: Strategic Substitutes and Complements

PART 6

541

566

PRICING IN INPUT MARKETS

CHAPTER 16 Labor Markets

571

Allocation of Time

573 573

A Mathematical Analysis of Labor Supply

576

Market Supply Curve for Labor 580 Labor Market Equilibrium 581 Monopsony in the Labor Market Labor Unions 586 Summary

584

589

Problems 590 Suggestions for Further Reading

CHAPTER 17 Capital and Time

594

595

Capital and the Rate of Return Determining the Rate of Return

595 597

The Firm’s Demand for Capital 604 Present Discounted Value Approach to Investment Decisions Natural Resource Pricing Summary 614 Problems

611

614

Suggestions for Further Reading 618 Appendix: The Mathematics of Compound Interest

619

606

xvii

xviii

Contents

PART 7

MARKET FAILURE

CHAPTER 18 Asymmetric Information

625

627

Complex Contracts as a Response to Asymmetric Information Principal-Agent Model

629

Hidden Actions 630 Owner-Manager Relationship Moral Hazard in Insurance Hidden Types 642 Nonlinear Pricing

632 637

642

Adverse Selection in Insurance Market Signaling 657 Auctions Summary

659 663

Problems

663

650

Suggestions for Further Reading 666 Extensions: Nonlinear Pricing with a Continuum of Types

CHAPTER 19 Externalities and Public Goods Deﬁning Externalities

670

670

Externalities and Allocative Inefﬁciency 672 Solutions to the Externality Problem 675 Attributes of Public Goods

679

Public Goods and Resource Allocation 680 Lindahl Pricing of Public Goods 684 Voting and Resource Allocation A Simple Political Model 690 Voting Mechanisms Summary Problems

687

692

694 694

Suggestions for Further Reading Extensions: Pollution Abatement

Brief Answers to Queries

698 699

701

Solutions to Odd-Numbered Problems Glossary of Frequently Used Terms Index

727

627

711 721

667

Preface

The 10th edition of Microeconomic Theory: Basic Principles and Extensions represents both a continuation of a highly successful treatment of microeconomics at a relatively advanced level and a major change from the past. This change, of course, is that Chris Snyder has joined me as a co-author. His insights have improved all sections of the book, especially with respect to its coverage of game theory, industrial organization, and models of imperfect information. Hence in many ways this is a new book, although on matters of style and pedagogy it retains much of what has made it successful for more than 35 years. This basic approach is to focus on building intuition about economic models while providing students with the mathematical tools needed to go further in their studies. The text also seeks to facilitate that linkage by providing many numerical examples, advanced problems, and extended discussions of empirical implementation—all of which are intended to show students how microeconomic theory is used today. New developments have made the ﬁeld more exciting than ever, and I hope this edition manages to capture that excitement.

NEW TO THE TENTH EDITION The primary change to this edition has been the inclusion of three entirely new chapters written by Chris Snyder: an extended and more advanced treatment of basic game theory concepts (Chapter 8); a thoroughly reworked and expanded chapter on models used in industrial organization theory (Chapter 15); and a completely new chapter on asymmetric information that focuses on the principal– agent problem and modern contract theory (Chapter 18). The importance of these additions to the overall quality of the text cannot be overstated. Because the topics covered in these new chapters constitute some of the most important growth areas in microeconomics, the book is now well positioned for many years into the future. Several other chapters of the book have undergone major revisions for this edition. A signiﬁcant amount of material has been added to the chapter on mathematical background (Chapter 2); new topics include: an expanded coverage of integration, basic models of dynamic optimization, and a brief introduction to mathematical statistics. The material on uncertainty and risk aversion has been thoroughly revised and updated (Chapter 7). Much of the theory of the ﬁrm, especially of the ﬁrm’s demands for inputs, has been expanded (Chapters 9–11). xix

xx

Preface

The chapter on general equilibrium modeling (Chapter 13) has been thoroughly reworked with the goal of providing students with more details about how computable general equilibrium models actually work. The chapter on capital and time (Chapter 17) has been signiﬁcantly expanded to include more on optimal savings behavior and on resource allocation over time. Numerous minor changes have also been made in the coverage and organization of the book to ensure that it continues to provide clear and up-to-date coverage of all of the topics examined. Two modiﬁcations have been made to the text to enhance its linkage to more general economic literature. First, the problems have been categorized into two types: basic problems and analytical problems. Whereas the basic problems are intended to reinforce concepts from the text, the analytical problems are intended to allow the student to go further by showing them how to obtain results on their own. The number of such problems has been signiﬁcantly expanded in this edition. Many of the analytical problems provide references so that students who wish to pursue the topic can read more. A second modiﬁcation of the text has been to expand and rewrite many of the end-ofchapter Extensions. The common goal of these revised Extensions is to provide students better linkage between the theoretical material in the text and that material’s use in actual empirical applications. Therefore, many of the Extensions introduce the functional forms customarily used as well as some of the econometric issues faced by researchers when using available data. The Extensions are thus intended to show students the importance of joining microeconomic theory and econometric practice.

SUPPLEMENTS TO THE TEXT The thoroughly revised ancillaries for this edition include the following. The Solutions Manual and Test Bank (by the text authors). The Solutions Manual contains comments and solutions to all problems and is available to all adopting instructors in both print and electronic versions. The Solutions Manual and Test Bank may be downloaded only by qualiﬁed instructors at the textbook support Web site (www.thomsonedu.com/economics/nicholson). PowerPoint Lecture Presentation Slides (by Linda Ghent, Eastern Illinois University). PowerPoint slides for each chapter of the text provide a thorough set of outlines for classroom use or for students as a study aid. Instructors and students may download these slides from the book’s Web site (www.thomsonedu.com/economics/ nicholson).

ONLINE RESOURCES Thomson South-Western provides students and instructors with a set of valuable online resources that are an effective complement to this text. Each new copy of the book comes with a registration card that provides access to Economic Applications and InfoTrac College Edition.

Economic Applications The purchase of this new textbook includes complimentary access to South-Western’s Economic Applications (EconApps) Web site. The EconApps Web site includes a suite of

Preface

regularly updated Web features for economics students and instructors: EconDebate Online, EconNews Online, EconData Online, and EconLinks Online. These resources can help students deepen their understanding of economic concepts by analyzing current news stories, policy debates, and economic data. EconApps can also help instructors develop assignments, case studies, and examples based on real-world issues. EconDebates Online provides current coverage of economics policy debates; it includes a primer on the issues, links to background information, and commentaries. EconNews Online summarizes recent economics news stories and offers questions for further discussion. EconData Online presents current and historical economic data with accompanying commentary, analysis, and exercises. EconLinks Online offers a navigation partner for exploring economics on the Web via a list of key topic links. Students buying a used book can purchase access to the EconApps site at http://econapps .swlearning.com.

InfoTrac College Edition The purchase of this new textbook also comes with four months of access to InfoTrac. This powerful and searchable online database provides access to full text articles from more than a thousand different publications ranging from the popular press to scholarly journals. Instructors can search topics and select readings for students, and students can search articles and readings for homework assignments and projects. The publications cover a variety of topics and include articles that range from current events to theoretical developments. InfoTrac College Edition offers instructors and students the ability to integrate scholarship and applications of economics into the learning process.

ACKNOWLEDGMENTS In preparation for undertaking this revision, we received very helpful reviews from: Tibor Besedes, Louisiana State University Elaine P. Catilina, American University Yi Deng, Southern Methodist University Silke Forbes, University of California–San Diego Joseph P. Hughes, Rutgers University Qihong Liu, University of Oklahoma Ragan Petrie, Georgia State University We have usually tried to follow their good advice, but of course none of these individuals bears any responsibility for the ﬁnal outcome. This edition of the book is the ﬁrst that was written with my co-author, Chris Snyder of Dartmouth College. I have been very pleased with the working relationship we have developed and with Chris’s friendship. I hope many more editions will follow. I am also indebted to the team at Thomson South-Western and especially to Susan Smart for once again bringing her organizing and cajoling skills to this edition. During her temporary absence from the project, we were completely lost.

xxi

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Preface

Copyediting this manuscript was, I know, a real chore. Those at Newgen-Austin did a great job of penetrating our messy manuscripts to obtain something that actually makes sense. The design of the text by Michelle Kunkler succeeded in achieving two seemingly irreconcilable goals—making the text both compact and easy to read. Cliff Kallemeyn did a ﬁne job of keeping the production on track; I especially appreciated the way he coordinated the copyediting and page production processes. As always, my Amherst College colleagues and students deserve some of the credit for this new edition. Frank Westhoff has been my most faithful user of this text over many years. This time (with his permission, I think) I actually lifted some of his work on general equilibrium to signiﬁcantly improve that portion of the text. To the list of former students—Mark Bruni, Eric Budish, Adrian Dillon, David Macoy, Tatyana Mamut, Katie Merrill, Jordan Milev, Doug Norton, and Jeff Rodman—whose efforts are still evident I can now add the name of Anoop Menon, who helped me solve problems when I ran out of patience with the algebra. As always, special thanks again go to my wife Susan; after seeing twenty editions of my microeconomics texts come and go, she must surely hope that even this good thing must eventually come to an end. My children (Kate, David, Tory, and Paul) all seem to be living happy and productive lives despite a severe lack of microeconomic education. As the next generation (Beth, Sarah, David, Sophia, and Abby) grows older, perhaps they will seek enlightenment—at least to the extent of wondering what the books dedicated to them are all about. Walter Nicholson Amherst, Massachusetts June 2007

It was a privilege to collaborate with Walter on this tenth edition. I used this textbook in the ﬁrst course I ever taught, as a graduate instructor at MIT, and I have enjoyed using it in my microeconomics courses in the thirteen years since. I have always appreciated the text’s ambitious coverage of the concepts and methods used by professional economists as well as its accessibility to students, which is enhanced by numerous elegant examples together with Walter’s lucid prose. It was a challenge to maintain this high standard with my contribution—although this was made easier by Walter’s suggestions, patience, and example, for which I am grateful. I encourage teachers and students to e-mail me with any comments on the text ([email protected]). I would like to add my wholehearted thanks to those whom Walter acknowledged for contributing to the book. I also thank Gretchen Otto and her colleagues at Newgen–Austin as well as Matt Darnell for carefully copyediting my portion of the revision. I thank Dartmouth College for providing the resources and environment that greatly facilitated writing the book. I thank my colleagues in the economics department for helpful discussions and understanding. Committing to such an extensive project is in some sense a family decision. I am indebted to my wife, Maura, for accommodating the many late nights that were required and for listening to my monotonous progress reports. I thank my daughters, Clare, Tess, and Meg, for their good behavior, which expedited the writing process. Christopher Snyder Hanover, New Hampshire June 2007

P A R T

Introduction CHAPTER 1 Economic Models CHAPTER 2 Mathematics for Microeconomics

This part contains only two chapters. Chapter 1 examines the general philosophy of how economists build models of economic behavior. Chapter 2 then reviews some of the mathematical tools used in the construction of these models. The mathematical tools from Chapter 2 will be used throughout the remainder of this book.

1

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CHAPTER

1 Economic Models The main goal of this book is to introduce you to the most important models that economists use to explain the behavior of consumers, ﬁrms, and markets. These models are central to the study of all areas of economics. Therefore, it is essential to understand both the need for such models and the basic framework used to develop them. The goal of this chapter is to begin this process by outlining some of the conceptual issues that determine the ways in which economists study practically every question that interests them.

THEORETICAL MODELS A modern economy is a complicated entity. Thousands of ﬁrms engage in producing millions of different goods. Many millions of people work in all sorts of occupations and make decisions about which of these goods to buy. Let’s use peanuts as an example. Peanuts must be harvested at the right time and shipped to processors who turn them into peanut butter, peanut oil, peanut brittle, and numerous other peanut delicacies. These processors, in turn, must make certain that their products arrive at thousands of retail outlets in the proper quantities to meet demand. Because it would be impossible to describe the features of even these peanut markets in complete detail, economists have chosen to abstract from the complexities of the real world and develop rather simple models that capture the “essentials.” Just as a road map is helpful even though it does not record every house or every store, economic models of, say, the market for peanuts are also useful even though they do not record every minute feature of the peanut economy. In this book we will study the most widely used economic models. We will see that, even though these models often make heroic abstractions from the complexities of the real world, they nonetheless capture essential features that are common to all economic activities. The use of models is widespread in the physical and social sciences. In physics, the notion of a “perfect” vacuum or an “ideal” gas is an abstraction that permits scientists to study real-world phenomena in simpliﬁed settings. In chemistry, the idea of an atom or a molecule is actually a simpliﬁed model of the structure of matter. Architects use mock-up models to plan buildings. Television repairers refer to wiring diagrams to locate problems. Economists’ models perform similar functions. They provide simpliﬁed portraits of the way individuals make decisions, the way ﬁrms behave, and the way in which these two groups interact to establish markets.

VERIFICATION OF ECONOMIC MODELS Of course, not all models prove to be “good.” For example, the earth-centered model of planetary motion devised by Ptolemy was eventually discarded because it proved incapable of accurately explaining how the planets move around the sun. An important purpose of scientiﬁc investigation is to sort out the “bad” models from the “good.” Two general methods have 3

4

Part 1

Introduction

been used for verifying economic models: (1) a direct approach, which seeks to establish the validity of the basic assumptions on which a model is based; and (2) an indirect approach, which attempts to conﬁrm validity by showing that a simpliﬁed model correctly predicts real-world events. To illustrate the basic differences between the two approaches, let’s brieﬂy examine a model that we will use extensively in later chapters of this book—the model of a ﬁrm that seeks to maximize proﬁts.

The proﬁt-maximization model The model of a ﬁrm seeking to maximize proﬁts is obviously a simpliﬁcation of reality. It ignores the personal motivations of the ﬁrm’s managers and does not consider conﬂicts among them. It assumes that proﬁts are the only relevant goal of the ﬁrm; other possible goals, such as obtaining power or prestige, are treated as unimportant. The model also assumes that the ﬁrm has sufﬁcient information about its costs and the nature of the market to which it sells to discover its proﬁt-maximizing options. Most real-world ﬁrms, of course, do not have this information readily available. Yet, such shortcomings in the model are not necessarily serious. No model can exactly describe reality. The real question is whether this simple model has any claim to being a good one.

Testing assumptions One test of the model of a proﬁt-maximizing ﬁrm investigates its basic assumption: Do ﬁrms really seek maximum proﬁts? Some economists have examined this question by sending questionnaires to executives, asking them to specify the goals they pursue. The results of such studies have been varied. Businesspeople often mention goals other than proﬁts or claim they only do “the best they can” to increase proﬁts given their limited information. On the other hand, most respondents also mention a strong “interest” in proﬁts and express the view that proﬁt maximization is an appropriate goal. Testing the proﬁt-maximizing model by testing its assumptions has therefore provided inconclusive results.

Testing predictions Some economists, most notably Milton Friedman, deny that a model can be tested by inquiring into the “reality” of its assumptions.1 They argue that all theoretical models are based on “unrealistic” assumptions; the very nature of theorizing demands that we make certain abstractions. These economists conclude that the only way to determine the validity of a model is to see whether it is capable of predicting and explaining real-world events. The ultimate test of an economic model comes when it is confronted with data from the economy itself. Friedman provides an important illustration of that principle. He asks what kind of a theory one should use to explain the shots expert pool players will make. He argues that the laws of velocity, momentum, and angles from theoretical physics would be a suitable model. Pool players shoot shots as if they follow these laws. But most players asked whether they precisely understand the physical principles behind the game of pool will undoubtedly answer that they do not. Nonetheless, Friedman argues, the physical laws provide very accurate predictions and therefore should be accepted as appropriate theoretical models of how experts play pool. A test of the proﬁt-maximization model, then, would be provided by predicting the behavior of real-world ﬁrms by assuming that these ﬁrms behave as if they were maximizing proﬁts. (See Example 1.1 later in this chapter.) If these predictions are reasonably in accord with reality, we may accept the proﬁt-maximization hypothesis. However, we would reject 1

See M. Friedman, Essays in Positive Economics (Chicago: University of Chicago Press, 1953), chap. 1. For an alternative view stressing the importance of using “realistic” assumptions, see H. A. Simon, “Rational Decision Making in Business Organizations,” American Economic Review 69, no. 4 (September 1979): 493– 513.

Chapter 1 Economic Models

the model if real-world data seem inconsistent with it. Hence, the ultimate test of either theory is its ability to predict real-world events.

Importance of empirical analysis The primary concern of this book is the construction of theoretical models. But the goal of such models is always to learn something about the real world. Although the inclusion of a lengthy set of applied examples would needlessly expand an already bulky book,2 the Extensions included at the end of many chapters are intended to provide a transition between the theory presented here and the ways in which that theory is actually applied in empirical studies.

GENERAL FEATURES OF ECONOMIC MODELS The number of economic models in current use is, of course, very large. Speciﬁc assumptions used and the degree of detail provided vary greatly depending on the problem being addressed. The models employed to explain the overall level of economic activity in the United States, for example, must be considerably more aggregated and complex than those that seek to interpret the pricing of Arizona strawberries. Despite this variety, however, practically all economic models incorporate three common elements: (1) the ceteris paribus (other things the same) assumption; (2) the supposition that economic decision makers seek to optimize something; and (3) a careful distinction between “positive” and “normative” questions. Because we will encounter these elements throughout this book, it may be helpful at the outset to brieﬂy describe the philosophy behind each of them.

The ceteris paribus assumption As in most sciences, models used in economics attempt to portray relatively simple relationships. A model of the market for wheat, for example, might seek to explain wheat prices with a small number of quantiﬁable variables, such as wages of farmworkers, rainfall, and consumer incomes. This parsimony in model speciﬁcation permits the study of wheat pricing in a simpliﬁed setting in which it is possible to understand how the speciﬁc forces operate. Although any researcher will recognize that many “outside” forces (presence of wheat diseases, changes in the prices of fertilizers or of tractors, or shifts in consumer attitudes about eating bread) affect the price of wheat, these other forces are held constant in the construction of the model. It is important to recognize that economists are not assuming that other factors do not affect wheat prices; rather, such other variables are assumed to be unchanged during the period of study. In this way, the effect of only a few forces can be studied in a simpliﬁed setting. Such ceteris paribus (other things equal) assumptions are used in all economic modeling. Use of the ceteris paribus assumption does pose some difﬁculties for the veriﬁcation of economic models from real-world data. In other sciences, such problems may not be so severe because of the ability to conduct controlled experiments. For example, a physicist who wishes to test a model of the force of gravity probably would not do so by dropping objects from the Empire State Building. Experiments conducted in that way would be subject to too many extraneous forces (wind currents, particles in the air, variations in temperature, and so forth) to permit a precise test of the theory. Rather, the physicist would conduct experiments in a laboratory, using a partial vacuum in which most other forces could be controlled or eliminated. In this way, the theory could be veriﬁed in a simple setting, without considering all the other forces that affect falling bodies in the real world. 2

For an intermediate-level text containing an extensive set of real-world applications, see W. Nicholson and C. Snyder, Intermediate Microeconomics and Its Application, 10th ed. (Mason, OH: Thomson/Southwestern, 2007).

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6

Part 1

Introduction

With a few notable exceptions, economists have not been able to conduct controlled experiments to test their models. Instead, economists have been forced to rely on various statistical methods to control for other forces when testing their theories. Although these statistical methods are as valid in principle as the controlled experiment methods used by other scientists, in practice they raise a number of thorny issues. For that reason, the limitations and precise meaning of the ceteris paribus assumption in economics are subject to greater controversy than in the laboratory sciences.

Optimization assumptions Many economic models start from the assumption that the economic actors being studied are rationally pursuing some goal. We brieﬂy discussed such an assumption when investigating the notion of ﬁrms maximizing proﬁts. Example 1.1 shows how that model can be used to make testable predictions. Other examples we will encounter in this book include consumers maximizing their own well-being (utility), ﬁrms minimizing costs, and government regulators attempting to maximize public welfare. Although, as we will show, all of these assumptions are unrealistic, all have won widespread acceptance as good starting places for developing economic models. There seem to be two reasons for this acceptance. First, the optimization assumptions are very useful for generating precise, solvable models, primarily because such models can draw on a variety of mathematical techniques suitable for optimization problems. Many of these techniques, together with the logic behind them, are reviewed in Chapter 2. A second reason for the popularity of optimization models concerns their apparent empirical validity. As some of our Extensions show, such models seem to be fairly good at explaining reality. In all, then, optimization models have come to occupy a prominent position in modern economic theory.

EXAMPLE 1.1 Proﬁt Maximization The proﬁt-maximization hypothesis provides a good illustration of how optimization assumptions can be used to generate empirically testable propositions about economic behavior. Suppose that a ﬁrm can sell all the output that it wishes at a price of p per unit and that the total costs of production, C, depend on the amount produced, q. Then, proﬁts are given by profits ¼ π ¼ pq C ðqÞ:

(1:1)

Maximization of proﬁts consists of ﬁnding that value of q which maximizes the proﬁt expression in Equation 1.1. This is a simple problem in calculus. Differentiation of Equation 1.1 and setting that derivative equal to 0 give the following ﬁrst-order condition for a maximum: dπ ¼ p C 0 ðqÞ ¼ 0 dq

or

p ¼ C 0 ðqÞ:

(1:2)

In words, the proﬁt-maximizing output level (q ) is found by selecting that output level for which price is equal to marginal cost, C 0 ðqÞ. This result should be familiar to you from your introductory economics course. Notice that in this derivation the price for the ﬁrm’s output is treated as a constant because the ﬁrm is a price taker. Equation 1.2 is only the ﬁrst-order condition for a maximum. Taking account of the second-order condition can help us to derive a testable implication of this model. The secondorder condition for a maximum is that at q it must be the case that d 2π ¼ C 00 ðqÞ < 0 dq 2

or

C 00 ðq Þ > 0:

(1:3)

Chapter 1 Economic Models

That is, marginal cost must be increasing at q for this to be a true point of maximum proﬁts. Our model can now be used to “predict” how a ﬁrm will react to a change in price. To do so, we differentiate Equation 1.2 with respect to price (p), assuming that the ﬁrm continues to choose a proﬁt-maximizing level of q: d½ p C 0 ðq Þ ¼ 0 dq ¼ 0: (1:4) ¼ 1 C 00 ðq Þ dp dp Rearranging terms a bit gives dq 1 ¼ 00 > 0: dp C ðq Þ

(1:5)

Here the ﬁnal inequality again reﬂects the fact that marginal cost must be increasing if q is to be a true maximum. This then is one of the testable propositions of the proﬁt-maximization hypothesis—if other things do not change, a price-taking ﬁrm should respond to an increase in price by increasing output. On the other hand, if ﬁrms respond to increases in price by reducing output, there must be something wrong with our model. Although this is a very simple model, it reﬂects the way we will proceed throughout much of this book. Speciﬁcally, the fact that the primary implication of the model is derived by calculus, and consists of showing what sign a derivative should have, is the kind of result we will see many times. QUERY: In general terms, how would the implications of this model be changed if the price a ﬁrm obtains for its output were a function of how much it sold? That is, how would the model work if the price-taking assumption were abandoned?

Positive-normative distinction A ﬁnal feature of most economic models is the attempt to differentiate carefully between “positive” and “normative” questions. So far we have been concerned primarily with positive economic theories. Such theories take the real world as an object to be studied, attempting to explain those economic phenomena that are observed. Positive economics seeks to determine how resources are in fact allocated in an economy. A somewhat different use of economic theory is normative analysis, taking a deﬁnite stance about what should be done. Under the heading of normative analysis, economists have a great deal to say about how resources should be allocated. For example, an economist engaged in positive analysis might investigate how prices are determined in the U.S. health-care economy. The economist also might want to measure the costs and beneﬁts of devoting even more resources to health care. But when he or she speciﬁcally advocates that more resources should be allocated to health care, the analysis becomes normative. Some economists believe that the only proper economic analysis is positive analysis. Drawing an analogy with the physical sciences, they argue that “scientiﬁc” economics should concern itself only with the description (and possibly prediction) of real-world economic events. To take moral positions and to plead for special interests are considered to be outside the competence of an economist acting as such. Other economists, however, believe strict application of the positive-normative distinction to economic matters is inappropriate. They believe that the study of economics necessarily involves the researchers’ own views about ethics, morality, and fairness. According to these economists, searching for scientiﬁc “objectivity” in such circumstances is hopeless. Despite some ambiguity, this book adopts a mainly positivist tone, leaving normative concerns for you to decide for yourself.

7

8

Part 1

Introduction

DEVELOPMENT OF THE ECONOMIC THEORY OF VALUE Because economic activity has been a central feature of all societies, it is surprising that these activities were not studied in any detail until recently. For the most part, economic phenomena were treated as a basic aspect of human behavior that was not sufﬁciently interesting to deserve speciﬁc attention. It is, of course, true that individuals have always studied economic activities with a view toward making some kind of personal gain. Roman traders were not above making proﬁts on their transactions. But investigations into the basic nature of these activities did not begin in any depth until the eighteenth century.3 Because this book is about economic theory as it stands today, rather than the history of economic thought, our discussion of the evolution of economic theory will be brief. Only one area of economic study will be examined in its historical setting: the theory of value.

Early economic thoughts on value The theory of value, not surprisingly, concerns the determinants of the “value” of a commodity. This subject is at the center of modern microeconomic theory and is closely intertwined with the fundamental economic problem of allocating scarce resources to alternative uses. The logical place to start is with a deﬁnition of the word “value.” Unfortunately, the meaning of this term has not been consistent throughout the development of the subject. Today we regard value as being synonymous with the price of a commodity.4 Earlier philosopher-economists, however, made a distinction between the market price of a commodity and its value. The term “value” was then thought of as being, in some sense, synonymous with “importance,” “essentiality,” or (at times) “godliness.” Because “price” and “value” were separate concepts, they could differ, and most early economic discussions centered on these divergences. For example, St. Thomas Aquinas believed value to be divinely determined. Since prices were set by humans, it was possible for the price of a commodity to differ from its value. A person accused of charging a price in excess of a good’s value was guilty of charging an “unjust” price. For example, St. Thomas believed the “just” rate of interest to be zero. Any lender who demanded a payment for the use of money was charging an unjust price and could be—and sometimes was—prosecuted by church ofﬁcials.

The founding of modern economics During the latter part of the eighteenth century, philosophers began to take a more scientiﬁc approach to economic questions. The 1776 publication of The Wealth of Nations by Adam Smith (1723–1790) is generally considered the beginning of modern economics. In his vast, all-encompassing work, Smith laid the foundation for thinking about market forces in an ordered and systematic way. Still, Smith and his immediate successors, such as David Ricardo (1772–1823), continued to distinguish between value and price. To Smith, for example, the value of a commodity meant its “value in use,” whereas the price represented its “value in exchange.” The distinction between these two concepts was illustrated by the famous waterdiamond paradox. Water, which obviously has great value in use, has little value in exchange (it has a low price); diamonds are of little practical use but have a great value in exchange. The paradox with which early economists struggled derives from the observation that some very useful items have low prices whereas certain nonessential items have high prices.

3 For a detailed treatment of early economic thought, see the classic work by J. A. Schumpeter, History of Economic Analysis (New York: Oxford University Press, 1954), pt. II, chaps. 1–3.

This is not completely true when “externalities” are involved and a distinction must be made between private and social value (see Chapter 19).

4

Chapter 1 Economic Models

Labor theory of exchange value Neither Smith nor Ricardo ever satisfactorily resolved the water-diamond paradox. The concept of value in use was left for philosophers to debate, while economists turned their attention to explaining the determinants of value in exchange (that is, to explaining relative prices). One obvious possible explanation is that exchange values of goods are determined by what it costs to produce them. Costs of production are primarily inﬂuenced by labor costs—at least this was so in the time of Smith and Ricardo—and therefore it was a short step to embrace a labor theory of value. For example, to paraphrase an example from Smith, if catching a deer takes twice the number of labor hours as catching a beaver, then one deer should exchange for two beavers. In other words, the price of a deer should be twice that of a beaver. Similarly, diamonds are relatively costly because their production requires substantial labor input. To students with even a passing knowledge of what we now call the law of supply and demand, Smith’s and Ricardo’s explanation must seem incomplete. Didn’t they recognize the effects of demand on price? The answer to this question is both yes and no. They did observe periods of rapidly rising and falling relative prices and attributed such changes to demand shifts. However, they regarded these changes as abnormalities that produced only a temporary divergence of market price from labor value. Because they had not really developed a theory of value in use, they were unwilling to assign demand any more than a transient role in determining relative prices. Rather, long-run exchange values were assumed to be determined solely by labor costs of production.

The marginalist revolution Between 1850 and 1880, economists became increasingly aware that to construct an adequate alternative to the labor theory of value, they had to come to devise a theory of value in use. During the 1870s, several economists discovered that it is not the total usefulness of a commodity that helps to determine its exchange value, but rather the usefulness of the last unit consumed. For example, water is certainly very useful—it is necessary for all life. But, because water is relatively plentiful, consuming one more pint (ceteris paribus) has a relatively low value to people. These “marginalists” redeﬁned the concept of value in use from an idea of overall usefulness to one of marginal, or incremental, usefulness—the usefulness of an additional unit of a commodity. The concept of the demand for an incremental unit of output was now contrasted to Smith’s and Ricardo’s analysis of production costs to derive a comprehensive picture of price determination.5

Marshallian supply-demand synthesis The clearest statement of these marginal principles was presented by the English economist Alfred Marshall (1842–1924) in his Principles of Economics, published in 1890. Marshall showed that demand and supply simultaneously operate to determine price. As Marshall noted, just as you cannot tell which blade of a scissors does the cutting, so too you cannot say that either demand or supply alone determines price. That analysis is illustrated by the famous Marshallian cross shown in Figure 1.1. In the diagram the quantity of a good purchased per period is shown on the horizontal axis and its price appears on the vertical axis. The curve DD represents the quantity of the good demanded per period at each possible price. The curve is negatively sloped to reﬂect the marginalist principle that as quantity increases, people are

Ricardo had earlier provided an important ﬁrst step in marginal analysis in his discussion of rent. Ricardo theorized that as the production of corn increased, land of inferior quality would be used and this would cause the price of corn to rise. In his argument Ricardo implicitly recognized that it is the marginal cost—the cost of producing an additional unit—that is relevant to pricing. Notice that Ricardo implicitly held other inputs constant when discussing diminishing land productivity; that is, he employed one version of the ceteris paribus assumption.

5

9

10

Part 1

FIGURE 1.1

Introduction

The Marshallian Supply-Demand Cross Marshall theorized that demand and supply interact to determine the equilibrium price (p) and the quantity (q ) that will be traded in the market. He concluded that it is not possible to say that either demand or supply alone determines price or therefore that either costs or usefulness to buyers alone determines exchange value.

Price D S

p* D

S

q*

Quantity per period

willing to pay less for the last unit purchased. It is the value of this last unit that sets the price for all units purchased. The curve SS shows how (marginal) production costs rise as more output is produced. This reﬂects the increasing cost of producing one more unit as total output expands. In other words, the upward slope of the SS curve reﬂects increasing marginal costs, just as the downward slope of the DD curve reﬂects decreasing marginal value. The two curves intersect at p, q . This is an equilibrium point—both buyers and sellers are content with the quantity being traded and the price at which it is traded. If one of the curves should shift, the equilibrium point would shift to a new location. Thus price and quantity are simultaneously determined by the joint operation of supply and demand.

Paradox resolved Marshall’s model resolves the water-diamond paradox. Prices reﬂect both the marginal evaluation that demanders place on goods and the marginal costs of producing the goods. Viewed in this way, there is no paradox. Water is low in price because it has both a low marginal value and a low marginal cost of production. On the other hand, diamonds are high in price because they have both a high marginal value (because people are willing to pay quite a bit for one more) and a high marginal cost of production. This basic model of supply and demand lies behind much of the analysis presented in this book.

General equilibrium models Although the Marshallian model is an extremely useful and versatile tool, it is a partial equilibrium model, looking at only one market at a time. For some questions, this narrowing of perspective gives valuable insights and analytical simplicity. For other, broader questions, such a narrow viewpoint may prevent the discovery of important relationships among markets. To answer more general questions we must have a model of the whole economy that suitably mirrors the connections among various markets and economic agents. The French economist Leon Walras (1831–1910), building on a long Continental tradition in such analysis, created the basis for modern investigations into those broad questions. His method of representing the

Chapter 1 Economic Models

economy by a large number of simultaneous equations forms the basis for understanding the interrelationships implicit in general equilibrium analysis. Walras recognized that one cannot talk about a single market in isolation; what is needed is a model that permits the effects of a change in one market to be followed through other markets. EXAMPLE 1.2 Supply-Demand Equilibrium Although graphical presentations are adequate for some purposes, economists often use algebraic representations of their models to both clarify their arguments and make them more precise. As an elementary example, suppose we wished to study the market for peanuts and, on the basis of statistical analysis of historical data, concluded that the quantity of peanuts demanded each week (q, measured in bushels) depended on the price of peanuts (p, measured in dollars per bushel) according to the equation quantity demanded ¼ qD ¼ 1,000 100p:

(1:6)

Because this equation for qD contains only the single independent variable p, we are implicitly holding constant all other factors that might affect the demand for peanuts. Equation 1.6 indicates that, if other things do not change, at a price of $5 per bushel people will demand 500 bushels of peanuts, whereas at a price of $4 per bushel they will demand 600 bushels. The negative coefﬁcient for p in Equation 1.6 reﬂects the marginalist principle that a lower price will cause people to buy more peanuts. To complete this simple model of pricing, suppose that the quantity of peanuts supplied also depends on price: quantity supplied ¼ qS ¼ 125 þ 125p:

(1:7)

Here the positive coefﬁcient of price also reﬂects the marginal principle that a higher price will call forth increased supply—primarily because (as we saw in Example 1.1) it permits ﬁrms to incur higher marginal costs of production without incurring losses on the additional units produced. Equilibrium price determination. Equation 1.6 and 1.7 therefore reﬂect our model of price determination in the market for peanuts. An equilibrium price can be found by setting quantity demanded equal to quantity supplied: q D ¼ qS

(1:8)

1,000 100p ¼ 125 þ 125p

(1:9)

225p ¼ 1,125,

(1:10)

p ¼ 5:

(1:11)

or or so

At a price of $5 per bushel, this market is in equilibrium: at this price people want to purchase 500 bushels, and that is exactly what peanut producers are willing to supply. This equilibrium is pictured graphically as the intersection of D and S in Figure 1.2. A more general model. In order to illustrate how this supply-demand model might be used, let’s adopt a more general notation. Suppose now that the demand and supply functions are given by (continued)

11

12

Part 1

Introduction

EXAMPLE 1.2 CONTINUED FIGURE 1.2

Changing Supply-Demand Equilibria The initial supply-demand equilibrium is illustrated by the intersection of D and S (p ¼ 5, q ¼ 500). When demand shifts to qD 0 ¼ 1,450 100p (denoted as D 0), the equilibrium shifts to p ¼ 7, q ¼ 750. Price ($)

D′

14.5

S D

10

7 5

S

0

500

750

qD ¼ a þ bp

D

D′

1000

1450

and

qS ¼ c þ dp

Quantity per period (bushels)

(1:12)

where a and c are constants that can be used to shift the demand and supply curves, respectively, and b (<0) and d (>0) represent demanders’ and suppliers’ reactions to price. Equilibrium in this market requires q D ¼ qS

or

a þ bp ¼ c þ dp:

(1:13)

So, equilibrium price is given by6 p ¼

ac : d b

(1:14)

6 Equation 1.14 is sometimes called the “reduced form” for the supply-demand structural model of Equations 1.12 and 1.13. It shows that the equilibrium value for the endogenous variable p ultimately depends only on the exogenous factors in the model (a and c) and on the behavioral parameters b and d. A similar equation can be calculated for equilibrium quantity.

Chapter 1 Economic Models

Notice that, in our prior example, a ¼ 1,000, b ¼ 100, c ¼ 125, and d ¼ 125, so p ¼

1,000 þ 125 1,125 ¼ ¼ 5: 125 þ 100 225

(1:15)

With this more general formulation, however, we can pose questions about how the equilibrium price might change if either the demand or supply curve shifted. For example, differentiation of Equation 1.14 shows that dp 1 ¼ > 0, da d b dp 1 ¼ < 0: dc d b

(1:16)

That is, an increase in demand (an increase in a) increases equilibrium price whereas an increase in supply (an increase in c) reduces price. This is exactly what a graphical analysis of supply and demand curves would show. For example, Figure 1.2 shows that when the constant term, a, in the demand equation increases to 1450, equilibrium price increases to p ¼ 7 ½¼ ð1,450 þ 125Þ=225. QUERY: How might you use Equation 1.16 to “predict” how each unit increase in the constant a affects p ? Does this equation correctly predict the increase in p when the constant a increases from 1,000 to 1,450?

For example, suppose that the demand for peanuts were to increase. This would cause the price of peanuts to increase. Marshallian analysis would seek to understand the size of this increase by looking at conditions of supply and demand in the peanut market. General equilibrium analysis would look not only at that market but also at repercussions in other markets. A rise in the price of peanuts would increase costs for peanut butter makers, which would, in turn, affect the supply curve for peanut butter. Similarly, the rising price of peanuts might mean higher land prices for peanut farmers, which would affect the demand curves for all products that they buy. The demand curves for automobiles, furniture, and trips to Europe would all shift out, and that might create additional incomes for the providers of those products. Consequently, the effects of the initial increase in demand for peanuts eventually would spread throughout the economy. General equilibrium analysis attempts to develop models that permit us to examine such effects in a simpliﬁed setting. Several models of this type are described in Chapter 13.

Production possibility frontier Here we brieﬂy introduce some general equilibrium ideas by using another graph you should remember from introductory economics—the production possibility frontier. This graph shows the various amounts of two goods that an economy can produce using its available resources during some period (say, one week). Because the production possibility frontier shows two goods, rather than the single good in Marshall’s model, it is used as a basic building block for general equilibrium models. Figure 1.3 shows the production possibility frontier for two goods, food and clothing. The graph illustrates the supply of these goods by showing the combinations that can be produced with this economy’s resources. For example, 10 pounds of food and 3 units of clothing could be produced, or 4 pounds of food and 12 units of clothing. Many other combinations of food and clothing could also be produced. The production possibility frontier shows all of them. Combinations of food and clothing outside the frontier cannot be produced because not enough resources are available. The production possibility frontier

13

14

Part 1

FIGURE 1.3

Introduction

Production Possibility Frontier The production possibility frontier shows the different combinations of two goods that can be produced from a certain amount of scarce resources. It also shows the opportunity cost of producing more of one good as the amount of the other good that cannot then be produced. The opportunity cost at two different levels of clothing production can be seen by comparing points A and B.

Quantity of food per week Opportunity cost of clothing = 12 pound of food A

10 9.5

Opportunity cost of clothing = 2 pounds of food B 4 2

0

3

4

12 13

Quantity of clothing per week

reminds us of the basic economic fact that resources are scarce—there are not enough resources available to produce all we might want of every good. This scarcity means that we must choose how much of each good to produce. Figure 1.3 makes clear that each choice has its costs. For example, if this economy produces 10 pounds of food and 3 units of clothing at point A, producing 1 more unit of clothing would “cost” 12 pound of food—increasing the output of clothing by 1 unit means the production of food would have to decrease by 12 pound. So, the opportunity cost of 1 unit of clothing at point A is 12 pound of food. On the other hand, if the economy initially produces 4 pounds of food and 12 units of clothing at point B, it would cost 2 pounds of food to produce 1 more unit of clothing. The opportunity cost of 1 more unit of clothing at point B has increased to 2 pounds of food. Because more units of clothing are produced at point B than at point A, both Ricardo’s and Marshall’s ideas of increasing incremental costs suggest that the opportunity cost of an additional unit of clothing will be higher at point B than at point A. This effect is shown by Figure 1.3. The production possibility frontier provides two general equilibrium insights that are not clear in Marshall’s supply and demand model of a single market. First, the graph shows that producing more of one good means producing less of another good because resources are scarce. Economists often (perhaps too often!) use the expression “there is no such thing as a free lunch” to explain that every economic action has opportunity costs. Second, the production possibility frontier shows that opportunity costs depend on how much of each good is produced. The frontier is like a supply curve for two goods: it shows the opportunity cost of producing more of one good as the decrease in the amount of the second good. The production possibility frontier is therefore a particularly useful tool for studying several markets at the same time.

Chapter 1 Economic Models

EXAMPLE 1.3 The Production Possibility Frontier and Economic Ineﬃciency General equilibrium models are good tools for evaluating the efﬁciency of various economic arrangements. As we will see in Chapter 13, such models have been used to assess a wide variety of policies such as trade agreements, tax structures, and environmental regulation. In this simple example, we explore the idea of efﬁciency in its most elementary form. Suppose that an economy produces two goods, x and y, using labor as the only input.The (where lx is the quantity of labor used in x production function for good x is x ¼ l 0:5 x production) and the production function for good y is y ¼ 2l 0:5 y . Total labor available is constrained by lx þ ly 200. Construction of the production possibility frontier in this economy is extremely simple: lx þ ly ¼ x 2 þ 0:25y 2 200

(1:17)

if the economy is to be producing as much as possible (which, after all, is why it’s called a “frontier”). Equation 1.17 shows that the frontier here has the shape of a quarter ellipse—its concavity derives from the diminishing returns exhibited by each production function. Opportunity cost. Assuming this economy is on the frontier, the opportunity cost of good y in terms of good x can be derived by solving for y as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (1:18) y 2 ¼ 800 4x 2 or y ¼ 800 4x 2 ¼ ½800 4x 2 0:5 and then differentiating this expression: dy 4x ¼ 0:5½800 4x 2 0:5 ð8xÞ ¼ : dx y

(1:19)

Suppose, for example, labor is equally allocated between the two goods. Then x ¼ 10, y ¼ 20, and dy=dx ¼ 4ð10Þ=20 ¼ 2. With this allocation of labor, each unit increase in x output would require a reduction in y of 2 units. This can be veriﬁed by considering a slightly different allocation, lx ¼ 101 and ly ¼ 99. Now production is x ¼ 10:05 and y ¼ 19:9. Moving to this alternative allocation would have Dy ð19:9 20Þ 0:1 ¼ ¼ ¼ 2, Dx ð10:05 10Þ 0:05 which is precisely what was derived from the calculus approach. Concavity. Equation 1.19 clearly illustrates the concavity of the production possibility frontier. The slope of the frontier becomes steeper (more negative) as x output increases and y output falls. For example, if labor is allocated so that lx ¼ 144 and ly ¼ 56, then outputs are x ¼ 12 and y 15 and so dy=dx ¼ 4ð12Þ=15 ¼ 3:2. With expanded x production, the opportunity cost of one more unit of x increases from 2 to 3.2 units of y. Ineﬃciency. If an economy operates inside its production possibility frontier, it is operating inefﬁciently. Moving outward to the frontier could increase the output of both goods. In this book we will explore many reasons for such inefﬁciency. These usually derive from a failure of some market to perform correctly. For the purposes of this illustration, let’s assume that the labor market in this economy does not work well and that 20 workers are permanently unemployed. Now the production possibility frontier becomes x 2 þ 0:25y 2 ¼ 180,

(1:20) (continued)

15

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Part 1

Introduction

EXAMPLE 1.3 CONTINUED and the output combinations we described previously are no longer feasible. For example, if x ¼ 10 then y output is now y 17:9. The loss of about 2.1 units of y is a measure of the cost of the labor market inefﬁciency. Alternatively, if the labor supply of 180 were allocated evenly between the production of the two goods then we would have x 9:5 and y 19, and the inefﬁciency would show up in both goods’ production—more of both goods could be produced if the labor market inefﬁciency were resolved. QUERY: How would the inefﬁciency cost of labor market imperfections be measured solely in terms of x production in this model? How would it be measured solely in terms of y production? What would you need to know in order to assign a single number to the efﬁciency cost of the imperfection when labor is equally allocated to the two goods?

Welfare economics In addition to their use in examining positive questions about how the economy operates, the tools used in general equilibrium analysis have also been applied to the study of normative questions about the welfare properties of various economic arrangements. Although such questions were a major focus of the great eighteenth- and nineteenth-century economists (Smith, Ricardo, Marx, Marshall, and so forth), perhaps the most signiﬁcant advances in their study were made by the British economist Francis Y. Edgeworth (1848–1926) and the Italian economist Vilfredo Pareto (1848–1923) in the early years of the twentieth century. These economists helped to provide a precise deﬁnition for the concept of “economic efﬁciency” and to demonstrate the conditions under which markets will be able to achieve that goal. By clarifying the relationship between the allocation pricing of resources, they provided some support for the idea, ﬁrst enunciated by Adam Smith, that properly functioning markets provide an “invisible hand” that helps allocate resources efﬁciently. Later sections of this book focus on some of these welfare issues.

MODERN DEVELOPMENTS Research activity in economics expanded rapidly in the years following World War II. A major purpose of this book is to summarize much of this research. By illustrating how economists have tried to develop models to explain increasingly complex aspects of economic behavior, this book seeks to help you recognize some of the remaining unanswered questions.

The mathematical foundations of economic models A major postwar development in microeconomic theory was the clariﬁcation and formalization of the basic assumptions that are made about individuals and ﬁrms. The ﬁrst landmark in this development was the 1947 publication of Paul Samuelson’s Foundations of Economic Analysis, in which the author (the ﬁrst American Nobel Prize winner in economics) laid out a number of models of optimizing behavior.7 Samuelson demonstrated the importance of basing behavioral models on well-speciﬁed mathematical postulates so that various optimization techniques from mathematics could be applied. The power of his approach made it inescapably clear that mathematics had become an integral part of modern economics. In Chapter 2 of this book we review some of the mathematical concepts most often used in microeconomics. 7

Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947).

Chapter 1 Economic Models

17

New tools for studying markets A second feature that has been incorporated into this book is the presentation of a number of new tools for explaining market equilibria. These include techniques for describing pricing in single markets, such as increasingly sophisticated models of monopolistic pricing or models of the strategic relationships among ﬁrms that use game theory. They also include general equilibrium tools for simultaneously exploring relationships among many markets. As we shall see, all of these new techniques help to provide a more complete and realistic picture of how markets operate.

The economics of uncertainty and information A ﬁnal major theoretical advance during the postwar period was the incorporation of uncertainty and imperfect information into economic models. Some of the basic assumptions used to study behavior in uncertain situations were originally developed in the 1940s in connection with the theory of games. Later developments showed how these ideas could be used to explain why individuals tend to be adverse to risk and how they might gather information in order to reduce the uncertainties they face. In this book, problems of uncertainty and information enter the analysis on many occasions.

Computers and empirical analysis One ﬁnal aspect of the postwar development of microeconomics should be mentioned—the increasing use of computers to analyze economic data and build economic models. As computers have become able to handle larger amounts of information and carry out complex mathematical manipulations, economists’ ability to test their theories has dramatically improved. Whereas previous generations had to be content with rudimentary tabular or graphical analyses of realworld data, today’s economists have available a wide variety of sophisticated techniques together with extensive microeconomic data with which to test their models. To examine these techniques and some of their limitations would be beyond the scope and purpose of this book. But, Extensions at the end of most chapters are intended to help you start reading about some of these applications.

SUMMARY This chapter provided background on how economists approach the study of the allocation of resources. Much of the material discussed here should be familiar to you from introductory economics. In many respects, the study of economics represents acquiring increasingly sophisticated tools for addressing the same basic problems. The purpose of this book (and, indeed, of most upper-level books on economics) is to provide you with more of these tools. As a starting place, this chapter reminded you of the following points: •

Economics is the study of how scarce resources are allocated among alternative uses. Economists seek to develop simple models to help understand that process. Many of these models have a mathematical basis because the use of mathematics offers a precise shorthand for stating the models and exploring their consequences.

•

The most commonly used economic model is the supply-demand model ﬁrst thoroughly developed by

Alfred Marshall in the latter part of the nineteenth century. This model shows how observed prices can be taken to represent an equilibrium balancing of the production costs incurred by ﬁrms and the willingness of demanders to pay for those costs. •

Marshall’s model of equilibrium is only “partial”—that is, it looks only at one market at a time. To look at many markets together requires an expanded set of general equilibrium tools.

•

Testing the validity of an economic model is perhaps the most difﬁcult task economists face. Occasionally, a model’s validity can be appraised by asking whether it is based on “reasonable” assumptions. More often, however, models are judged by how well they can explain economic events in the real world.

18

Part 1

Introduction

SUGGESTIONS FOR FURTHER READING On Methodology Blaug, Mark, and John Pencavel. The Methodology of Economics: Or How Economists Explain, 2nd ed. Cambridge: Cambridge University Press, 1992. A revised and expanded version of a classic study on economic methodology. Ties the discussion to more general issues in the philosophy of science.

Marx, K. Capital. New York: Modern Library, 1906. Full development of labor theory of value. Discussion of “transformation problem” provides a (perhaps faulty) start for general equilibrium analysis. Presents fundamental criticisms of institution of private property.

Ricardo, D. Principles of Political Economy and Taxation. London: J. M. Dent & Sons, 1911.

Boland, Lawrence E. “A Critique of Friedman’s Critics.” Journal of Economic Literature (June 1979): 503– 22.

Very analytical, tightly written work. Pioneer in developing careful analysis of policy questions, especially trade-related issues. Discusses ﬁrst basic notions of marginalism.

Good summary of criticisms of positive approaches to economics and of the role of empirical veriﬁcation of assumptions.

Smith, A. The Wealth of Nations. New York: Modern Library, 1937.

Friedman, Milton. “The Methodology of Positive Economics.” In Essays in Positive Economics, pp. 3– 43. Chicago: University of Chicago Press, 1953. Basic statement of Friedman’s positivist views.

Harrod, Roy F. “Scope and Method in Economics.” Economic Journal 48 (1938): 383– 412.

First great economics classic. Very long and detailed, but Smith had the ﬁrst word on practically every economic matter. This edition has helpful marginal notes.

Walras, L. Elements of Pure Economics. Translated by W. Jaffé. Homewood, IL: Richard D. Irwin, 1954. Beginnings of general equilibrium theory. Rather diﬃcult reading.

Classic statement of appropriate role for economic modeling.

Hausman, David M., and Michael S. McPherson. Economic Analysis, Moral Philosophy, and Public Policy, 2nd ed. Cambridge: Cambridge University Press, 2006. The authors stress their belief that consideration of issues in moral philosophy can improve economic analysis.

McCloskey, Donald N. If You’re So Smart: The Narrative of Economic Expertise. Chicago: University of Chicago Press, 1990. Discussion of McCloskey’s view that economic persuasion depends on rhetoric as much as on science. For an interchange on this topic, see also the articles in the Journal of Economic Literature, June 1995.

Sen, Amartya. On Ethics and Economics. Oxford: Blackwell Reprints, 1989. The author seeks to bridge the gap between economics and ethical studies. This is a reprint of a classic study on this topic.

Primary Sources on the History of Economics Edgeworth, F. Y. Mathematical Psychics. London: Kegan Paul, 1881. Initial investigations of welfare economics, including rudimentary notions of economic eﬃciency and the contract curve.

Marshall, A. Principles of Economics, 8th ed. London: Macmillan & Co., 1920. Complete summary of neoclassical view. A long-running, popular text. Detailed mathematical appendix.

Secondary Sources on the History of Economics Backhouse, Roger E. The Ordinary Business of Life: The History of Economics from the Ancient World to the 21st Century. Princeton, NJ: Princeton University Press, 2002. An iconoclastic history. Quite good on the earliest economic ideas, but some blind spots on recent uses of mathematics and econometrics.

Blaug, Mark. Economic Theory in Retrospect, 5th ed. Cambridge: Cambridge University Press, 1997. Very complete summary stressing analytical issues. Excellent “Readers’ Guides” to the classics in each chapter.

Heilbroner, Robert L. The Worldly Philosophers, 7th ed. New York: Simon & Schuster, 1999. Fascinating, easy-to-read biographies of leading economists. Chapters on Utopian Socialists and Thorstein Veblen highly recommended.

Keynes, John M. Essays in Biography. New York: W. W. Norton, 1963. Essays on many famous persons (Lloyd George, Winston Churchill, Leon Trotsky) and on several economists (Malthus, Marshall, Edgeworth, F. P. Ramsey, and Jevons). Shows the true gift of Keynes as a writer.

Schumpeter, J. A. History of Economic Analysis. New York: Oxford University Press, 1954. Encyclopedic treatment. Covers all the famous and many not-so-famous economists. Also brieﬂy summarizes concurrent developments in other branches of the social sciences.

CHAPTER

2 Mathematics for Microeconomics Microeconomic models are constructed using a wide variety of mathematical techniques. In this chapter we provide a brief summary of some of the most important techniques that you will encounter in this book. A major portion of the chapter concerns mathematical procedures for ﬁnding the optimal value of some function. Because we will frequently adopt the assumption that an economic actor seeks to maximize or minimize some function, we will encounter these procedures (most of which are based on diﬀerential calculus) many times. After our detailed discussion of the calculus of optimization, we turn to four topics that are covered more brieﬂy. First, we look at a few special types of functions that arise in economic problems. Knowledge of properties of these functions can often be very helpful in solving economic problems. Next, we provide a brief summary of integral calculus. Although integration is used in this book far less frequently than is diﬀerentiation, we will nevertheless encounter several situations where we will want to employ integrals to measure areas that are important to economic theory or to add up outcomes that occur over time or across many individuals. One particular use of integration is to examine problems in which the objective is to maximize a stream of outcomes over time. Our third added topic focuses on techniques to be used for such problems in dynamic optimization. Finally, Chapter 2 concludes with a brief summary of mathematical statistics, which will be particularly useful in our study of economic behavior in uncertain situations.

MAXIMIZATION OF A FUNCTION OF ONE VARIABLE Let’s start our study of optimization with a simple example. Suppose that a manager of a ﬁrm desires to maximize1 the proﬁts received from selling a particular good. Suppose also that the proﬁts ðπÞ received depend only on the quantity ðqÞ of the good sold. Mathematically, π ¼ f ðqÞ:

(2.1)

Figure 2.1 shows a possible relationship between π and q. Clearly, to achieve maximum proﬁts, the manager should produce output q , which yields proﬁts π . If a graph such as that of Figure 2.1 were available, this would seem to be a simple matter to be accomplished with a ruler. Suppose, however, as is more likely, the manager does not have such an accurate picture of the market. He or she may then try varying q to see where a maximum proﬁt is obtained. For example, by starting at q1 , proﬁts from sales would be π1 . Next, the manager may try output q2 , observing that proﬁts have increased to π2 . The commonsense idea that proﬁts have increased in response to an increase in q can be stated formally as π2 π1 >0 q2 q1

or

∆π > 0, ∆q

(2.2)

1

Here we will generally explore maximization problems. A virtually identical approach would be taken to study minimization problems because maximization of f ðxÞ is equivalent to minimizing f ðxÞ.

19

20

Part 1 Introduction

FIGURE 2.1

Hypothetical Relationship between Quantity Produced and Proﬁts If a manager wishes to produce the level of output that maximizes proﬁts, then q should be produced. Notice that at q , dπ=dq ¼ 0. π π* π2

π = f(q)

π3 π1 q1

q2

q*

q3

Quantity

where the ∆ notation is used to mean “the change in” π or q. As long as ∆π=∆q is positive, proﬁts are increasing and the manager will continue to increase output. For increases in output to the right of q , however, ∆π=∆q will be negative, and the manager will realize that a mistake has been made.

Derivatives As you probably know, the limit of ∆π=∆q for very small changes in q is called the derivative of the function, π ¼ f ðqÞ, and is denoted by dπ=dq or df =dq or f 0 ðqÞ. More formally, the derivative of a function π ¼ f ðqÞ at the point q1 is deﬁned as dπ df f ðq1 þ hÞ f ðq1 Þ ¼ ¼ lim : dq dq h!0 h

(2.3)

Notice that the value of this ratio obviously depends on the point q1 that is chosen.

Value of the derivative at a point A notational convention should be mentioned: Sometimes one wishes to note explicitly the point at which the derivative is to be evaluated. For example, the evaluation of the derivative at the point q ¼ q1 could be denoted by dπ : (2.4) dq q¼q1 At other times, one is interested in the value of dπ=dq for all possible values of q and no explicit mention of a particular point of evaluation is made. In the example of Figure 2.1, dπ > 0, dq q¼q 1

whereas

dπ < 0: dq q¼q3

What is the value of dπ=dq at q ? It would seem to be 0, because the value is positive for values of q less than q and negative for values of q greater than q . The derivative is the slope of the curve in question; this slope is positive to the left of q and negative to the right of q . At the point q , the slope of f ðqÞ is 0.

Chapter 2 Mathematics for Microeconomics

First-order condition for a maximum This result is quite general. For a function of one variable to attain its maximum value at some point, the derivative at that point (if it exists) must be 0. Hence, if a manager could estimate the function f ðqÞ from some sort of real-world data, it would theoretically be possible to ﬁnd the point where df =dq ¼ 0. At this optimal point (say, q ), df ¼ 0: (2.5) dq q¼q

Second-order conditions An unsuspecting manager could be tricked, however, by a naive application of this ﬁrstderivative rule alone. For example, suppose that the proﬁt function looks like that shown in either Figure 2.2a or 2.2b. If the proﬁt function is that shown in Figure 2.2a, the manager, by producing where dπ=dq ¼ 0, will choose point q a . This point in fact yields minimum, not maximum, proﬁts for the manager. Similarly, if the proﬁt function is that shown in Figure 2.2, the manager will choose point q b , which, although it yields a proﬁt greater than that for any output lower than q b , is certainly inferior to any output greater than q b . These situations illustrate the mathematical fact that dπ=dq ¼ 0 is a necessary condition for a maximum, but not a sufﬁcient condition. To ensure that the chosen point is indeed a maximum point, a second condition must be imposed. Intuitively, this additional condition is clear: The proﬁt available by producing either a bit more or a bit less than q must be smaller than that available from q . If this is not true, the manager can do better than q . Mathematically, this means that dπ=dq must be greater

FIGURE 2.2 Two Proﬁt Functions That Give Misleading Results If the First Derivative Rule Is Applied Uncritically In (a), the application of the ﬁrst derivative rule would result in point q a being chosen. This point is in fact a point of minimum proﬁts. Similarly, in (b), output level q b would be recommended by the ﬁrst derivative rule, but this point is inferior to all outputs greater than q b . This demonstrates graphically that ﬁnding a point at which the derivative is equal to 0 is a necessary, but not a sufﬁcient, condition for a function to attain its maximum value. π

π

π*b π*a

q*a (a)

q*b

Quantity (b)

Quantity

21

22

Part 1 Introduction

than 0 for q < q and must be less than 0 for q > q . Therefore, at q , dπ=dq must be decreasing. Another way of saying this is that the derivative of dπ=dq must be negative at q .

Second derivatives The derivative of a derivative is called a second derivative and is denoted by d 2π d 2f or or f ðq Þ: 2 dq dq 2 The additional condition for q to represent a (local) maximum is therefore d 2 π ¼ f ″ðq Þ < 0, dq 2 q¼q q¼q

(2.6)

where the notation is again a reminder that this second derivative is to be evaluated at q . Hence, although Equation 2.5 ðdπ=dq ¼ 0Þ is a necessary condition for a maximum, that equation must be combined with Equation 2.6 ðd 2 π=dq 2 < 0Þ to ensure that the point is a local maximum for the function. Equations 2.5 and 2.6 together are therefore sufﬁcient conditions for such a maximum. Of course, it is possible that by a series of trials the manager may be able to decide on q by relying on market information rather than on mathematical reasoning (remember Friedman’s pool-player analogy). In this book we shall be less interested in how the point is discovered than in its properties and how the point changes when conditions change. A mathematical development will be very helpful in answering these questions.

Rules for ﬁnding derivatives Here are a few familiar rules for taking derivatives. We will use these at many places in this book. 1. If b is a constant, then db ¼ 0: dx 2. If b is a constant, then d½bf ðxÞ ¼ bf 0 ðx Þ: dx 3. If b is a constant, then dx b ¼ bx b1 : dx d ln x 1 ¼ dx x where ln signiﬁes the logarithm to the base e ð¼ 2:71828Þ. da x 5. ¼ a x ln a for any constant a dx A particular case of this rule is de x =dx ¼ e x .

4.

Now suppose that f ðxÞ and gðxÞ are two functions of x and that f 0 ðxÞ and g 0 ðxÞ exist. Then: 6. d½ f ðxÞ þ gðxÞ ¼ f 0 ðx Þ þ g 0 ðx Þ: dx

Chapter 2 Mathematics for Microeconomics

7. d½ f ðxÞ ⋅ gðxÞ ¼ f ðx Þg 0 ðx Þ þ f 0 ðx Þg ðx Þ: dx 0 0 8. d½ f ðxÞ=gðxÞ ¼ f ðxÞgðxÞ f ðxÞg ðxÞ , 2 dx ½ gðxÞ

provided that gðxÞ 6¼ 0. Finally, if y ¼ f ðxÞ and x ¼ gðzÞ and if both f 0 ðxÞ and g 0 ðzÞ exist, then 9. dy ¼ dy ⋅ dx ¼ df ⋅ dg : dz dx dz dx dz This result is called the chain rule. It provides a convenient way to study how one variable ðzÞ affects another variable ðyÞ solely through its inﬂuence on some intermediate variable ðxÞ. Some examples are ax ax dðaxÞ 10. de ¼ de ¼ e ax ⋅ a ¼ ae ax : ⋅ dx dðaxÞ dx 11. d½lnðaxÞ ¼ d½lnðaxÞ ⋅ dðaxÞ ¼ 1 ⋅ a ¼ 1 : dx dðaxÞ dx ax x 2 2 2 12. d½lnðx Þ ¼ d½lnðx Þ ⋅ dðx Þ ¼ 1 ⋅ 2x ¼ 2 : dx dðx 2 Þ dx x2 x

FUNCTIONS OF SEVERAL VARIABLES Economic problems seldom involve functions of only a single variable. Most goals of interest to economic agents depend on several variables, and trade-offs must be made among these variables. For example, the utility an individual receives from activities as a consumer depends on the amount of each good consumed. For a ﬁrm’s production function, the amount produced depends on the quantity of labor, capital, and land devoted to production. In these circumstances, this dependence of one variable ðyÞ on a series of other variables ðx1 , x2 , …, xn Þ is denoted by y ¼ f ðx1 , x2 , …, xn Þ:

(2.7)

Partial derivatives We are interested in the point at which y reaches a maximum and in the trade-offs that must be made to reach that point. It is again convenient to picture the agent as changing the variables at his or her disposal (the x’s) in order to locate a maximum. Unfortunately, for a function of several variables, the idea of the derivative is not well-deﬁned. Just as the steepness of ascent when climbing a mountain depends on which direction you go, so does the slope (or derivative) of the function depend on the direction in which it is taken. Usually, the only directional slopes of interest are those that are obtained by increasing one of the x’s while holding all the other variables constant (the analogy for mountain climbing might be to measure slopes only in a north-south or east-west direction). These directional slopes are called partial derivatives. The partial derivative of y with respect to (that is, in the direction of ) x1 is denoted by ∂y ∂x1

or

∂f ∂x1

or

fx1

or

f1 :

It is understood that in calculating this derivative all of the other x’s are held constant. Again it should be emphasized that the numerical value of this slope depends on the value of x1 and on the (preassigned) values of x2 , …, xn .

23

24

Part 1 Introduction

EXAMPLE 2.1 Proﬁt Maximization Suppose that the relationship between proﬁts ðπÞ and quantity produced ðqÞ is given by πðqÞ ¼ 1,000q 5q 2 :

(2.8)

A graph of this function would resemble the parabola shown in Figure 2.1. The value of q that maximizes proﬁts can be found by differentiation: dπ ¼ 1,000 10q ¼ 0, dq

(2.9)

q ¼ 100:

(2.10)

so

At q ¼ 100, Equation 2.8 shows that proﬁts are 50,000—the largest value possible. If, for example, the ﬁrm opted to produce q ¼ 50, proﬁts would be 37,500. At q ¼ 200, proﬁts are precisely 0. That q ¼ 100 is a “global” maximum can be shown by noting that the second derivative of the proﬁt function is 10 (see Equation 2.9). Hence, the rate of increase in proﬁts is always decreasing—up to q ¼ 100 this rate of increase is still positive, but beyond that point it becomes negative. In this example, q ¼ 100 is the only local maximum value for the function π. With more complex functions, however, there may be several such maxima. QUERY: Suppose that a ﬁrm’s p output ðqÞ is determined by the amount of labor ðlÞ it hires ﬃﬃ according to the function q ¼ 2 l . Suppose also that the ﬁrm can hire all of the labor it wants at $10 per unitpand ﬃﬃ sells its output at $50 per unit. Proﬁts are therefore a function of l given by πðl Þ ¼ 100 l 10l . How much labor should this ﬁrm hire in order to maximize proﬁts, and what will those proﬁts be?

A somewhat more formal deﬁnition of the partial derivative is _ _ _ _ ∂f f ðx1 þh; x 2 , …, x n Þ f ðx1 , x 2 , …, x n Þ ¼ lim , h→0 ∂x1 x_ , …, x_ h 2

(2.11)

n

where the _ is intended to indicate that x2 , …, xn are all held constant at the preassigned _ notation values x 2 , …, x n so the effect of changing x1 only can be studied. Partial derivatives with respect to the other variables ðx2 , …, xn Þ would be calculated in a similar way.

Calculating partial derivatives It is easy to calculate partial derivatives. The calculation proceeds as for the usual derivative by treating x2 , …, xn as constants (which indeed they are in the deﬁnition of a partial derivative). Consider the following examples. 1. If y ¼ f x1 , x2 ¼ ax 21 þ bx1 x2 þ cx 22 , then ∂f ¼ f1 ¼ 2ax1 þ bx2 ∂x1 and ∂f ¼ f2 ¼ bx1 þ 2cx2 : ∂x2

Chapter 2 Mathematics for Microeconomics

Notice that ∂f =∂x1 is in general a function of both x1 and x2 and therefore its value will depend on the particular values assigned to these variables. It also depends on the parameters a, b, and c, which do not change as x1 and x2 change. 2. If y ¼ f ðx1 , x2 Þ ¼ e ax1 þbx2 , then ∂f ¼ f1 ¼ ae ax1 þbx2 ∂x1 and ∂f ¼ f2 ¼ be ax1 þbx2 : ∂x2 3. If y ¼ f ðx1 , x2 Þ ¼ a ln x1 þ b ln x2 , then ∂f a ¼ f1 ¼ ∂x1 x1 and ∂f b ¼ f2 ¼ : ∂x2 x2 Notice here that the treatment of x2 as a constant in the derivation of ∂f =∂x1 causes the term b ln x2 to disappear upon differentiation because it does not change when x1 changes. In this case, unlike our previous examples, the size of the effect of x1 on y is independent of the value of x2 . In other cases, the effect of x1 on y will depend on the level of x2 .

Partial derivatives and the ceteris paribus assumption In Chapter 1, we described the way in which economists use the ceteris paribus assumption in their models to hold constant a variety of outside inﬂuences so the particular relationship being studied can be explored in a simpliﬁed setting. Partial derivatives are a precise mathematical way of representing this approach; that is, they show how changes in one variable affect some outcome when other inﬂuences are held constant—exactly what economists need for their models. For example, Marshall’s demand curve shows the relationship between price ðpÞ and quantity ðqÞ demanded when other factors are held constant. Using partial derivatives, we could represent the slope of this curve by ∂q=∂p to indicate the ceteris paribus assumptions that are in effect. The fundamental law of demand—that price and quantity move in opposite directions when other factors do not change—is therefore reﬂected by the mathematical statement “∂q=∂p < 0.” Again, the use of a partial derivative serves as a reminder of the ceteris paribus assumptions that surround the law of demand.

Partial derivatives and units of measurement In mathematics relatively little attention is paid to how variables are measured. In fact, most often no explicit mention is made of the issue. But the variables used in economics usually refer to real-world magnitudes and therefore we must be concerned with how they are measured. Perhaps the most important consequence of choosing units of measurement is that the partial derivatives often used to summarize economic behavior will reﬂect these units. For example, if q represents the quantity of gasoline demanded by all U.S. consumers during a given year (measured in billions of gallons) and p represents the price in dollars per gallon, then ∂q=∂p will measure the change in demand (in billions of gallons per year) for a dollar per gallon change in price. The numerical size of this derivative depends on how q and p are measured. A decision to measure consumption in millions of gallons per year would multiply

25

26

Part 1 Introduction

the size of the derivative by 1,000, whereas a decision to measure price in cents per gallon would reduce it by a factor of 100. The dependence of the numerical size of partial derivatives on the chosen units of measurement poses problems for economists. Although many economic theories make predictions about the sign (direction) of partial derivatives, any predictions about the numerical magnitude of such derivatives would be contingent on how authors chose to measure their variables. Making comparisons among studies could prove practically impossible, especially given the wide variety of measuring systems in use around the world. For this reason, economists have chosen to adopt a different, unit-free way to measure quantitative impacts.

Elasticity—A general deﬁnition Economists use elasticities to summarize virtually all of the quantitative impacts that are of interest to them. Because such measures focus on the proportional effect of a change in one variable on another, they are unit-free—the units “cancel out” when the elasticity is calculated. Suppose, for example, that y is a function of x and, possibly, other variables. Then the elasticity of y with respect to x (denoted as ey, x ) is deﬁned as ∆y y ∆y x ∂y x ey , x ¼ (2.12) ¼ ⋅ ¼ ⋅ : ∆x ∆x y ∂x y x Notice that, no matter how the variables y and x are measured, the units of measurement cancel out because they appear in both a numerator and a denominator. Notice also that, because y and x are positive in most economic situations, the elasticity ey, x and the partial derivative ∂y=∂x will have the same sign. Hence, theoretical predictions about the direction of certain derivatives will also apply to their related elasticities. Speciﬁc applications of the elasticity concept will be encountered throughout this book. These include ones with which you should be familiar, such as the market price elasticity of demand or supply. But many new concepts that can be expressed most clearly in elasticity terms will also be introduced. EXAMPLE 2.2 Elasticity and Functional Form The deﬁnition in Equation 2.12 makes clear that elasticity should be evaluated at a speciﬁc point on a function. In general the value of this parameter would be expected to vary across different ranges of the function. This observation is most clearly shown in the case where y is a linear function of x of the form y ¼ a þ bx þ other terms: In this case, ey , x ¼

∂y x x x , ⋅ ¼ b⋅ ¼ b⋅ ∂x y y a þ bx þ …

(2.13)

which makes clear that ey, x is not constant. Hence, for linear functions it is especially important to note the point at which elasticity is to be computed. If the functional relationship between y and x is of the exponential form y ¼ ax b then the elasticity is a constant, independent of where it is measured: ey , x ¼

∂y x x b1 ⋅ b ¼ b: ⋅ ¼ abx ∂x y ax

Chapter 2 Mathematics for Microeconomics

A logarithmic transformation of this equation also provides a very convenient alternative deﬁnition of elasticity. Because ln y ¼ ln a þ b ln x, we have ∂ ln y : (2.14) ∂ ln x Hence, elasticities can be calculated through “logarithmic differentiation.” As we shall see, this is frequently the easiest way to proceed in making such calculations. ey , x ¼ b ¼

QUERY: Are there any functional forms in addition to the exponential that have a constant elasticity, at least over some range?

Second-order partial derivatives The partial derivative of a partial derivative is directly analogous to the second derivative of a function of one variable and is called a second-order partial derivative. This may be written as ∂ð∂f =∂xi Þ ∂xj or more simply as ∂2 f ¼ fij : ∂xj ∂xi For the examples above: 1.

∂2 f ¼ f11 ¼ 2a ∂x1 ∂x1 f12 ¼ b f21 ¼ b f22 ¼ 2c:

2. f11 ¼ a 2 e ax1 þbx2 f12 ¼ abe ax1 þbx2 f21 ¼ abe ax1 þbx2 f22 ¼ b 2 e ax1 þbx2 a x 21 f12 ¼ 0

3. f11 ¼

f21 ¼ 0 b f22 ¼ 2 : x2

(2.15)

27

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Part 1 Introduction

Young’s theorem These examples illustrate the mathematical result that, under quite general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter. That is, fij ¼ fji

(2.16)

for any pair of variables xi , xj . This result is sometimes called “Young’s theorem.” For an intuitive explanation of the theorem, we can return to our mountain-climbing analogy. In this example, the theorem states that the gain in elevation a hiker experiences depends on the directions and distances traveled, but not on the order in which these occur. That is, the gain in altitude is independent of the actual path taken as long as the hiker proceeds from one set of map coordinates to another. He or she may, for example, go one mile north, then one mile east or proceed in the opposite order by ﬁrst going one mile east, then one mile north. In either case, the gain in elevation is the same since in both cases the hiker is moving from one speciﬁc place to another. In later chapters we will make good use of this result because it provides a very convenient way of showing some of the predictions that economic models make about behavior.2

Uses of second-order partials Second-order partial derivatives will play an important role in many of the economic theories that are developed throughout this book. Probably the most important examples relate to the “own” second-order partial, fii . This function shows how the marginal inﬂuence of xi on y ði:e:, ∂y=∂xi Þ changes as the value of xi increases. A negative value for fii is the mathematical way of indicating the economic idea of diminishing marginal effectiveness. Similarly, the cross-partial fij indicates how the marginal effectiveness of xi changes as xj increases. The sign of this effect could be either positive or negative. Young’s theorem indicates that, in general, such cross-effects are symmetric. More generally, the second-order partial derivatives of a function provide information about the curvature of the function. Later in this chapter we will see how such information plays an important role in determining whether various second-order conditions for a maximum are satisﬁed.

MAXIMIZATION OF FUNCTIONS OF SEVERAL VARIABLES Using partial derivatives, we can now discuss how to ﬁnd the maximum value for a function of several variables. To understand the mathematics used in solving this problem, an analogy to the one-variable case is helpful. In this one-variable case, we can picture an agent varying x by a small amount, dx, and observing the change in y, dy. This change is given by dy ¼ f 0 ðxÞdx:

(2.17)

The identity in Equation 2.17 records the fact that the change in y is equal to the change in x times the slope of the function. This formula is equivalent to the point-slope formula used for linear equations in basic algebra. As before, the necessary condition for a maximum is that dy ¼ 0 for small changes in x around the optimal point. Otherwise, y could be increased by suitable changes in x. But because dx does not necessarily equal 0 in Equation 2.17, dy ¼ 0 must imply that at the desired point, f 0 ðxÞ ¼ 0. This is another way of obtaining the ﬁrst-order condition for a maximum that we already derived. 2

Young’s theorem implies that the matrix of the second-order partial derivatives of a function is symmetric. This symmetry offers a number of economic insights. For a brief introduction to the matrix concepts used in economics, see the Extensions to this chapter.

Chapter 2 Mathematics for Microeconomics

Using this analogy, let’s look at the decisions made by an economic agent who must choose the levels of several variables. Suppose that this agent wishes to ﬁnd a set of x’s that will maximize the value of y ¼ f ðx1 , x2 , …, xn Þ. The agent might consider changing only one of the x’s, say x1 , while holding all the others constant. The change in y (that is, dy) that would result from this change in x1 is given by dy ¼

∂f dx ¼ f1 dx1 : ∂x1 1

This says that the change in y is equal to the change in x1 times the slope measured in the x1 direction. Using the mountain analogy again, the gain in altitude a climber heading north would achieve is given by the distance northward traveled times the slope of the mountain measured in a northward direction.

Total diﬀerential If all the x’s are varied by a small amount, the total effect on y will be the sum of effects such as that shown above. Therefore the total change in y is deﬁned to be ∂f ∂f ∂f dx1 þ dx2 þ … þ dx ∂x1 ∂x2 ∂xn n ¼ f1 dx1 þ f2 dx2 þ … þ fn dxn :

dy ¼

(2.18)

This expression is called the total differential of f and is directly analogous to the expression for the single-variable case given in Equation 2.17. The equation is intuitively sensible: The total change in y is the sum of changes brought about by varying each of the x’s.3

First-order condition for a maximum A necessary condition for a maximum (or a minimum) of the function f ðx1 , x2 , …, xn Þ is that dy ¼ 0 for any combination of small changes in the x’s. The only way this can happen is if, at the point being considered, f1 ¼ f2 ¼ … ¼ fn ¼ 0:

(2.19)

A point where Equations 2.19 hold is called a critical point. Equations 2.19 are the necessary conditions for a local maximum. To see this intuitively, note that if one of the partials (say, fi ) were greater (or less) than 0, then y could be increased by increasing (or decreasing) xi . An economic agent then could ﬁnd this maximal point by ﬁnding the spot where y does not respond to very small movements in any of the x’s. This is an extremely important result for economic analysis. It says that any activity (that is, the x’s) should be pushed to the point where its “marginal” contribution to the objective (that is, y) is 0. To stop short of that point would fail to maximize y.

3

The total differential in Equation 2.18 can be used to derive the chain rule as it applies to functions of several variables. Suppose that y ¼ f ðx1 , x2 Þ and that x1 ¼ gðzÞ and x2 ¼ hðzÞ. If all of these functions are differentiable, then it is possible to calculate the effects of a change in z on y. The total differential of y is dy ¼ f1 dx1 þ f2 dx2 : Dividing this equation by dz gives dy dx dx dg dh ¼ f1 1 þ f 2 2 ¼ f 1 þ f2 : dz dz dz dz dz Hence, calculating the effect of z on y requires calculating how z affects both of the determinants of y (that is, x1 and x2 ). If y depends on more than two variables, an analogous result holds. This result acts as a reminder to be rather careful to include all possible effects when calculating derivatives of functions of several variables.

29

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Part 1 Introduction

EXAMPLE 2.3 Finding a Maximum Suppose that y is a function of x1 and x2 given by y ¼ ðx1 1Þ2 ðx2 2Þ2 þ 10

(2.20)

or y ¼ x 21 þ 2x1 x 22 þ 4x2 þ 5: For example, y might represent an individual’s health (measured on a scale of 0 to 10), and x1 and x2 might be daily dosages of two health-enhancing drugs. We wish to ﬁnd values for x1 and x2 that make y as large as possible. Taking the partial derivatives of y with respect to x1 and x2 and applying the necessary conditions given by Equations 2.19 yields ∂y ¼ 2x1 þ 2 ¼ 0, ∂x1 ∂y ¼ 2x2 þ 4 ¼ 0 ∂x2

(2.21)

or x 1 ¼ 1, x 2 ¼ 2: The function is therefore at a critical point when x1 ¼ 1, x2 ¼ 2. At that point, y ¼ 10 is the best health status possible. A bit of experimentation provides convincing evidence that this is the greatest value y can have. For example, if x1 ¼ x2 ¼ 0, then y ¼ 5, or if x1 ¼ x2 ¼ 1, then y ¼ 9. Values of x1 and x2 larger than 1 and 2, respectively, reduce y because the negative quadratic terms in Equation 2.20 become large. Consequently, the point found by applying the necessary conditions is in fact a local (and global) maximum.4 QUERY: Suppose y took on a ﬁxed value (say, 5). What would the relationship implied between x1 and x2 look like? How about for y ¼ 7? Or y ¼ 10? (These graphs are contour lines of the function and will be examined in more detail in several later chapters. See also Problem 2.1.)

Second-order conditions Again, however, the conditions of Equations 2.19 are not sufﬁcient to ensure a maximum. This can be illustrated by returning to an already overworked analogy: All hilltops are (more or less) ﬂat, but not every ﬂat place is a hilltop. A second-order condition similar to Equation 2.6 is needed to ensure that the point found by applying Equations 2.19 is a local maximum. Intuitively, for a local maximum, y should be decreasing for any small changes in the x’s away from the critical point. As in the single-variable case, this necessarily involves looking at the second-order partial derivatives of the function f . These second-order partials must obey certain restrictions (analogous to the restriction that was derived in the singlevariable case) if the critical point found by applying Equations 2.19 is to be a local maximum. Later in this chapter we will look at these restrictions.

More formally, the point x1 ¼ 1, x2 ¼ 2 is a global maximum because the function described by Equation 2.20 is concave (see our discussion later in this chapter).

4

Chapter 2 Mathematics for Microeconomics

IMPLICIT FUNCTIONS Although mathematical equations are often written with a “dependent” variable (y) as a function of one or more independent variables (x), this is not the only way to write such a relationship. As a trivial example, the equation y ¼ mx þ b

(2.22)

y mx b ¼ 0

(2.23)

f ðx, y, m, bÞ ¼ 0,

(2.24)

can also be written as

or, even more generally, as

where this functional notation indicates a relationship between x and y that also depends on the slope (m) and intercept (b) parameters of the function, which do not change. Functions written in these forms are sometimes called implicit functions because the relationships between the variables and parameters are implicitly present in the equation rather than being explicitly calculated as, say, y as a function of x and the parameters m and b. Often it is a simple matter to translate from implicit functions to explicit ones. For example, the implicit function x þ 2y 4 ¼ 0

(2.25)

x ¼ 2y þ 4

(2.26)

x þ 2: 2

(2.27)

can easily be “solved” for x as or for y as y¼

Derivatives from implicit functions In many circumstances it is helpful to compute derivatives directly from implicit functions without solving for one of the variables directly. For example, the implicit function f ðx, yÞ ¼ 0 has a total differential of 0 ¼ fx dx þ fy dy, so dy f ¼ x: fy dx

(2.28)

Hence, the implicit derivative dy=dx can be found as the negative of the ratio of the partial derivatives of the implicit function, providing fy 6¼ 0. EXAMPLE 2.4 A Production Possibility Frontier—Again In Example 1.3 we examined a production possibility frontier for two goods of the form x 2 þ 0:25y 2 ¼ 200

(2.29)

f ðx, yÞ ¼ x 2 þ 0:25y 2 200 ¼ 0:

(2.30)

or, written implicitly,

Hence, (continued)

31

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Part 1 Introduction

EXAMPLE 2.4 CONTINUED fx ¼ 2x, fy ¼ 0:5y, and, by Equation 2.28, the opportunity cost trade-off between x and y is dy fx 2x 4x ¼ ¼ , ¼ fy dx 0:5y y

(2.31)

which is precisely the result we obtained earlier, with considerably less work. QUERY: Why does the trade-off between x and y here depend only on the ratio of x to y and not on the size of the labor force as reﬂected by the 200 constant?

Implicit function theorem It may not always be possible to solve implicit functions of the form gðx, yÞ ¼ 0 for unique explicit functions of the form y ¼ f ðxÞ. Mathematicians have analyzed the conditions under which a given implicit function can be solved explicitly with one variable being a function of other variables and various parameters. Although we will not investigate these conditions here, they involve requirements on the various partial derivatives of the function that are sufﬁcient to ensure that there is indeed a unique relationship between the dependent and independent variables.5 In many economic applications, these derivative conditions are precisely those required to ensure that the second-order conditions for a maximum (or a minimum) hold. Hence, in these cases, we will assert that the implicit function theorem holds and that it is therefore possible to solve explicitly for trade-offs among the variables involved.

THE ENVELOPE THEOREM One major application of the implicit function theorem, which will be used many times in this book, is called the envelope theorem; it concerns how the optimal value for a particular function changes when a parameter of the function changes. Because many of the economic problems we will be studying concern the effects of changing a parameter (for example, the effects that changing the market price of a commodity will have on an individual’s purchases), this is a type of calculation we will frequently make. The envelope theorem often provides a nice shortcut.

A speciﬁc example Perhaps the easiest way to understand the envelope theorem is through an example. Suppose y is a function of a single variable ðxÞ and a parameter ðaÞ given by y ¼ x 2 þ ax:

(2.32)

For different values of the parameter a, this function represents a family of inverted parabolas. If a is assigned a speciﬁc value, Equation 2.32 is a function of x only, and the value of x that maximizes y can be calculated. For example, if a ¼ 1, then x ¼ 12 and, for these values of x and a, y ¼ 14 (its maximal value). Similarly, if a ¼ 2, then x ¼ 1 and y ¼ 1. Hence, an increase 5

For a detailed discussion of the implicit function theorem in various contexts, see Carl P. Simon and Lawrence Blume, Mathematics for Economists (New York: W. W. Norton, 1994), chap. 15.

Chapter 2 Mathematics for Microeconomics

of 1 in the value of the parameter a has increased the maximum value of y by 34. In Table 2.1, integral values of a between 0 and 6 are used to calculate the optimal values for x and the associated values of the objective, y. Notice that as a increases, the maximal value for y also increases. This is also illustrated in Figure 2.3, which shows that the relationship between a and y is quadratic. Now we wish to calculate explicitly how y changes as the parameter a changes. TABLE 2.1

Optimal Values of y and x for Alternative Values of a in y ¼ x 2 þ ax

Value of a

Value of x

Value of y

0

0

0

1

1 2

1 4

2

1

1

3

3 2

9 4

4

2

4

5

5 2

25 4

6

3

9

FIGURE 2.3

Illustration of the Envelope Theorem

The envelope theorem states that the slope of the relationship between y (the maximum value of y ) and the parameter a can be found by calculating the slope of the auxiliary relationship found by substituting the respective optimal values for x into the objective function and calculating ∂y=∂a. y*

10 y* = f(a) 9 8 7 6 5 4 3 2 1

0

1

2

3

4

5

6

a

33

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Part 1 Introduction

A direct, time-consuming approach The envelope theorem states that there are two equivalent ways we can make this calculation. First, we can calculate the slope of the function in Figure 2.3 directly. To do so, we must solve Equation 2.32 for the optimal value of x for any value of a: dy ¼ 2x þ a ¼ 0; dx hence, x ¼

a : 2

Substituting this value of x in Equation 2.32 gives y ¼ ðx Þ2 þ aðx Þ a 2 a ¼ þa 2 2 a2 a2 a2 þ ¼ , 4 2 4 and this is precisely the relationship shown in Figure 2.3. From the previous equation, it is easy to see that ¼

dy 2a a ¼ ¼ (2.33) da 4 2 and, for example, at a ¼ 2, dy =da ¼ 1. That is, near a ¼ 2 the marginal impact of increasing a is to increase y by the same amount. Near a ¼ 6, any small increase in a will increase y by three times this change. Table 2.1 illustrates this result.

The envelope shortcut Arriving at this conclusion was a bit complicated. We had to ﬁnd the optimal value of x for each value of a and then substitute this value for x into the equation for y. In more general cases this may be quite burdensome since it requires repeatedly maximizing the objective function. The envelope theorem, providing an alternative approach, states that for small changes in a, dy =da can be computed by holding x constant at its optimal value and simply calculating ∂y=∂a from the objective function directly. Proceeding in this way gives

and at x we have

∂y ¼ x, ∂a

(2.34)

∂y a ¼ x ¼ : (2.35) ∂a 2 This is precisely the result obtained earlier. The reason that the two approaches yield identical results is illustrated in Figure 2.3. The tangents shown in the ﬁgure report values of y for a ﬁxed x . The tangents’ slopes are ∂y=∂a. Clearly, at y this slope gives the value we seek. This result is quite general, and we will use it at several places in this book to simplify our analysis. To summarize, the envelope theorem states that the change in the optimal value of a function with respect to a parameter of that function can be found by partially differentiating the objective function while holding x constant at its optimal value. That is,

Chapter 2 Mathematics for Microeconomics

dy ∂y ¼ x ¼ x ða Þ , (2.36) da ∂a where the notation provides a reminder that ∂y=∂a must be computed at that value of x that is optimal for the speciﬁc value of the parameter a being examined.

Many-variable case An analogous envelope theorem holds for the case where y is a function of several variables. Suppose that y depends on a set of x’s ðx1 , …, xn Þ and on a particular parameter of interest, say, a: y ¼ f ðx1 , …, xn , aÞ:

(2.37)

Finding an optimal value for y would consist of solving n ﬁrst-order equations of the form ∂y ¼ 0 ði ¼ 1, …, n Þ, (2.38) ∂xi and a solution to this process would yield optimal values for these x’s x , x , …, x that 1

2

n

would implicitly depend on the parameter a. Assuming the second-order conditions are met, the implicit function theorem would apply in this case and ensure that we could solve each x i as a function of the parameter a: x ¼ x ðaÞ, 1

x 2 ¼ .. . x ¼ n

1

x 2 ðaÞ,

(2.39)

x n ðaÞ:

Substituting these functions into our original objective (Equation 2.37) yields an expression in which the optimal value of y (say, y ) depends on the parameter a both directly and indirectly through the effect of a on the x ’s: y ¼ f ½x 1 ðaÞ, x 2 ðaÞ, …, x n ðaÞ, a: Totally differentiating this expression with respect to a yields dy ∂f dx1 ∂f dx2 ∂f dxn ∂f þ þ…þ þ ¼ : ⋅ ⋅ ⋅ da ∂x1 da ∂x2 da ∂xn da ∂a

(2.40)

But, because of the ﬁrst-order conditions all of these terms except the last are equal to 0 if the x’s are at their optimal values. Hence, again we have the envelope result: dy ∂f ¼ , (2.41) da ∂a where this derivative is to be evaluated at the optimal values for the x’s. EXAMPLE 2.5 The Envelope Theorem: Health Status Revisited Earlier, in Example 2.3, we examined the maximum values for the health status function y ¼ ðx1 1Þ2 ðx2 2Þ2 þ 10

(2.42)

x 1 ¼ 1, x 2 ¼ 2,

(2.43)

and found that

(continued)

35

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Part 1 Introduction

EXAMPLE 2.5 CONTINUED and y ¼ 10: Suppose now we use the arbitrary parameter a instead of the constant 10 in Equation 2.42. Here a might represent a measure of the best possible health for a person, but this value would obviously vary from person to person. Hence, y ¼ f ðx1 , x2 , aÞ ¼ ðx1 1Þ2 ðx2 2Þ2 þ a:

(2.44) In this case the optimal values for x1 and x2 do not depend on a (they are always x 1 ¼ 1, x 2 ¼ 2), so at those optimal values we have y ¼ a

(2.45)

and

dy ¼ 1: (2.46) da People with “naturally better health” will have concomitantly higher values for y , providing they choose x1 and x2 optimally. But this is precisely what the envelope theorem indicates, because dy ∂f ¼ ¼1 (2.47) da ∂a from Equation 2.44. Increasing the parameter a simply increases the optimal value for y by an identical amount (again, assuming the dosages of x1 and x2 are correctly chosen). QUERY: Suppose we focused instead on the optimal dosage for x1 in Equation 2.42—that is, suppose we used a general parameter, say b, instead of 1. Explain in words and using mathematics why ∂y =∂b would necessarily be 0 in this case.

CONSTRAINED MAXIMIZATION So far we have focused our attention on ﬁnding the maximum value of a function without restricting the choices of the x’s available. In most economic problems, however, not all values for the x’s are feasible. In many situations, for example, it is required that all the x’s be positive. This would be true for the problem faced by the manager choosing output to maximize proﬁts; a negative output would have no meaning. In other instances the x’s may be constrained by economic considerations. For example, in choosing the items to consume, an individual is not able to choose any quantities desired. Rather, choices are constrained by the amount of purchasing power available; that is, by this person’s budget constraint. Such constraints may lower the maximum value for the function being maximized. Because we are not able to choose freely among all the x’s, y may not be as large as it could be. The constraints would be “nonbinding” if we could obtain the same level of y with or without imposing the constraint.

Lagrangian multiplier method One method for solving constrained maximization problems is the Lagrangian multiplier method, which involves a clever mathematical trick that also turns out to have a useful economic interpretation. The rationale of this method is quite simple, although no rigorous

Chapter 2 Mathematics for Microeconomics

presentation will be attempted here.6 In a prior section, the necessary conditions for a local maximum were discussed. We showed that at the optimal point all the partial derivatives of f must be 0. There are therefore n equations (fi ¼ 0 for i ¼ 1, …, n) in n unknowns (the x’s). Generally, these equations can be solved for the optimal x’s. When the x’s are constrained, however, there is at least one additional equation (the constraint) but no additional variables. The set of equations therefore is overdetermined. The Lagrangian technique introduces an additional variable (the Lagrangian multiplier), which not only helps to solve the problem at hand (because there are now n þ 1 equations in n þ 1 unknowns), but also has an interpretation that is useful in a variety of economic circumstances.

The formal problem More speciﬁcally, suppose that we wish to ﬁnd the values of x1 , x2 , …, xn that maximize y ¼ f ðx1 , x2 , …, xn Þ,

(2.48)

subject to a constraint that permits only certain values of the x’s to be used. A general way of writing that constraint is (2.49) gðx1 , x2 , …, xn Þ ¼ 0, where the function7 g represents the relationship that must hold among all the x’s.

First-order conditions The Lagrangian multiplier method starts with setting up the expression ℒ ¼ f ðx1 , x2 , …, xn Þ þ λgðx1 , x2 , …, xn Þ,

(2.50)

where λ is an additional variable called the Lagrangian multiplier. Later we will interpret this new variable. First, however, notice that when the constraint holds, ℒ and f have the same value [because gðx1 , x2 , …, xn Þ ¼ 0]. Consequently, if we restrict our attention only to values of the x’s that satisfy the constraint, ﬁnding the constrained maximum value of f is equivalent to ﬁnding a critical value of ℒ. Let us proceed then to do so, treating λ also as a variable (in addition to the x’s). From Equation 2.50, the conditions for a critical point are: ∂ℒ ¼ f1 þ λg1 ¼ 0, ∂x1 ∂ℒ ¼ f2 þ λg2 ¼ 0, ∂x2 .. .

(2.51)

∂ℒ ¼ fn þ λgn ¼ 0, ∂xn ∂ℒ ¼ gðx1 , x2 , …, xn Þ ¼ 0: ∂λ Equations 2.51 are then the conditions for a critical point for the function ℒ. Notice that there are n þ 1 equations (one for each x and a ﬁnal one for λ) in n þ 1 unknowns. The equations can generally be solved for x1 , x2 , …, xn , and λ. Such a solution will have two 6 For a detailed presentation, see A. K. Dixit, Optimization in Economic Theory, 2nd ed. (Oxford: Oxford University Press, 1990), chap. 2.

As we pointed out earlier, any function of x1 , x2 , …, xn can be written in this implicit way. For example, the constraint x1 þ x2 ¼ 10 could be written 10 x1 x2 ¼ 0. In later chapters, we will usually follow this procedure in dealing with constraints. Often the constraints we examine will be linear.

7

37

38

Part 1 Introduction

properties: (1) the x’s will obey the constraint because the last equation in 2.51 imposes that condition; and (2) among all those values of x’s that satisfy the constraint, those that also solve Equations 2.51 will make ℒ (and hence f ) as large as possible (assuming second-order conditions are met). The Lagrangian multiplier method therefore provides a way to ﬁnd a solution to the constrained maximization problem we posed at the outset.8 The solution to Equations 2.51 will usually differ from that in the unconstrained case (see Equations 2.19). Rather than proceeding to the point where the marginal contribution of each x is 0, Equations 2.51 require us to stop short because of the constraint. Only if the constraint were ineffective (in which case, as we show below, λ would be 0) would the constrained and unconstrained equations (and their respective solutions) agree. These revised marginal conditions have economic interpretations in many different situations.

Interpretation of the Lagrangian multiplier So far we have used the Lagrangian multiplier (λ) only as a mathematical “trick” to arrive at the solution we wanted. In fact, that variable also has an important economic interpretation, which will be central to our analysis at many points in this book. To develop this interpretation, rewrite the ﬁrst n equations of 2.51 as f1 f f ¼ 2 ¼ … ¼ n ¼ λ: g1 g2 gn

(2.52)

In other words, at the maximum point, the ratio of fi to gi is the same for every xi . The numerators in Equations 2.52 are the marginal contributions of each x to the function f . They show the marginal beneﬁt that one more unit of xi will have for the function that is being maximized (that is, for f ). A complete interpretation of the denominators in Equations 2.52 is probably best left until we encounter these ratios in actual economic applications. There we will see that these usually have a “marginal cost” interpretation. That is, they reﬂect the added burden on the constraint of using slightly more xi . As a simple illustration, suppose the constraint required that total spending on x1 and x2 be given by a ﬁxed dollar amount, F . Hence, the constraint would be p1 x1 þ p2 x2 ¼ F (where pi is the per unit cost of xi ). Using our present terminology, this constraint would be written in implicit form as gðx1 , x2 Þ ¼ F p1 x1 p2 x2 ¼ 0:

(2.53)

gi ¼ pi

(2.54)

In this situation, then, and the derivative gi does indeed reﬂect the per unit, marginal cost of using xi . Practically all of the optimization problems we will encounter in later chapters have a similar interpretation for the denominators in Equations 2.52.

Lagrangian multiplier as a beneﬁt-cost ratio Now we can give Equations 2.52 an intuitive interpretation. They indicate that, at the optimal choices for the x’s, the ratio of the marginal beneﬁt of increasing xi to the marginal cost of increasing xi should be the same for every x. To see that this is an obvious condition 8

Strictly speaking, these are the necessary conditions for an interior local maximum. In some economic problems, it is necessary to amend these conditions (in fairly obvious ways) to take account of the possibility that some of the x’s may be on the boundary of the region of permissible x’s. For example, if all of the x’s are required to be nonnegative, it may be that the conditions of Equations 2.51 will not hold exactly, because these may require negative x’s. We look at this situation later in this chapter.

Chapter 2 Mathematics for Microeconomics

for a maximum, suppose that it were not true: Suppose that the “beneﬁt-cost ratio” were higher for x1 than for x2 . In this case, slightly more x1 should be used in order to achieve a maximum. Consider using more x1 but giving up just enough x2 to keep g (the constraint) constant. Hence, the marginal cost of the additional x1 used would equal the cost saved by using less x2 . But because the beneﬁt-cost ratio (the amount of beneﬁt per unit of cost) is greater for x1 than for x2 , the additional beneﬁts from using more x1 would exceed the loss in beneﬁts from using less x2 . The use of more x1 and appropriately less x2 would then increase y because x1 provides more “bang for your buck.” Only if the marginal beneﬁt–marginal cost ratios are equal for all the x’s will there be a local maximum, one in which no small changes in the x’s can increase the objective. Concrete applications of this basic principle are developed in many places in this book. The result is fundamental for the microeconomic theory of optimizing behavior. The Lagrangian multiplier (λ) can also be interpreted in light of this discussion. λ is the common beneﬁt-cost ratio for all the x’s. That is, λ¼

marginal benefit of xi marginal cost of xi

(2.55)

for every xi . If the constraint were relaxed slightly, it would not matter exactly which x is changed (indeed, all the x’s could be altered), because, at the margin, each promises the same ratio of beneﬁts to costs. The Lagrangian multiplier then provides a measure of how such an overall relaxation of the constraint would affect the value of y. In essence, λ assigns a “shadow price” to the constraint. A high λ indicates that y could be increased substantially by relaxing the constraint, because each x has a high beneﬁt-cost ratio. A low value of λ, on the other hand, indicates that there is not much to be gained by relaxing the constraint. If the constraint is not binding at all, λ will have a value of 0, thereby indicating that the constraint is not restricting the value of y. In such a case, ﬁnding the maximum value of y subject to the constraint would be identical to ﬁnding an unconstrained maximum. The shadow price of the constraint is 0. This interpretation of λ can also be shown using the envelope theorem as described later in this chapter.9

Duality This discussion shows that there is a clear relationship between the problem of maximizing a function subject to constraints and the problem of assigning values to constraints. This reﬂects what is called the mathematical principle of “duality”: Any constrained maximization problem has an associated dual problem in constrained minimization that focuses attention on the constraints in the original (primal) problem. For example, to jump a bit ahead of our story, economists assume that individuals maximize their utility, subject to a budget constraint. This is the consumer’s primal problem. The dual problem for the consumer is to minimize the expenditure needed to achieve a given level of utility. Or, a ﬁrm’s primal problem may be to minimize the total cost of inputs used to produce a given level of output, whereas the dual problem is to maximize output for a given cost of inputs purchased. Many similar examples will be developed in later chapters. Each illustrates that there are always two ways to look at any constrained optimization problem. Sometimes taking a frontal attack by analyzing the primal problem can lead to greater insights. In other instances, the “back door” approach of examining the dual problem may be more instructive. Whichever route is taken, the results will generally, though not always, be identical, so the choice made will mainly be a matter of convenience.

9

The discussion in the text concerns problems involving a single constraint. In general, one can handle m constraints ðm < nÞ by simply introducing m new variables (Lagrangian multipliers) and proceeding in an analogous way to that discussed above.

39

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Part 1 Introduction

EXAMPLE 2.6 Constrained Maximization: Health Status Yet Again Let’s return once more to our (perhaps tedious) health maximization problem. As before, the individual’s goal is to maximize y ¼ x 21 þ 2x1 x 22 þ 4x2 þ 5, but now assume that choices of x1 and x2 are constrained by the fact that he or she can only tolerate one drug dose per day. That is, x1 þ x 2 ¼ 1

(2.56)

or 1 x1 x2 ¼ 0: Notice that the original optimal point ðx1 ¼ 1, x2 ¼ 2Þ is no longer attainable because of the constraint on possible dosages: other values must be found. To do so, we ﬁrst set up the Lagrangian expression: ℒ ¼ x 21 þ 2x1 x 22 þ 4x2 þ 5 þ λð1 x1 x2 Þ:

(2.57)

Differentiation of ℒ with respect to x1 , x2 , and λ yields the following necessary condition for a constrained maximum: ∂ℒ ¼ 2x1 þ 2 λ ¼ 0, ∂x1 ∂ℒ ¼ 2x2 þ 4 λ ¼ 0, ∂x2

(2.58)

∂ℒ ¼ 1 x1 x2 ¼ 0: ∂λ These equations must now be solved for the optimal values of x1 , x2 , and λ. Using the ﬁrst and second equations gives 2x1 þ 2 ¼ λ ¼ 2x2 þ 4 or x1 ¼ x2 1:

(2.59)

Substitution of this value for x1 into the constraint yields the solution: x2 ¼ 1, x1 ¼ 0:

(2.60)

In words, if this person can tolerate only one dose of drugs, he or she should opt for taking only the second drug. By using either of the ﬁrst two equations, it is easy to complete our solution by showing that λ ¼ 2:

(2.61)

This, then, is the solution to the constrained maximum problem. If x1 ¼ 0, x2 ¼ 1, then y takes on the value 8. Constraining the values of x1 and x2 to sum to 1 has reduced the maximum value of health status, y, from 10 to 8. QUERY: Suppose this individual could tolerate two doses per day. Would you expect y to increase? Would increases in tolerance beyond three doses per day have any effect on y?

Chapter 2 Mathematics for Microeconomics

EXAMPLE 2.7 Optimal Fences and Constrained Maximization Suppose a farmer had a certain length of fence, P , and wished to enclose the largest possible rectangular area. What shape area should the farmer choose? This is clearly a problem in constrained maximization. To solve it, let x be the length of one side of the rectangle and y be the length of the other side. The problem then is to choose x and y so as to maximize the area of the ﬁeld (given by A ¼ x ⋅ y), subject to the constraint that the perimeter is ﬁxed at P ¼ 2x þ 2y. Setting up the Lagrangian expression gives ℒ ¼ x ⋅ y þ λðP 2x 2yÞ,

(2.62)

where λ is an unknown Lagrangian multiplier. The ﬁrst-order conditions for a maximum are ∂ℒ ¼ y 2λ ¼ 0, ∂x ∂ℒ (2.63) ¼ x 2λ ¼ 0, ∂y ∂ℒ ¼ P 2x 2y ¼ 0: ∂λ The three equations in 2.63 must be solved simultaneously for x, y, and λ. The ﬁrst two equations say that y=2 ¼ x=2 ¼ λ, showing that x must be equal to y (the ﬁeld should be square). They also imply that x and y should be chosen so that the ratio of marginal beneﬁts to marginal cost is the same for both variables. The beneﬁt (in terms of area) of one more unit of x is given by y (area is increased by 1 y), and the marginal cost (in terms of perimeter) is 2 (the available perimeter is reduced by 2 for each unit that the length of side x is increased). The maximum conditions state that this ratio should be equal for each of the variables. Since we have shown that x ¼ y, we can use the constraint to show that x¼y¼

P , 4

(2.64)

and, because y ¼ 2λ, λ¼

P : 8

(2.65)

Interpretation of the Lagrangian Multiplier. If the farmer were interested in knowing how much more ﬁeld could be fenced by adding an extra yard of fence, the Lagrangian multiplier suggests that he or she could ﬁnd out by dividing the present perimeter by 8. Some speciﬁc numbers might make this clear. Suppose that the ﬁeld currently has a perimeter of 400 yards. If the farmer has planned “optimally,” the ﬁeld will be a square with 100 yards ð¼ P =4Þ on a side. The enclosed area will be 10,000 square yards. Suppose now that the perimeter (that is, the available fence) were enlarged by one yard. Equation 2.65 would then “predict” that the total area would be increased by approximately 50 ð¼ P =8Þ square yards. That this is indeed the case can be shown as follows: Because the perimeter is now 401 yards, each side of the square will be 401=4 yards. The total area of the ﬁeld is therefore ð401=4Þ2, which, according to the author’s calculator, works out to be 10,050.06 square yards. Hence, the “prediction” of a 50square-yard increase that is provided by the Lagrangian multiplier proves to be remarkably close. As in all constrained maximization problems, here the Lagrangian multiplier provides useful information about the implicit value of the constraint. (continued)

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EXAMPLE 2.7 CONTINUED Duality. The dual of this constrained maximization problem is that for a given area of a rectangular ﬁeld, the farmer wishes to minimize the fence required to surround it. Mathematically, the problem is to minimize P ¼ 2x þ 2y,

(2.66)

A ¼ x ⋅ y:

(2.67)

ℒD ¼ 2x þ 2y þ λD ðA x ⋅ yÞ

(2.68)

subject to the constraint Setting up the Lagrangian expression (where the D denotes the dual concept) yields the following ﬁrst-order conditions for a minimum: ∂ℒD ¼ 2 λD ⋅ y ¼ 0, ∂x ∂ℒD ¼ 2 λD ⋅ x ¼ 0, ∂y ∂ℒD ¼ A x ⋅ y ¼ 0: ∂λD Solving these equations as before yields the result pﬃﬃﬃﬃ x ¼ y ¼ A:

(2.69)

(2.70)

Again, the ﬁeld should be square if the length of fence is to be minimized. The value of the Lagrangian multiplier in this problem is λD ¼

2 2 2 ¼ ¼ pﬃﬃﬃﬃ : y x A

(2.71)

As before, this Lagrangian multiplier indicates the relationship between the objective (minimizing fence) and the constraint (needing to surround the ﬁeld). If the ﬁeld were 10,000 square yards, as we saw before, 400 yards of fence would be needed. pﬃﬃﬃﬃ Increasing the ﬁeld by one square yard would require about .02 more yards of fence (¼ 2= A ¼ 2=100). The reader may wish to ﬁre up his or her calculator to show this is indeed the case—a fence 100.005 yards on each side will exactly enclose 10,001 square yards. Here, as in most duality problems, the value of the Lagrangian in the dual is the reciprocal of the value for the Lagrangian in the primal problem. Both provide the same information, although in a somewhat different form. QUERY: An implicit constraint here is that the farmer’s ﬁeld be rectangular. If this constraint were not imposed, what shape ﬁeld would enclose maximal area? How would you prove that?

ENVELOPE THEOREM IN CONSTRAINED MAXIMIZATION PROBLEMS The envelope theorem, which we discussed previously in connection with unconstrained maximization problems, also has important applications in constrained maximization problems. Here we will provide only a brief presentation of the theorem. In later chapters we will look at a number of applications.

Chapter 2 Mathematics for Microeconomics

Suppose we seek the maximum value of y ¼ f ðx1 , …, xn ; aÞ,

(2.72)

gðx1 , …, xn ; aÞ ¼ 0,

(2.73)

subject to the constraint

where we have made explicit the dependence of the functions f and g on some parameter a. As we have shown, one way to solve this problem is to set up the Lagrangian expression ℒ ¼ f ðx1 , …, xn ; aÞ þ λgðx1 , …, xn ; aÞ

(2.74)

and solve the ﬁrst-order conditions (see Equations 2.51) for the optimal, constrained values x 1 , …, x n . Alternatively, it can be shown that dy ∂ℒ ¼ ðx , …, x n ; aÞ: (2.75) da ∂a 1 That is, the change in the maximal value of y that results when the parameter a changes (and all of the x’s are recalculated to new optimal values) can be found by partially differentiating the Lagrangian expression (Equation 2.74) and evaluating the resultant partial derivative at the optimal point.10 Hence, the Lagrangian expression plays the same role in applying the envelope theorem to constrained problems as does the objective function alone in unconstrained problems. As a simple exercise, the reader may wish to show that this result holds for the problem of fencing a rectangular ﬁeld described in Example 2.7.11

INEQUALITY CONSTRAINTS In some economic problems the constraints need not hold exactly. For example, an individual’s budget constraint requires that he or she spend no more than a certain amount per period, but it is at least possible to spend less than this amount. Inequality constraints also arise in the values permitted for some variables in economic problems. Usually, for example, economic variables must be nonnegative (though they can take on the value of zero). In this section we will show how the Lagrangian technique can be adapted to such circumstances. Although we will encounter only a few problems later in the text that require this mathematics, development here will illustrate a few general principles that are quite consistent with economic intuition.

A two-variable example In order to avoid much cumbersome notation, we will explore inequality constraints only for the simple case involving two choice variables. The results derived are readily generalized. Suppose that we seek to maximize y ¼ f ðx1 , x2 Þ subject to three inequality constraints:

10

For a more complete discussion of the envelope theorem in constrained maximization problems, see Eugene Silberberg and Wing Suen, The Structure of Economics: A Mathematical Analysis, 3rd ed. (Boston: Irwin/McGraw-Hill, 2001), pp. 159–61.

11

For the primal problem, the perimeter P is the parameter of principal interest. By solving for the optimal values of x and y and substituting into the expression for the area ðAÞ of the ﬁeld, it is easy to show that dA=dP ¼ P=8. Differentiation of the Lagrangian expression (Equation 2.62) yields ∂ℒ=∂P ¼ λ and, at the optimal values of x and y, dA=dP ¼ ∂ℒ=∂P ¼ λ ¼ P =8. The envelope theorem in this case then offers further proof that the Lagrangian multiplier can be used to assign an implicit value to the constraint.

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1: g x1 , x2 0; 2: x1 0; 3: x2 0:

and

(2.76)

Hence, we are allowing for the possibility that the constraint we introduced before need not hold exactly (a person need not spend all of his or her income) and for the fact that both of the x’s must be nonnegative (as in most economic problems).

Slack variables One way to solve this optimization problem is to introduce three new variables ða, b, and cÞ that convert the inequality constraints in Equation 2.76 into equalities. To ensure that the inequalities continue to hold, we will square these new variables, ensuring that the resulting values are positive. Using this procedure, the inequality constraints become 1: g x1 , x2 a 2 ¼ 0; 2: x1 b 2 ¼ 0;

and

(2.77)

3: x2 c 2 ¼ 0: Any solution that obeys these three equality constraints will also obey the inequality constraints. It will also turn out that the optimal values for a, b, and c will provide several insights into the nature of the solutions to a problem of this type.

Solution by the method of Lagrange By converting the original problem involving inequalities into one involving equalities, we are now in a position to use Lagrangian methods to solve it. Because there are three constraints, we must introduce three Lagrangian multipliers: λ1, λ2, and λ3. The full Lagrangian expression is ℒ ¼ f ðx1 , x2 Þ þ λ1 ½gðx1 , x2 Þ a 2 þ λ2 ðx1 b 2 Þ þ λ3 ðx2 c 2 Þ:

(2.78)

We wish to ﬁnd the values of x1 , x2 , a, b, c, λ1, λ2, and λ3 that constitute a critical point for this expression. This will necessitate eight ﬁrst-order conditions: ∂ℒ ∂x1 ∂ℒ ∂x2 ∂ℒ ∂a ∂ℒ ∂b ∂ℒ ∂c ∂ℒ ∂λ1 ∂ℒ ∂λ2 ∂ℒ ∂λ3

¼ f1 þ λ1 g1 þ λ2 ¼ 0, ¼ f2 þ λ1 g2 þ λ3 ¼ 0, ¼ 2aλ1 ¼ 0, ¼ 2bλ2 ¼ 0, (2.79) ¼ 2cλ3 ¼ 0, ¼ gðx1 , x2 Þ a 2 ¼ 0, ¼ x1 b 2 ¼ 0, ¼ x2 c 2 ¼ 0,

In many ways these conditions resemble those we derived earlier for the case of a single equality constraint (see Equation 2.51). For example, the ﬁnal three conditions merely repeat the three

Chapter 2 Mathematics for Microeconomics

revised constraints. This ensures that any solution will obey these conditions. The ﬁrst two equations also resemble the optimal conditions developed earlier. If λ2 and λ3 were 0, the conditions would in fact be identical. But the presence of the additional Lagrangian multipliers in the expressions shows that the customary optimality conditions may not hold exactly here.

Complementary slackness The three equations involving the variables a, b, and c provide the most important insights into the nature of solutions to problems involving inequality constraints. For example, the third line in Equation 2.79 implies that, in the optimal solution, either λ1 or a must be 0.12 In the second case ða ¼ 0Þ, the constraint gðx1 , x2 Þ ¼ 0 holds exactly and the calculated value of λ1 indicates its relative importance to the objective function, f . On the other hand, if a 6¼ 0, then λ1 ¼ 0 and this shows that the availability of some slackness in the constraint implies that its value to the objective is 0. In the consumer context, this means that if a person does not spend all his or her income, even more income would do nothing to raise his or her well-being. Similar complementary slackness relationships also hold for the choice variables x1 and x2 . For example, the fourth line in Equation 2.79 requires that the optimal solution have either b or λ2 be 0. If λ2 ¼ 0 then the optimal solution has x1 > 0, and this choice variable meets the precise beneﬁt-cost test that f1 þ λ1 g1 ¼ 0. Alternatively, solutions where b ¼ 0 have x1 ¼ 0, and also require that λ2 > 0. So, such solutions do not involve any use of x1 because that variable does not meet the beneﬁt-cost test as shown by the ﬁrst line of Equation 2.79, which implies that f1 þ λ1 g1 < 0. An identical result holds for the choice variable x2 . These results, which are sometimes called Kuhn-Tucker conditions after their discoverers, show that the solutions to optimization problems involving inequality constraints will differ from similar problems involving equality constraints in rather simple ways. Hence, we cannot go far wrong by working primarily with constraints involving equalities and assuming that we can rely on intuition to state what would happen if the problems actually involved inequalities. That is the general approach we will take in this book.13

SECOND-ORDER CONDITIONS So far our discussion of optimization has focused primarily on necessary (ﬁrst-order) conditions for ﬁnding a maximum. That is indeed the practice we will follow throughout much of this book because, as we shall see, most economic problems involve functions for which the second-order conditions for a maximum are also satisﬁed. In this section we give a brief analysis of the connection between second-order conditions for a maximum and the related curvature conditions that functions must have to ensure that these hold. The economic explanations for these curvature conditions will be discussed throughout the text.

Functions of one variable First consider the case in which the objective, y, is a function of only a single variable, x. That is, y ¼ f ðxÞ:

(2.80)

12

We will not examine the degenerate case where both of these variables are 0.

13

The situation can become much more complex when calculus cannot be relied upon to give a solution, perhaps because some of the functions in a problem are not differentiable. For a discussion, see Avinask K. Dixit, Optimization in Economic Theory, 2nd ed. (Oxford: Oxford University Press, 1990).

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A necessary condition for this function to attain its maximum value at some point is that dy (2.81) ¼ f 0 ðx Þ ¼ 0 dx at that point. To ensure that the point is indeed a maximum, we must have y decreasing for movements away from it. We already know (by Equation 2.81) that for small changes in x, the value of y does not change; what we need to check is whether y is increasing before that “plateau” is reached and declining thereafter. We have already derived an expression for the change in yðdyÞ, which is given by the total differential dy ¼ f 0 ðxÞdx:

(2.82)

What we now require is that dy be decreasing for small increases in the value of x. The differential of Equation 2.82 is given by d½ f 0 ðxÞdx 00 00 2 (2.83) d ðdy Þ ¼ d 2 y ¼ ⋅ dx ¼ f ðx Þdx ⋅ dx ¼ f ðx Þdx : dx But d 2y < 0 implies that f 00 ðxÞdx 2 < 0,

(2.84)

2

and since dx must be positive (because anything squared is positive), we have f 00 ðxÞ < 0

(2.85)

as the required second-order condition. In words, this condition requires that the function f have a concave shape at the critical point (contrast Figures 2.1 and 2.2). Similar curvature conditions will be encountered throughout this section. EXAMPLE 2.8 Proﬁt Maximization Again In Example 2.1 we considered the problem of ﬁnding the maximum of the function π ¼ 1,000q 5q 2 :

(2.86)

The ﬁrst-order condition for a maximum requires dπ ¼ 1,000 10q ¼ 0 dq

(2.87)

q ¼ 100:

(2.88)

or

The second derivative of the function is given by d 2π ¼ 10 < 0, dq 2

(2.89)

and hence the point q ¼ 100 obeys the sufﬁcient conditions for a local maximum. QUERY: Here the second derivative is negative not only at the optimal point; it is always negative. What does that imply about the optimal point? How should the fact that the second derivative is a constant be interpreted?

Chapter 2 Mathematics for Microeconomics

Functions of two variables As a second case, we consider y as a function of two independent variables: y ¼ f ðx1 , x2 Þ:

(2.90)

A necessary condition for such a function to attain its maximum value is that its partial derivatives, in both the x1 and the x2 directions, be 0. That is, ∂y ¼ f1 ¼ 0, ∂x1 (2.91) ∂y ¼ f2 ¼ 0: ∂x2 A point that satisﬁes these conditions will be a “ﬂat” spot on the function (a point where dy ¼ 0) and therefore will be a candidate for a maximum. To ensure that the point is a local maximum, y must diminish for movements in any direction away from the critical point: In pictorial terms there is only one way to leave a true mountaintop, and that is to go down.

An intuitive argument Before describing the mathematical properties required of such a point, an intuitive approach may be helpful. If we consider only movements in the x1 direction, the required condition is clear: The slope in the x1 direction (that is, the partial derivative f1 ) must be diminishing at the critical point. This is a direct application of our discussion of the single-variable case. It shows that, for a maximum, the second partial derivative in the x1 direction must be negative. An identical argument holds for movements only in the x2 direction. Hence, both own second partial derivatives ð f11 and f22 Þ must be negative for a local maximum. In our mountain analogy, if attention is conﬁned only to north-south or east-west movements, the slope of the mountain must be diminishing as we cross its summit—the slope must change from positive to negative. The particular complexity that arises in the two-variable case involves movements through the optimal point that are not solely in the x1 or x2 directions (say, movements from northeast to southwest). In such cases, the second-order partial derivatives do not provide complete information about how the slope is changing near the critical point. Conditions must also be placed on the cross-partial derivative ð f12 ¼ f21 Þ to ensure that dy is decreasing for movements through the critical point in any direction. As we shall see, those conditions amount to requiring that the own second-order partial derivatives be sufﬁciently negative so as to counterbalance any possible “perverse” cross-partial derivatives that may exist. Intuitively, if the mountain falls away steeply enough in the north-south and east-west directions, relatively minor failures to do so in other directions can be compensated for.

A formal analysis We now proceed to make these points more formally. What we wish to discover are the conditions that must be placed on the second partial derivatives of the function f to ensure that d 2 y is negative for movements in any direction through the critical point. Recall ﬁrst that the total differential of the function is given by dy ¼ f1 dx1 þ f2 dx2 :

(2.92)

The differential of that function is given by d 2 y ¼ ðf11 dx1 þ f12 dx2 Þdx1 þ ð f21 dx1 þ f22 dx2 Þdx2

(2.93)

d 2 y ¼ f11 dx 21 þ f12 dx2 dx1 þ f21 dx1 dx2 þ f22 dx 22 :

(2.94)

or

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Because, by Young’s theorem, f12 ¼ f21 , we can arrange terms to get d 2 y ¼ f11 dx 21 þ 2f12 dx1 dx2 þ f22 dx 22 :

(2.95)

For Equation 2.95 to be unambiguously negative for any change in the x’s (that is, for any choices of dx1 and dx2 ), it is obviously necessary that f11 and f22 be negative. If, for example, dx2 ¼ 0, then d 2 y ¼ f11 dx 21

(2.96)

f11 < 0:

(2.97)

and d 2 y < 0 implies An identical argument can be made for f22 by setting dx1 ¼ 0. If neither dx1 nor dx2 is 0, we then must consider the cross partial, f12 , in deciding whether or not d 2 y is unambiguously negative. Relatively simple algebra can be used to show that the required condition is14 f11 f22 f 212 > 0:

(2.98)

Concave functions Intuitively, what Equation 2.98 requires is that the own second partial derivatives ð f11 and f22 Þ be sufﬁciently negative so that their product (which is positive) will outweigh any possible perverse effects from the cross-partial derivatives ð f12 ¼ f21 Þ. Functions that obey such a condition are called concave functions. In three dimensions, such functions resemble inverted teacups (for an illustration, see Example 2.10). This image makes it clear that a ﬂat spot on such a function is indeed a true maximum because the function always slopes downward from such a spot. More generally, concave functions have the property that they always lie below any plane that is tangent to them—the plane deﬁned by the maximum value of the function is simply a special case of this property. EXAMPLE 2.9 Second-Order Conditions: Health Status for the Last Time In Example 2.3 we considered the health status function y ¼ f x1 , x2 ¼ x 21 þ 2x1 x 22 þ 4x2 þ 5:

(2.99)

The ﬁrst-order conditions for a maximum are f1 ¼ 2x1 þ 2 ¼ 0, f2 ¼ 2x2 þ 4 ¼ 0

(2.100)

or x 1 ¼ 1, x 2 ¼ 2:

(2.101)

The proof proceeds by adding and subtracting the term ð f12 dx2 Þ2 =f11 to Equation 2.95 and factoring. But this approach is only applicable to this special case. A more easily generalized approach that uses matrix algebra recognizes that Equation 2.95 is a “Quadratic Form” in dx1 and dx2 , and that Equations 2.97 and 2.98 amount to requiring that the Hessian matrix

f11 f12 f21 f22

14

be “negative deﬁnite.” In particular, Equation 2.98 requires that the determinant of this Hessian be positive. For a discussion, see the Extensions to this chapter.

Chapter 2 Mathematics for Microeconomics

The second-order partial derivatives for Equation 2.99 are f11 ¼ 2, f22 ¼ 2, f12 ¼ 0:

(2.102)

These derivatives clearly obey Equations 2.97 and 2.98, so both necessary and sufﬁcient conditions for a local maximum are satisﬁed.15 QUERY: Describe the concave shape of the health status function and indicate why it has only a single global maximum value.

Constrained maximization As another illustration of second-order conditions, consider the problem of choosing x1 and x2 to maximize (2.103) y ¼ f ðx1 , x2 Þ, subject to the linear constraint c b1 x1 b2 x2 ¼ 0

(2.104)

(where c, b1 , b2 are constant parameters in the problem). This problem is of a type that will be frequently encountered in this book and is a special case of the constrained maximum problems that we examined earlier. There we showed that the ﬁrst-order conditions for a maximum may be derived by setting up the Lagrangian expression ℒ ¼ f ðx1 , x2 Þ þ λðc b1 x1 b2 x2 Þ:

(2.105)

Partial differentiation with respect to x1 , x2 , and λ yields the familiar results: f1 λb1 ¼ 0, f2 λb2 ¼ 0,

(2.106)

c b1 x1 b2 x2 ¼ 0: These equations can in general be solved for the optimal values of x1 , x2 , and λ. To ensure that the point derived in that way is a local maximum, we must again examine movements away from the critical points by using the “second” total differential: d 2 y ¼ f11 dx 21 þ 2f12 dx1 dx2 þ f22 dx 22 :

(2.107)

In this case, however, not all possible small changes in the x’s are permissible. Only those values of x1 and x2 that continue to satisfy the constraint can be considered valid alternatives to the critical point. To examine such changes, we must calculate the total differential of the constraint: b1 dx1 b2 dx2 ¼ 0

(2.108)

or dx2 ¼

15

b1 dx : b2 1

(2.109)

Notice that Equations 2.102 obey the sufﬁcient conditions not only at the critical point but also for all possible choices of x1 and x2 . That is, the function is concave. In more complex examples this need not be the case: The second-order conditions need be satisﬁed only at the critical point for a local maximum to occur.

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This equation shows the relative changes in x1 and x2 that are allowable in considering movements from the critical point. To proceed further on this problem, we need to use the ﬁrst-order conditions. The ﬁrst two of these imply f1 b 1 ¼ , f2 b 2

(2.110)

and combining this result with Equation 2.109 yields dx2 ¼

f1 dx : f2 1

(2.111)

We now substitute this expression for dx2 in Equation 2.107 to demonstrate the conditions that must hold for d 2 y to be negative: 2 f f d 2 y ¼ f11 dx 21 þ 2f12 dx1 1 dx1 þ f22 1 dx1 f2 f2 2 f f ¼ f11 dx 21 2f12 1 dx 21 þ f22 21 dx 21 : (2.112) f2 f2 Combining terms and putting each over a common denominator gives d 2 y ¼ ð f11 f 22 2f12 f1 f2 þ f22 f 21 Þ

dx 21 : f 22

(2.113)

Consequently, for d 2 y < 0, it must be the case that f11 f 22 2f12 f1 f2 þ f22 f 21 < 0:

(2.114)

Quasi-concave functions Although Equation 2.114 appears to be little more than an inordinately complex mass of mathematical symbols, in fact the condition is an important one. It characterizes a set of functions termed quasi-concave functions. These functions have the property that the set of all points for which such a function takes on a value greater than any speciﬁc constant is a convex set (that is, any two points in the set can be joined by a line contained completely within the set). Many economic models are characterized by such functions and, as we will see in considerable detail in Chapter 3, in these cases the condition for quasi-concavity has a relatively simple economic interpretation. Problems 2.9 and 2.10 examine two speciﬁc quasi-concave functions that we will frequently encounter in this book. Example 2.10 shows the relationship between concave and quasi-concave functions. EXAMPLE 2.10 Concave and Quasi-Concave Functions The differences between concave and quasi-concave functions can be illustrated with the function16 y ¼ f ðx1 , x2 Þ ¼ ðx1 ⋅ x2 Þk ,

(2.115)

where the x’s take on only positive values, and the parameter k can take on a variety of positive values.

16 This function is a special case of the Cobb-Douglas function. See also Problem 2.10 and the Extensions to this chapter for more details on this function.

Chapter 2 Mathematics for Microeconomics

No matter what value k takes, this function is quasi-concave. One way to show this is to look at the “level curves” of the function by setting y equal to a speciﬁc value, say c. In this case y ¼ c ¼ ðx1 x2 Þk

or

x1 x2 ¼ c 1=k ¼ c 0 :

(2.116)

But this is just the equation of a standard rectangular hyperbola. Clearly the set of points for which y takes on values larger than c is convex because it is bounded by this hyperbola. A more mathematical way to show quasi-concavity would apply Equation 2.114 to this function. Although the algebra of doing this is a bit messy, it may be worth the struggle. The various components of Equation 2.114 are: f1 ¼ kx 1k1 x k2 , f2 ¼ kx k1 x 2k1 , f11 ¼ kðk 1Þx 1k2 x k2 ,

(2.117)

f22 ¼ kðk 1Þx k1 x 2k2 , f12 ¼ k 2 x 1k1 x 2k1 : So, f11 f 22 2f12 f1 f2 þ f22 f 21 ¼ k 3 ðk 1Þx 3k2 x 3k2 2k4 x 3k2 x 3k2 1 1 2 2 þ k 3 ðk 1Þx 3k2 x 3k2 1 2 ¼ 2k3 x 3k2 x 3k2 ð1Þ, 1 2

(2.118)

which is clearly negative, as is required for quasi-concavity. Whether or not the function f is concave depends on the value of k. If k < 0:5 the function is indeed concave. An intuitive way to see this is to consider only points where x1 ¼ x2 . For these points, y ¼ ðx 21 Þk ¼ x 2k 1 ,

(2.119)

which, for k < 0:5, is concave. Alternatively, for k > 0:5, this function is convex. A more deﬁnitive proof makes use of the partial derivatives from Equation 2.117. In this case the condition for concavity can be expressed as x 2k2 k4 x 12k2 x 2k2 f11 f22 f 212 ¼ k 2 ðk 1Þ2 x 2k2 1 2 2 ¼ x 2k2 x 22k2 ½k2 ðk 1Þ2 k4 1 x 22k1 ½k2 ð2k þ 1Þ, ¼ x 2k1 1

(2.120)

and this expression is positive (as is required for concavity) for ð2k þ 1Þ > 0

or

k < 0:5:

On the other hand, the function is convex for k > 0:5.

A graphic illustration. Figure 2.4 provides three-dimensional illustrations of three speciﬁc examples of this function: for k ¼ 0:2, k ¼ 0:5, and k ¼ 1. Notice that in all three cases the level curves of the function have hyperbolic, convex shapes. That is, for any ﬁxed value of y the functions are quite similar. This shows the quasi-concavity of the function. The primary differences among the functions are illustrated by the way in which the value of y increases as (continued)

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EXAMPLE 2.10 CONTINUED both x’s increase together. In Figure 2.4a (when k ¼ 0:2), the increase in y slows as the x’s increase. This gives the function a rounded, teacuplike shape that indicates its concavity. For k ¼ 0:5, y appears to increase linearly with increases in both of the x’s. This is the borderline between concavity and convexity. Finally, when k ¼ 1 (as in Figure 2.4c), simultaneous increases in the values of both of the x’s increase y very rapidly. The spine of the function looks convex to reﬂect such increasing returns.

FIGURE 2.4

Concave and Quasi-Concave Functions In all three cases these functions are quasi-concave. For a ﬁxed y, their level curves are convex. But only for k ¼ 0:2 is the function strictly concave. The case k ¼ 1:0 clearly shows nonconcavity because the function is not below its tangent plane.

(a) k = 0.2

(b) k = 0.5

(c) k = 1.0

Chapter 2 Mathematics for Microeconomics

A careful look at Figure 2.4a suggests that any function that is concave will also be quasiconcave. You are asked to prove that this is indeed the case in Problem 2.8. This example shows that the converse of this statement is not true—quasi-concave functions need not necessarily be concave. Most functions we will encounter in this book will also illustrate this fact; most will be quasi-concave but not necessarily concave. QUERY: Explain why the functions illustrated both in Figure 2.4a and 2.4c would have maximum values if the x’s were subject to a linear constraint, but only the graph in Figure 2.4a would have an unconstrained maximum.

HOMOGENEOUS FUNCTIONS Many of the functions that arise naturally out of economic theory have additional mathematical properties. One particularly important set of properties relates to how the functions behave when all (or most) of their arguments are increased proportionally. Such situations arise when we ask questions such as what would happen if all prices increased by 10 percent or how would a ﬁrm’s output change if it doubled all of the inputs that it uses. Thinking about these questions leads naturally to the concept of homogeneous functions. Speciﬁcally, a function f ðx1 , x2 , …, xn Þ is said to be homogeneous of degree k if f ðtx1 , tx2 , …, txn Þ ¼ t k f ðx1 , x2 , …, xn Þ:

(2.121)

The most important examples of homogeneous functions are those for which k ¼ 1 or k ¼ 0. In words, when a function is homogeneous of degree one, a doubling of all of its arguments doubles the value of the function itself. For functions that are homogeneous of degree 0, a doubling of all of its arguments leaves the value of the function unchanged. Functions may also be homogeneous for changes in only certain subsets of their arguments— that is, a doubling of some of the x’s may double the value of the function if the other arguments of the function are held constant. Usually, however, homogeneity applies to changes in all of the arguments in a function.

Homogeneity and derivatives If a function is homogeneous of degree k and can be differentiated, the partial derivatives of the function will be homogeneous of degree k 1. A proof of this follows directly from the deﬁnition of homogeneity. For example, differentiating Equation 2.121 with respect to its ﬁrst argument gives ∂f ðtx1 , …, txn Þ k ∂f ðx1 , …, xn Þ ⋅t ¼ t ∂x1 ∂x1 or f1 ðtx1 , …, txn Þ ¼ t k1 f1 ðx1 , …, xn Þ,

(2.122)

which shows that f1 meets the deﬁnition for homogeneity of degree k 1. Because marginal ideas are so prevalent in microeconomic theory, this property shows that some important properties of marginal effects can be inferred from the properties of the underlying function itself.

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Euler’s theorem Another useful feature of homogeneous functions can be shown by differentiating the deﬁnition for homogeneity with respect to the proportionality factor, t . In this case, we differentiate the right side of Equation 2.121 ﬁrst: kt k1 f ðx , …, x Þ ¼ x f ðtx , …, tx Þ þ … þ x f ðtx , …, tx Þ: 1

1

n

1 1

1

n

n n

1

n

If we let t ¼ 1, this equation becomes kf ðx1 , …, xn Þ ¼ x1 f1 ðx1 , …, xn Þ þ … þ xn fn ðx1 , …, xn Þ:

(2.123)

This equation is termed Euler’s theorem (after the mathematician who also discovered the constant e) for homogeneous functions. It shows that, for a homogeneous function, there is a deﬁnite relationship between the values of the function and the values of its partial derivatives. Several important economic relationships among functions are based on this observation.

Homothetic functions A homothetic function is one that is formed by taking a monotonic transformation of a homogeneous function.17 Monotonic transformations, by deﬁnition, preserve the order of the relationship between the arguments of a function and the value of that function. If certain sets of x’s yield larger values for f , they will also yield larger values for a monotonic transformation of f . Because monotonic transformations may take many forms, however, they would not be expected to preserve an exact mathematical relationship such as that embodied in homogeneous functions. Consider, for example, the function f ðx, yÞ ¼ x ⋅ y. Clearly this function is homogeneous of degree 2—a doubling of its two arguments will multiply the value of the function by 4. But the monotonic transformation, F , that simply adds 1 to f [that is, F ðf Þ ¼ f þ 1 ¼ xy þ 1] is not homogeneous at all. Hence, except in special cases, homothetic functions do not possess the homogeneity properties of their underlying functions. Homothetic functions do, however, preserve one nice feature of homogeneous functions. This property is that the implicit trade-offs among the variables in a function depend only on the ratios of those variables, not on their absolute values. Here we show this for the simple two-variable, implicit function f ðx, yÞ ¼ 0. It will be easier to demonstrate more general cases when we get to the economics of the matter later in this book. Equation 2.28 showed that the implicit trade-off between x and y for a two-variable function is given by dy f ¼ x: fy dx If we assume f is homogeneous of degree k, its partial derivatives will be homogeneous of degree k 1 and the implicit trade-off between x and y is dy t k1 f ðtx, tyÞ f ðtx, tyÞ ¼ x : ¼ k1 x fy ðtx, tyÞ dx t fy ðtx, tyÞ Now let t ¼ 1=y and Equation 2.124 becomes dy f ðx=y, 1Þ , ¼ x dx fy ðx=y, 1Þ

(2.124)

(2.125)

which shows that the trade-off depends only on the ratio of x to y. Now if we apply any monotonic transformation, F (with F 0 > 0), to the original homogeneous function f , we have 17 Because a limiting case of a monotonic transformation is to leave the function unchanged, all homogeneous functions are also homothetic.

Chapter 2 Mathematics for Microeconomics

dy F 0 f ðx=y, 1Þ f ðx=y, 1Þ ¼ x , ¼ 0x dx F fy ðx=y, 1Þ fy ðx=y, 1Þ

(2.126)

and this shows both that the trade-off is unaffected by the monotonic transformation and that it remains a function only of the ratio of x to y. In Chapter 3 (and elsewhere) this property will make it very convenient to discuss some theoretical results with simple twodimensional graphs, for which we need not consider the overall levels of key variables, but only their ratios. EXAMPLE 2.11 Cardinal and Ordinal Properties In applied economics it is sometimes important to know the exact numerical relationship among variables. For example, in the study of production, one might wish to know precisely how much extra output would be produced by hiring another worker. This is a question about the “cardinal” (i.e., numerical) properties of the production function. In other cases, one may only care about the order in which various points are ranked. In the theory of utility, for example, we assume that people can rank bundles of goods and will choose the bundle with the highest ranking, but that there are no unique numerical values assigned to these rankings. Mathematically, ordinal properties of functions are preserved by any monotonic transformation because, by deﬁnition, a monotonic transformation preserves order. Usually, however, cardinal properties are not preserved by arbitrary monotonic transformations. These distinctions are illustrated by the functions we examined in Example 2.10. There we studied monotonic transformations of the function f ðx1 , x2 Þ ¼ ðx1 x2 Þk

(2.127)

by considering various values of the parameter k. We showed that quasi-concavity (an ordinal property) was preserved for all values of k. Hence, when approaching problems that focus on maximizing or minimizing such a function subject to linear constraints we need not worry about precisely which transformation is used. On the other hand, the function in Equation 2.127 is concave (a cardinal property) only for a narrow range of values of k. Many monotonic transformations destroy the concavity of f . The function in Equation 2.127 also can be used to illustrate the difference between homogeneous and homothetic functions. A proportional increase in the two arguments of f would yield f ðtx1 , tx2 Þ ¼ t 2k x1 x2 ¼ t 2k f ðx1 , x2 Þ:

(2.128)

Hence, the degree of homogeneity for this function depends on k—that is, the degree of homogeneity is not preserved independently of which monotonic transformation is used. Alternatively, the function in Equation 2.127 is homothetic because dx2 f kx k1 x k x ¼ 1 ¼ 1k k12 ¼ 2 : dx1 f2 x1 kx 1 x 2

(2.129)

That is, the trade-off between x2 and x1 depends only on the ratio of these two variables and is unaffected by the value of k. Hence, homotheticity is an ordinal property. As we shall see, this property is quite convenient when developing graphical arguments about economic propositions. QUERY: How would the discussion in this example be changed if we considered monotonic transformations of the form f ðx1 , x2 , kÞ ¼ x1 x2 þ k for various values of k?

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INTEGRATION Integration is another of the tools of calculus that ﬁnds a number of applications in microeconomic theory. The technique is used both to calculate areas that measure various economic outcomes and, more generally, to provide a way of summing up outcomes that occur over time or across individuals. Our treatment of the topic here necessarily must be brief, so readers desiring a more complete background should consult the references at the end of this chapter.

Anti-derivatives Formally, integration is the inverse of differentiation. When you are asked to calculate the integral of a function, f ðxÞ, you are being asked to ﬁnd a function that has f ðxÞ as its derivative. If we call this “anti-derivative” F ðxÞ, this function is supposed to have the property that dF ðxÞ ¼ F 0 ðxÞ ¼ f ðxÞ: dx If such a function exists then we denote it as F ðxÞ ¼

(2.130)

∫ f ðxÞ dx:

(2.131)

The precise reason for this rather odd-looking notation will be described in detail later. First, let’s look at a few examples. If f ðxÞ ¼ x then F ðxÞ ¼

∫ f ðxÞ dx ¼ ∫ x dx ¼ x2 þ C , 2

(2.132)

where C is an arbitrary “constant of integration” that disappears upon differentiation. The correctness of this result can be easily veriﬁed: F 0 ðxÞ ¼

dðx 2 =2 þ C Þ ¼ x þ 0 ¼ x: dx

(2.133)

Calculating anti-derivatives Calculation of anti-derivatives can be extremely simple, or difﬁcult, or agonizing, or impossible, depending on the particular f ðxÞ speciﬁed. Here we will look at three simple methods for making such calculations, but, as you might expect, these will not always work. 1. Creative guesswork. Probably the most common way of ﬁnding integrals (antiderivatives) is to work backwards by asking “what function will yield f ðxÞ as its derivative?” Here are a few obvious examples:

∫ F ðxÞ ¼ ∫ ax bx þ þ cx þ C , F ðxÞ ¼ ∫ðax þ bx þ cÞ dx ¼ 3 2 F ðxÞ ¼ ∫e dx ¼ e þ C , a þ C, F ðxÞ ¼ ∫a dx ¼ ln a 1 dx ¼ lnðjxjÞ þ C , F ðxÞ ¼ ∫ x F ðxÞ ¼ ∫ðln xÞ dx ¼ x ln x x þ C :

F ðxÞ ¼

x3 þ C, 3 nþ1 x þ C, x n dx ¼ nþ1 x 2 dx ¼

2

x

x

x

x

3

2

(2.134)

Chapter 2 Mathematics for Microeconomics

You should use differentiation to check that all of these obey the property that F 0 ðxÞ ¼ f ðxÞ. Notice that in every case the integral includes a constant of integration because anti-derivatives are unique only up to an additive constant which would become zero upon differentiation. For many purposes, the results in Equation 2.134 (or trivial generalizations of them) will be sufﬁcient for our purposes in this book. Nevertheless, here are two more methods that may work when intuition fails. 2. Change of variable. A clever redeﬁnition of variables may sometimes make a function much easier to integrate. For example, it is not at all obvious what the integral of 2x=ð1 þ x 2 Þ is. But, if we let y ¼ 1 þ x 2 , then dy ¼ 2xdx and

∫ 1 þ2xx

2

dx ¼

∫ 1y dy ¼ lnðjyjÞ ¼ lnðj1 þ x jÞ: 2

(2.135)

The key to this procedure is in breaking the original function into a term in y and a term in dy. It takes a lot of practice to see patterns for which this will work. 3. Integration by parts. A similar method for ﬁnding integrals makes use of the differential expression duv ¼ udv þ vdu for any two functions u and v. Integration of this differential yields

∫ duv ¼ uv ¼ ∫ u dv þ ∫ v du

or

∫ u dv ¼ uv ∫ v du:

(2.136)

Here the strategy is to deﬁne functions u and v in a way that the unknown integral on the left can be calculated by the difference between the two known expressions on the right. For example, it is by no means obvious what the integral of xe x is. But we can deﬁne u ¼ x (so du ¼ dx) and dv ¼ e x dx (so v ¼ e x ). Hence we now have

∫ xe dx ¼ ∫ u dv ¼ uv ∫ v du ¼ xe ∫ e dx ¼ ðx 1Þe þ C : x

x

x

x

(2.137)

Again, only practice can suggest useful patterns in the ways in which u and v can be deﬁned.

Deﬁnite integrals The integrals we have been discussing so far are “indeﬁnite” integrals—they provide only a general function that is the anti-derivative of another function. A somewhat different, though related, approach uses integration to sum up the area under a graph of a function over some deﬁned interval. Figure 2.5 illustrates this process. We wish to know the area under the function f ðxÞ from x ¼ a to x ¼ b. One way to do this would be to partition the interval into narrow slivers of xð∆xÞ and sum up the areas of the rectangles shown in the ﬁgure. That is: X f ðxi Þ∆xi , (2.138) area under f ðxÞ i

where the notation is intended to indicate that the height of each rectangle is approximated by the value of f ðxÞ for a value of x in the interval. Taking this process to the limit by shrinking the size of the ∆x intervals yields an exact measure of the area we want and is denoted by: x¼b

area under f ðxÞ ¼

∫ f ðxÞ dx:

(2.139)

x¼a

This then explains the origin of the oddly shaped integral sign—it is a stylized S, indicating “sum.” As we shall see, integrating is a very general way of summing the values of a continuous function over some interval.

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FIGURE 2.5

Deﬁnite Integrals Show the Areas under the Graph of a Function Deﬁnite integrals measure the area under a curve by summing rectangular areas as shown in the graph. The dimension of each rectangle is f ðx Þdx.

f(x)

f(x)

a

b

x

Fundamental theorem of calculus Evaluating the integral in Equation 2.139 is very simple if we know the anti-derivative of f ðxÞ, say, F ðxÞ. In this case we have x¼b

area under f ðxÞ ¼

∫ f ðxÞ dx ¼ F ðbÞ F ðaÞ:

(2.140)

x¼a

That is, all we need do is calculate the anti-derivative of f ðxÞ and subtract the value of this function at the lower limit of integration from its value at the upper limit of integration. This result is sometimes termed the “fundamental theorem of calculus” because it directly ties together the two principal tools of calculus, derivatives and integrals. In Example 2.12, we show that this result is much more general than simply a way to measure areas. It can be used to illustrate one of the primary conceptual principles of economics—the distinction between “stocks” and “ﬂows.” EXAMPLE 2.12 Stocks and Flows The deﬁnite integral provides a useful way for summing up any function that is providing a continuous ﬂow over time. For example, suppose that net population increase (births minus deaths) for a country can be approximated by the function f ðt Þ ¼ 1,000e 0:02t . Hence, the net population change is growing at the rate of 2 percent per year—it is 1,000 new people in year 0, 1,020 new people in the ﬁrst year, 1,041 in the second year, and so forth. Suppose we wish to know how much in total the population will increase within 50 years. This might be a tedious calculation without calculus, but using the fundamental theorem of calculus provides an easy answer:

Chapter 2 Mathematics for Microeconomics t ¼50

increase in population ¼

t ¼50

∫ f ðt Þ dt ¼ ∫ 1,000e

t ¼0

¼

t ¼0

1,000e 0:02t 0:02

0:02t

50 dt ¼ F ðt Þ 0

50 ¼ 1,000e 50,000 ¼ 85,914 0:02

(2:141)

0

[where the notation jba indicates that the expression is to be evaluated as F ðbÞ F ðaÞ]. Hence, the conclusion is that the population will grow by nearly 86,000 people over the next 50 years. Notice how the fundamental theorem of calculus ties together a “ﬂow” concept, net population increase (which is measured as an amount per year), with a “stock” concept, total population (which is measured at a speciﬁc date and does not have a time dimension). Note also that the 86,000 calculation refers only to the total increase between year zero and year ﬁfty. In order to know the actual total population at any date we would have to add the number of people in the population at year zero. That would be similar to choosing a constant of integration in this speciﬁc problem. Now consider an application with more economic content. Suppose that total costs for a particular ﬁrm are given by CðqÞ ¼ 0:1q 2 þ 500 (where q represents output during some period). Here the term 0:1q 2 represents variable costs (costs that vary with output) whereas the 500 ﬁgure represents ﬁxed costs. Marginal costs for this production process can be found through differentiation—MC ¼ dCðqÞ=dq ¼ 0:2q—hence, marginal costs are increasing with q and ﬁxed costs drop out upon differentiation. What are the total costs associated with producing, say, q ¼ 100? One way to answer this question is to use the total cost function directly: Cð100Þ ¼ 0:1ð100Þ2 þ 500 ¼ 1,500. An alternative way would be to integrate marginal cost over the range 0 to 100 to get total variable cost: q¼100

variable cost ¼

∫

100 0:2q dq ¼ 0:1q ¼ 1,000 0 ¼ 1,000, 2

q¼0

(2.142)

0

to which we would have to add ﬁxed costs of 500 (the constant of integration in this problem) to get total costs. Of course, this method of arriving at total cost is much more cumbersome than just using the equation for total cost directly. But the derivation does show that total variable cost between any two output levels can be found through integration as the area below the marginal cost curve—a conclusion that we will ﬁnd useful in some graphical applications. QUERY: How would you calculate the total variable cost associated with expanding output from 100 to 110? Explain why ﬁxed costs do not enter into this calculation.

Diﬀerentiating a deﬁnite integral Occasionally we will wish to differentiate a deﬁnite integral—usually in the context of seeking to maximize the value of this integral. Although performing such differentiations can sometimes be rather complex, there are a few rules that should make the process easier. 1. Differentiation with respect to the variable of integration. This is a trick question, but instructive nonetheless. A deﬁnite integral has a constant value; hence its derivative is zero. That is: d∫ba f ðxÞ dx ¼ 0: (2.143) dx The summing process required for integration has already been accomplished once we write down a deﬁnite integral. It does not matter whether the variable of integration is x or t or

59

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anything else. The value of this integrated sum will not change when the variable x changes, no matter what x is (but see rule 3 below). 2. Differentiation with respect to the upper bound of integration. Changing the upper bound of integration will obviously change the value of a deﬁnite integral. In this case, we must make a distinction between the variable determining the upper bound of integration (say, x) and the variable of integration (say, t ). The result then is a simple application of the fundamental theorem of calculus. For example: d∫xa f ðt Þdt d½F ðxÞ F ðaÞ ¼ ¼ f ðxÞ 0 ¼ f ðxÞ, (2.144) dx dx where F ðxÞ is the antiderivative of f ðxÞ. By referring back to Figure 2.5 we can see why this conclusion makes sense—we are asking how the value of the deﬁnite integral changes if x increases slightly. Obviously, the answer is that the value of the integral increases by the height of f ðxÞ (notice that this value will ultimately depend on the speciﬁed value of x). If the upper bound of integration is a function of x, this result can be generalized using the chain rule: gðxÞ

d∫a

f ðt Þ dt d½F ð gðxÞÞ F ðaÞ d½F ð gðxÞÞ dgðxÞ ¼ ¼ ¼f ¼ fg 0 ðxÞ, dðxÞ dx dx dx

(2.145)

where, again, the speciﬁc value for this derivative would depend on the value of x assumed. Finally, notice that differentiation with respect to a lower bound of integration just changes the sign of this expression: d∫bgðxÞ f ðt Þ dt dx

¼

d½F ðbÞ F ð gðxÞÞ dF ð gðxÞÞ ¼ ¼ fg 0 ðxÞ: dx dx

(2.146)

3. Differentiation with respect to another relevant variable. In some cases we may wish to integrate an expression that is a function of several variables. In general, this can involve multiple integrals and differentiation can become quite complicated. But there is one simple case that should be mentioned. Suppose that we have a function of two variables, f ðx, yÞ, and that we wish to integrate this function with respect to the variable x. The speciﬁc value for this integral will obviously depend on the value of y and we might even ask how that value changes when y changes. In this case, it is possible to “differentiate through the integral sign” to obtain a result. That is: d∫ba f ðx, yÞ dx ¼ dy

b

∫ f ðx, yÞ dx: y

(2.147)

a

This expression shows that we can ﬁrst partially differentiate f ðx, yÞ with respect to y before proceeding to compute the value of the deﬁnite integral. Of course, the resulting value may still depend on the speciﬁc value that is assigned to y, but often it will yield more economic insights than the original problem does. Some further examples of using deﬁnite integrals are found in Problem 2.8.

DYNAMIC OPTIMIZATION Some optimization problems that arise in microeconomics involve multiple periods.18 We are interested in ﬁnding the optimal time path for a variable or set of variables that succeeds in optimizing some goal. For example, an individual may wish to choose a path of lifetime 18 Throughout this section we treat dynamic optimization problems as occurring over time. In other contexts, the same techniques can be used to solve optimization problems that occur across a continuum of ﬁrms or individuals when the optimal choices for one agent affect what is optimal for others.

Chapter 2 Mathematics for Microeconomics

consumptions that maximizes his or her utility. Or a ﬁrm may seek a path for input and output choices that maximizes the present value of all future proﬁts. The particular feature of such problems that makes them difﬁcult is that decisions made in one period affect outcomes in later periods. Hence, one must explicitly take account of this interrelationship in choosing optimal paths. If decisions in one period did not affect later periods, the problem would not have a “dynamic” structure—one could just proceed to optimize decisions in each period without regard for what comes next. Here, however, we wish to explicitly allow for dynamic considerations.

The optimal control problem Mathematicians and economists have developed many techniques for solving problems in dynamic optimization. The references at the end of this chapter provide broad introductions to these methods. Here, however, we will be concerned with only one such method that has many similarities to the optimization techniques discussed earlier in this chapter—the optimal control problem. The framework of the problem is relatively simple. A decision maker wishes to ﬁnd the optimal time path for some variable xðt Þ over a speciﬁed time interval ½t0 , t1 . Changes in x are governed by a differential equation: dxðt Þ ¼ g½xðt Þ, cðt Þ, t , (2.148) dt where the variable cðt Þ is used to “control” the change in xðt Þ. In each period of time, the decision maker derives value from x and c according to the function f ½xðt Þ, cðt Þ, t and his or her goal to optimize ∫tt10 f ½xðt Þ, cðt Þ, t dt . Often this problem will also be subject to “endpoint” constraints on the variable x. These might be written as xðt0 Þ ¼ x0 and xðt1 Þ ¼ x1 . Notice how this problem is “dynamic.” Any decision about how much to change x this period will affect not only the future value of x, it will also affect future values of the outcome function f . The problem then is how to keep xðt Þ on its optimal path. Economic intuition can help to solve this problem. Suppose that we just focused on the function f and chose x and c to maximize it at each instant of time. There are two difﬁculties with this “myopic” approach. First, we are not really free to “choose” x at any time. Rather, the value of x will be determined by its initial value x0 and by its history of changes as given by Equation 2.148. A second problem with this myopic approach is that it disregards the dynamic nature of the problem by not asking how this period’s decisions affect the future. We need some way to reﬂect the dynamics of this problem in a single period’s decisions. Assigning the correct value (price) to x at each instant of time will do just that. Because this implicit price will have many similarities to the Lagrangian multipliers studied earlier in this chapter, we will call it λðt Þ. The value of x is treated as a function of time because the importance of x can obviously change over time.

The maximum principle Now let’s look at the decision maker’s problem at a single point in time. He or she must be concerned with both the current value of the objective function f ½xðt Þ, cðtÞ, t and with the implied change in the value of xðt Þ . Because the current value of xðtÞ is given by λðt Þxðt Þ, the instantaneous rate of change of this value is given by: d½λðt Þxðt Þ dxðt Þ dλðt Þ ¼ λðt Þ þ xðt Þ , dt dt dt

(2.149)

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and so at any time t a comprehensive measure of the value of concern19 to the decision maker is dλðt Þ : (2.150) dt This comprehensive value represents both the current beneﬁts being received and the instantaneous change in the value of x. Now we can ask what conditions must hold for xðtÞ and cðt Þ to optimize this expression.20 That is: H ¼ f ½xðt Þ, cðt Þ, t þ λðt Þg½xðt Þ, cðt Þ, t þ xðt Þ

∂H ¼ fc þ λgc ¼ 0 or fc ¼ λgc ; ∂c (2.151) ∂H ∂λðt Þ ∂λðt Þ ¼ fx þ λgx þ ¼ 0 or fx þ λgx ¼ : ∂x dt ∂t These are then the two optimality conditions for this dynamic problem. They are usually referred to as the “maximum principle.” This solution to the optimal control problem was ﬁrst proposed by the Russian mathematician L. S. Pontryagin and his colleagues in the early 1960s. Although the logic of the maximum principle can best be illustrated by the economic applications we will encounter later in this book, a brief summary of the intuition behind them may be helpful. The ﬁrst condition asks about the optimal choice of c. It suggests that, at the margin, the gain from c in terms of the function f must be balanced by the losses from c in terms of the value of its ability to change x. That is, present gains must be weighed against future costs. The second condition relates to the characteristics that an optimal time path of xðt Þ should have. It implies that, at the margin, any net gains from more current x (either in terms of f or in terms of the accompanying value of changes in x) must be balanced by changes in the implied value of x itself. That is, the net current gain from more x must be weighed against the declining future value of x. EXAMPLE 2.13 Allocating a Fixed Supply As an extremely simple illustration of the maximum principle, assume that someone has inherited 1,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 20 years. How should this be done to maximize the utility from doing so? Suppose that this person’s utility function for wine is given by u½cðtÞ ¼ ln cðt Þ. Hence the utility from wine drinking exhibits diminishing marginal utility ðu 0 > 0, u 00 < 0Þ. This person’s goal is to maximize 20

20

0

0

∫ u½cðt Þ dt ¼ ∫ ln cðt Þ dt :

(2.152)

Let xðt Þ represent the number of bottles of wine remaining at time t . This series is constrained by xð0Þ ¼ 1,000 and xð20Þ ¼ 0. The differential equation determining the evolution of xðt Þ takes the simple form:21 19

We denote this current value expression by H to suggest its similarity to the Hamiltonian expression used in formal dynamic optimization theory. Usually the Hamiltonian does not have the ﬁnal term in Equation 2.150, however.

20 Notice that the variable x is not really a choice variable here—its value is determined by history. Differentiation with respect to x can be regarded as implicitly asking the question: “If xðt Þ were optimal, what characteristics would it have?” 21

The simple form of this differential equation (where dx=dt depends only on the value of the control variable, c) means that this problem is identical to one explored using the “calculus of variations” approach to dynamic optimization. In such a case, one can substitute dx=dt into the function f and the ﬁrst-order conditions for a maximum can be compressed into

Chapter 2 Mathematics for Microeconomics

dxðt Þ ¼ cðt Þ: (2.153) dt That is, each instant’s consumption just reduces the stock of remaining bottles. The current value Hamiltonian expression for this problem is H ¼ ln cðt Þ þ λ½cðt Þ þ xðt Þ

dλ , dt

(2.154)

and the ﬁrst-order conditions for a maximum are ∂H 1 ¼ λ ¼ 0, ∂c c (2.155) ∂H dλ ¼ ¼ 0: ∂x dt The second of these conditions requires that λ (the implicit value of wine) be constant over time. This makes intuitive sense: because consuming a bottle of wine always reduces the available stock by one bottle, any solution where the value of wine differed over time would provide an incentive to change behavior by drinking more wine when it is cheap and less when it is expensive. Combining this second condition for a maximum with the ﬁrst condition implies that cðt Þ itself must be constant over time. If cðt Þ ¼ k, the number of bottles remaining at any time will be xðt Þ ¼ 1,000 kt . If k ¼ 50, the system will obey the end point constraints xð0Þ ¼ 1000 and xð20Þ ¼ 0. Of course, in this problem you could probably guess that the optimum plan would be to drink the wine at the rate of 50 bottles per year for 20 years because diminishing marginal utility suggests one does not want to drink excessively in any period. The maximum principle conﬁrms this intuition. More complicated utility. Now let’s take a more complicated utility function that may yield more interesting results. Suppose that the utility of consuming wine at any date, t , is given by

½cðt Þγ =γ if γ 6¼ 0, γ < 1; (2.156) u½cðt Þ ¼ ln cðt Þ if γ ¼ 0: Assume also that the consumer discounts future consumption at the rate δ. Hence this person’s goal is to maximize 20

∫

20

u½cðt Þ dt ¼

0

∫

e δt

0

½cðt Þγ dt γ

(2.157)

subject to the following constraints: dxðt Þ ¼ cðt Þ, dt xð0Þ ¼ 1,000,

(2.158)

xð20Þ ¼ 0: Setting up the current value Hamiltonian expression yields H ¼ e δt

½cðt Þγ dλðt Þ þ λðcÞ þ xðt Þ , γ dt

(2.159)

and the maximum principle requires that (continued) the single equation fx ¼ dfdx=dt =dt , which is termed the “Euler equation.” In Chapter 17 we will encounter many Euler equations.

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Part 1 Introduction

EXAMPLE 2.13 CONTINUED ∂H ¼ e δt ½cðt Þγ1 λ ¼ 0 and ∂c (2.160) ∂H dλ ¼ 0þ0þ ¼ 0: ∂x dt Hence, we can again conclude that the implicit value of the wine stock (λ) should be constant over time (call this constant k) and that e δt ½cðt Þγ1 ¼ k

or

cðt Þ ¼ k1=ðγ1Þ e δt =ðγ1Þ :

(2.161)

So, optimal wine consumption should fall over time in order to compensate for the fact that future consumption is being discounted in the consumer’s mind. If, for example, we let δ ¼ 0:1 and γ ¼ 1 (“reasonable” values, as we will show in later chapters), then cðt Þ ¼ k0:5 e 0:05t

(2.162)

Now we must do a bit more work in choosing k to satisfy the endpoint constraints. We want 20

∫ 0

20

cðt Þ dt ¼

∫

20 k0:5 e 0:05t dt ¼ 20k0:5 e 0:05t 0

0

(2.163)

¼ 20k 0:5 ðe 1 1Þ ¼ 12:64k0:5 ¼ 1,000:

Finally, then, we have the optimal consumption plan as cðt Þ 79e 0:05t :

(2.164)

This consumption plan requires that wine consumption start out fairly high and decline at a continuous rate of 5 percent per year. Because consumption is continuously declining, we must use integration to calculate wine consumption in any particular year ðxÞ as follows: x x x 0:05t 0:05t cðt Þ dt ¼ 79e dt ¼ 1,580e consumption in year x (2.165) x1

∫

x1

∫

x1

¼ 1,580ðe 0:05ðx1Þ e 0:05x Þ: If x ¼ 1, consumption is about 77 bottles in this ﬁrst year. Consumption then declines smoothly, ending with about 30 bottles being consumed in the 20th year. QUERY: Our ﬁrst illustration was just an example of the second in which δ ¼ γ ¼ 0. Explain how alternative values of these parameters will affect the path of optimal wine consumption. Explain your results intuitively (for more on optimal consumption over time, see Chapter 17).

MATHEMATICAL STATISTICS In recent years microeconomic theory has increasingly focused on issues raised by uncertainty and imperfect information. To understand much of this literature, it is important to have a good background in mathematical statistics. The purpose of this section is, therefore, to summarize a few of the statistical principles that we will encounter at various places in this book.

Chapter 2 Mathematics for Microeconomics

Random variables and probability density functions A random variable describes (in numerical form) the outcomes from an experiment that is subject to chance. For example, we might ﬂip a coin and observe whether it lands heads or tails. If we call this random variable x, we can denote the possible outcomes (“realizations”) of the variable as:

1 if coin is heads, x¼ 0 if coin is tails: Notice that, prior to the ﬂip of the coin, x can be either 1 or 0. Only after the uncertainty is resolved (that is, after the coin is ﬂipped) do we know what the value of x is.22

Discrete and continuous random variables The outcomes from a random experiment may be either a ﬁnite number of possibilities or a continuum of possibilities. For example, recording the number that comes up on a single die is a random variable with six outcomes. With two dice, we could either record the sum of the faces (in which case there are 12 outcomes, some of which are more likely than others) or we could record a two-digit number, one for the value of each die (in which case there would be 36 equally likely outcomes). These are examples of discrete random variables. Alternatively, a continuous random variable may take on any value in a given range of real numbers. For example, we could view the outdoor temperature tomorrow as a continuous variable (assuming temperatures can be measured very ﬁnely) ranging from, say, 50°C to +50°C. Of course, some of these temperatures would be very unlikely to occur, but in principle the precisely measured temperature could be anywhere between these two bounds. Similarly, we could view tomorrow’s percentage change in the value of a particular stock index as taking on all values between 100% and, say, +1,000%. Again, of course, percentage changes around 0% would be considerably more likely to occur than would be the extreme values.

Probability density functions For any random variable, its probability density function (PDF) shows the probability that each speciﬁc outcome will occur. For a discrete random variable, deﬁning such a function poses no particular difﬁculties. In the coin ﬂip case, for example, the PDF [denoted by f ðxÞ] would be given by f ðx ¼ 1Þ ¼ 0:5, f ðx ¼ 0Þ ¼ 0:5:

(2.166)

For the roll of a single die, the PDF would be: f ðx ¼ 1Þ ¼ 1=6, f ðx ¼ 2Þ ¼ 1=6, f ðx ¼ 3Þ ¼ 1=6, f ðx ¼ 4Þ ¼ 1=6,

(2.167)

f ðx ¼ 5Þ ¼ 1=6, f ðx ¼ 6Þ ¼ 1=6:

Sometimes random variables are denoted by xe to make a distinction between variables whose outcome is subject to random chance and (nonrandom) algebraic variables. This notational device can be useful for keeping track of what is random and what is not in a particular problem and we will use it in some cases. When there is no ambiguity, however, we will not employ this special notation.

22

65

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Part 1 Introduction

Notice that in both of these cases the probabilities speciﬁed by the PDF sum to 1.0. This is because, by deﬁnition, one of the outcomes of the random experiment must occur. More generally, if we denote all of the outcomes for a discrete random variable by xi for i ¼ 1, …, n, then we must have: n X f ðxi Þ ¼ 1: (2.168) i¼1

For a continuous random variable we must be careful in deﬁning the PDF concept. Because such a random variable takes on a continuum of values, if we were to assign any nonzero value as the probability for a speciﬁc outcome (i.e., a temperature of +25.53470°C), we could quickly have sums of probabilities that are inﬁnitely large. Hence, for a continuous random variable we deﬁne the PDF f ðxÞ as a function with the property that the probability that x falls in a particular small interval dx is given by the area of f ðxÞdx. Using this convention, the property that the probabilities from a random experiment must sum to 1.0 is stated as follows: þ∞

∫ f ðxÞ dx ¼ 1:0:

(2.169)

∞

A few important PDFs Most any function will do as a probability density function provided that f ðxÞ 0 and the function sums (or integrates) to 1.0. The trick, of course, is to ﬁnd functions that mirror random experiments that occur in the real world. Here we look at four such functions that we will ﬁnd useful in various places in this book. Graphs for all four of these functions are shown in Figure 2.6. 1. Binomial distribution. This is the most basic discrete distribution. Usually x is assumed to take on only two values, 1 and 0. The PDF for the binomial is given by: f ðx ¼ 1Þ ¼ p, f ðx ¼ 0Þ ¼ 1 p, where

(2.170)

0 < p < 1:

The coin ﬂip example is obviously a special case of the binomial where p ¼ 0:5. 2. Uniform distribution. This is the simplest continuous PDF. It assumes that the possible values of the variable x occur in a deﬁned interval and that each value is equally likely. That is: 1 ba f ðxÞ ¼ 0

f ðxÞ ¼

for a x b;

(2.171)

for x < a or x > b:

Notice that here the probabilities integrate to 1.0: þ∞

∫

∞

b

f ðxÞ dx ¼

∫ a

b 1 x b a ba dx ¼ ¼ ¼ 1:0: ¼ ba b a a b a b a b a

(2.172)

3. Exponential distribution. This is a continuous distribution for which the probabilities decline at a smooth exponential rate as x increases. Formally:

λx if x > 0, λe (2.173) f ðxÞ ¼ 0 if x 0,

67

Chapter 2 Mathematics for Microeconomics

FIGURE 2.6

Four Common Probability Density Functions

Random variables that have these PDFs are widely used. Each graph indicates the expected value of the PDF shown.

f(x)

f(x)

P 1–P 1 b–a

0

P

1

x

a

(a) Binomial

a+b 2

b

x

(b) Uniform f(x)

f(x)

λ 1/√––– 2π

1/λ (c) Exponential

x

0 (d) Normal

where λ is a positive constant. Again, it is easy to show that this function integrates to 1.0: þ∞

∫

∞

∞

f ðxÞ dx ¼

∫

∞ λe λx dx ¼ e λx ¼ 0 ð1Þ ¼ 1:0: 0

(2.174)

0

4. Normal distribution. The Normal (or Gaussian) distribution is the most important in mathematical statistics. It’s importance stems largely from the central limit theorem, which states that the distribution of any sum of independent random variables will increasingly

x

68

Part 1 Introduction

approximate the Normal distribution as the number of such variables increase. Because sample averages can be regarded as sums of independent random variables, this theorem says that any sample average will have a Normal distribution no matter what the distribution of the population from which the sample is selected. Hence, it may often be appropriate to assume a random variable has a Normal distribution if it can be thought of as some sort of average. The mathematical form for the Normal PDF is 1 2 f ðxÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ e x =2 , 2π

(2.175)

and this is deﬁned for all real values of x. Although the function may look complicated, a few of its properties can be easily described. First, the function is symmetric around zero (because of the x 2 term). Second, the function is asymptotic to zero as x becomes very large or pvery ﬃﬃﬃﬃﬃﬃ small. Third, the function reaches its maximal value at x ¼ 0. This value is 1= 2π 0:4. Finally, the graph of this function has a general “bell shape”—a shape used throughout the study of statistics. Integration of this function pﬃﬃﬃﬃﬃﬃ is relatively tricky (though easy in polar coordinates). The presence of the constant 1= 2π is needed if the function is to integrate to 1.0.

Expected value The expected value of a random variable is the numerical value that the random variable might be expected to have, on average.23 It is the “center of gravity” of the probability density function. For a discrete random variable that takes on the values x1 , x2 , …, xn , the expected value is deﬁned as EðxÞ ¼

n X

xi f ðxi Þ:

(2.176)

i¼1

That is, each outcome is weighted by the probability that it will occur and the result is summed over all possible outcomes. For a continuous random variable, Equation 2.176 is readily generalized as þ∞

EðxÞ ¼

∫ x f ðxÞ dx:

(2.177)

∞

Again, in this integration, each value of x is weighted by the probability that this value will occur. The concept of expected value can be generalized to include the expected value of any function of a random variable [say, gðxÞ]. In the continuous case, for example, we would write þ∞

E½ gðxÞ ¼

∫ gðxÞf ðxÞ dx:

(2.178)

∞

23

The expected value of a random variable is sometimes referred to as the mean of that variable. In the study of sampling this can sometimes lead to confusion between the expected value of a random variable and the separate concept of the sample arithmetic average.

Chapter 2 Mathematics for Microeconomics

As a special case, consider a linear function y ¼ ax þ b. Then þ∞

EðyÞ ¼ Eðax þ bÞ ¼

∫ ðax þ bÞf ðxÞ dx

∞ þ∞

¼ a

þ∞

∫ xf ðxÞ dx þ b ∫ f ðxÞ dx ¼ aEðxÞ þ b:

∞

(2:179)

∞

Sometimes expected values are phrased in terms of the cumulative distribution function (CDF) F ðxÞ, deﬁned as x

F ðxÞ ¼

∫ f ðt Þ dt :

(2.180)

∞

That is, F ðxÞ represents the probability that the random variable t is less than or equal to x. With this notation, the expected value of gðxÞ is deﬁned as þ∞

E½ gðxÞ ¼

∫ gðxÞ dF ðxÞ:

(2.181)

∞

Because of the fundamental theorem of calculus, Equation 2.181 and Equation 2.178 mean exactly the same thing. EXAMPLE 2.14 Expected Values of a Few Random Variables The expected values of each of the random variables with the simple PDFs introduced earlier are easy to calculate. All of these expected values are indicated on the graphs of the functions’ PDFs in Figure 2.6. 1. Binomial. In this case: EðxÞ ¼ 1 ⋅ f ðx ¼ 1Þ þ 0 ⋅ f ðx ¼ 0Þ ¼ 1 ⋅ p þ 0 ⋅ ð1 pÞ ¼ p:

(2.182)

For the coin ﬂip case (where p ¼ 0:5), this says that EðxÞ ¼ p ¼ 0:5—the expected value of this random variable is, as you might have guessed, one half. 2. Uniform. For this continuous random variable, b b x x2 b2 a2 bþa ¼ dx ¼ : (2.183) EðxÞ ¼ ¼ 2ðb aÞ 2ðb aÞ 2ðb aÞ ba 2

∫

a

a

Again, as you might have guessed, the expected value of the uniform distribution is precisely halfway between a and b. 3. Exponential. For this case of declining probabilities: ∞

EðxÞ ¼

∫ 0

∞ 1 1 xλe λx dx ¼ xe λx e λx ¼ , 0 λ λ

(2.184)

where the integration follows from the integration by parts example shown earlier in this chapter (Equation 2.137). Notice here that the faster the probabilities decline, the lower is the expected value of x. For example, if λ ¼ 0:5 then EðxÞ ¼ 2, whereas if λ ¼ 0:05 then EðxÞ ¼ 20. (continued)

69

70

Part 1 Introduction

EXAMPLE 2.14 CONTINUED 4. Normal. Because the Normal PDF is symmetric around zero, it seems clear that EðxÞ ¼ 0. A formal proof uses a change of variable integration by letting u ¼ x 2 =2 ðdu ¼ xdxÞ: þ∞

∫

∞

þ∞

1 1 2 pﬃﬃﬃﬃﬃﬃﬃ xe x =2 dx ¼ pﬃﬃﬃﬃﬃﬃﬃ 2π 2π

∫

∞

þ∞ 1 1 2 e u du ¼ pﬃﬃﬃﬃﬃﬃﬃ ½e x =2 ¼ pﬃﬃﬃﬃﬃﬃﬃ ½0 0 ¼ 0: ∞ 2π 2π

(2.185)

Of course, the expected value of a normally distributed random variable (or of any random variable) may be altered by a linear transformation, as shown in Equation 2.179. QUERY: A linear transformation changes a random variable’s expected value in a very predictable way—if y ¼ ax þ b, then EðyÞ ¼ aEðxÞ þ b. Hence, for this transformation [say, hðxÞ] we have E½hðxÞ ¼ h½EðxÞ. Suppose instead that x were transformed by a concave function, say gðxÞ with g 0 > 0 and g 00 < 0. How would E½ gðxÞ compare to g½EðxÞ? Note: This is an illustration of Jensen’s inequality, a concept we will pursue in detail in Chapter 7. See also Problem 2.13.

Variance and standard deviation The expected value of a random variable is a measure of central tendency. On the other hand, the variance of a random variable [denoted by σ2x or VarðxÞ] is a measure of dispersion. Speciﬁcally, the variance is deﬁned as the “expected squared deviation” of a random variable from its expected value. Formally: þ∞

VarðxÞ ¼

σ2x

2

¼ E½ðx EðxÞÞ ¼

∫ ðx EðxÞÞ f ðxÞ dx: 2

(2.186)

∞

Somewhat imprecisely, the variance measures the “typical” squared deviation from the central value of a random variable. In making the calculation, deviations from the expected value are squared so that positive and negative deviations from the expected value will both contribute to this measure of dispersion. After the calculation is made, the squaring process can be reversed to yield a measure of dispersion that is in the original units in which the random variable was measured. This square root of the variance is called the “standard pﬃﬃﬃﬃﬃ deviation” and is denoted as σx ð¼ σ2x Þ. The wording of the term effectively conveys its meaning: σx is indeed the typical (“standard”) deviation of a random variable from its expected value. When a random variable is subject to a linear transformation, its variance and standard deviation will be changed in a fairly obvious way. If y ¼ ax þ b, then þ∞

σ2y

¼

þ∞

∫ ½ax þ b Eðax þ bÞ f ðxÞ dx ¼ ∫ a ½x EðxÞ f ðxÞ dx ¼ a σ : 2

∞

2

2

2 2 x

(2.187)

∞

Hence, addition of a constant to a random variable does not change its variance, whereas multiplication by a constant multiplies the variance by the square of the constant. It is clear therefore that multiplying a variable by a constant multiplies its standard deviation by that constant: σax ¼ aσx .

Chapter 2 Mathematics for Microeconomics

EXAMPLE 2.15 Variances and Standard Deviations for Simple Random Variables Knowing the variances and standard deviations of the four simple random variables we have been looking at can sometimes be quite useful in economic applications. 1. Binomial. The variance of the binomial can be calculated by applying the deﬁnition in its discrete analog: n X σ2x ¼ ðxi EðxÞÞ2 f ðxi Þ ¼ ð1 pÞ2 ⋅ p þ ð0 pÞ2 ð1 pÞ i¼1

¼ ð1 pÞðp p 2 þ p 2 Þ ¼ pð1 pÞ: (2:188) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Hence, σx ¼ pð1 pÞ. One implication of this result is that a binomial variable has the largest variance and standard deviation when p ¼ 0:5, in which case σ2x ¼ 0:25 and σx ¼ 0:5. Because of the relatively ﬂat parabolic shape of pð1 pÞ, modest deviations of p from 0.5 do not change this variance substantially. 2. Uniform. Calculating the variance of the uniform distribution yields a mildly interesting result: b aþb 2 1 aþb 3 1 x dx ¼ x 2 ba 2 3ðb aÞa a " # 1 ðb aÞ3 ða bÞ3 ðb aÞ2 : ¼ ¼ 8 8 12 3ðb aÞ b

σ2x

¼

∫

(2:189)

This is one of the few places where the number 12 has any use in mathematics other than in measuring quantities of oranges or doughnuts. 3. Exponential. Integrating the variance formula for the exponential is relatively laborious. Fortunately, the result is quite simple; for the exponential, it turns out that σ2x ¼ 1=λ2 and σx ¼ 1=λ. Hence, the mean and standard deviation are the same for the exponential distribution—it is a “one-parameter distribution.” 4. Normal. In this case also, the integration can be burdensome. But again the result is simple: for the Normal distribution, σ2x ¼ σx ¼ 1. Areas below the Normal curve can be readily calculated and tables of these are available in any statistics text. Two useful facts about the Normal PDF are: þ1

∫ f ðxÞ dx 0:68

þ2

and

1

∫ f ðxÞ dx 0:95:

(2.190)

2

That is, the probability is about two thirds that a Normal variable will be within 1 standard deviation of the expected value and “most of the time” (i.e., with probability 0.95) it will be within 2 standard deviations. Standardizing the Normal. If the random variable x has a standard Normal PDF, it will have an expected value of 0 and a standard deviation of 1. However, a simple linear transformation can be used to give this random variable any desired expected value (μ) and standard deviation (σ). Consider the transformation y ¼ σx þ μ. Now EðyÞ ¼ σEðxÞ þ μ ¼ μ

and

VarðyÞ ¼ σ2y ¼ σ2 VarðxÞ ¼ σ2 :

(2.191)

Reversing this process can be used to “standardize” any Normally distributed random variable (y) with an arbitrary expected value (μ) and standard deviation (σ) (this is sometimes denoted (continued)

71

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Part 1 Introduction

EXAMPLE 2.15 CONTINUED as y ∼ N ðμ, σÞ) by using z ¼ ðy μÞ=σ. For example, SAT scores (y) are distributed Normally with an expected value of 500 points and a standard deviation of 100 points (that is, y ∼ N ð500, 100Þ). Hence, z ¼ ðy 500Þ=100 has a standard Normal distribution with expected value 0 and standard deviation 1. Equation 2.190 shows that approximately 68 percent of all scores lie between 400 and 600 points and 95 percent of all scores lie between 300 and 700 points. QUERY: Suppose that the random variable x is distributed uniformly along the interval [0, 12]. What are the mean and standard deviation of x? What fraction of the x distribution is within 1 standard deviation of the mean? What fraction of the distribution is within 2 standard deviations of the expected value? Explain why this differs from the fractions computed for the Normal distribution.

Covariance Some economic problems involve two or more random variables. For example, an investor may consider allocating his or her wealth among several assets the returns on which are taken to be random. Although the concepts of expected value, variance, and so forth carry over more or less directly when looking at a single random variable in such cases, it is also necessary to consider the relationship between the variables to get a complete picture. The concept of covariance is used to quantify this relationship. Before providing a deﬁnition, however, we will need to develop some background. Consider a case with two continuous random variables, x and y. The probability density function for these two variables, denoted by f ðx, yÞ, has the property that the probability associated with a set of outcomes in a small area (with dimensions dxdy) is given by f ðx, yÞdxdy. To be a proper PDF, it must be the case that: þ∞ þ∞

f ðx, yÞ 0

and

∫ ∫ f ðx, yÞ dx dy ¼ 1:

(2.192)

∞ ∞

The single-variable measures we have already introduced can be developed in this twovariable context by “integrating out” the other variable. That is, þ∞ þ∞

EðxÞ ¼

∫ ∫ xf ðx, yÞ dy dx

and

∞ ∞

(2.193)

þ∞ þ∞

VarðxÞ ¼

∫ ∫ ½x EðxÞ f ðx, yÞ dy dx: 2

∞ ∞

In this way, the parameters describing the random variable x are measured over all possible outcomes for y after taking into account the likelihood of those various outcomes. In this context, the covariance between x and y seeks to measure the direction of association between the variables. Speciﬁcally the covariance between x and y [denoted as Covðx, yÞ] is deﬁned as þ∞ þ∞

Covðx, yÞ ¼

∫ ∫ x EðxÞ

∞ ∞

y EðyÞ f ðx, yÞ dx dy:

(2.194)

Chapter 2 Mathematics for Microeconomics

The covariance between two random variables may be positive, negative, or zero. If values of x that are greater than EðxÞ tend to occur relatively frequently with values of y that are greater than EðyÞ (and similarly, if low values of x tend to occur together with low values of y ), then the covariance will be positive. In this case, values of x and y tend to move in the same direction. Alternatively, if high values of x tend to be associated with low values for y (and vice versa), the covariance will be negative. Two random variables are deﬁned to be independent if the probability of any particular value of, say, x is not affected by the particular value of y that might occur (and vice versa).24 In mathematical terms, this means that the PDF must have the property that f ðx, yÞ ¼ gðxÞhðyÞ—that is, the joint probability density function can be expressed as the product of two single-variable PDFs. If x and y are independent, their covariance will be zero: þ∞ þ∞

Covðx, yÞ ¼

∫ ∫ ½x EðxÞ½y EðyÞ gðxÞhðyÞ dx dy

∞ ∞

¼

þ∞

þ∞

∞

∞

∫ ½x EðxÞ gðxÞ dx ⋅ ∫ ½y EðyÞhðyÞ dy ¼ 0 ⋅ 0 ¼ 0:

(2.195)

The converse of this statement is not necessarily true, however. A zero covariance does not necessarily imply statistical independence. Finally, the covariance concept is crucial for understanding the variance of sums or differences of random variables. Although the expected value of a sum of two random variables is (as one might guess) the sum of their expected values: þ∞ þ∞

Eðx þ yÞ ¼

∫ ∫ ðx þ yÞf ðx, yÞ dx dy

∞ ∞ þ∞

¼

þ∞

∫ xf ðx, yÞ dy dx þ ∫ yf ðx, yÞ dx dy ¼ EðxÞ þ EðyÞ,

∞

(2.196)

∞

the relationship for the variance of such a sum is more complicated. Using the deﬁnitions we have developed yields þ∞ þ∞

Varðx þ yÞ ¼

∫ ∫ ½x þ y Eðx þ yÞ f ðx, yÞ dx dy 2

∞ ∞ þ∞ þ∞

¼

∫ ∫ ½x EðxÞ þ y EðyÞ f ðx, yÞ dx dy 2

∞ ∞ þ∞ þ∞

¼

∫ ∫ ½x EðxÞ þ ½y EðyÞ þ 2½x EðxÞ½y EðyÞ f ðx, yÞ dx dy 2

2

∞ ∞

¼ VarðxÞ þ VarðyÞ þ 2 Covðx, yÞ:

(2.197)

Hence, if x and y are independent then Varðx þ yÞ ¼ VarðxÞ þ VarðyÞ. The variance of the sum will be greater than the sum of the variances if the two random variables have a positive covariance and will be less than the sum of the variances if they have a negative covariance. Problems 2.13 and 2.14 provide further details on statistical issues that arise in microeconomic theory. 24

A formal deﬁnition relies on the concept of conditional probability. The conditional probability of an event B given that A has occurred (written P ðBjAÞ is deﬁned as P ðBjAÞ ¼ P ðA and BÞ=PðAÞ; B and A are deﬁned to be independent if P ðBjAÞ ¼ P ðBÞ. In this case, P ðA and BÞ ¼ P ðAÞ ⋅ P ðBÞ.

73

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Part 1 Introduction

SUMMARY Despite the formidable appearance of some parts of this chapter, this is not a book on mathematics. Rather, the intention here was to gather together a variety of tools that will be used to develop economic models throughout the remainder of the text. Material in this chapter will then be useful as a handy reference. One way to summarize the mathematical tools introduced in this chapter is by stressing again the economic lessons that these tools illustrate: •

Using mathematics provides a convenient, shorthand way for economists to develop their models. Implications of various economic assumptions can be studied in a simpliﬁed setting through the use of such mathematical tools.

•

The mathematical concept of the derivatives of a function is widely used in economic models because economists are often interested in how marginal changes in one variable affect another variable. Partial derivatives are especially useful for this purpose because they are deﬁned to represent such marginal changes when all other factors are held constant.

•

•

•

The mathematics of optimization is an important tool for the development of models that assume that economic agents rationally pursue some goal. In the unconstrained case, the ﬁrst-order conditions state that any activity that contributes to the agent’s goal should be expanded up to the point at which the marginal contribution of further expansion is zero. In mathematical terms, the ﬁrst-order condition for an optimum requires that all partial derivatives be zero. Most economic optimization problems involve constraints on the choices agents can make. In this case the ﬁrst-order conditions for a maximum suggest that each activity be operated at a level at which the ratio of the marginal beneﬁt–of the activity to its marginal cost is the same for all activities actually used. This common marginal beneﬁt– marginal cost ratio is also equal to the Lagrangian multiplier, which is often introduced to help solve constrained optimization problems. The Lagrangian multiplier can also be interpreted as the implicit value (or shadow price) of the constraint. The implicit function theorem is a useful mathematical device for illustrating the dependence of the choices that result from an optimization problem on the parameters

of that problem (for example, market prices). The envelope theorem is useful for examining how these optimal choices change when the problem’s parameters (prices) change. •

Some optimization problems may involve constraints that are inequalities rather than equalities. Solutions to these problems often illustrate “complementary slackness.” That is, either the constraints hold with equality and their related Lagrangian multipliers are nonzero, or the constraints are strict inequalities and their related Lagrangian multipliers are zero. Again this illustrates how the Lagrangian multiplier implies something about the “importance” of constraints.

•

The ﬁrst-order conditions shown in this chapter are only the necessary conditions for a local maximum or minimum. One must also check second-order conditions that require that certain curvature conditions be met.

•

Certain types of functions occur in many economic problems. Quasi-concave functions (those functions for which the level curves form convex sets) obey the secondorder conditions of constrained maximum or minimum problems when the constraints are linear. Homothetic functions have the useful property that implicit trade-offs among the variables of the function depend only on the ratios of these variables.

•

Integral calculus is often used in economics both as a way of describing areas below graphs and as a way of summing results over time. Techniques that involve various ways of differentiating integrals play an important role in the theory of optimizing behavior.

•

Many economic problems are dynamic in that decisions at one date affect decisions and outcomes at later dates. The mathematics for solving such dynamic optimization problems is often a straightforward generalization of Lagrangian methods.

•

Concepts from mathematical statistics are often used in studying the economics of uncertainty and information. The most fundamental concept is the notion of a random variable and its associated probability density function. Parameters of this distribution, such as its expected value or its variance, also play important roles in many economic models.

Chapter 2 Mathematics for Microeconomics

PROBLEMS 2.1 Suppose U ðx, yÞ ¼ 4x 2 þ 3y 2 . a. Calculate ∂U =∂x, ∂U =∂y. b. Evaluate these partial derivatives at x ¼ 1, y ¼ 2. c. Write the total differential for U . d. Calculate dy=dx for dU ¼ 0—that is, what is the implied trade-off between x and y holding U constant? e. Show U ¼ 16 when x ¼ 1, y ¼ 2. f. In what ratio must x and y change to hold U constant at 16 for movements away from x ¼ 1, y ¼ 2? g. More generally, what is the shape of the U ¼ 16 contour line for this function? What is the slope of that line?

2.2 Suppose a ﬁrm’s total revenues depend on the amount produced ðqÞ according to the function R ¼ 70q q 2 : Total costs also depend on q: C ¼ q 2 þ 30q þ 100: a. What level of output should the ﬁrm produce in order to maximize proﬁts (R C)? What will proﬁts be? b. Show that the second-order conditions for a maximum are satisﬁed at the output level found in part (a). c. Does the solution calculated here obey the “marginal revenue equals marginal cost” rule? Explain.

2.3 Suppose that f ðx, yÞ ¼ xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways: by substitution and by using the Lagrangian multiplier method.

2.4 The dual problem to the one described in Problem 2.3 is minimize x þ y subject to xy ¼ 0:25: Solve this problem using the Lagrangian technique. Then compare the value you get for the Lagrangian multiplier to the value you got in Problem 2.3. Explain the relationship between the two solutions.

2.5 The height of a ball that is thrown straight up with a certain force is a function of the time (t ) from which it is released given by f ðt Þ ¼ 0:5gt 2 þ 40t (where g is a constant determined by gravity).

75

76

Part 1 Introduction a. How does the value of t at which the height of the ball is at a maximum depend on the parameter g? b. Use your answer to part (a) to describe how maximum height changes as the parameter g changes. c. Use the envelope theorem to answer part (b) directly. d. On the Earth g ¼ 32, but this value varies somewhat around the globe. If two locations had gravitational constants that differed by 0.1, what would be the difference in the maximum height of a ball tossed in the two places?

2.6 A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long. Now cut a smaller square that is t feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a traylike structure with no top. a. Show that the volume of oil that can be held by this tray is given by V ¼ t ðx 2t Þð3x 2t Þ ¼ 3tx 2 8t 2 x þ 4t 3 : b. How should t be chosen so as to maximize V for any given value of x? c. Is there a value of x that maximizes the volume of oil that can be carried? d. Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation 3x 2 4t 2 ¼ 1,000,000 (because the builder can return the cut-out squares for credit). How does the solution to this constrained maximum problem compare to the solutions described in parts (b) and (c)?

2.7 Consider the following constrained maximization problem: maximize y ¼ x1 þ 5 ln x2 subject to k x1 x2 ¼ 0, where k is a constant that can be assigned any speciﬁc value. a. Show that if k ¼ 10, this problem can be solved as one involving only equality constraints. b. Show that solving this problem for k ¼ 4 requires that x1 ¼ 1. c. If the x’s in this problem must be nonnegative, what is the optimal solution when k ¼ 4? d. What is the solution for this problem when k ¼ 20? What do you conclude by comparing this solution to the solution for part (a)? Note: This problem involves what is called a “quasi-linear function.” Such functions provide important examples of some types of behavior in consumer theory—as we shall see.

2.8 Suppose that a ﬁrm has a marginal cost function given by MCðqÞ ¼ q þ 1. a. What is this ﬁrm’s total cost function? Explain why total costs are known only up to a constant of integration, which represents ﬁxed costs. b. As you may know from an earlier economics course, if a ﬁrm takes price (p) as given in its decisions then it will produce that output for which p ¼ MCðqÞ. If the ﬁrm follows this proﬁtmaximizing rule, how much will it produce when p ¼ 15? Assuming that the ﬁrm is just breaking even at this price, what are ﬁxed costs?

Chapter 2 Mathematics for Microeconomics c. How much will proﬁts for this ﬁrm increase if price increases to 20? d. Show that, if we continue to assume proﬁt maximization, then this ﬁrm’s proﬁts can be expressed solely as a function of the price it receives for its output. e. Show that the increase in proﬁts from p ¼ 15 to p ¼ 20 can be calculated in two ways: (i) directly from the equation derived in part (d); and (ii) by integrating the inverse marginal cost function ½MC 1 ðpÞ ¼ p 1 from p ¼ 15 to p ¼ 20. Explain this result intuitively using the envelope theorem.

Analytical Problems 2.9 Concave and quasi-concave functions Show that if f ðx1 , x2 Þ is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (deﬁning quasi-concavity) to Equation 2.98 (deﬁning concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasi-concave functions necessarily concave? If not, give a counterexample.

2.10 The Cobb-Douglas function One of the most important functions we will encounter in this book is the Cobb-Douglas function: y ¼ ðx1 Þα ðx2 Þβ , where α and β are positive constants that are each less than 1. a. Show that this function is quasi-concave using a “brute force” method by applying Equation 2.114. b. Show that the Cobb-Douglas function is quasi-concave by showing that any contour line of the form y ¼ c (where c is any positive constant) is convex and therefore that the set of points for which y > c is a convex set. c. Show that if α þ β > 1 then the Cobb-Douglas function is not concave (thereby illustrating again that not all quasi-concave functions are concave). Note: The Cobb-Douglas function is discussed further in the Extensions to this chapter.

2.11 The power function Another function we will encounter often in this book is the “power function”: y ¼ xδ, where 0 δ 1 (at times we will also examine this function for cases where δ can be negative, too, in which case we will use the form y ¼ x δ =δ to ensure that the derivatives have the proper sign). a. Show that this function is concave (and therefore also, by the result of Problem 2.9, quasi-concave). Notice that the δ ¼ 1 is a special case and that the function is “strictly” concave only for δ < 1. b. Show that the multivariate form of the power function y ¼ f ðx1 , x2 Þ ¼ ðx1 Þδ þ ðx2 Þδ is also concave (and quasi-concave). Explain why, in this case, the fact that f12 ¼ f21 ¼ 0 makes the determination of concavity especially simple. c. One way to incorporate “scale” effects into the function described in part (b) is to use the monotonic transformation gðx1 , x2 Þ ¼ y γ ¼ ½ðx1 Þδ þ ðx2 Þδ γ , where γ is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave?

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Part 1 Introduction

2.12 Taylor approximations Taylor’s theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables. a. Any continuous and differentiable function of a single variable, f ðxÞ, can be approximated near the point a by the formula f ðxÞ ¼ f ðaÞ þ f 0 ðaÞðx aÞ þ 0:5f 00 ðaÞðx aÞ2 þ terms in f 000 , f 0000 , …: Using only the ﬁrst three of these terms results in a quadratic Taylor approximation. Use this approximation together with the deﬁnition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point a. b. The quadratic Taylor approximation for any function of two variables, f ðx, yÞ, near the point ða, bÞ is given by f ðx, yÞ ¼ f ða, bÞ þ f1 ða, bÞðx aÞ þ f2 ða, bÞðy bÞ þ 0:5½ f11 ða,bÞðx aÞ2 þ 2f12 ða, bÞðx aÞðy bÞ þ f22 ðy bÞ2 : Use this approximation to show that any concave function (as deﬁned by Equation 2.98) must lie on or below its tangent plane at (a, b).

2.13 More on expected value Because the expected value concept plays an important role in many economic theories, it may be useful to summarize a few more properties of this statistical measure. Throughout this problem, x is assumed to be a continuous random variable with probability density function f ðxÞ. a. (Jensen’s inequality) Suppose that gðxÞ is a concave function. Show that E½ gðxÞ g½EðxÞ. Hint: Construct the tangent to gðxÞ at the point EðxÞ. This tangent will have the form c þ dx gðxÞ for all values of x and c þ dEðxÞ ¼ g½EðxÞ where c and d are constants. b. Use the procedure from part (a) to show that if gðxÞ is a convex function then E½ gðxÞ g½EðxÞ. c. Suppose x takes on only nonnegative values—that is, 0 x ∞. Use integration by parts to show that ∞

EðxÞ ¼

∫½1 F ðxÞ dx, 0

where F ðxÞ is the cumulative distribution function for x [that is, F ðxÞ ¼ ∫x0 f ðt Þ dt ]. d. (Markov’s inequality) Show that if x takes on only positive values then the following inequality holds: EðxÞ P ðx t Þ : t Hint: EðxÞ ¼ ∫∞0 xf ðxÞ dx ¼ ∫t0 xf ðxÞ dx þ ∫∞t xf ðxÞ dx: e. Consider the probability density function f ðxÞ ¼ 2x 3 for x 1. (1) Show that this is a proper PDF. (2) Calculate F ðxÞ for this PDF. (3) Use the results of part (c) to calculate EðxÞ for this PDF. (4) Show that Markov’s inequality holds for this function. f. The concept of conditional expected value is useful in some economic problems. We denote the expected value of x conditional on the occurrence of some event, A, as EðxjAÞ. To compute this value we need to know the PDF for x given that A has occurred [denoted by f ðxjAÞ]. With this

Chapter 2 Mathematics for Microeconomics

79

notation, EðxjAÞ ¼ ∫þ∞ ∞ xf ðxjAÞ dx. Perhaps the easiest way to understand these relationships is with an example. Let f ðxÞ ¼

x2 for 1 x 2: 3

(1) Show that this is a proper PDF. (2) Calculate EðxÞ. (3) Calculate the probability that 1 x 0. (4) Consider the event 0 x 2, and call this event A. What is f ðxjAÞ? (5) Calculate EðxjAÞ. (6) Explain your results intuitively.

2.14 More on variances and covariances This problem presents a few useful mathematical facts about variances and covariances. a. Show that VarðxÞ ¼ Eðx 2 Þ ½EðxÞ2 . b. Show that the result in part (a) can be generalized as Covðx, yÞ ¼ EðxyÞ EðxÞEðyÞ. Note: If Covðx, yÞ ¼ 0, then EðxyÞ ¼ EðxÞEðyÞ. c. Show that Varðax byÞ ¼ a 2 VarðxÞ þ b 2 VarðyÞ 2ab Covðx, yÞ. d. Assume that two independent random variables, x and y, are characterized by EðxÞ ¼ EðyÞ and VarðxÞ ¼ VarðyÞ . Show that Eð0:5x þ 0:5yÞ ¼ EðxÞ. Then use part (c) to show that Varð0:5x þ 0:5yÞ ¼ 0:5 VarðxÞ. Describe why this fact provides the rationale for diversiﬁcation of assets.

SUGGESTIONS FOR FURTHER READING Dadkhan, Kamran. Foundations of Mathematical and Computational Economics. Mason, OH: Thomson/SouthWestern, 2007.

Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947. Mathematical Appendix A.

This is a good introduction to many calculus techniques. The book shows how many mathematical questions can be approached using popular software programs such as Matlab or Excel.

A basic reference. Mathematical Appendix A provides an advanced treatment of necessary and suﬃcient conditions for a maximum.

Dixit, A. K. Optimization in Economic Theory, 2nd ed. New York: Oxford University Press, 1990. A complete and modern treatment of optimization techniques. Uses relatively advanced analytical methods.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGrawHill, 2001. A mathematical microeconomics text that stresses the observable predictions of economic theory. The text makes extensive use of the envelope theorem.

Hoy, Michael, John Livernois, Chris McKenna, Ray Rees, and Thanasis Stengos. Mathematics for Economists, 2nd ed. Cambridge, MA: MIT Press, 2001.

Simon, Carl P., and Lawrence Blume. Mathematics for Economists. New York: W. W. Norton, 1994.

A complete introduction to most of the mathematics covered in microeconomics courses. The strength of the book is its presentation of many worked-out examples, most of which are based on microeconomic theory.

A very useful text covering most areas of mathematics relevant to economists. Treatment is at a relatively high level. Two topics discussed better here than elsewhere are diﬀerential equations and basic point-set topology.

Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995.

Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000.

Encyclopedic treatment of mathematical microeconomics. Extensive mathematical appendices cover relatively high-level topics in analysis.

An indispensable tool for mathematical review. Contains 32 chapters covering most of the mathematical tools that economists use.

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Discussions are very brief, so this is not the place to encounter new concepts for the ﬁrst time.

Taylor, Angus E., and W. Robert Mann. Advanced Calculus, 3rd ed. New York: John Wiley, 1983, pp. 183–95. A comprehensive calculus text with a good discussion of the Lagrangian technique.

Thomas, George B., and Ross L. Finney. Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, 1992. Basic calculus text with excellent coverage of diﬀerentiation techniques.

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Chapter 2 Mathematics for Microeconomics

EXTENSIONS Second-Order Conditions and Matrix Algebra A is 2 2, then the ﬁrst leading principal minor is a11 and the second is a11 a22 a21 a12 .

The second-order conditions described in Chapter 2 can be written in very compact ways by using matrix algebra. In this extension, we look brieﬂy at that notation. We return to this notation at a few other places in the extensions and problems for later chapters.

6. An n n square matrix, A, is positive deﬁnite if all of its leading principal minors are positive. The matrix is negative deﬁnite if its principal minors alternate in sign starting with a minus.1

Matrix algebra background

7. A particularly useful symmetric matrix is the Hessian matrix formed by all of the secondorder partial derivatives of a function. If f is a continuous and twice differentiable function of n variables, then its Hessian is given by 2 3 f11 f12 … f1n 6 f21 f22 … f2n 7 6 7 Hð f Þ ¼ 6 . 7: 4 .. 5 fn1 fn2 … fnn

The extensions presented here assume some general familiarity with matrix algebra. A succinct reminder of these principles might include: 1. An n k matrix, A, is a rectangular array of terms of the form 2 3 a11 a12 … a1k h i 6 a21 a22 … a2k 7 6 7 A ¼ aij ¼ 6 . 7: 4 .. 5 an1 an2 … ank Here i ¼ 1, n; j ¼ 1, k. Matrices can be added, subtracted, or multiplied providing their dimensions are conformable. 2. If n ¼ k, then A is a square matrix. A square matrix is symmetric if aij ¼ aji . The identity matrix, In, is an n þ n square matrix where aij ¼ 1 if i ¼ j and aij ¼ 0 if i 6¼ j . 3. The determinant of a square matrix (denoted by jAj) is a scalar (i.e., a single term) found by suitably multiplying together all of the terms in the matrix. If A is 2 2, jAj ¼ a11 a22 a21 a12 :

1 3 Example: If A ¼ then 5 2

Using these notational ideas, we can now examine again some of the second-order conditions derived in Chapter 2.

E2.1 Concave and convex functions A concave function is one that is always below (or on) any tangent to it. Alternatively, a convex function is always above (or on) any tangent. The concavity or convexity of any function is determined by its second derivative(s). For a function of a single variable, f ðxÞ, the requirement is straightforward. Using the Taylor approximation at any point (x0 ) f ðx0 þ dxÞ ¼ f ðx0 Þ þ f 0 ðx0 Þdx þ f 00 ðx0 Þ þ higher-order terms:

Assuming that the higher-order terms are 0, we have f ðx0 þ dxÞ f ðx0 Þ þ f 0 ðx0 Þdx

jAj ¼ 2 15 ¼ 13: 4. The inverse of an n n square matrix, A, is another n n matrix, A 1 , such that AA

1

¼ In :

Not every square matrix has an inverse. A necessary and sufﬁcient condition for the existence of A1 is that jAj 6¼ 0. 5. The leading principal minors of an n n square matrix A are the series of determinants of the ﬁrst p rows and columns of A, where p ¼ 1, n. If

dx 2 2

if f 00 ðx0 Þ 0 and f ðx0 þ dxÞ f ðx0 Þ þ f 0 ðx0 Þdx if f 00 ðx0 Þ 0. Because the expressions on the right of these inequalities are in fact the equation of the tangent to the function at x0 , it is clear that the

1

If some of the determinants in this deﬁnition are 0 then the matrix is said to be positive semideﬁnite or negative semideﬁnite.

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Part 1 Introduction

function is (locally) concave if f 00 ðx0 Þ 0 and (locally) convex if f 00 ðx0 Þ 0. Extending this intuitive idea to many dimensions is cumbersome in terms of functional notation, but relatively simple when matrix algebra is used. Concavity requires that the Hessian matrix be negative deﬁnite whereas convexity requires that this matrix be positive deﬁnite. As in the single variable case, these conditions amount to requiring that the function move consistently away from any tangent to it no matter what direction is taken.2 If f ðx1 , x2 Þ is a function of two variables, the Hessian is given by

f11 f12 : H¼ f21 f22

Hence, the Hessian for this function is

aða 1Þx a2 y b abx a1 y b1 : H¼ abx a1 y b1 bðb 1Þx a y b2 The ﬁrst leading principal minor of this Hessian is H1 ¼ aða 1Þx a2 y b < 0 and so the function will be concave, providing H2 ¼ aða 1ÞðbÞðb 1Þx 2a2 y 2b2 a 2 b 2 x 2a2 y 2b2 ¼ abð1a bÞx 2a2 y 2b2 > 0: This condition clearly holds if a þ b < 1. That is, in production function terminology, the function must exhibit diminishing returns to scale to be concave. Geometrically, the function must turn downward as both inputs are increased together.

This is negative deﬁnite if f11 < 0

and

f11 f22 f21 f12 > 0,

which is precisely the condition described in Equation 2.98. Generalizations to functions of three or more variables follow the same matrix pattern. Example 1 For the health status function in Chapter 2 (Equation 2.20), the Hessian is given by

2 0 H¼ , 0 2 and the ﬁrst and second leading principal minors are H1 ¼ 2 < 0 and H2 ¼ ð2Þð2Þ 0 ¼ 4 > 0: Hence, the function is concave. Example 2 The Cobb-Douglas function x a y b where a, b 2 ð0, 1Þ is used to illustrate utility functions and production functions in many places in this text. The ﬁrst- and second-order derivatives of the function are y , fx ¼ ax a b1 fy ¼ bx y , a1 b

E2.2 Maximization As we saw in Chapter 2, the ﬁrst-order conditions for an unconstrained maximum of a function of many variables requires ﬁnding a point at which the partial derivatives are zero. If the function is concave it will be below its tangent plane at this point and therefore the point will be a true maximum.3 Because the health status function is concave, for example, the ﬁrstorder conditions for a maximum are also sufﬁcient.

E2.3 Constrained maxima When the x’s in a maximization or minimization problem are subject to constraints, these constraints have to be taken into account in stating second-order conditions. Again, matrix algebra provides a compact (if not very intuitive) way of denoting these conditions. The notation involves adding rows and columns of the Hessian matrix for the unconstrained problem and then checking the properties of this augmented matrix. Speciﬁcally, we wish to maximize f ðx1 , …, xn Þ subject to the constraint4 gðx1 , …, xn Þ ¼ 0:

fxx ¼ aða 1Þx a2 y b , fyy ¼ bðb 1Þx a y b2 : 3 This will be a “local” maximum if the function is concave only in a region, or “global” if the function is concave everywhere. 4

2 A proof using the multivariable version of Taylor’s approximation is provided in Simon and Blume (1994), chap. 21.

Here we look only at the case of a single constraint. Generalization to many constraints is conceptually straightforward but notationally complex. For a concise statement see Sydsaeter, Strom, and Berck (2000), p. 93.

Chapter 2 Mathematics for Microeconomics

We saw in Chapter 2 that the ﬁrst-order conditions for a maximum are of the form fi þ λgi ¼ 0, where λ is the Lagrangian multiplier for this problem. Second-order conditions for a maximum are based on the augmented (“bordered”) Hessian5 3 2 0 g1 g2 … gn 6 g1 f11 f12 f1n 7 7 6 6 g2 f21 f22 f2n 7 Hb ¼ 6 7: 7 6 .. 5 4 . … gn fn1 fn2 fnn For a maximum, (1)Hb must be negative deﬁnite— that is, the leading principal minors of Hb must follow the pattern + + and so forth, starting with the second such minor.6 The second-order conditions for minimum require that (1)Hb be positive deﬁnite—that is, all of the leading principal minors of Hb (except the ﬁrst) should be negative. Example The Lagrangian for the constrained health status problem (Example 2.6) is ℒ ¼ x 21 þ 2x1 x 22 þ 4x2 þ 5 þ λð1 x1 x2 Þ, and the bordered Hessian for this problem is 2 3 0 1 1 Hb ¼ 4 1 2 0 5: 1 0 2 The second leading principal minor here is

0 1 ¼ 1, Hb2 ¼ 1 2 and the third is 2

0 1

1

3

7 6 05 Hb3 ¼ 4 1 2 1 0 2 ¼ 0 þ 0 þ 0 ð2Þ 0 ð2Þ ¼ 4, so the leading principal minors of the Hb have the required pattern and the point x2 ¼ 1, x1 ¼ 0, is a constrained maximum. 5 Notice that, if gij ¼ 0 for all i and j , then Hb can be regarded as the simple Hessian associated with the Lagrangian expression given in Equation 2.50, which is a function of the n þ 1 variables λ, x1 , …, xn . 6

Notice that the ﬁrst leading principal minor of Hb is 0.

83

Example In the optimal fence problem (Example 2.7), the bordered Hessian is 2 3 0 2 2 0 15 Hb ¼ 4 2 2 1 0 and Hb2 ¼ 4, Hb3 ¼ 8, so again the leading principal minors have the sign pattern required for a maximum.

E2.4 Quasi-concavity If the constraint g is linear, then the second-order conditions explored in Extension 2.3 can be related solely to the shape of the function to be optimized, f . In this case the constraint can be written as gðx1 , …, xn Þ ¼ c b1 x1 b2 x2 … bn xn ¼ 0, and the ﬁrst-order conditions for a maximum are fi ¼ λbi ,

i ¼ 1, …, n:

Using the conditions, it is clear that the bordered Hessian Hb and the matrix 2 3 f2 … fn 0 f1 6 f1 f11 f12 f1n 7 7 H0 ¼ 6 4f f21 f22 f2n 5 2 fn fn1 fn2 … fnn have the same leading principal minors except for a (positive) constant of proportionality.7 The conditions for a maximum of f subject to a linear constraint will be satisﬁed provided H0 follows the same sign conventions as Hb—that is, (1)H0 must be negative deﬁnite. A function f for which H0 does follow this pattern is called quasi-concave. As we shall see, f has the property that the set of points x for which f ðxÞ c (where c is any constant) is convex. For such a function, the necessary conditions for a maximum are also sufﬁcient. Example For the fences problem, f ðx, yÞ ¼ xy and H0 is given by

7 This can be shown by noting that multiplying a row (or a column) of a matrix by a constant multiplies the determinant by that constant.

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Part 1 Introduction

2

0 H ¼4y x 0

y 0 1

3 x 1 5: 0

So H02 ¼ y 2 < 0, H03 ¼ 2xy > 0, and the function is quasi-concave.8 Example More generally, if f is a function of only two variables, then quasi-concavity requires that

Since f ðx, yÞ ¼ xy is a form of a Cobb-Douglas function that is not concave, this shows that not every quasi-concave function is concave. Notice that a monotonic function of f (such as f 1=3 ) would be concave, however. 8

H02 ¼ ðf1 Þ2 < 0 and H03 ¼ f11 f 22 f22 f 21 þ 2f1 f2 f12 > 0, which is precisely the condition stated in Equation 2.114. Hence, we have a fairly simple way of determining quasi-concavity.

References Simon, C. P., and L. Blume. Mathematics for Economists. New York: W.W. Norton, 1994. Sydsaeter, R., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000.

P A R T

Choice and Demand CHAPTER 3 Preferences and Utility CHAPTER 4 Utility Maximization and Choice CHAPTER 5 Income and Substitution Effects CHAPTER 6 Demand Relationships among Goods CHAPTER 7 Uncertainty and Information CHAPTER 8 Strategy and Game Theory

In Part 2 we will investigate the economic theory of choice. One goal of this examination is to develop the notion of demand in a formal way so that it can be used in later sections of the text when we turn to the study of markets. A more general goal of this part is to illustrate the theory economists use to explain how individuals make choices in a wide variety of contexts. Part 2 begins with a description of the way economists model individual preferences, which are usually referred to by the formal term utility. Chapter 3 shows how economists are able to conceptualize utility in a mathematical way. This permits the development of “indiﬀerence curves,” which show the various exchanges that individuals are willing to make voluntarily. The utility concept is used in Chapter 4 to illustrate the theory of choice. The fundamental hypothesis of the chapter is that people faced with limited incomes will make economic choices in such a way as to achieve as much utility as possible. Chapter 4 uses mathematical and intuitive analyses to indicate the insights that this hypothesis provides about economic behavior. Chapters 5 and 6 use the model of utility maximization to investigate how individuals will respond to changes in their circumstances. Chapter 5 is primarily concerned with responses to changes in the price of a commodity, an analysis that leads directly to the demand curve notion. Chapter 6 applies this type of analysis to developing an understanding of demand relationships among diﬀerent goods. The ﬁnal two chapters in this part look at individual behavior in uncertain situations. In Chapter 7 we describe why people generally dislike risks and are willing to pay something to avoid taking them. Chapter 8 then looks at uncertainties that arise when two or more people ﬁnd themselves in a “game” in which they must make strategic choices. The equilibrium notions we develop in studying such games are widely used throughout economics.

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CHAPTER

3 Preferences and Utility In this chapter we look at the way in which economists characterize individuals’ preferences. We begin with a fairly abstract discussion of the “preference relation,” but quickly turn to the economists’ primary tool for studying individual choices—the utility function. We look at some general characteristics of that function and a few simple examples of speciﬁc utility functions we will encounter throughout this book.

AXIOMS OF RATIONAL CHOICE One way to begin an analysis of individuals’ choices is to state a basic set of postulates, or axioms, that characterize “rational” behavior. These begin with the concept of “preference”: An individual who reports that “A is preferred to B” is taken to mean that all things considered, he or she feels better off under situation A than under situation B. The preference relation is assumed to have three basic properties as follows. I. Completeness. If A and B are any two situations, the individual can always specify exactly one of the following three possibilities: 1. “A is preferred to B,” 2. “B is preferred to A,” or 3. “A and B are equally attractive.” Consequently, people are assumed not to be paralyzed by indecision: They completely understand and can always make up their minds about the desirability of any two alternatives. The assumption also rules out the possibility that an individual can report both that A is preferred to B and that B is preferred to A. II. Transitivity. If an individual reports that “A is preferred to B” and “B is preferred to C,” then he or she must also report that “A is preferred to C.” This assumption states that the individual’s choices are internally consistent. Such an assumption can be subjected to empirical study. Generally, such studies conclude that a person’s choices are indeed transitive, but this conclusion must be modiﬁed in cases where the individual may not fully understand the consequences of the choices he or she is making. Because, for the most part, we will assume choices are fully informed (but see the discussion of uncertainty in Chapter 7 and elsewhere), the transitivity property seems to be an appropriate assumption to make about preferences. III. Continuity. If an individual reports “A is preferred to B,” then situations suitably “close to” A must also be preferred to B. This rather technical assumption is required if we wish to analyze individuals’ responses to relatively small changes in income and prices. The purpose of the assumption is to rule out certain kinds of discontinuous, knife-edge preferences that pose problems for a mathematical development of the theory of choice. Assuming continuity does 87

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not seem to risk missing types of economic behavior that are important in the real world.

UTILITY Given the assumptions of completeness, transitivity, and continuity, it is possible to show formally that people are able to rank all possible situations from the least desirable to the most.1 Following the terminology introduced by the nineteenth-century political theorist Jeremy Bentham, economists call this ranking utility.2 We also will follow Bentham by saying that more desirable situations offer more utility than do less desirable ones. That is, if a person prefers situation A to situation B, we would say that the utility assigned to option A, denoted by U ðAÞ, exceeds the utility assigned to B, U ðBÞ.

Nonuniqueness of utility measures We might even attach numbers to these utility rankings; however, these numbers will not be unique. Any set of numbers we arbitrarily assign that accurately reﬂects the original preference ordering will imply the same set of choices. It makes no difference whether we say that U ðAÞ ¼ 5 and U ðBÞ ¼ 4, or that U ðAÞ ¼ 1,000,000 and U ðBÞ ¼ 0:5. In both cases the numbers imply that A is preferred to B. In technical terms, our notion of utility is deﬁned only up to an order-preserving (“monotonic”) transformation.3 Any set of numbers that accurately reﬂects a person’s preference ordering will do. Consequently, it makes no sense to ask “how much more is A preferred than B?” since that question has no unique answer. Surveys that ask people to rank their “happiness” on a scale of 1 to 10 could just as well use a scale of 7 to 1,000,000. We can only hope that a person who reports he or she is a “6” on the scale one day and a “7” on the next day is indeed happier on the second day. Utility rankings are therefore like the ordinal rankings of restaurants or movies using one, two, three, or four stars. They simply record the relative desirability of commodity bundles. This lack of uniqueness in the assignment of utility numbers also implies that it is not possible to compare utilities of different people. If one person reports that a steak dinner provides a utility of “5” and another reports that the same dinner offers a utility of “100,” we cannot say which individual values the dinner more because they could be using very different scales. Similarly, we have no way of measuring whether a move from situation A to situation B provides more utility to one person or another. Nonetheless, as we will see, economists can say quite a bit about utility rankings by examining what people voluntarily choose to do.

The ceteris paribus assumption Because utility refers to overall satisfaction, such a measure clearly is affected by a variety of factors. A person’s utility is affected not only by his or her consumption of physical commodities, but also by psychological attitudes, peer group pressures, personal experiences, and the 1

These properties and their connection to representation of preferences by a utility function are discussed in detail in Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (New York: Oxford University Press, 1995).

2

J. Bentham, Introduction to the Principles of Morals and Legislation (London: Hafner, 1848).

We can denote this idea mathematically by saying that any numerical utility ranking ðU Þ can be transformed into another set of numbers by the function F providing that F ðU Þ is order preserving. This can be ensured if F 0ðU Þ > 0. For example, the transformation F ðU Þ ¼ U 2 is order preserving as is the transformation F ðU Þ ¼ ln U . At some places in the text and problems we will find it convenient to make such transformations in order to make a particular utility ranking easier to analyze. 3

Chapter 3 Preferences and Utility

general cultural environment. Although economists do have a general interest in examining such inﬂuences, a narrowing of focus is usually necessary. Consequently, a common practice is to devote attention exclusively to choices among quantiﬁable options (for example, the relative quantities of food and shelter bought, the number of hours worked per week, or the votes among speciﬁc taxing formulas) while holding constant the other things that affect behavior. This ceteris paribus (other things being equal) assumption is invoked in all economic analyses of utility-maximizing choices so as to make the analysis of choices manageable within a simpliﬁed setting.

Utility from consumption of goods As an important example of the ceteris paribus assumption, consider an individual’s problem of choosing, at a single point in time, among n consumption goods x1 , x2 , …, xn : We shall assume that the individual’s ranking of these goods can be represented by a utility function of the form utility ¼ U ðx1 , x2 , …, xn ; other thingsÞ,

(3.1)

where the x’s refer to the quantities of the goods that might be chosen and the “other things” notation is used as a reminder that many aspects of individual welfare are being held constant in the analysis. Quite often it is easier to write Equation 3.1 as utility ¼ U ðx1 , x2 , …, xn Þ

(3.2)

or, if only two goods are being considered, as utility ¼ U ðx, yÞ,

(3.20 )

where it is clear that everything is being held constant (that is, outside the frame of analysis) except the goods actually referred to in the utility function. It would be tedious to remind you at each step what is being held constant in the analysis, but it should be remembered that some form of the ceteris paribus assumption will always be in effect.

Arguments of utility functions The utility function notation is used to indicate how an individual ranks the particular arguments of the function being considered. In the most common case, the utility function (Equation 3.2) will be used to represent how an individual ranks certain bundles of goods that might be purchased at one point in time. On occasion we will use other arguments in the utility function, and it is best to clear up certain conventions at the outset. For example, it may be useful to talk about the utility an individual receives from real wealth ðW Þ. Therefore, we shall use the notation utility ¼ U ðW Þ.

(3.3)

Unless the individual is a rather peculiar, Scrooge-type person, wealth in its own right gives no direct utility. Rather, it is only when wealth is spent on consumption goods that any utility results. For this reason, Equation 3.3 will be taken to mean that the utility from wealth is in fact derived by spending that wealth in such a way as to yield as much utility as possible. Two other arguments of utility functions will be used in later chapters. In Chapter 16 we will be concerned with the individual’s labor-leisure choice and will therefore have to consider the presence of leisure in the utility function. A function of the form utility ¼ U ðc, hÞ

(3.4)

will be used. Here, c represents consumption and h represents hours of nonwork time (that is, leisure) during a particular time period.

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In Chapter 17 we will be interested in the individual’s consumption decisions in different time periods. In that chapter we will use a utility function of the form utility ¼ U ðc1 , c2 Þ,

(3.5)

where c1 is consumption in this period and c2 is consumption in the next period. By changing the arguments of the utility function, therefore, we will be able to focus on specific aspects of an individual’s choices in a variety of simplified settings. In summary then, we start our examination of individual behavior with the following deﬁnition. DEFINITION

Utility. Individuals’ preferences are assumed to be represented by a utility function of the form U ðx1 , x2 , …, xn Þ,

(3.6)

where x1 , x2 , …, xn are the quantities of each of n goods that might be consumed in a period. This function is unique only up to an order-preserving transformation.

Economic goods In this representation the variables are taken to be “goods”; that is, whatever economic quantities they represent, we assume that more of any particular xi during some period is preferred to less. We assume this is true of every good, be it a simple consumption item such as a hot dog or a complex aggregate such as wealth or leisure. We have pictured this convention for a two-good utility function in Figure 3.1. There, all consumption bundles in the shaded area are FIGURE 3.1

More of a Good Is Preferred to Less The shaded area represents those combinations of x and y that are unambiguously preferred to the combination x , y . Ceteris paribus, individuals prefer more of any good rather than less. Combinations identiﬁed by “?” involve ambiguous changes in welfare because they contain more of one good and less of the other.

Quantity of y

? Preferred to x*, y* y* Worse than x*, y*

?

x*

Quantity of x

Chapter 3 Preferences and Utility

preferred to the bundle x , y because any bundle in the shaded area provides more of at least one of the goods. By our deﬁnition of “goods,” then, bundles of goods in the shaded area are ranked higher than x , y . Similarly, bundles in the area marked “worse” are clearly inferior to x , y , since they contain less of at least one of the goods and no more of the other. Bundles in the two areas indicated by question marks are difﬁcult to compare to x , y because they contain more of one of the goods and less of the other. Movements into these areas involve trade-offs between the two goods.

TRADES AND SUBSTITUTION Most economic activity involves voluntary trading between individuals. When someone buys, say, a loaf of bread, he or she is voluntarily giving up one thing (money) for something else (bread) that is of greater value to that individual. To examine this kind of voluntary transaction, we need to develop a formal apparatus for illustrating trades in the utility function context.

Indifference curves and the marginal rate of substitution To discuss such voluntary trades, we develop the idea of an indifference curve. In Figure 3.2, the curve U1 represents all the alternative combinations of x and y for which an individual is equally well off (remember again that all other arguments of the utility function are being

FIGURE 3.2

A Single Indifference Curve

The curve U1 represents those combinations of x and y from which the individual derives the same utility. The slope of this curve represents the rate at which the individual is willing to trade x for y while remaining equally well off. This slope (or, more properly, the negative of the slope) is termed the marginal rate of substitution. In the ﬁgure, the indifference curve is drawn on the assumption of a diminishing marginal rate of substitution.

Quantity of y U1

y1 y2

U1

x1

x2

Quantity of x

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Part 2 Choice and Demand

held constant). This person is equally happy consuming, for example, either the combination of goods x1 , y1 or the combination x2 , y2 . This curve representing all the consumption bundles that the individual ranks equally is called an indifference curve. DEFINITION

Indifference curve. An indifference curve (or, in many dimensions, an indifference surface) shows a set of consumption bundles about which the individual is indifferent. That is, the bundles all provide the same level of utility. The slope of the indifference curve in Figure 3.2 is negative, showing that if the individual is forced to give up some y, he or she must be compensated by an additional amount of x to remain indifferent between the two bundles of goods. The curve is also drawn so that the slope increases as x increases (that is, the slope starts at negative inﬁnity and increases toward zero). This is a graphical representation of the assumption that people become progressively less willing to trade away y to get more x. In mathematical terms, the absolute value of this slope diminishes as x increases. Hence, we have the following deﬁnition.

DEFINITION

Marginal rate of substitution. The negative of the slope of an indifference curve ðU1 Þ at some point is termed the marginal rate of substitution (MRS) at that point. That is, dy , (3.7) MRS ¼ dx U ¼U1 where the notation indicates that the slope is to be calculated along the U1 indifference curve. The slope of U1 and the MRS therefore tell us something about the trades this person will voluntarily make. At a point such as x1 , y1 , the person has quite a lot of y and is willing to trade away a signiﬁcant amount to get one more x. The indifference curve at x1 , y1 is therefore rather steep. This is a situation where the person has, say, many hamburgers ðyÞ and little to drink with them (x). This person would gladly give up a few burgers (say, 5) to quench his or her thirst with one more drink. At x2 , y2 , on the other hand, the indifference curve is ﬂatter. Here, this person has quite a few drinks and is willing to give up relatively few burgers (say, 1) to get another soft drink. Consequently, the MRS diminishes between x1 , y1 and x2 , y2 . The changing slope of U1 shows how the particular consumption bundle available inﬂuences the trades this person will freely make.

Indifference curve map In Figure 3.2 only one indifference curve was drawn. The x, y quadrant, however, is densely packed with such curves, each corresponding to a different level of utility. Because every bundle of goods can be ranked and yields some level of utility, each point in Figure 3.2 must have an indifference curve passing through it. Indifference curves are similar to contour lines on a map in that they represent lines of equal “altitude” of utility. In Figure 3.3 several indifference curves are shown to indicate that there are inﬁnitely many in the plane. The level of utility represented by these curves increases as we move in a northeast direction; the utility of curve U1 is less than that of U2 , which is less than that of U3 . This is because of the assumption made in Figure 3.1: More of a good is preferred to less. As was discussed earlier, there is no unique way to assign numbers to these utility levels. The curves only show that the combinations of goods on U3 are preferred to those on U2 , which are preferred to those on U1 .

Chapter 3 Preferences and Utility

FIGURE 3.3

There Are Infinitely Many Indifference Curves in the x–y Plane

There is an indifference curve passing through each point in the x–y plane. Each of these curves records combinations of x and y from which the individual receives a certain level of satisfaction. Movements in a northeast direction represent movements to higher levels of satisfaction.

Quantity of y U1 U 2 U 3

Increasing utility

U3 U2 U1 Quantity of x

Indifference curves and transitivity As an exercise in examining the relationship between consistent preferences and the representation of preferences by utility functions, consider the following question: Can any two of an individual’s indifference curves intersect? Two such intersecting curves are shown in Figure 3.4. We wish to know if they violate our basic axioms of rationality. Using our map analogy, there would seem to be something wrong at point E, where “altitude” is equal to two different numbers, U1 and U2 . But no point can be both 100 and 200 feet above sea level. To proceed formally, let us analyze the bundles of goods represented by points A, B, C, and D. By the assumption of nonsatiation (i.e., more of a good always increases utility), “A is preferred to B” and “C is preferred to D.” But this person is equally satisﬁed with B and C (they lie on the same indifference curve), so the axiom of transitivity implies that A must be preferred to D. But that cannot be true, because A and D are on the same indifference curve and are by deﬁnition regarded as equally desirable. This contradiction shows that indifference curves cannot intersect. Therefore we should always draw indifference curve maps as they appear in Figure 3.3.

Convexity of indifference curves An alternative way of stating the principle of a diminishing marginal rate of substitution uses the mathematical notion of a convex set. A set of points is said to be convex if any two points within the set can be joined by a straight line that is contained completely within the set. The assumption of a diminishing MRS is equivalent to the assumption that all combinations of x and y

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Part 2 Choice and Demand

FIGURE 3.4

Intersecting Indifference Curves Imply Inconsistent Preferences Combinations A and D lie on the same indifference curve and therefore are equally desirable. But the axiom of transitivity can be used to show that A is preferred to D. Hence, intersecting indifference curves are not consistent with rational preferences.

Quantity of y

C D E A U1 B

U2

Quantity of x

that are preferred or indifferent to a particular combination x , y form a convex set.4 This is illustrated in Figure 3.5a, where all combinations preferred or indifferent to x , y are in the shaded area. Any two of these combinations—say, x1 , y1 and x2 , y2 —can be joined by a straight line also contained in the shaded area. In Figure 3.5b this is not true. A line joining x1 , y1 and x2 , y2 passes outside the shaded area. Therefore, the indifference curve through x , y in Figure 3.5b does not obey the assumption of a diminishing MRS, because the set of points preferred or indifferent to x , y is not convex.

Convexity and balance in consumption By using the notion of convexity, we can show that individuals prefer some balance in their consumption. Suppose that an individual is indifferent between the combinations x1 , y1 and x2 , y2 . If the indifference curve is strictly convex, then the combination ðx1 þ x2 Þ=2, ðy1 þ y2 Þ=2 will be preferred to either of the initial combinations.5 Intuitively, “well-balanced” bundles of commodities are preferred to bundles that are heavily weighted toward one commodity. This is illustrated in Figure 3.6. Because the indifference curve is assumed to be convex, all points on the straight line joining ðx1 , y1 Þ and ðx2 , y2 Þ are preferred to these initial points. This therefore will be true of the point ðx1 þ x2 Þ=2, ðy1 þ y2 Þ=2, which lies at the midpoint of such a line. 4

This definition is equivalent to assuming that the utility function is quasi-concave. Such functions were discussed in Chapter 2, and we shall return to examine them in the next section. Sometimes the term strict quasi-concavity is used to rule out the possibility of indifference curves having linear segments. We generally will assume strict quasi-concavity, but in a few places we will illustrate the complications posed by linear portions of indifference curves.

5

In the case in which the indifference curve has a linear segment, the individual will be indifferent among all three combinations.

The Notion of Convexity as an Alternative Definition of a Diminishing MRS

FIGURE 3.5

In (a) the indifference curve is convex (any line joining two points above U1 is also above U1 ). In (b) this is not the case, and the curve shown here does not everywhere have a diminishing MRS.

Quantity of y

Quantity of y U1

U1

y1 y1

y*

y* y2 U1 x1

x*

x2

Quantity of x

(a)

FIGURE 3.6

y2

U1 x1

x*

x2

Quantity of x

(b)

Balanced Bundles of Goods Are Preferred to Extreme Bundles

If indifference curves are convex (if they obey the assumption of a diminishing MRS), then the line joining any two points that are indifferent will contain points preferred to either of the initial combinations. Intuitively, balanced bundles are preferred to unbalanced ones.

Quantity of y U1

y1 y1 + y 2 2

y2

U1

x1

x1 + x2 2

x2

Quantity of x

96

Part 2 Choice and Demand

Indeed, any proportional combination of the two indifferent bundles of goods will be preferred to the initial bundles, because it will represent a more balanced combination. Thus, strict convexity is equivalent to the assumption of a diminishing MRS. Both assumptions rule out the possibility of an indifference curve being straight over any portion of its length. EXAMPLE 3.1 Utility and the MRS Suppose a person’s ranking of hamburgers ðyÞ and soft drinks ðxÞ could be represented by the utility function pﬃﬃﬃﬃﬃﬃﬃﬃﬃ utility ¼ x ⋅ y . (3.8) An indifference curve for this function is found by identifying that set of combinations of x and y for which utility has the same value. Suppose we arbitrarily set utility equal to 10. Then the equation for this indifference curve is pﬃﬃﬃﬃﬃﬃﬃﬃﬃ utility ¼ 10 ¼ x ⋅ y . (3.9) Because squaring this function is order preserving, the indifference curve is also represented by 100 ¼ x ⋅ y,

(3.10)

which is easier to graph. In Figure 3.7 we show this indifference curve; it is a familiar rectangular hyperbola. One way to calculate the MRS is to solve Equation 3.10 for y, y ¼ 100=x, FIGURE 3.7

Indifference Curve for Utility ¼

(3.11)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ x⋅y

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ This indifference curve illustrates the function 10 ¼ U ¼ x ⋅ y . At point A (5, 20), the MRS is 4, implying that this person is willing to trade 4y for an additional x. At point B (20, 5), however, the MRS is 0.25, implying a greatly reduced willingness to trade.

Quantity of y

A

20

C

12.5

B

5

U = 10 0

5

12.5

20

Quantity of x

Chapter 3 Preferences and Utility

and then use the deﬁnition (Equation 3.7): MRS ¼ dy=dx ðalong U1 Þ ¼ 100=x 2 .

(3.12)

Clearly this MRS declines as x increases. At a point such as A on the indifference curve with a lot of hamburgers (say, x ¼ 5, y ¼ 20), the slope is steep so the MRS is high: MRS at ð5, 20Þ ¼ 100=x 2 ¼ 100=25 ¼ 4.

(3.13)

Here the person is willing to give up 4 hamburgers to get 1 more soft drink. On the other hand, at B where there are relatively few hamburgers (here x ¼ 20, y ¼ 5), the slope is flat and the MRS is low: MRS at ð20, 5Þ ¼ 100=x 2 ¼ 100=400 ¼ 0:25.

(3.14)

Now he or she will only give up one quarter of a hamburger for another soft drink. Notice also how convexity of the indifference curve U1 is illustrated by this numerical example. Point C is midway between points A and B; at C this person has 12.5 hamburgers and 12.5 soft drinks. Here utility is given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ (3.15) utility ¼ x ⋅ y ¼ ð12:5Þ2 ¼ 12:5, which clearly exceeds the utility along U1 (which was assumed to be 10). QUERY: From our derivation here, it appears that the MRS depends only on the quantity of x consumed. Why is this misleading? How does the quantity of y implicitly enter into Equations 3.13 and 3.14?

A MATHEMATICAL DERIVATION A mathematical derivation provides additional insights about the shape of indifference curves and the nature of preferences. In this section we provide such a derivation for the case of a utility function involving only two goods. This will allow us to compare the mathematics to the two-dimensional indifference curve map. The case of many goods will be taken up at the end of the chapter, but it will turn out that this more complicated case really adds very little.

The MRS and marginal utility If the utility a person receives from two goods is represented by U ðx, yÞ, we can write the total differential of this function as dU ¼

∂U ∂U ⋅ dx þ ⋅ dy. ∂x ∂y

(3.16)

Along any particular indifference curve dU ¼ 0, a simple manipulation of Equation 3.16 yields dy ∂U =∂x . (3.17) ¼ MRS ¼ dx U ¼constant ∂U =∂y In words, the MRS of x for y is equal to the ratio of the marginal utility of x (that is, ∂U =∂x) to the marginal utility of y ð∂U =∂yÞ. This result makes intuitive sense. Suppose that a person’s utility were actually measurable in, say, units called “utils.” Assume also that this person consumes only two goods, food ðxÞ and clothing (y), and that each extra unit of food provides 6 utils whereas each extra unit of clothing provides 2 utils. Then Equation 3.17 would mean that

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Part 2 Choice and Demand

dy 6 utils ¼ 3, MRS ¼ ¼ dx U ¼constant 2 utils so this person is willing to trade away 3 units of clothing to get 1 more unit of food. This trade would result in no net change in utility because the gains and losses would be precisely offsetting. Notice that the units in which utility is measured (what we have, for lack of a better word, called “utils”) cancel out in making this calculation. Although marginal utility is obviously affected by the units in which utility is measured, the MRS is independent of that choice.6

The convexity of indifference curves In Chapter 1 we described how the assumption of diminishing marginal utility was used by Marshall to solve the water-diamond paradox. Marshall theorized that it is the marginal valuation that an individual places on a good that determines its value: It is the amount that an individual is willing to pay for one more pint of water that determines the price of water. Because it might be thought that this marginal value declines as the quantity of water that is consumed increases, Marshall showed why water has a low exchange value. Intuitively, it seems that the assumption of a decreasing marginal utility of a good is related to the assumption of a decreasing MRS; both concepts seem to refer to the same commonsense idea of an individual becoming relatively satiated with a good as more of it is consumed. Unfortunately, the two concepts are quite different. (See Problem 3.3.) Technically, the assumption of a diminishing MRS is equivalent to requiring that the utility function be quasi-concave. This requirement is related in a rather complex way to the assumption that each good encounters diminishing marginal utility (that is, that fii is negative for each good).7 But that is to be expected because the concept of diminishing marginal utility is not independent of how

0

More formally, let F ðU Þ be any arbitrary order-preserving transformation of U (that is, F ðU Þ > 0). Then, for the transformed utility function,

6

0

MRS ¼

∂F =∂x F ðU Þ∂U =∂x ¼ 0 ∂F =∂y F ðU Þ∂U =∂y ∂U =∂x , ¼ ∂U =∂y 0

which is the MRS for the original function U . That the F ðU Þ terms cancel out shows that the MRS is independent of how utility is measured. 7

We have shown that if utility is given by U ¼ f ðx, yÞ, then MRS ¼

fx f dy ¼ 1¼ . fy f2 dx

The assumption of a diminishing MRS means that dMRS=dx < 0, but dMRS f ð f þ f12 ⋅ dy=dxÞ f1 ð f21 þ f22 ⋅ dy=dxÞ : ¼ 2 11 dx f 22 Using the fact that f1 =f2 ¼ dy=dx, we have dMRS f ½ f f12 ð f1 =f2 Þ f1 ½ f21 f22 ð f1 =f2 Þ : ¼ 2 11 dx f 22 Combining terms and recognizing that f12 ¼ f21 yields dMRS f f 2f1 f12 þ ð f22 f 21 Þ=f2 ¼ 2 11 dx f 22 or, multiplying numerator and denominator by f2 , dMRS f 2 f 2f1 f2 f12 þ f 21 f22 . ¼ 2 11 dx f 32

Chapter 3 Preferences and Utility

utility itself is measured, whereas the convexity of indifference curves is indeed independent of such measurement. EXAMPLE 3.2 Showing Convexity of Indifference Curves Calculation of the MRS for speciﬁc utility functions is frequently a good shortcut for showing convexity of indifference curves. In particular, the process can be much simpler than applying the deﬁnition of quasi-concavity, though it is more difﬁcult to generalize to more than two goods. Here we look at how Equation 3.17 can be used for three different utility functions (for more practice, see Problem 3.1). pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1. U ðx, yÞ ¼ x ⋅ y . This example just repeats the case illustrated in Example 3.1. One shortcut to applying Equation 3.17 that can simplify the algebra is to take the logarithm of this utility function. Because taking logs is order preserving, this will not alter the MRS to be calculated. So, let U ðx, yÞ ¼ ln½U ðx, yÞ ¼ 0:5 ln x þ 0:5 ln y. (3.18) Applying Equation 3.17 yields MRS ¼

∂U =∂x 0:5=x y ¼ , ¼ ∂U =∂y 0:5=y x

(3.19)

which seems to be a much simpler approach than we used previously.8 Clearly this MRS is diminishing as x increases and y decreases. The indifference curves are therefore convex. 2. U ðx, yÞ ¼ x þ xy þ y. In this case there is no advantage to transforming this utility function. Applying Equation 3.17 yields ∂U =∂x 1 þ y ¼ . (3.20) MRS ¼ ∂U =∂y 1 þ x Again, this ratio clearly decreases as x increases and y decreases, so the indifference curves for this function are convex. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3. U ðx, yÞ ¼ x 2 þ y 2 . For this example it is easier to use the transformation U ðx, yÞ ¼ ½U ðx, yÞ2 ¼ x 2 þ y 2 .

(3.21)

Because this is the equation for a quarter-circle, we should begin to suspect that there (continued) If we assume that f2 > 0 (that marginal utility is positive), then the MRS will diminish as long as f 22 f11 2f1 f2 f12 þ f 21 f22 < 0. Notice that diminishing marginal utility ( f11 < 0 and f22 < 0) will not ensure this inequality. One must also be concerned with the f12 term. That is, one must know how decreases in y affect the marginal utility of x. In general it is not possible to predict the sign of that term. The condition required for a diminishing MRS is precisely that discussed in Chapter 2 to ensure that the function f is strictly quasi-concave. The condition shows that the necessary conditions for a maximum of f subject to a linear constraint are also sufficient. We will use this result in Chapter 4 and elsewhere. 8 In Example 3.1 we looked at the U ¼ 10 indifference curve. So, for that curve, y ¼ 100=x and the MRS in Equation 3.19 would be MRS ¼ 100=x 2 as calculated before.

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Part 2 Choice and Demand

EXAMPLE 3.2 CONTINUED might be some problems with the indifference curves for this utility function. These suspicions are confirmed by again applying the definition of the MRS to yield ∂U =∂x 2x x MRS ¼ ¼ ¼ . (3.22) ∂U =∂y 2y y For this function, it is clear that, as x increases and y decreases, the MRS increases ! Hence the indifference curves are concave, not convex, and this is clearly not a quasiconcave function. QUERY: Does a doubling of x and y change the MRS in each of these three examples? That is, does the MRS depend only on the ratio of x to y, not on the absolute scale of purchases? (See also Example 3.3.)

UTILITY FUNCTIONS FOR SPECIFIC PREFERENCES Individuals’ rankings of commodity bundles and the utility functions implied by these rankings are unobservable. All we can learn about people’s preferences must come from the behavior we observe when they respond to changes in income, prices, and other factors. It is nevertheless useful to examine a few of the forms particular utility functions might take, because such an examination may offer insights into observed behavior and (more to the point) understanding the properties of such functions can be of some help in solving problems. Here we will examine four speciﬁc examples of utility functions for two goods. Indifference curve maps for these functions are illustrated in the four panels of Figure 3.8. As should be visually apparent, these cover quite a few possible shapes. Even greater variety is possible once we move to functions for three or more goods, and some of these possibilities are mentioned in later chapters.

Cobb-Douglas utility Figure 3.8a shows the familiar shape of an indifference curve. One commonly used utility function that generates such curves has the form utility ¼ U ðx, yÞ ¼ x α y β ,

(3.23)

where α and β are positive constants. In Examples 3.1 and 3.2, we studied a particular case of this function for which α ¼ β ¼ 0:5. The more general case presented in Equation 3.23 is termed a Cobb-Douglas utility function, after two researchers who used such a function for their detailed study of production relationships in the U.S. economy (see Chapter 7). In general, the relative sizes of α and β indicate the relative importance of the two goods to this individual. Since utility is unique only up to a monotonic transformation, it is often convenient to normalize these parameters so that α þ β ¼ 1.

Perfect substitutes The linear indifference curves in Figure 3.8b are generated by a utility function of the form utility ¼ U ðx, yÞ ¼ αx þ βy,

(3.24)

Chapter 3 Preferences and Utility

FIGURE 3.8

Examples of Utility Functions

The four indifference curve maps illustrate alternative degrees of substitutability of x for y. The Cobb-Douglas and CES functions (drawn here for relatively low substitutability) fall between the extremes of perfect substitution (panel b) and no substitution (panel c).

Quantity of y

Quantity of y

U2 U1

U2 U1

U0

U0

Quantity of x (a) Cobb-Douglas

Quantity of x (b) Perfect substitutes

Quantity of y

Quantity of y

U2 U1 U2 U1 U0

U0

Quantity of x (c) Perfect complements

Quantity of x (d) CES

where, again, α and β are positive constants. That the indifference curves for this function are straight lines should be readily apparent: Any particular level curve can be calculated by setting U ðx, yÞ equal to a constant that, given the linear form of the function, clearly speciﬁes a straight line. The linear nature of these indifference curves gave rise to the term perfect substitutes to describe the implied relationship between x and y. Because the MRS is constant (and equal to α=β) along the entire indifference curve, our previous notions of a diminishing MRS do not apply in this case. A person with these preferences would be willing to give up the same amount of y to get one more x no matter how much x was being consumed. Such a situation might describe the relationship between different brands of what is essentially the same product. For example, many people (including the author) do not care where they buy gasoline. A gallon of gas is a gallon of gas in spite of the best efforts

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of the Exxon and Shell advertising departments to convince me otherwise. Given this fact, I am always willing to give up 10 gallons of Exxon in exchange for 10 gallons of Shell because it does not matter to me which I use or where I got my last tankful. Indeed, as we will see in the next chapter, one implication of such a relationship is that I will buy all my gas from the least expensive seller. Because I do not experience a diminishing MRS of Exxon for Shell, I have no reason to seek a balance among the gasoline types I use.

Perfect complements A situation directly opposite to the case of perfect substitutes is illustrated by the L-shaped indifference curves in Figure 3.8c. These preferences would apply to goods that “go together”—coffee and cream, peanut butter and jelly, and cream cheese and lox are familiar examples. The indifference curves shown in Figure 3.8c imply that these pairs of goods will be used in the ﬁxed proportional relationship represented by the vertices of the curves. A person who prefers 1 ounce of cream with 8 ounces of coffee will want 2 ounces of cream with 16 ounces of coffee. Extra coffee without cream is of no value to this person, just as extra cream would be of no value without coffee. Only by choosing the goods together can utility be increased. These concepts can be formalized by examining the mathematical form of the utility function that generates these L-shaped indifference curves: utility ¼ U ðx, yÞ ¼ minðαx, βyÞ.

(3.25)

Here α and β are positive parameters, and the operator “min” means that utility is given by the smaller of the two terms in the parentheses. In the coffee-cream example, if we let ounces of coffee be represented by x and ounces of cream by y, utility would be given by utility ¼ U ðx, yÞ ¼ minðx, 8yÞ.

(3.26)

Now 8 ounces of coffee and 1 ounce of cream provide 8 units of utility. But 16 ounces of coffee and 1 ounce of cream still provide only 8 units of utility because min(16, 8) ¼ 8. The extra coffee without cream is of no value, as shown by the horizontal section of the indifference curves for movement away from a vertex; utility does not increase when only x increases (with y constant). Only if coffee and cream are both doubled (to 16 and 2, respectively) will utility increase to 16. More generally, neither of the two goods will be in excess only if αx ¼ βy.

(3.27)

y=x ¼ α=β,

(3.28)

Hence which shows the fixed proportional relationship between the two goods that must occur if choices are to be at the vertices of the indifference curves.

CES utility The three speciﬁc utility functions illustrated so far are special cases of the more general constant elasticity of substitution function (CES), which takes the form utility ¼ U ðx, yÞ ¼

xδ yδ þ , δ δ

(3.29)

where δ 1, δ 6¼ 0, and utility ¼ U ðx, yÞ ¼ ln x þ ln y

(3.30)

Chapter 3 Preferences and Utility

when δ ¼ 0. It is obvious that the case of perfect substitutes corresponds to the limiting case, δ ¼ 1, in Equation 3.29 and that the Cobb-Douglas9 case corresponds to δ ¼ 0 in Equation 3.30. Less obvious is that the case of ﬁxed proportions corresponds to δ ¼ ∞ in Equation 3.29, but that result can also be shown using a limits argument. The use of the term “elasticity of substitution” for this function derives from the notion that the possibilities illustrated in Figure 3.8 correspond to various values for the substitution parameter, σ, which for this function is given by σ ¼ 1=ð1 δÞ. For perfect substitutes, then, σ ¼ ∞, and the ﬁxed proportions case has σ ¼ 0.10 Because the CES function allows us to explore all of these cases, and many cases in between, it will prove quite useful for illustrating the degree of substitutability present in various economic relationships. The speciﬁc shape of the CES function illustrated in Figure 3.8a is for the case δ ¼ 1. That is, utility ¼ x 1 y 1 ¼

1 1 . x y

(3.31)

For this situation, σ ¼ 1=ð1 δÞ ¼ 1=2 and, as the graph shows, these sharply curved indifference curves apparently fall between the Cobb-Douglas and fixed proportion cases. The negative signs in this utility function may seem strange, but the marginal utilities of both x and y are positive and diminishing, as would be expected. This explains why δ must appear in the denominators in Equation 3.29. In the particular case of Equation 3.31, utility increases from ∞ (when x ¼ y ¼ 0) toward 0 as x and y increase. This is an odd utility scale, perhaps, but perfectly acceptable. EXAMPLE 3.3 Homothetic Preferences All of the utility functions described in Figure 3.8 are homothetic (see Chapter 2). That is, the marginal rate of substitution for these functions depends only on the ratio of the amounts of the two goods, not on the total quantities of the goods. This fact is obvious for the case of the perfect substitutes (when the MRS is the same at every point) and the case of perfect complements (where the MRS is inﬁnite for y=x > α=β, undeﬁned when y=x ¼ α=β, and zero when y=x < α=β). For the general Cobb-Douglas function, the MRS can be found as MRS ¼

∂U =∂x αx α1 y β α y ¼ ⋅ , ¼ ∂U =∂y βx α y β1 β x

(3.32)

which clearly depends only on the ratio y=x. Showing that the CES function is also homothetic is left as an exercise (see Problem 3.12). The importance of homothetic functions is that one indifference curve is much like another. Slopes of the curves depend only on the ratio y=x, not on how far the curve is from the origin. Indifference curves for higher utility are simple copies of those for lower utility. Hence, we can study the behavior of an individual who has homothetic preferences by looking only at one indifference curve or at a few nearby curves without fearing that our results would change dramatically at very different levels of utility. (continued) 9

The CES function could easily be generalized to allow for differing weights to be attached to the two goods. Since the main use of the function is to examine substitution questions, we usually will not make that generalization. In some of the applications of the CES function, we will also omit the denominators of the function because these constitute only a scale factor when δ is positive. For negative values of δ, however, the denominator is needed to ensure that marginal utility is positive.

10

The elasticity of substitution concept is discussed in more detail in connection with production functions in Chapter 9.

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EXAMPLE 3.3 CONTINUED QUERY: How might you deﬁne homothetic functions geometrically? What would the locus of all points with a particular MRS look like on an individual’s indifference curve map?

EXAMPLE 3.4 Nonhomothetic Preferences Although all of the indifference curve maps in Figure 3.8 exhibit homothetic preferences, this need not always be true. Consider the quasi-linear utility function utility ¼ U ðx, yÞ ¼ x þ ln y.

(3.33)

For this function, good y exhibits diminishing marginal utility, but good x does not. The MRS can be computed as MRS ¼

∂U =∂x 1 ¼ ¼ y. ∂U =∂y 1=y

(3.34)

The MRS diminishes as the chosen quantity of y decreases, but it is independent of the quantity of x consumed. Because x has a constant marginal utility, a person’s willingness to give up y to get one more unit of x depends only on how much y he or she has. Contrary to the homothetic case, then, a doubling of both x and y doubles the MRS rather than leaving it unchanged. QUERY: What does the indifference curve map for the utility function in Equation 3.33 look like? Why might this approximate a situation where y is a speciﬁc good and x represents everything else?

THE MANY-GOOD CASE All of the concepts we have studied so far for the case of two goods can be generalized to situations where utility is a function of arbitrarily many goods. In this section, we will brieﬂy explore those generalizations. Although this examination will not add much to what we have already shown, considering peoples’ preferences for many goods can be quite important in applied economics, as we will see in later chapters.

The MRS with many goods Suppose utility is a function of n goods given by utility ¼ U ðx1 , x2 , …, xn Þ.

(3.35)

The total differential of this expression is dU ¼

∂U ∂U ∂U dx þ dx þ … þ dx ∂x1 1 ∂x2 2 ∂xn n

(3.36)

and, as before, we can find the MRS between any two goods by setting dU ¼ 0. In this derivation, we also hold constant quantities of all of the goods other than those two. Hence we have

Chapter 3 Preferences and Utility

dU ¼ 0 ¼

∂U ∂U dxi þ dx ; ∂xi ∂xj j

105

(3.37)

after some algebraic manipulation, we get MRSðxi for xj Þ ¼

dxj dxi

¼

∂U =∂xi , ∂U =∂xj

(3.38)

which is precisely what we got in Equation 3.17. Whether this concept is as useful as it is in two dimensions is open to question, however. With only two goods, asking how a person would trade one for the other is an interesting question—a transaction we might actually observe. With many goods, however, it seems unlikely that a person would simply trade one good for another while holding all other goods constant. Rather, it seems more likely that an event (such as a price increase) that caused a person to want to reduce, say, the quantity of cornflakes ðxi Þ consumed would also cause him or her to change the quantities consumed of many other goods such as milk, sugar, Cheerios, spoons, and so forth. As we shall see in Chapter 6, this entire reallocation process can best be studied by looking at the entire utility function as represented in Equation 3.35. Still, the notion of making trade-offs between only two goods will prove useful as a way of conceptualizing the utility maximization process that we will take up next.

Multigood indifference surfaces Generalizing the concept of indifference curves to multiple dimensions poses no major mathematical difﬁculties. We simply deﬁne an indifference surface as being the set of points in n dimensions that satisfy the equation U ðx1 , x2 , …, xn Þ ¼ k,

(3.39)

where k is any preassigned constant. If the utility function is quasi-concave, the set of points for which U k will be convex; that is, all of the points on a line joining any two points on the U ¼ k indifference surface will also have U k. It is this property that we will find most useful in later applications. Unfortunately, however, the mathematical conditions that ensure quasi-concavity in many dimensions are not especially intuitive (see the Extensions to Chapter 2), and visualizing many dimensions is virtually impossible. Hence, when intuition is required, we will usually revert to two-good examples.

SUMMARY In this chapter we have described the way in which economists formalize individuals’ preferences about the goods they choose. We drew several conclusions about such preferences that will play a central role in our analysis of the theory of choice in the following chapters: •

•

If individuals obey certain basic behavioral postulates in their preferences among goods, they will be able to rank all commodity bundles, and that ranking can be represented by a utility function. In making choices, individuals will behave as if they were maximizing this function. Utility functions for two goods can be illustrated by an indifference curve map. Each indifference curve contour

on this map shows all the commodity bundles that yield a given level of utility. •

The negative of the slope of an indifference curve is deﬁned to be the marginal rate of substitution (MRS). This shows the rate at which an individual would willingly give up an amount of one good (y) if he or she were compensated by receiving one more unit of another good (x).

•

The assumption that the MRS decreases as x is substituted for y in consumption is consistent with the notion that individuals prefer some balance in their consumption choices. If the MRS is always decreasing, individuals

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Part 2 Choice and Demand

will have strictly convex indifference curves. That is, their utility function will be strictly quasi-concave. •

A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods. Here we examined the Cobb-Douglas function, the linear function (perfect substitutes), the ﬁxed proportions function (perfect complements), and the CES function (which includes the other three as special cases).

•

It is a simple matter mathematically to generalize from two-good examples to many goods. And, as we shall see, studying peoples’ choices among many goods can yield many insights. But the mathematics of many goods is not especially intuitive, so we will primarily rely on twogood cases to build such intuition.

PROBLEMS 3.1 Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether the MRS declines as x increases). a. U ðx, yÞ ¼ 3x þ y. pﬃﬃﬃﬃﬃﬃﬃﬃﬃ b. U ðx, yÞ ¼ x ⋅ y : pﬃﬃﬃ c. U ðx, yÞ ¼ x þ y: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d. U ðx, yÞ ¼ x 2 y 2 : xy . e. U ðx, y Þ ¼ x þy

3.2 In footnote 7 we showed that, in order for a utility function for two goods to have a strictly diminishing MRS (that is, to be strictly quasi-concave), the following condition must hold: f 22 f11 2f1 f2 f12 þ f 21 f22 < 0: Use this condition to check the convexity of the indifference curves for each of the utility functions in Problem 3.1. Describe any shortcuts you discover in this process.

3.3 Consider the following utility functions: a. U ðx, yÞ ¼ xy. b. U ðx, yÞ ¼ x 2 y 2 . c. U ðx, yÞ ¼ ln x þ ln y. Show that each of these has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

3.4 As we saw in Figure 3.5, one way to show convexity of indifference curves is to show that, for any two pointsðx1 , y1 Þ andðx2 , y2 Þ on an indifference curve that promises U ¼ k, the utility associated with the point

x1 þx2 2

,

y1 þy2 2

is at least as great as k. Use this approach to discuss the convexity of the indifference

curves for the following three functions. Be sure to graph your results. a. U ðx, yÞ ¼ minðx, yÞ. b. U ðx, yÞ ¼ maxðx, yÞ. c. U ðx, yÞ ¼ x þ y.

Chapter 3 Preferences and Utility

3.5 The Phillie Phanatic always eats his ballpark franks in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items and any extra amount of a single item without the other constituents is worthless. a. What form does PP’s utility function for these four goods have? b. How might we simplify matters by considering PP’s utility to be a function of only one good? What is that good? c. Suppose foot-long hot dogs cost $1.00 each, buns cost $0.50 each, mustard costs $0.05 per ounce, and pickle relish costs $0.15 per ounce. How much does the good deﬁned in part (b) cost? d. If the price of foot-long hot dogs increases by 50 percent (to $1.50 each), what is the percentage increase in the price of the good? e. How would a 50 percent increase in the price of a bun affect the price of the good? Why is your answer different from part (d)? f. If the government wanted to raise $1.00 by taxing the goods that PP buys, how should it spread this tax over the four goods so as to minimize the utility cost to PP?

3.6 Many advertising slogans seem to be asserting something about people’s preferences. How would you capture the following slogans with a mathematical utility function? a. Promise margarine is just as good as butter. b. Things go better with Coke. c. You can’t eat just one Pringle’s potato chip. d. Krispy Kreme glazed doughnuts are just better than Dunkin’. e. Miller Brewing advises us to drink (beer) “responsibly.” [What would “irresponsible” drinking be?]

3.7 a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form U ðx, yÞ ¼ x α y β , where α and β are positive constants, what is the utility function for this consumer? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?

3.8 Find utility functions given each of the following indifference curves [deﬁned by U (⋅) ¼ C]: a. z ¼

C 1=δ x α=δ y β=δ

.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b. y ¼ 0:5 x 2 4ðx 2 C Þ 0:5x: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 4 4xðx 2 y C Þ y 2 : c. z ¼ 2x 2x

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Analytical Problems 3.9 Initial endowments Suppose that a person has initial of the two goods that provide utility to him or her. These _ amounts _ initial amounts are given by x and y . a. Graph these initial amounts on this person’s indifference curve map. b. If this person can trade x for y (or vice versa) with other people, what kinds of trades would he or she voluntarily make? What_kinds _ would not be made? How do these trades relate to this person’s MRS at the point ( x , y )? c. Suppose this person is relatively happy with the initial amounts in his or her possession and will only consider trades that increase utility by at least amount k. How would you illustrate this on the indifference curve map?

3.10 Cobb-Douglas utility Example 3.3 shows that the MRS for the Cobb-Douglas function U ðx, yÞ ¼ x α y β is given by MRS ¼

α y . β x

a. Does this result depend on whether α þ β ¼ 1? Does this sum have any relevance to the theory of choice? b. For commodity bundles for which y ¼ x, how does the MRS depend on the values of α and β? Develop an intuitive explanation of why, if α > β, MRS > 1. Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of x and y that exceed minimal subsistence levels given by x0 , y0 . In this case, U ðx, yÞ ¼ ðx x0 Þα ðy y0 Þβ . Is this function homothetic? (For a further discussion, see the Extensions to Chapter 4.)

3.11 Independent marginal utilities Two goods have independent marginal utilities if ∂2 U ∂2 U ¼ ¼ 0. ∂y∂x ∂x∂y Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing MRS. Provide an example to show that the converse of this statement is not true.

3.12 CES utility a. Show that the CES function α

xδ yδ þβ δ δ

is homothetic. How does the MRS depend on the ratio y=x? b. Show that your results from part (a) agree with our discussion of the cases δ ¼ 1 (perfect substitutes) and δ ¼ 0 (Cobb-Douglas). c. Show that the MRS is strictly diminishing for all values of δ < 1. d. Show that if x ¼ y, the MRS for this function depends only on the relative sizes of α and β.

Chapter 3 Preferences and Utility

109

e. Calculate the MRS for this function when y=x ¼ 0:9 and y=x ¼ 1:1 for the two cases δ ¼ 0:5 and δ ¼ 1. What do you conclude about the extent to which the MRS changes in the vicinity of x ¼ y? How would you interpret this geometrically?

3.13 The quasi-linear function Consider the function U ðx, yÞ ¼ x þ ln y. This is a function that is used relatively frequently in economic modeling as it has some useful properties. a. Find the MRS of the function. Now, interpret the result. b. Conﬁrm that the function is quasi-concave. c. Find the equation for an indifference curve for this function. d. Compare the marginal utility of x and y. How do you interpret these functions? How might consumers choose between x and y as they try to increase their utility by, for example, consuming more when their income increases? (We will look at this “income effect” in detail in the Chapter 5 problems.) e. Considering how the utility changes as the quantities of the two goods increase, describe some situations where this function might be useful.

3.14 Utility functions and preferences Imagine two goods that, when consumed individually, give increasing utility with increasing amounts consumed (they are individually monotonic) but that, when consumed together, detract from the utility that the other one gives. (One could think of milk and orange juice, which are ﬁne individually but which, when consumed together, yield considerable disutility.) a. Propose a functional form for the utility function for the two goods just described. b. Find the MRS between the two goods with your functional form. c. Which (if any) of the general assumptions that we make regarding preferences and utility functions does your functional form violate?

SUGGESTIONS FOR FURTHER READING Aleskerov, Fuad, and Bernard Monjardet. Utility Maximization, Choice, and Preference. Berlin: Springer-Verlag, 2002.

Kreps, David M. Notes on the Theory of Choice. London: Westview Press, 1988.

A complete study of preference theory. Covers a variety of threshold models and models of “context-dependent” decision making.

Good discussion of the foundations of preference theory. Most of the focus of the book is on utility in uncertain situations.

Jehle, G. R., and P. J. Reny. Advanced Microeconomic Theory, 2nd ed. Boston: Addison Wesley/Longman, 2001.

Mas-Colell, Andrea, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995.

Chapter 2 has a good proof of the existence of utility functions when basic axioms of rationality hold.

Kreps, David M. A Course in Microeconomic Theory. Princeton, NJ: Princeton University Press, 1990. Chapter 1 covers preference theory in some detail. Good discussion of quasi-concavity.

Chapters 2 and 3 provide a detailed development of preference relations and their representation by utility functions.

Stigler, G. “The Development of Utility Theory.” Journal of Political Economy 59, pts. 1–2 (August/October 1950): 307–27, 373–96. A lucid and complete survey of the history of utility theory. Has many interesting insights and asides.

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EXTENSIONS Special Preferences The utility function concept is a quite general one that can be adapted to a large number of special circumstances. Discovery of ingenious functional forms that reﬂect the essential aspects of some problem can provide a number of insights that would not be readily apparent with a more literary approach. Here we look at four aspects of preferences that economists have tried to model: (1) threshold effects; (2) quality; (3) habits and addiction; and (4) second-party preferences. In Chapters 7 and 17, we illustrate a number of additional ways of capturing aspects of preferences.

E3.1 Threshold effects The model of utility that we developed in this chapter implies an individual will always prefer commodity bundle A to bundle B, provided U ðAÞ > U ðBÞ. There may be events that will cause people to shift quickly from consuming bundle A to consuming B. In many cases, however, such a lightning-quick response seems unlikely. People may in fact be “set in their ways” and may require a rather large change in circumstances to change what they do. For example, individuals may not have especially strong opinions about what precise brand of toothpaste they choose and may stick with what they know despite a proliferation of new (and perhaps better) brands. Similarly, people may stick with an old favorite TV show even though it has declined in quality. One way to capture such behavior is to assume individuals make decisions as if they faced thresholds of preference. In such a situation, commodity bundle A might be chosen over B only when U ðAÞ > U ðBÞ þ ε,

(i)

where ε is the threshold that must be overcome. With this specification, then, indifference curves may be rather thick and even fuzzy, rather than the distinct contour lines shown in this chapter. Threshold models of this type are used extensively in marketing. The theory behind such models is presented in detail in Aleskerov and Monjardet (2002). There, the authors consider a number of ways of specifying the threshold so that it might depend on the characteristics of the bundles being considered or on other contextual variables. Alternative fuels Vedenov, Dufﬁeld, and Wetzstein (2006) use the threshold idea to examine the conditions under which individuals will shift from gasoline to other fuels

(primarily ethanol) for powering their cars. The authors point out that the main disadvantage of using gasoline in recent years has been the excessive price volatility of the product relative to other fuels. They conclude that switching to ethanol blends is efﬁcient (especially during periods of increased gasoline price volatility), provided that the blends do not decrease fuel efﬁciency.

E3.2 Quality Because many consumption items differ widely in quality, economists have an interest in incorporating such differences into models of choice. One approach is simply to regard items of different quality as totally separate goods that are relatively close substitutes. But this approach can be unwieldy because of the large number of goods involved. An alternative approach focuses on quality as a direct item of choice. Utility might in this case be reﬂected by utility ¼ U ðq, Q Þ,

(ii)

where q is the quantity consumed and Q is the quality of that consumption. Although this approach permits some examination of quality-quantity trade-offs, it encounters difficulty when the quantity consumed of a commodity (e.g., wine) consists of a variety of qualities. Quality might then be defined as an average (see Theil,1 1952), but that approach may not be appropriate when the quality of new goods is changing rapidly (as in the case of personal computers, for example). A more general approach (originally suggested by Lancaster, 1971) focuses on a well-defined set of attributes of goods and assumes that those attributes provide utility. If a good q provides two such attributes, a1 and a2 , then utility might be written as utility ¼ U ½q, a1 ðqÞ, a2 ðqÞ,

(iii)

and utility improvements might arise either because this individual chooses a larger quantity of the good or because a given quantity yields a higher level of valuable attributes. Personal computers This is the practice followed by economists who study demand in such rapidly changing industries as personal 1

Theil also suggests measuring quality by looking at correlations between changes in consumption and the income elasticities of various goods.

Chapter 3 Preferences and Utility

computers. In this case it would clearly be incorrect to focus only on the quantity of personal computers purchased each year, since new machines are much better than old ones (and, presumably, provide more utility). For example, Berndt, Griliches, and Rappaport (1995) ﬁnd that personal computer quality has been rising about 30 percent per year over a relatively long period of time, primarily because of improved attributes such as faster processors or better hard drives. A person who spends, say, $2,000 for a personal computer today buys much more utility than did a similar consumer 5 years ago.

E3.3 Habits and addiction Because consumption occurs over time, there is the possibility that decisions made in one period will affect utility in later periods. Habits are formed when individuals discover they enjoy using a commodity in one period and this increases their consumption in subsequent periods. An extreme case is addiction (be it to drugs, cigarettes, or Marx Brothers movies), where past consumption signiﬁcantly increases the utility of present consumption. One way to portray these ideas mathematically is to assume that utility in period t depends on consumption in period t and the total of all prior consumption of the habit-forming good (say, X ):

111

studying cigarette smoking and other addictive behavior. They show that reductions in smoking early in life can have very large effects on eventual cigarette consumption because of the dynamics in individuals’ utility functions. Whether addictive behavior is “rational” has been extensively studied by economists. For example, Gruber and Koszegi (2001) show that smoking can be approached as a rational, though time-inconsistent,2 choice.

E3.4 Second-party preferences Individuals clearly care about the well-being of other individuals. Phenomena such as making charitable contributions or making bequests to children cannot be understood without recognizing the interdependence that exists among people. Second-party preferences can be incorporated into the utility function of person i, say, by utility ¼ Ui ðxi , yi , Uj Þ,

(vi)

i¼1

where Uj is the utility of someone else. If ∂Ui =∂Uj > 0 then this person will engage in altruistic behavior, whereas if ∂Ui =∂Uj < 0 then he or she will demonstrate the malevolent behavior associated with envy. The usual case of ∂Ui =∂Uj ¼ 0 is then simply a middle ground between these alternative preference types. Gary Becker has been a pioneer in the study of these possibilities and has written on a variety of topics, including the general theory of social interactions (1976) and the importance of altruism in the theory of the family (1981).

In empirical applications, however, data on all past levels of consumption usually do not exist. It is therefore common to model habits using only data on current consumption (xt ) and on consumption in the previous period (xt 1). A common way to proceed is to assume that utility is given by

Evolutionary biology and genetics Biologists have suggested a particular form for the utility function in Equation iv, drawn from the theory of genetics. In this case X r j Uj , (vii) utility ¼ Ui ðxi , yi Þ þ

utility ¼ Ut ðxt , yt , st Þ,

(iv)

where st ¼

∞ X

xt i .

utility ¼ Ut ðx t , yt Þ,

j

(v)

where x t is some simple function of xt and xt 1 , such as x t ¼ xt xt1 or x t ¼ xt =xt 1 . Such functions imply that, ceteris paribus, the higher is xt 1 , the more xt will be chosen in the current period. Modeling habits These approaches to modeling habits have been applied to a wide variety of topics. Stigler and Becker (1977) use such models to explain why people develop a “taste” for going to operas or playing golf. Becker, Grossman, and Murphy (1994) adapt the models to

where rj measures closeness of the genetic relationship between person i and person j . For parents and children, for example, rj ¼ 0:5, whereas for cousins rj ¼ 0:125. Bergstrom (1996) describes a few of the conclusions about evolutionary behavior that biologists have drawn from this particular functional form.

2

For more on time inconsistency, see Chapter 17.

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References Aleskerov, Fuad, and Bernard Monjardet. Utility Maximization, Choice, and Preference. Berlin: Springer-Verlag, 2002. Becker, Gary S. The Economic Approach to Human Behavior. Chicago: University of Chicago Press, 1976. ———. A Treatise on the Family. Cambridge, MA: Harvard University Press, 1981. Becker, Gary S., Michael Grossman, and Kevin M. Murphy. “An Empirical Analysis of Cigarette Addiction.” American Economic Review (June 1994): 396–418. Bergstrom, Theodore C. “Economics in a Family Way.” Journal of Economic Literature (December 1996): 1903–34. Berndt, Ernst R., Zvi Griliches, and Neal J. Rappaport. “Econometric Estimates of Price Indexes for Personal

Computers in the 1990s.” Journal of Econometrics (July 1995): 243–68. Gruber, Jonathan, and Botond Koszegi. “Is Addiction ‘Rational’? Theory and Evidence.” Quarterly Journal of Economics (November 2001): 1261–1303. Lancaster, Kelvin J. Consumer Demand: A New Approach. New York: Columbia University Press, 1971. Stigler, George J., and Gary S. Becker. “De Gustibus Non Est Disputandum.” American Economic Review (March 1977): 76–90. Theil, Henri. “Qualities, Prices, and Budget Enquiries.” Review of Economic Studies (April 1952): 129–47. Vedenov, Dmitry V., James A. Duffield, and Michael E. Wetzstein. “Entry of Alternative Fuels in a Volatile U.S. Gasoline Market.” Journal of Agricultural and Resource Economics (April 2006): 1–13.

CHAPTER

4 Utility Maximization and Choice In this chapter we examine the basic model of choice that economists use to explain individuals’ behavior. That model assumes that individuals who are constrained by limited incomes will behave as if they are using their purchasing power in such a way as to achieve the highest utility possible. That is, individuals are assumed to behave as if they maximize utility subject to a budget constraint. Although the speciﬁc applications of this model are quite varied, as we will show, all of them are based on the same fundamental mathematical model, and all arrive at the same general conclusion: To maximize utility, individuals will choose bundles of commodities for which the rate of trade-off between any two goods (the MRS) is equal to the ratio of the goods’ market prices. Market prices convey information about opportunity costs to individuals, and this information plays an important role in affecting the choices actually made.

Utility maximization and lightning calculations Before starting a formal study of the theory of choice, it may be appropriate to dispose of two complaints noneconomists often make about the approach we will take. First is the charge that no real person can make the kinds of “lightning calculations” required for utility maximization. According to this complaint, when moving down a supermarket aisle, people just grab what is available with no real pattern or purpose to their actions. Economists are not persuaded by this complaint. They doubt that people behave randomly (everyone, after all, is bound by some sort of budget constraint), and they view the lightning calculation charge as misplaced. Recall, again, Friedman’s pool player from Chapter 1. The pool player also cannot make the lightning calculations required to plan a shot according to the laws of physics, but those laws still predict the player’s behavior. So too, as we shall see, the utility-maximization model predicts many aspects of behavior even though no one carries around a computer with his or her utility function programmed into it. To be precise, economists assume that people behave as if they made such calculations, so the complaint that the calculations cannot possibly be made is largely irrelevant.

Altruism and selfishness A second complaint against our model of choice is that it appears to be extremely selﬁsh; no one, according to this complaint, has such solely self-centered goals. Although economists are probably more ready to accept self-interest as a motivating force than are other, more Utopian thinkers (Adam Smith observed, “We are not ready to suspect any person of being deﬁcient in selﬁshness”1), this charge is also misplaced. Nothing in the utility-maximization model prevents individuals from deriving satisfaction from philanthropy or generally “doing good.” These activities also can be assumed to provide utility. Indeed, economists have used the utility-maximization model extensively to study such issues as donating time and money 1

Adam Smith, The Theory of Moral Sentiments (1759; reprint, New Rochelle, NY: Arlington House, 1969), p. 446.

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to charity, leaving bequests to children, or even giving blood. One need not take a position on whether such activities are selﬁsh or selﬂess since economists doubt people would undertake them if they were against their own best interests, broadly conceived.

AN INITIAL SURVEY The general results of our examination of utility maximization can be stated succinctly as follows. OPTIMIZATION PRINCIPLE

Utility maximization To maximize utility, given a ﬁxed amount of income to spend, an individual will buy those quantities of goods that exhaust his or her total income and for which the psychic rate of trade-off between any two goods (the MRS) is equal to the rate at which the goods can be traded one for the other in the marketplace. That spending all one’s income is required for utility maximization is obvious. Because extra goods provide extra utility (there is no satiation) and because there is no other use for income, to leave any unspent would be to fail to maximize utility. Throwing money away is not a utility-maximizing activity. The condition specifying equality of trade-off rates requires a bit more explanation. Because the rate at which one good can be traded for another in the market is given by the ratio of their prices, this result can be restated to say that the individual will equate the MRS (of x for y) to the ratio of the price of x to the price of y ðpx =py Þ. This equating of a personal trade-off rate to a market-determined trade-off rate is a result common to all individual utility-maximization problems (and to many other types of maximization problems). It will occur again and again throughout this text.

A numerical illustration To see the intuitive reasoning behind this result, assume that it were not true that an individual had equated the MRS to the ratio of the prices of goods. Speciﬁcally, suppose that the individual’s MRS is equal to 1 and that he or she is willing to trade 1 unit of x for 1 unit of y and remain equally well off. Assume also that the price of x is $2 per unit and of y is $1 per unit. It is easy to show that this person can be made better off. Suppose this person reduces x consumption by 1 unit and trades it in the market for 2 units of y. Only 1 extra unit of y was needed to keep this person as happy as before the trade—the second unit of y is a net addition to well-being. Therefore, the individual’s spending could not have been allocated optimally in the ﬁrst place. A similar method of reasoning can be used whenever the MRS and the price ratio px =py differ. The condition for maximum utility must be the equality of these two magnitudes.

THE TWO-GOOD CASE: A GRAPHICAL ANALYSIS This discussion seems eminently reasonable, but it can hardly be called a proof. Rather, we must now show the result in a rigorous manner and, at the same time, illustrate several other important attributes of the maximization process. First we take a graphic analysis; then we take a more mathematical approach.

Budget constraint Assume that the individual has I dollars to allocate between good x and good y. If px is the price of good x and py is the price of good y, then the individual is constrained by

Chapter 4

FIGURE 4.1

Utility Maximization and Choice

The Individual’s Budget Constraint for Two Goods

Those combinations of x and y that the individual can afford are shown in the shaded triangle. If, as we usually assume, the individual prefers more rather than less of every good, the outer boundary of this triangle is the relevant constraint where all of the available funds are spent either on x or on y. The slope of this straight-line boundary is given by px =py .

Quantity of y I py

I = pxx + pyy

0

I px

Quantity of x

px x þ py y I .

(4.1)

That is, no more than I can be spent on the two goods in question. This budget constraint is shown graphically in Figure 4.1. This person can afford to choose only combinations of x and y in the shaded triangle of the figure. If all of I is spent on good x, it will buy I =px units of x. Similarly, if all is spent on y, it will buy I =py units of y. The slope of the constraint is easily seen to be px =py . This slope shows how y can be traded for x in the market. If px ¼ 2 and py ¼ 1, then 2 units of y will trade for 1 unit of x.

First-order conditions for a maximum This budget constraint can be imposed on this person’s indifference curve map to show the utility-maximization process. Figure 4.2 illustrates this procedure. The individual would be irrational to choose a point such as A; he or she can get to a higher utility level just by spending more of his or her income. The assumption of nonsatiation implies that a person should spend all of his or her income in order to receive maximum utility. Similarly, by reallocating expenditures, the individual can do better than point B. Point D is out of the question because income is not large enough to purchase D. It is clear that the position of maximum utility is at point C, where the combination x , y is chosen. This is the only point on indifference curve U2 that can be bought with I dollars; no higher utility level can be

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FIGURE 4.2

A Graphical Demonstration of Utility Maximization Point C represents the highest utility level that can be reached by the individual, given the budget constraint. The combination x , y is therefore the rational way for the individual to allocate purchasing power. Only for this combination of goods will two conditions hold: All available funds will be spent, and the individual’s psychic rate of trade-off (MRS) will be equal to the rate at which the goods can be traded in the market ðpx =py Þ.

Quantity of y U1 U 2 U 3

B

D

I = pxx + pyy

C

y* A

U3 U1 0

x*

U2 Quantity of x

bought. C is a point of tangency between the budget constraint and the indifference curve. Therefore, at C we have px ¼ slope of indifference curve slope of budget constraint ¼ py dy ¼ (4.2) dx U ¼ constant or

px dy ¼ ¼ MRS ðof x for yÞ. py dx U ¼ constant

(4.3)

Our intuitive result is proved: for a utility maximum, all income should be spent and the MRS should equal the ratio of the prices of the goods. It is obvious from the diagram that if this condition is not fulfilled, the individual could be made better off by reallocating expenditures.

Second-order conditions for a maximum The tangency rule is only a necessary condition for a maximum. To see that it is not a sufﬁcient condition, consider the indifference curve map shown in Figure 4.3. Here, a point

Chapter 4

Utility Maximization and Choice

FIGURE 4.3 Example of an Indifference Curve Map for Which the Tangency Condition Does Not Ensure a Maximum If indifference curves do not obey the assumption of a diminishing MRS, not all points of tangency (points for which MRS px =py Þ may truly be points of maximum utility. In this example, tangency point C is inferior to many other points that can also be purchased with the available funds. In order that the necessary conditions for a maximum (that is, the tangency conditions) also be sufﬁcient, one usually assumes that the MRS is diminishing; that is, the utility function is strictly quasi-concave.

Quantity of y U1

U2

U3

A

I = pxx + pyy C U3 B

U2 U1 Quantity of x

of tangency ðCÞ is inferior to a point of nontangency ðBÞ. Indeed, the true maximum is at another point of tangency ðAÞ. The failure of the tangency condition to produce an unambiguous maximum can be attributed to the shape of the indifference curves in Figure 4.3. If the indifference curves are shaped like those in Figure 4.2, no such problem can arise. But we have already shown that “normally” shaped indifference curves result from the assumption of a diminishing MRS. Therefore, if the MRS is assumed to be diminishing, the condition of tangency is both a necessary and sufﬁcient condition for a maximum.2 Without this assumption, one would have to be careful in applying the tangency rule.

Corner solutions The utility-maximization problem illustrated in Figure 4.2 resulted in an “interior” maximum, in which positive amounts of both goods were consumed. In some situations individuals’ preferences may be such that they can obtain maximum utility by choosing to consume

2

In mathematical terms, because the assumption of a diminishing MRS is equivalent to assuming quasi-concavity, the necessary conditions for a maximum subject to a linear constraint are also sufficient, as we showed in Chapter 2.

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FIGURE 4.4

Corner Solution for Utility Maximization With the preferences represented by this set of indifference curves, utility maximization occurs at E, where 0 amounts of good y are consumed. The ﬁrst-order conditions for a maximum must be modiﬁed somewhat to accommodate this possibility.

Quantity of y U1

U2

U3

E x*

Quantity of x

no amount of one of the goods. If someone does not like hamburgers very much, there is no reason to allocate any income to their purchase. This possibility is reﬂected in Figure 4.4. There, utility is maximized at E, where x ¼ x and y ¼ 0, so any point on the budget constraint where positive amounts of y are consumed yields a lower utility than does point E. Notice that at E the budget constraint is not precisely tangent to the indifference curve U2 . Instead, at the optimal point the budget constraint is ﬂatter than U2 , indicating that the rate at which x can be traded for y in the market is lower than the individual’s psychic trade-off rate (the MRS). At prevailing market prices the individual is more than willing to trade away y to get extra x. Because it is impossible in this problem to consume negative amounts of y, however, the physical limit for this process is the X-axis, along which purchases of y are 0. Hence, as this discussion makes clear, it is necessary to amend the ﬁrst-order conditions for a utility maximum a bit to allow for corner solutions of the type shown in Figure 4.4. Following our discussion of the general n-good case, we will use the mathematics from Chapter 2 to show how this can be accomplished.

THE n-GOOD CASE The results derived graphically in the case of two goods carry over directly to the case of n goods. Again it can be shown that for an interior utility maximum, the MRS between any two goods must equal the ratio of the prices of these goods. To study this more general case, however, it is best to use some mathematics.

Chapter 4

Utility Maximization and Choice

First-order conditions With n goods, the individual’s objective is to maximize utility from these n goods: utility ¼ U ðx1 , x2 , …, xn Þ, subject to the budget constraint

(4.4)

3

I ¼ p1 x1 þ p2 x2 þ … þ pn xn

(4.5)

I p1 x1 p2 x2 … pn xn ¼ 0.

(4.6)

or Following the techniques developed in Chapter 2 for maximizing a function subject to a constraint, we set up the Lagrangian expression ℒ ¼ U ðx1 , x2 , …, xn Þ þ λðI p1 x1 p2 x2 … pn xn Þ. (4.7) Setting the partial derivatives of ℒ (with respect to x1 , x2 , …, xn and λ) equal to 0 yields n þ 1 equations representing the necessary conditions for an interior maximum: ∂ℒ ∂U ¼ λp1 ¼ 0, ∂x1 ∂x1 ∂ℒ ∂U ¼ λp2 ¼ 0, ∂x2 ∂x2 .. (4.8) . ∂ℒ ∂U ¼ λpn ¼ 0, ∂xn ∂xn ∂ℒ ¼ I p1 x1 p2 x2 … pn xn ¼ 0. ∂λ These n þ 1 equations can, in principle, be solved for the optimal x1 , x2 , …, xn and for λ (see Examples 4.1 and 4.2 to be convinced that such a solution is possible). Equations 4.8 are necessary but not sufﬁcient for a maximum. The second-order conditions that ensure a maximum are relatively complex and must be stated in matrix terms (see the Extensions to Chapter 2). However, the assumption of strict quasi-concavity (a diminishing MRS in the two-good case) is sufﬁcient to ensure that any point obeying Equations 4.8 is in fact a true maximum.

Implications of first-order conditions The ﬁrst-order conditions represented by Equations 4.8 can be rewritten in a variety of interesting ways. For example, for any two goods, xi and xj , we have ∂U =∂xi pi ¼ . ∂U =∂xj pj

(4.9)

In Chapter 3 we showed that the ratio of the marginal utilities of two goods is equal to the marginal rate of substitution between them. Therefore, the conditions for an optimal allocation of income become p (4.10) MRSðxi for xj Þ ¼ i . pj This is exactly the result derived graphically earlier in this chapter; to maximize utility, the individual should equate the psychic rate of trade-off to the market trade-off rate. 3

Again, the budget constraint has been written as an equality because, given the assumption of nonsatiation, it is clear that the individual will spend all available income.

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Interpreting the Lagrangian multiplier Another result can be derived by solving Equations 4.8 for λ: λ¼

∂U =∂x1 ∂U =∂x2 … ∂U =∂xn ¼ ¼ ¼ p1 p2 pn

(4.11)

or λ¼

MUx1 p1

¼

MUx2 p2

¼…¼

MUxn pn

.

These equations state that, at the utility-maximizing point, each good purchased should yield the same marginal utility per dollar spent on that good. Each good therefore should have an identical (marginal) benefit-to-(marginal)-cost ratio. If this were not true, one good would promise more “marginal enjoyment per dollar” than some other good, and funds would not be optimally allocated. Although the reader is again warned against talking very conﬁdently about marginal utility, what Equation 4.11 says is that an extra dollar should yield the same “additional utility” no matter which good it is spent on. The common value for this extra utility is given by the Lagrangian multiplier for the consumer’s budget constraint (that is, by λ). Consequently, λ can be regarded as the marginal utility of an extra dollar of consumption expenditure (the marginal utility of “income”). One ﬁnal way to rewrite the necessary conditions for a maximum is MUxi (4.12) pi ¼ λ for every good i that is bought. To interpret this equation, consider a situation where a person’s marginal utility of income (λ) is constant over some range. Then variations in the price he or she must pay for good i ðpi Þ are directly proportional to the extra utility derived from that good. At the margin, therefore, the price of a good reflects an individual’s willingness to pay for one more unit. This is a result of considerable importance in applied welfare economics because willingness to pay can be inferred from market reactions to prices. In Chapter 5 we will see how this insight can be used to evaluate the welfare effects of price changes and, in later chapters, we will use this idea to discuss a variety of questions about the efficiency of resource allocation.

Corner solutions The ﬁrst-order conditions of Equations 4.8 hold exactly only for interior maxima for which some positive amount of each good is purchased. As discussed in Chapter 2, when corner solutions (such as those illustrated in Figure 4.4) arise, the conditions must be modiﬁed slightly.4 In this case, Equations 4.8 become ∂ℒ ∂U ¼ λpi 0 ði ¼ 1, …, nÞ (4.13) ∂xi ∂xi and, if ∂ℒ ∂U ¼ λpi < 0, ∂xi ∂xi

(4.14)

xi ¼ 0.

(4.15)

then

4

Formally, these conditions are called the “Kuhn-Tucker” conditions for nonlinear programming.

Chapter 4

Utility Maximization and Choice

To interpret these conditions, we can rewrite Equation 4.14 as ∂U =∂xi MUxi ¼ . (4.16) pi > λ λ Hence, the optimal conditions are as before, except that any good whose price ðpi Þ exceeds its marginal value to the consumer (MUxi =λ) will not be purchased (xi ¼ 0). Thus, the mathematical results conform to the commonsense idea that individuals will not purchase goods that they believe are not worth the money. Although corner solutions do not provide a major focus for our analysis in this book, the reader should keep in mind the possibilities for such solutions arising and the economic interpretation that can be attached to the optimal conditions in such cases. EXAMPLE 4.1 Cobb-Douglas Demand Functions As we showed in Chapter 3, the Cobb-Douglas utility function is given by U ðx, yÞ ¼ x α y β ,

(4.17)

where, for convenience, we assume α þ β ¼ 1. We can now solve for the utility-maximizing values of x and y for any prices (px , py ) and income (I ). Setting up the Lagrangian expression 5

ℒ ¼ x α y β þ λðI px x py yÞ

(4.18)

yields the first-order conditions ∂ℒ ¼ αx α1 y β λpx ¼ 0, ∂x ∂ℒ ¼ βx α y β1 λpy ¼ 0, ∂y ∂ℒ ¼ I px x py y ¼ 0. ∂λ Taking the ratio of the first two terms shows that αy px ¼ , βx py

(4.19)

(4.20)

or β 1α (4.21) p x¼ px x, α x α where the final equation follows because α þ β ¼ 1. Substitution of this first-order condition in Equation 4.21 into the budget constraint gives 1α 1α 1 (4.22) px x ¼ px x 1 þ ¼ px x; I ¼ px x þ py y ¼ px x þ α α α py y ¼

solving for x yields x ¼

αI , px

(4.23) (continued)

5

Notice that the exponents in the Cobb-Douglas utility function can always be normalized to sum to 1 because U 1=ðαþβÞ is a monotonic transformation.

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Part 2 Choice and Demand

EXAMPLE 4.1 CONTINUED and a similar set of manipulations would give y ¼

βI . py

(4.24)

These results show that an individual whose utility function is given by Equation 4.17 will always choose to allocate α proportion of his or her income to buying good x (i.e., px x=I ¼ α) and β proportion to buying good y ðpy y=I ¼ βÞ. Although this feature of the Cobb-Douglas function often makes it very easy to work out simple problems, it does suggest that the function has limits in its ability to explain actual consumption behavior. Because the share of income devoted to particular goods often changes significantly in response to changing economic conditions, a more general functional form may provide insights not provided by the Cobb-Douglas function. We illustrate a few possibilities in Example 4.2, and the general topic of budget shares is taken up in more detail in the Extensions to this chapter. Numerical example. First, however, let’s look at a speciﬁc numerical example for the CobbDouglas case. Suppose that x sells for $1 and y sells for $4 and that total income is $8. Succinctly then, assume that px ¼ 1, py ¼ 4, I ¼ 8. Suppose also that α ¼ β ¼ 0:5 so that this individual splits his or her income equally between these two goods. Now the demand Equations 4.23 and 4.24 imply x ¼ αI =px ¼ 0:5I =px ¼ 0:5ð8Þ=1 ¼ 4, (4.25) y ¼ βI =p ¼ 0:5I =p ¼ 0:5ð8Þ=4 ¼ 1, y

y

and, at these optimal choices, utility ¼ x 0:5 y 0:5 ¼ ð4Þ0:5 ð1Þ0:5 ¼ 2.

(4.26)

Notice also that we can compute the value for the Lagrangian multiplier associated with this income allocation by using Equation 4.19: λ ¼ αx α1 y β =px ¼ 0:5ð4Þ0:5 ð1Þ0:5 =1 ¼ 0:25.

(4.27)

This value implies that each small change in income will increase utility by about one-fourth of that amount. Suppose, for example, that this person had 1 percent more income ($8.08). In this case he or she would choose x ¼ 4:04 and y ¼ 1:01, and utility would be 4:040:5 ⋅ 1:010:5 ¼ 2:02. Hence, a $0.08 increase in income increases utility by 0.02, as predicted by the fact that λ ¼ 0:25. QUERY: Would a change in py affect the quantity of x demanded in Equation 4.23? Explain your answer mathematically. Also develop an intuitive explanation based on the notion that the share of income devoted to good y is given by the parameter of the utility function, β.

EXAMPLE 4.2 CES Demand To illustrate cases in which budget shares are responsive to economic circumstances, let’s look at three speciﬁc examples of the CES function. Case 1: δ ¼ 0:5. In this case, utility is U ðx, yÞ ¼ x 0:5 þ y 0:5 .

(4.28)

Chapter 4

Utility Maximization and Choice

Setting up the Lagrangian expression ℒ ¼ x 0:5 þ y 0:5 þ λðI px x py yÞ

(4.29)

yields the following first-order conditions for a maximum: ∂ℒ=∂x ¼ 0:5x 0:5 λpx ¼ 0, ∂ℒ=∂y ¼ 0:5y 0:5 λpy ¼ 0,

(4.30)

∂ℒ=∂λ ¼ I px x py y ¼ 0. Division of the first two of these shows that ðy=xÞ0:5 ¼ px =py .

(4.31)

By substituting this into the budget constraint and doing some messy algebraic manipulation, we can derive the demand functions associated with this utility function: (4.32) x ¼ I =p ½1 þ ðp =p Þ, x

x

y

y ¼ I =py ½1 þ ðpy =px Þ.

(4.33)

Price responsiveness. In these demand functions notice that the share of income spent on, say, good x—that is, px x=I ¼ 1=½1 þ ðpx =py Þ—is not a constant; it depends on the price ratio px =py . The higher is the relative price of x, the smaller will be the share of income spent on that good. In other words, the demand for x is so responsive to its own price that a rise in the price reduces total spending on x. That the demand for x is very price responsive can also be illustrated by comparing the implied exponent on px in the demand function given by Equation 4.32 (2) to that from Equation 4.23 (1). In Chapter 5 we will discuss this observation more fully when we examine the elasticity concept in detail. Case 2: δ ¼ 1. Alternatively, let’s look at a demand function with less substitutability6 than the Cobb-Douglas. If δ ¼ 1, the utility function is given by U ðx, yÞ ¼ x 1 y 1 ,

(4.34)

and it is easy to show that the first-order conditions for a maximum require y=x ¼ ð px =py Þ0:5 .

(4.35)

Again, substitution of this condition into the budget constraint, together with some messy algebra, yields the demand functions x ¼ I =px ½1 þ ðpy =px Þ0:5 , (4.36) y ¼ I =py ½1 þ ð px =py Þ0:5 . That these demand functions are less price responsive can be seen in two ways. First, now the share of income spent on good x—that is, px x=I ¼ 1=½1 þ ðpy =px Þ0:5 —responds positively to increases in px . As the price of x rises, this individual cuts back only modestly on good x, so total spending on that good rises. That the demand functions in Equations 4.36 are less price responsive than the Cobb-Douglas is also illustrated by the relatively small exponents of each good’s own price ð0:5Þ. (continued)

6

One way to measure substitutability is by the elasticity of substitution, which for the CES function is given by σ ¼ 1=ð1 δÞ. Here δ ¼ 0:5 implies σ ¼ 2, δ ¼ 0 (the Cobb-Douglas) implies σ ¼ 1, and δ ¼ 1 implies σ ¼ 0:5. See also the discussion of the CES function in connection with the theory of production in Chapter 9.

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EXAMPLE 4.2 CONTINUED Case 3: δ ¼ ∞. This is the important case in which x and y must be consumed in ﬁxed proportions. Suppose, for example, that each unit of y must be consumed together with exactly 4 units of x. The utility function that represents this situation is U ðx, yÞ ¼ minðx, 4yÞ.

(4.37)

In this situation, a utility-maximizing person will choose only combinations of the two goods for which x ¼ 4y; that is, utility maximization implies that this person will choose to be at a vertex of his or her L-shaped indifference curves. Substituting this condition into the budget constraint yields x (4.38) I ¼ px x þ py y ¼ px x þ py ¼ ðpx þ 0:25py Þx. 4 Hence I x ¼ , (4.39) px þ 0:25py and similar substitutions yield y ¼

I . 4px þ py

(4.40)

In this case, the share of a person’s budget devoted to, say, good x rises rapidly as the price of x increases because x and y must be consumed in fixed proportions. For example, if we use the values assumed in Example 4.1 (px ¼ 1, py ¼ 4, I ¼ 8), Equations 4.39 and 4.40 would predict x ¼ 4, y ¼ 1, and, as before, half of the individual’s income would be spent on each good. If we instead use px ¼ 2, py ¼ 4, and I ¼ 8 then x ¼ 8=3, y ¼ 2=3, and this person spends two thirds ½ px x=I ¼ ð2 ⋅ 8=3Þ=8 ¼ 2=3 of his or her income on good x. Trying a few other numbers suggests that the share of income devoted to good x approaches 1 as the price of x increases.7 QUERY: Do changes in income affect expenditure shares in any of the CES functions discussed here? How is the behavior of expenditure shares related to the homothetic nature of this function?

INDIRECT UTILITY FUNCTION Examples 4.1 and 4.2 illustrate the principle that it is often possible to manipulate the ﬁrstorder conditions for a constrained utility-maximization problem to solve for the optimal values of x1 , x2 , …, xn . These optimal values in general will depend on the prices of all the goods and on the individual’s income. That is, x ¼ x ðp , p , …, p , I Þ, 1

1

1

2

n

x 2 ¼ x2 ðp1 , p2 , …, pn , I Þ, .. . x n ¼ xn ðp1 , p2 , …, pn , I Þ.

(4.41)

In the next chapter we will analyze in more detail this set of demand functions, which show the dependence of the quantity of each xi demanded on p1 , p2 , …, pn and I . Here we use 7

These relationships for the CES function are pursued in more detail in Problem 4.9 and in Extension E4.3.

Chapter 4

Utility Maximization and Choice

the optimal values of the x’s from Equations 4.42 to substitute in the original utility function to yield (4.42) maximum utility ¼ U ðx , x , …, x Þ 1

2

n

¼ V ðp1 , p2 , …, pn , I Þ .

(4.43)

In words: because of the individual’s desire to maximize utility given a budget constraint, the optimal level of utility obtainable will depend indirectly on the prices of the goods being bought and the individual’s income. This dependence is reﬂected by the indirect utility function V . If either prices or income were to change, the level of utility that could be attained would also be affected. Sometimes, in both consumer theory and many other contexts, it is possible to use this indirect approach to study how changes in economic circumstances affect various kinds of outcomes, such as utility or (later in this book) ﬁrms’ costs.

THE LUMP SUM PRINCIPLE Many economic insights stem from the recognition that utility ultimately depends on the income of individuals and on the prices they face. One of the most important of these is the so-called lump sum principle that illustrates the superiority of taxes on a person’s general purchasing power to taxes on speciﬁc goods. A related insight is that general income grants to low-income people will raise utility more than will a similar amount of money spent subsidizing speciﬁc goods. The intuition behind this result derives directly from the utility-maximization hypothesis; an income tax or subsidy leaves the individual free to decide how to allocate whatever ﬁnal income he or she has. On the other hand, taxes or subsidies on speciﬁc goods both change a person’s purchasing power and distort his or her choices because of the artiﬁcial prices incorporated in such schemes. Hence, general income taxes and subsidies are to be preferred if efﬁciency is an important criterion in social policy. The lump sum principle as it applies to taxation is illustrated in Figure 4.5. Initially this person has an income of I and is choosing to consume the combination x , y . A tax on good x would raise its price, and the utility-maximizing choice would shift to combination x1 , y1 . Tax collections would be t ⋅ x1 (where t is the tax rate imposed on good x). Alternatively, an income tax that shifted the budget constraint inward to I 0 would also collect this same amount of revenue.8 But the utility provided by the income tax ðU2 Þ exceeds that provided by the tax on x alone ðU1 Þ. Hence, we have shown that the utility burden of the income tax is smaller. A similar argument can be used to illustrate the superiority of income grants to subsidies on speciﬁc goods. EXAMPLE 4.3 Indirect Utility and the Lump Sum Principle In this example we use the notion of an indirect utility function to illustrate the lump sum principle as it applies to taxation. First we have to derive indirect utility functions for two illustrative cases. (continued)

Because I ¼ ðpx þ t Þx1 þ py y1 , we have I 0 ¼ I tx1 ¼ px x1 þ py y1 , which shows that the budget constraint with an equal-size income tax also passes through the point x1 , y1 .

8

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EXAMPLE 4.3 CONTINUED Case 1: Cobb-Douglas. In Example 4.1 we showed that, for the Cobb-Douglas utility function with α ¼ β ¼ 0:5, optimal purchases are I , x ¼ 2px (4.44) I y ¼ ⋅ 2py So the indirect utility function in this case is V ðpx , py , I Þ ¼ U ðx , y Þ ¼ ðx Þ0:5 ðy Þ0:5 ¼

I . 0:5 2p 0:5 x py

Notice that when px ¼ 1, py ¼ 4, and I ¼ 8 we have V ¼ 8=ð2 ⋅ 1 ⋅ 2Þ ¼ 2, which is the utility that we calculated before for this situation. Case 2: Fixed proportions. In the third case of Example 4.2 we found that I , x ¼ px þ 0:25py (4.46) I y ¼ ⋅ 4px þ py So, in this case indirect utility is given by I px þ 0:25py 4 I ¼ 4y ¼ ¼ ; 4px þ py px þ 0:25py

V ðpx , py , I Þ ¼ minðx , 4y Þ ¼ x ¼

(4.47)

with px ¼ 1, py ¼ 4, and I ¼ 8, indirect utility is given by V ¼ 4, which is what we calculated before. The lump sum principle. Consider ﬁrst using the Cobb-Douglas case to illustrate the lump sum principle. Suppose that a tax of $1 were imposed on good x. Equation 4.45 shows that indirect utility in this case would fall from 2 to 1:41 ½¼ 8=ð2 ⋅ 20:5 ⋅ 2Þ. Because this person chooses x ¼ 2 with the tax, total tax collections will be $2. An equal-revenue income tax would therefore reduce net income to $6, and indirect utility would be 1:5 ½¼ 6=ð2 ⋅ 1 ⋅ 2Þ. So the income tax is a clear improvement over the case where x alone is taxed. The tax on good x reduces utility for two reasons: it reduces a person’s purchasing power and it biases his or her choices away from good x. With income taxation, only the ﬁrst effect is felt and so the tax is more efﬁcient.9 The ﬁxed-proportions case supports this intuition. In that case, a $1 tax on good x would reduce indirect utility from 4 to 8=3 ½¼ 8=ð2 þ 1Þ. In this case x ¼ 8=3 and tax collections would be $8=3. An income tax that collected $8=3 would leave this consumer with $16=3 in net income, and that income would yield an indirect utility of V ¼ 8=3 ½¼ ð16=3Þ=ð1 þ 1Þ. Hence after-tax utility is the same under both the excise and income taxes. The reason the lump sum result does not hold in this case is that with ﬁxed-proportions utility, the excise tax does not distort choices because preferences are so rigid. QUERY: Both of the indirect utility functions illustrated here show that a doubling of income and all prices would leave indirect utility unchanged. Explain why you would expect this to be a property of all indirect utility functions.

9

This discussion assumes that there are no incentive effects of income taxation—probably not a very good assumption.

Chapter 4

FIGURE 4.5

Utility Maximization and Choice

The Lump Sum Principle of Taxation

A tax on good x would shift the utility-maximizing choice from x , y to x1 , y1 . An income tax that collected the same amount would shift the budget constraint to I 0 . Utility would be higher ðU2 Þ with the income tax than with the tax on x alone ðU1 Þ.

Quantity of y

y1

l′

y* y2 U3

U2

l

U1

x1

x2

x*

Quantity of x

EXPENDITURE MINIMIZATION In Chapter 2 we pointed out that many constrained maximum problems have associated “dual” constrained minimum problems. For the case of utility maximization, the associated dual minimization problem concerns allocating income in such a way as to achieve a given utility level with the minimal expenditure. This problem is clearly analogous to the primary utility-maximization problem, but the goals and constraints of the problems have been reversed. Figure 4.6 illustrates this dual expenditure-minimization problem. There, the individual must attain utility level U2 ; this is now the constraint in the problem. Three possible expenditure amounts (E1 , E2 , and E3 ) are shown as three “budget constraint” lines in the ﬁgure. Expenditure level E1 is clearly too small to achieve U2 , hence it cannot solve the dual problem. With expenditures given by E3 , the individual can reach U2 (at either of the two points B or C), but this is not the minimal expenditure level required. Rather, E2 clearly provides just enough total expenditures to reach U2 (at point A), and this is in fact the solution to the dual problem. By comparing Figures 4.2 and 4.6, it is obvious that both the primary utility-maximization approach and the dual expenditure-minimization approach yield the same solution ðx , y Þ; they are simply alternative ways of viewing the same process. Often the expenditure-minimization approach is more useful, however, because expenditures are directly observable, whereas utility is not.

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FIGURE 4.6

The Dual Expenditure-Minimization Problem The dual of the utility-maximization problem is to attain a given utility level ðU2 Þ with minimal expenditures. An expenditure level of E1 does not permit U2 to be reached, whereas E3 provides more spending power than is strictly necessary. With expenditure E2 , this person can just reach U2 by consuming x and y .

Quantity of y

B E3

E2 E1

A

y*

C U2

x*

Quantity of x

A mathematical statement More formally, the individual’s dual expenditure-minimization problem is to choose x1 , x2 , …, xn so as to minimize total expenditures ¼ E ¼ p1 x1 þ p2 x2 þ … þ pn xn , subject to the constraint

_ utility ¼ U ¼ U ðx1 , x2 , …, xn Þ.

(4.48)

(4.49)

The optimal amounts of x1 , x2 , …, xn chosen in this problem will depend _ on the prices of the various goods ðp1 , p2 , …, pn Þ and on the required utility level U2 . If any of the prices were to change or if the individual had a different utility “target,” then another commodity bundle would be optimal. This dependence can be summarized by an expenditure function. DEFINITION

Expenditure function. The individual’s expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices. That is, minimal expenditures ¼ Eðp1 , p2 , …, pn , U Þ.

(4.50)

This definition shows that the expenditure function and the indirect utility function are inverse functions of one another (compare Equations 4.43 and 4.50). Both depend on

Chapter 4

Utility Maximization and Choice

market prices but involve different constraints (income or utility). In the next chapter we will see how this relationship is quite useful in allowing us to examine the theory of how individuals respond to price changes. First, however, let’s look at two expenditure functions. EXAMPLE 4.4 Two Expenditure Functions There are two ways one might compute an expenditure function. The ﬁrst, most straightforward method would be to state the expenditure-minimization problem directly and apply the Lagrangian technique. Some of the problems at the end of this chapter ask you to do precisely that. Here, however, we will adopt a more streamlined procedure by taking advantage of the relationship between expenditure functions and indirect utility functions. Because these two functions are inverses of each other, calculation of one greatly facilitates the calculation of the other. We have already calculated indirect utility functions for two important cases in Example 4.3. Retrieving the related expenditure functions is simple algebra. Case 1: Cobb-Douglas utility. Equation 4.45 shows that the indirect utility function in the two-good, Cobb-Douglas case is I (4.51) V ðpx , py , I Þ ¼ 0:5 0:5 . 2p x p y If we now interchange the role of utility (which we will now treat as a constant denoted by U ) and income (which we will now term “expenditures,” E, and treat as a function of the parameters of this problem), then we have the expenditure function 0:5 Eðpx , py , U Þ ¼ 2p 0:5 x py U .

(4.52)

Checking this against our former results, now we use a utility target of U ¼ 2 with, again, px ¼ 1 and py ¼ 4. With these parameters, Equation 4.52 predicts that the required minimal expenditures are $8 ð¼ 2 ⋅ 10:5 ⋅ 40:5 ⋅ 2Þ. Not surprisingly, both the primal utilitymaximization problem and the dual expenditure-minimization problem are formally identical. Case 2: Fixed proportions. For the ﬁxed-proportions case, Equation 4.47 gave the indirect utility function as I . (4.53) V ð px , py , I Þ ¼ px þ 0:25py If we again switch the role of utility and expenditures, we quickly derive the expenditure function: Eð px , py , U Þ ¼ ðpx þ 0:25py ÞU .

(4.54)

A check of the hypothetical values used in Example 4.3 ðpx ¼ 1, py ¼ 4, U ¼ 4Þ again shows that it would cost $8 ½¼ ð1 þ 0:25 ⋅ 4Þ ⋅ 4 to reach the utility target of 4. Compensating for a price change. These expenditure functions allow us to investigate how a person might be compensated for a price change. Speciﬁcally, suppose that the price of good y were to rise from $4 to $5. This would clearly reduce a person’s utility, so we might ask what amount of monetary compensation would mitigate the harm. Because the expenditure function allows utility to be held constant, it provides a direct estimate of this amount. Speciﬁcally, in the Cobb-Douglas case, expenditures would have to be increased from $8 to (continued)

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EXAMPLE 4.4 CONTINUED $8:94 ð¼ 2 ⋅ 1 ⋅ 50:5 ⋅ 2Þ in order to provide enough extra purchasing power to precisely compensate for this price rise. With ﬁxed proportions, expenditures would have to be increased from $8 to $9 to compensate for the price increase. Hence, the compensations are about the same in these simple cases. There is one important difference between the two examples, however. In the ﬁxedproportions case, the $1 of extra compensation simply permits this person to return to his or her prior consumption bundle ðx ¼ 4, y ¼ 1Þ. That is the only way to restore utility to U ¼ 4 for this rigid person. In the Cobb-Douglas case, however, this person will not use the extra compensation to revert to his or her prior consumption bundle. Instead, utility maximization will require that the $8.94 be allocated so that x ¼ 4:47 and y ¼ 0:894. This will still provide a utility level of U ¼ 2, but this person will economize on the now more expensive good y. QUERY: How should a person be compensated for a price decline? What sort of compensation would be required if the price of good y fell from $4 to $3?

PROPERTIES OF EXPENDITURE FUNCTIONS Because expenditure functions are widely used in applied economics, it is useful to understand a few of the properties shared by all such functions. Here we look at three such properties. All of these follow directly from the fact that expenditure functions are based on individual utility maximization. 1. Homogeneity. For both of the functions illustrated in Example 4.4, a doubling of all prices will precisely double the value of required expenditures. Technically, these expenditure functions are “homogeneous of degree one” in all prices.10 This is a quite general property of expenditure functions. Because the individual’s budget constraint is linear in prices, any proportional increase in both prices and purchasing power will permit the person to buy the same utility-maximizing commodity bundle that was chosen before the price rise. In Chapter 5 we will see that, for this reason, demand functions are homogenous of degree 0 in all prices and income. 2. Expenditure functions are nondecreasing in prices. This property can be succinctly summarized by the mathematical statement ∂E 0 ∂pi

for every good i.

(4.55)

This seems intuitively obvious. Because the expenditure function reports the minimum expenditure necessary to reach a given utility level, an increase in any price must increase this minimum. More formally, suppose p1 takes on two values: pa1 and pb1 with pb1 > pa1 , where all other prices are unchanged between states a and b. Also, let x be the bundle of goods purchased in state a and y the bundle purchased in state b. By the definition of the expenditure function, both of these bundles of goods must

10 As described in Chapter 2, the function f ðx1 , x2 , …, xn Þ is said to be homogeneous of degree k if f ðtx1 , tx2 , …, txn Þ ¼ t k f ðx1 , x2 , …, xn Þ. In this case, k ¼ 1.

Chapter 4

Utility Maximization and Choice

yield the same target utility. Clearly bundle y costs more with state-b prices than it would with state-a prices. But we know that bundle x is the lowest-cost way to achieve the target utility level with state-a prices. Hence, expenditures on bundle y must be greater than or equal to those on bundle x. Similarly, a decline in a price must not increase expenditures. 3. Expenditure functions are concave in prices. In Chapter 2 we discussed concave functions as functions that always lie below tangents to them. Although the technical mathematical conditions that describe such functions are complicated, it is relatively simple to show how the concept applies to expenditure functions by considering the variation in a single price. Figure 4.7 shows an individual’s expenditures as a function of the single price, p1 . At the initial price, p1 , this person’s expenditures are given by Eðp1 , …Þ. Now consider prices higher or lower than p1 . If this person continued to buy the same bundle of goods, expenditures would increase or decrease linearly as this price changed. This would give rise to the pseudo expenditure function E pseudo in the ﬁgure. This line shows a level of expenditures that would allow this person to buy the original bundle of goods despite the changing value of p1 . If, as seems more likely, this person adjusted his or her purchases as p1 changed, we know (because of expenditure minimization) that actual expenditures would be less than these pseudo

FIGURE 4.7

Expenditure Functions Are Concave in Prices

At p1 this person spends Eðp1 , …Þ. If he or she continues to buy the same set of goods as p1 changes, then expenditures would be given by E pseudo. Because his or her consumption patterns will likely change as p1 changes, actual expenditures will be less than this.

E(p1, . . .)

E pseudo E(p1, . . .) E(p1*, . . .)

E(p1*, . . .)

p1

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Part 2 Choice and Demand

amounts. Hence, the actual expenditure function, E, will lie everywhere below E pseudo and the function will be concave.11 The concavity of the expenditure function is a useful property for a number of applications, especially those related to the construction of index numbers (see the Extensions to Chapter 5).

SUMMARY sumption of some goods is zero. In this case, the ratio of marginal utility to price for such a good will be below the common marginal beneﬁt–marginal cost ratio for goods actually bought.

In this chapter we explored the basic economic model of utility maximization subject to a budget constraint. Although we approached this problem in a variety of ways, all of these approaches lead to the same basic result. •

To reach a constrained maximum, an individual should spend all available income and should choose a commodity bundle such that the MRS between any two goods is equal to the ratio of those goods’ market prices. This basic tangency will result in the individual equating the ratios of the marginal utility to market price for every good that is actually consumed. Such a result is common to most constrained optimization problems.

•

The tangency conditions are only the ﬁrst-order conditions for a unique constrained maximum, however. To ensure that these conditions are also sufﬁcient, the individual’s indifference curve map must exhibit a diminishing MRS. In formal terms, the utility function must be strictly quasi-concave.

•

The tangency conditions must also be modiﬁed to allow for corner solutions in which the optimal level of con-

•

A consequence of the assumption of constrained utility maximization is that the individual’s optimal choices will depend implicitly on the parameters of his or her budget constraint. That is, the choices observed will be implicit functions of all prices and income. Utility will therefore also be an indirect function of these parameters.

•

The dual to the constrained utility-maximization problem is to minimize the expenditure required to reach a given utility target. Although this dual approach yields the same optimal solution as the primal constrained maximum problem, it also yields additional insight into the theory of choice. Speciﬁcally, this approach leads to expenditure functions in which the spending required to reach a given utility target depends on goods’ market prices. Expenditure functions are therefore, in principle, measurable.

PROBLEMS 4.1 Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (t ) and soda (s), and these provide him a utility of pﬃﬃﬃﬃ utility ¼ U ðt , sÞ ¼ ts . a. If Twinkies cost $0.10 each and soda costs $0.25 per cup, how should Paul spend the $1 his mother gives him in order to maximize his utility? b. If the school tries to discourage Twinkie consumption by raising the price to $0.40, by how much will Paul’s mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?

4.2 a. A young connoisseur has $600 to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux (wF ) at $40 per bottle and a less expensive 2005 California varietal wine (wC ) priced at $8. If her utility is

One result of concavity is that fii ¼ ∂2 E=∂p2i 0. This is precisely what Figure 4.7 shows.

11

Chapter 4 2=3

Utility Maximization and Choice

1=3

U ðwF , wC Þ ¼ w F w C , then how much of each wine should she purchase? b. When she arrived at the wine store, our young oenologist discovered that the price of the French Bordeaux had fallen to $20 a bottle because of a decline in the value of the franc. If the price of the California wine remains stable at $8 per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions? c. Explain why this wine fancier is better off in part (b) than in part (a). How would you put a monetary value on this utility increase?

4.3 a. On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according to the function U ðc, bÞ ¼ 20c c 2 þ 18b 3b 2 . How many cigars and glasses of brandy does he consume during an evening? (Cost is no object to J. P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to 5. How many glasses of brandy and cigars will he consume under these circumstances?

4.4 a. Mr. Odde Ball enjoys commodities x and y according to the utility function qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ U ðx, yÞ ¼ x 2 þ y 2 : Maximize Mr. Ball’s utility if px ¼ $3, py ¼ $4, and he has $50 to spend. Hint: It may be easier here to maximize U 2 rather than U . Why won’t this alter your results? b. Graph Mr. Ball’s indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball’s behavior? Have you found a true maximum?

4.5 Mr. A derives utility from martinis (m) in proportion to the number he drinks: U ðmÞ ¼ m. Mr. A is very particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin ( g) to one part vermouth (v). Hence, we can rewrite Mr. A’s utility function as g U ðmÞ ¼ U ð g, vÞ ¼ min , v . 2 a. Graph Mr. A’s indifference curve in terms of g and v for various levels of utility. Show that, regardless of the prices of the two ingredients, Mr. A will never alter the way he mixes martinis. b. Calculate the demand functions for g and v. c. Using the results from part (b), what is Mr. A’s indirect utility function? d. Calculate Mr. A’s expenditure function; for each level of utility, show spending as a function of pg and pv . Hint: Because this problem involves a ﬁxed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus.

4.6 Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)—according to the Cobb-Douglas utility function

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134

Part 2 Choice and Demand U ðx, y, zÞ ¼ x 0:5 y 0:5 ð1 þ zÞ0:5 . Suppose also that the prices for these goods are given by px ¼ 0:25, py ¼ 1, and pz ¼ 2 and that this consumer’s income is given by I ¼ 2. a. Show that, for z ¼ 0, maximization of utility results in the same optimal choices as in Example 4.1. Show also that any choice that results in z > 0 (even for a fractional z) reduces utility from this optimum. b. How do you explain the fact that z ¼ 0 is optimal here? c. How high would this individual’s income have to be in order for any z to be purchased?

4.7 The lump sum principle illustrated in Figure 4.5 applies to transfer policy as well as taxation. This problem examines this application of the principle. a. Use a graph similar to Figure 4.5 to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government. b. Use the Cobb-Douglas expenditure function presented in Equation 4.52 to calculate the extra purchasing power needed to raise this person’s utility from U ¼ 2 to U ¼ 3. c. Use Equation 4.52 again to estimate the degree to which good x must be subsidized in order to raise this person’s utility from U ¼ 2 to U ¼ 3. How much would this subsidy cost the government? How would this cost compare to the cost calculated in part (b)? d. Problem 4.10 asks you to compute an expenditure function for a more general Cobb-Douglas utility function than the one used in Example 4.4. Use that expenditure function to re-solve parts (b) and (c) here for the case α ¼ 0:3, a ﬁgure close to the fraction of income that lowincome people spend on food. e. How would your calculations in this problem have changed if we had used the expenditure function for the ﬁxed proportions case (Equation 4.54) instead?

4.8 Mr. Carr derives a lot of pleasure from driving under the wide blue skies. For the number of miles x that he drives, he receives utility U ðxÞ ¼ 500x x 2 . (Once he drives beyond a certain number of miles, weariness kicks in and the ride becomes less and less enjoyable.) Now, his car gives him a decent highway mileage of 25 miles to the gallon. But paying for gas, represented by y, induces disutility for Mr. Carr, shown by U ðyÞ ¼ 1, 000y. Mr. Carr is willing to spend up to $25 for leisurely driving every week. a. Find the optimum number of miles driven by Mr. Carr every week, given that the price of gas is $2.50 per gallon. b. How does that value change when the price of gas rises to $5.00 per gallon? c. Now, further assume that there is a probability of 0.001 that Mr. Carr will get a ﬂat tire every mile he drives. The disutility from a ﬂat tire is given by U ðzÞ ¼ 50,000z (where z is the number of ﬂat tires incurred), and each ﬂat tire costs $50 to replace. Find the distance driven that maximizes Mr. Carr’s utility after taking into account the expected likelihood of ﬂat tires (assume that the price of gas is $2.50 per gallon).

4.9 Suppose that we have a utility function involving two goods that is linear of the form U ðx, yÞ ¼ ax þ by. Calculate the expenditure function for this utility function. Hint: The expenditure function will have kinks at various price ratios.

Chapter 4

Utility Maximization and Choice

Analytical Problems 4.10 Cobb-Douglas utility In Example 4.1 we looked at the Cobb-Douglas utility function U ðx, yÞ ¼ x α y 1α , where 0 α 1. This problem illustrates a few more attributes of that function. a. Calculate the indirect utility function for this Cobb-Douglas case. b. Calculate the expenditure function for this case. c. Show explicitly how the compensation required to offset the effect of a rise in the price of x is related to the size of the exponent α.

4.11 CES utility The CES utility function we have used in this chapter is given by U ðx, yÞ ¼

xδ yδ þ . δ δ

a. Show that the ﬁrst-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion !1=ðδ1Þ x px . ¼ py y b. Show that the result in part (a) implies that individuals will allocate their funds equally between x and y for the Cobb-Douglas case (δ ¼ 0), as we have shown before in several problems. c. How does the ratio px x=py y depend on the value of δ? Explain your results intuitively. (For further details on this function, see Extension E4.3.) d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.

4.12 Stone-Geary utility Suppose individuals require a certain level of food (x) to remain alive. Let this amount be given by x0 . Once x0 is purchased, individuals obtain utility from food and other goods (y) of the form U ðx, yÞ ¼ ðx x0 Þα y β , where α þ β ¼ 1: a. Show that if I > px x0 then the individual will maximize utility by spending αðI px x0 Þ þ px x0 on good x and βðI px x0 Þ on good y. Interpret this result. b. How do the ratios px x=I and py y=I change as income increases in this problem? (See also Extension E4.2 for more on this utility function.)

4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U ðx, yÞ ¼ ðx δ þ y δ Þ1=δ [in this function the elasticity of substitution σ ¼ 1=ð1 δÞ]. a. Show that the indirect utility function for the utility function just given is V ¼ I ðp rx þ p ry Þ1=r , where r ¼ δ=ðδ 1Þ ¼ 1 σ.

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Part 2 Choice and Demand b. Show that the function derived in part (a) is homogeneous of degree 0 in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E ¼ V ð p rx þ p ry Þ1=r . f. Show that the function derived in part (e) is homogeneous of degree 1 in the goods’ prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.

SUGGESTIONS FOR FURTHER READING Barten, A. P., and Volker Böhm. “Consumer Theory.” In K. J. Arrow and M. D. Intriligator, Eds., Handbook of Mathematical Economics, vol. II. Amsterdam: North-Holland, 1982.

Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford: Oxford University Press, 1995.

Sections 10 and 11 have compact summaries of many of the concepts covered in this chapter.

Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947.

Deaton, A., and J. Muelbauer. Economics and Consumer Behavior. Cambridge: Cambridge University Press, 1980.

Chapter V and Appendix A provide a succinct analysis of the ﬁrst-order conditions for a utility maximum. The appendix provides good coverage of second-order conditions.

Section 2.5 provides a nice geometric treatment of duality concepts.

Dixit, A. K. Optimization in Economic Theory. Oxford: Oxford University Press, 1990. Chapter 2 provides several Lagrangian analyses focusing on the Cobb-Douglas utility function.

Hicks, J. R. Value and Capital. Oxford: Clarendon Press, 1946. Chapter II and the Mathematical Appendix provide some early suggestions of the importance of the expenditure function.

Chapter 3 contains a thorough analysis of utility and expenditure functions.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. A useful, though fairly difﬁcult, treatment of duality in consumer theory.

Theil, H. Theory and Measurement of Consumer Demand. Amsterdam: North-Holland, 1975. Good summary of basic theory of demand together with implications for empirical estimation.

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Utility Maximization and Choice

137

EXTENSIONS Budget Shares The nineteenth-century economist Ernst Engel was one of the ﬁrst social scientists to intensively study people’s actual spending patterns. He focused speciﬁcally on food consumption. His ﬁnding that the fraction of income spent on food declines as income increases has come to be known as Engel’s law and has been conﬁrmed in many studies. Engel’s law is such an empirical regularity that some economists have suggested measuring poverty by the fraction of income spent on food. Two other interesting applications are: (1) the study by Hayashi (1995) showing that the share of income devoted to foods favored by the elderly is much higher in two-generation households than in one-generation households; and (2) ﬁndings by Behrman (1989) from less-developed countries showing that people’s desires for a more varied diet as their incomes rise may in fact result in reducing the fraction of income spent on particular nutrients. In the remainder of this extension we look at some evidence on budget shares (denoted by si ¼ pi xi =I ) together with a bit more theory on the topic.

E4.1 The variability of budget shares Table E4.1 shows some recent budget share data from the United States. Engel’s law is clearly visible in the table: as income rises families spend a smaller proportion of their funds on food. Other important variations in the table include the declining share of income spent on health-care needs and the much larger share of income devoted to retirement plans by higher-income people. Interestingly, the shares of income devoted to shelter and transportation are relatively constant over the range of income shown in the table; apparently, high-income people buy bigger houses and cars. The variable income shares in Table E4.1 illustrate why the Cobb-Douglas utility function is not useful for detailed empirical studies of household behavior. When utility is given by U ðx, yÞ ¼ x α y β , the implied demand equations are x ¼ αI =px and y ¼ βI =py . Therefore, sx ¼ px x=I ¼ α and sy ¼ py y=I ¼ β,

other possible forms for the utility function that permit more flexibility.

E4.2 Linear expenditure system A generalization of the Cobb-Douglas function that incorporates the idea that certain minimal amounts of each good must be bought by an individual ðx0 , y0 Þ is the utility function U ðx, yÞ ¼ ðx x0 Þα ðy y0 Þβ

(ii)

for x x0 and y y0 , where again α þ β ¼ 1. Demand functions can be derived from this utility function in a way analogous to the Cobb-Douglas case by introducing the concept of supernumerary income ðI Þ, which represents the amount of purchasing power remaining after purchasing the minimum bundle (iii) I ¼ I px x0 py y0 . Using this notation, the demand functions are x ¼ ðpx x0 þ αI Þ=px , (iv) y ¼ ð p y þ βI Þ=p . y 0

y

In this case, then, the individual spends a constant fraction of supernumerary income on each good once the minimum bundle has been purchased. Manipulation of Equation iv yields the share equations sx ¼ α þ ðβpx x0 αpy y0 Þ=I , sy ¼ β þ ðαpy y0 βpx x0 Þ=I ,

(v)

which show that this demand system is not homothetic. Inspection of Equation v shows the unsurprising result that the budget share of a good is positively related to the minimal amount of that good needed and negatively related to the minimal amount of the other good required. Because the notion of necessary purchases seems to accord well with real-world observation, this linear expenditure system (LES), which was first developed by Stone (1954), is widely used in empirical studies. The utility function in Equation ii is also called a Stone-Geary utility function.

(i)

and budget shares are constant for all observed income levels and relative prices. Because of this shortcoming, economists have investigated a number of

Traditional purchases One of the most interesting uses of the LES is to examine how its notion of necessary purchases changes as conditions change. For example, Oczkowski and

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Part 2 Choice and Demand

TABLE E4.1

Budget Shares of U.S. Households, 2004 Annual Income $10,000–$14,999

$40,000–$49,999

Over $70,000

Food

15.3

14.3

11.8

Shelter

21.8

18.5

17.6

Utilities, fuel, and public services

10.2

7.7

5.4

Transportation

15.4

18.4

17.6

Health insurance

4.9

3.8

2.3

Other health-care expenses

4.4

2.9

2.4

Entertainment (including alcohol)

4.4

4.6

5.4

Tobacco

1.2

0.9

0.4

Education

2.5

1.1

2.6

Insurance and pensions

2.7

9.6

14.7

17.2

18.2

19.8

Expenditure Item

Other (apparel, personal care, other housing expenses, and misc.)

SOURCE: Consumer Expenditure Report, 2004, Bureau of Labor Statistics website: http://www.bls.gov.

Philip (1994) study how access to modern consumer goods may affect the share of income that individuals in transitional economies devote to traditional local items. They show that villagers of Papua, New Guinea, reduce such shares signiﬁcantly as outside goods become increasingly accessible. Hence, such improvements as better roads for moving goods provide one of the primary routes by which traditional cultural practices are undermined.

E4.3 CES utility In Chapter 3 we introduced the CES utility function xδ yδ þ (vi) δ δ for δ 1, δ 6¼ 0. The primary use of this function is to illustrate alternative substitution possibilities (as reflected in the value of the parameter δ). Budget shares implied by this utility function provide a U ðx, yÞ ¼

number of such insights. Manipulation of the firstorder conditions for a constrained utility maximum with the CES function yields the share equations sx ¼ 1=½1 þ ðpy =px ÞK , sy ¼ 1=½1 þ ðpx =py ÞK ,

(vii)

where K ¼ δ=ðδ 1Þ. The homothetic nature of the CES function is shown by the fact that these share expressions depend only on the price ratio, px =py . Behavior of the shares in response to changes in relative prices depends on the value of the parameter K . For the Cobb-Douglas case, δ ¼ 0 and so K ¼ 0 and sx ¼ sy ¼ 1=2. When δ > 0; substitution possibilities are great and K < 0. In this case, Equation vii shows that sx and px =py move in opposite directions. If px =py rises, the individual substitutes y for x to such an extent that sx falls. Alternatively, if δ < 0, then substitution possibilities are limited, K > 0, and sx and px =py move in the same

Chapter 4

direction. In this case, an increase in px =py causes only minor substitution of y for x, and sx actually rises because of the relatively higher price of good x. North American free trade CES demand functions are most often used in largescale computer models of general equilibrium (see Chapter 13) that economists use to evaluate the impact of major economic changes. Because the CES model stresses that shares respond to changes in relative prices, it is particularly appropriate for looking at innovations such as changes in tax policy or in international trade restrictions, where changes in relative prices are quite likely. One important recent area of such research has been on the impact of the North American Free Trade Agreement for Canada, Mexico, and the United States. In general, these models ﬁnd that all of the countries involved might be expected to gain from the agreement, but that Mexico’s gains may be the greatest because it is experiencing the greatest change in relative prices. Kehoe and Kehoe (1995) present a number of computable equilibrium models that economists have used in these examinations.1

Utility Maximization and Choice

139

This form approximates any expenditure function. For the function to be homogeneous of degree 1 in the prices, the parameters of the function must obey the constraints a1 þ a2 ¼ 1, b1 þ b2 ¼ 0, b2 þ b3 ¼ 0, and c1 þ c2 ¼ 0. Using the results of Equation viii shows that, for this function, c

c

c

c

sx ¼ a1 þ b1 ln px þ b2 ln py þ c1Vc0 p x1 p y2 , sy ¼ a2 þ b2 ln px þ b3 ln py þ c2Vc0 p x1 p y2 ⋅

(x)

Notice that, given the parameter restrictions, sx þ sy ¼ 1. Making use of the inverse relationship between indirect utility and expenditure functions and some additional algebraic manipulation will put these budget share equations into a simple form suitable for econometric estimation: sx ¼ a1 þ b1 ln px þ b2 ln py þ c1 ðE=pÞ, sy ¼ a2 þ b2 ln px þ b3 ln py þ c2 ðE=pÞ,

(xi)

where p is an index of prices defined by ln p ¼ a0 þ a1 ln px þ a2 ln py þ 0:5b1 ðln px Þ2 þ b2 ln px ln py þ 0:5b3 ðln py Þ2 . (xii)

E4.4 The almost ideal demand system An alternative way to study budget shares is to start from a speciﬁc expenditure function. This approach is especially convenient because the envelope theorem shows that budget shares can be derived directly from expenditure functions through logarithmic differentiation: ∂ lnEðpx ,py ,V Þ

1 ∂E ∂px ⋅ ⋅ ∂ ln px Eð px ,py ,V Þ ∂px ∂ ln px xp (viii) ¼ x ¼ sx . E Deaton and Muellbauer (1980) make extensive use of this relationship to study the characteristics of a particular class of expenditure functions that they term an almost ideal demand system (AIDS). Their expenditure function takes the form ln Eðpx , py , V Þ ¼ a0 þ a1 ln px þ a2 ln py ¼

þ 0:5b1 ðln px Þ2 þ b2 ln px ln py c

c

þ 0:5b3 ðln py Þ2 þ Vc0 p x1 p y2 . (ix)

1

The research on the North American Free Trade Agreement is discussed in more detail in the Extensions to Chapter 13

In other words, the AIDS share equations state that budget shares are linear in the logarithms of prices and in total real expenditures. In practice, simpler price indices are often substituted for the rather complex index given by Equation xii, although there is some controversy about this practice (see the Extensions to Chapter 5). British expenditure patterns Deaton and Muellbauer apply this demand system to the study of British expenditure patterns between 1954 and 1974. They ﬁnd that both food and housing have negative coefﬁcients of real expenditures, implying that the share of income devoted to these items falls (at least in Britain) as people get richer. The authors also ﬁnd signiﬁcant relative price effects in many of their share equations, and prices have especially large effects in explaining the share of expenditures devoted to transportation and communication. In applying the AIDS model to real-world data, the authors also encounter a variety of econometric difﬁculties, the most important of which is that many of the equations do not appear to obey the restrictions necessary for homogeneity. Addressing such issues has been a major topic for further research on this demand system.

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References Behrman, Jere R. “Is Variety the Spice of Life? Implications for Caloric Intake.” Review of Economics and Statistics (November 1989): 666–72. Deaton, Angus, and John Muellbauer. “An Almost Ideal Demand System.” American Economic Review (June 1980): 312–26. Hyashi, Fumio. “Is the Japanese Extended Family Altruistically Linked? A Test Based on Engel Curves.” Journal of Political Economy (June 1995): 661–74.

Kehoe, Patrick J., and Timothy J. Kehoe. Modeling North American Economic Integration. London: Kluwer Academic Publishers, 1995. Oczkowski, E., and N. E. Philip. “Household Expenditure Patterns and Access to Consumer Goods in a Transitional Economy.” Journal of Economic Development (June 1994): 165–83. Stone, R. “Linear Expenditure Systems and Demand Analysis.” Economic Journal (September 1954): 511–27.

CHAPTER

5 Income and Substitution Effects In this chapter we will use the utility-maximization model to study how the quantity of a good that an individual chooses is affected by a change in that good’s price. This examination allows us to construct the individual’s demand curve for the good. In the process we will provide a number of insights into the nature of this price response and into the kinds of assumptions that lie behind most analyses of demand.

DEMAND FUNCTIONS As we pointed out in Chapter 4, in principle it will usually be possible to solve the necessary conditions of a utility maximum for the optimal levels of x1 , x2 , …, xn (and λ, the Lagrangian multiplier) as functions of all prices and income. Mathematically, this can be expressed as n demand functions of the form x 1 ¼ x1 ðp1 , p2 , …, pn , I Þ, x 2 ¼ x2 ðp1 , p2 , …, pn , I Þ, (5.1) .. . x n ¼ xn ð p1 , p2 , …, pn , I Þ. If there are only two goods, x and y (the case we will usually be concerned with), this notation can be simplified a bit as x ¼ xðpx , py , I Þ, (5.2) y ¼ yð px , py , I Þ. Once we know the form of these demand functions and the values of all prices and income, we can “predict” how much of each good this person will choose to buy. The notation stresses that prices and income are “exogenous” to this process; that is, these are parameters over which the individual has no control at this stage of the analysis. Changes in the parameters will, of course, shift the budget constraint and cause this person to make different choices. That question is the focus of this chapter and the next. Specifically, in this chapter we will be looking at the partial derivatives ∂x=∂I and ∂x=∂px for any arbitrary good x. Chapter 6 will carry the discussion further by looking at “cross-price” effects of the form ∂x=∂py for any arbitrary pair of goods x and y.

Homogeneity A ﬁrst property of demand functions requires little mathematics. If we were to double all prices and income (indeed, if we were to multiply them all by any positive constant), then the optimal quantities demanded would remain unchanged. Doubling all prices and income changes only the units by which we count, not the “real” quantity of goods demanded. This 141

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result can be seen in a number of ways, although perhaps the easiest is through a graphic approach. Referring back to Figures 4.1 and 4.2, it is clear that doubling px , py , and I does not affect the graph of the budget constraint. Hence, x , y will still be the combination that is chosen. Further, px x þ py y ¼ I is the same constraint as 2px x þ 2py y ¼ 2I . Somewhat more technically, we can write this result as saying that, for any good xi , (5.3) x ¼ x ð p , p , …, p , I Þ ¼ x ðtp , tp , …, tp , tI Þ i

i

1

2

n

i

1

2

n

for any t > 0. Functions that obey the property illustrated in Equation 5.3 are said to be homogeneous of degree 0.1 Hence, we have shown that individual demand functions are homogeneous of degree 0 in all prices and income. Changing all prices and income in the same proportions will not affect the physical quantities of goods demanded. This result shows that (in theory) individuals’ demands will not be affected by a “pure” inflation during which all prices and incomes rise proportionally. They will continue to demand the same bundle of goods. Of course, if an inflation were not pure (that is, if some prices rose more rapidly than others), this would not be the case.

EXAMPLE 5.1 Homogeneity Homogeneity of demand is a direct result of the utility-maximization assumption. Demand functions derived from utility maximization will be homogeneous and, conversely, demand functions that are not homogeneous cannot reﬂect utility maximization (unless prices enter directly into the utility function itself, as they might for goods with snob appeal). If, for example, an individual’s utility for food ðxÞ and housing ðyÞ is given by utility ¼ U ðx, yÞ ¼ x 0:3 y 0:7 ,

(5.4)

then it is a simple matter (following the procedure used in Example 4.1) to derive the demand functions 0:3I , x ¼ px (5.5) 0:7I . y ¼ py These functions obviously exhibit homogeneity, since a doubling of all prices and income would leave x and y unaffected. If the individual’s preferences for x and y were reﬂected instead by the CES function U ðx, yÞ ¼ x 0:5 þ y 0:5 , then (as shown in Example 4.2) the demand functions are given by ! 1 I x ¼ ⋅ , 1 þ px =py px ! 1 I y ¼ ⋅ . 1 þ py =px py

(5.6)

(5.7)

As before, both of these demand functions are homogeneous of degree 0; a doubling of px , py , and I would leave x and y unaffected.

More generally, as we saw in Chapters 2 and 4, a function f ð x1 , x2 , …, xn Þ is said to be homogeneous of degree k if f ðtx1 , tx2 , …, txn Þ ¼ t k f ð x1 , x2 , …, xn Þ for any t > 0. The most common cases of homogeneous functions are k ¼ 0 and k ¼ 1. If f is homogeneous of degree 0, then doubling all of its arguments leaves f unchanged in value. If f is homogeneous of degree 1, then doubling all of its arguments will double the value of f . 1

Chapter 5 Income and Substitution Effects

QUERY: Do the demand functions derived in this example ensure that total spending on x and y will exhaust the individual’s income for any combination of px , py , and I ? Can you prove that this is the case?

CHANGES IN INCOME As a person’s purchasing power rises, it is natural to expect that the quantity of each good purchased will also increase. This situation is illustrated in Figure 5.1. As expenditures increase from I1 to I2 to I3 , the quantity of x demanded increases from x1 to x2 to x3 . Also, the quantity of y increases from y1 to y2 to y3 . Notice that the budget lines I1 , I2 , and I3 are all parallel, reﬂecting that only income is changing, not the relative prices of x and y. Because the ratio px =py stays constant, the utility-maximizing conditions also require that the MRS stay constant as the individual moves to higher levels of satisfaction. The MRS is therefore the same at point (x3 , y3 ) as at (x1 , y1 ).

Normal and inferior goods In Figure 5.1, both x and y increase as income increases—both ∂x=∂I and ∂y=∂I are positive. This might be considered the usual situation, and goods that have this property are called normal goods over the range of income change being observed.

FIGURE 5.1

Effect of an Increase in Income on the Quantities of x and y Chosen

As income increases from I1 to I2 to I3 , the optimal (utility-maximizing) choices of x and y are shown by the successively higher points of tangency. Observe that the budget constraint shifts in a parallel way because its slope (given by −px =py ) does not change. Quantity of y U1

U2

U3

y3 y2

U3 I3

y1 U2

I2 I1 x1

x2

U1 x3

Quantity of x

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Part 2 Choice and Demand

FIGURE 5.2

An Indifference Curve Map Exhibiting Inferiority In this diagram, good z is inferior because the quantity purchased actually declines as income increases. Here, y is a normal good (as it must be if there are only two goods available), and purchases of y increase as total expenditures increase.

Quantity of y

y3 U3 y2 U2 y1 I1 z 3 z 2 z1

I2

I3

U1

Quantity of z

For some goods, however, the quantity chosen may decrease as income increases in some ranges. Examples of such goods are rotgut whiskey, potatoes, and secondhand clothing. A good z for which ∂z=∂I is negative is called an inferior good. This phenomenon is illustrated in Figure 5.2. In this diagram, the good z is inferior because, for increases in income in the range shown, less of z is actually chosen. Notice that indifference curves do not have to be “oddly” shaped in order to exhibit inferiority; the curves corresponding to goods y and z in Figure 5.2 continue to obey the assumption of a diminishing MRS. Good z is inferior because of the way it relates to the other goods available (good y here), not because of a peculiarity unique to it. Hence, we have developed the following deﬁnitions. DEFINITION

Inferior and normal goods. A good xi for which ∂xi =∂I < 0 over some range of income changes is an inferior good in that range. If ∂xi =∂I 0 over some range of income variation then the good is a normal (or “noninferior”) good in that range.

CHANGES IN A GOOD’S PRICE The effect of a price change on the quantity of a good demanded is more complex to analyze than is the effect of a change in income. Geometrically, this is because changing a price involves changing not only the intercepts of the budget constraint but also its slope. Consequently, moving to the new utility-maximizing choice entails not only moving to another indifference curve but also changing the MRS. Therefore, when a price changes, two analytically different effects come into play. One of these is a substitution effect : even if

Chapter 5 Income and Substitution Effects

the individual were to stay on the same indifference curve, consumption patterns would be allocated so as to equate the MRS to the new price ratio. A second effect, the income effect, arises because a price change necessarily changes an individual’s “real” income. The individual cannot stay on the initial indifference curve and must move to a new one. We begin by analyzing these effects graphically. Then we will provide a mathematical development.

Graphical analysis of a fall in price Income and substitution effects are illustrated in Figure 5.3. This individual is initially maximizing utility (subject to total expenditures, I ) by consuming the combination x , y .

FIGURE 5.3 Demonstration of the Income and Substitution Effects of a Fall in the Price of x When the price of x falls from p1x to p2x , the utility-maximizing choice shifts from x , y to x , y . This movement can be broken down into two analytically different effects: ﬁrst, the substitution effect, involving a movement along the initial indifference curve to point B, where the MRS is equal to the new price ratio; and second, the income effect, entailing a movement to a higher level of utility because real income has increased. In the diagram, both the substitution and income effects cause more x to be bought when its price declines. Notice that point I =py is the same as before the price change; this is because py has not changed. Point I =py therefore appears on both the old and new budget constraints.

Quantity of y U1

U2

I py

I = px1x + pyy y** y*

I = p2x x + pyy

B

U2

U1 x*

xB

x**

Substitution Income effect effect Total increase in x

Quantity of x

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Part 2 Choice and Demand

The initial budget constraint is I ¼ p1x x þ py y. Now suppose that the price of x falls to p2x . The new budget constraint is given by the equation I ¼ p2x x þ py y in Figure 5.3. It is clear that the new position of maximum utility is at x , y , where the new budget line is tangent to the indifference curve U2 . The movement to this new point can be viewed as being composed of two effects. First, the change in the slope of the budget constraint would have motivated a move to point B, even if choices had been conﬁned to those on the original indifference curve U1 . The dashed line in Figure 5.3 has the same slope as the new budget constraint (I ¼ p2x x þ py y) but is drawn to be tangent to U1 because we are conceptually holding “real” income (that is, utility) constant. A relatively lower price for x causes a move from x , y to B if we do not allow this individual to be made better off as a result of the lower price. This movement is a graphic demonstration of the substitution effect. The further move from B to the optimal point x , y is analytically identical to the kind of change exhibited earlier for changes in income. Because the price of x has fallen, this person has a greater “real” income and can afford a utility level (U2 ) that is greater than that which could previously be attained. If x is a normal good, more of it will be chosen in response to this increase in purchasing power. This observation explains the origin of the term income effect for the movement. Overall then, the result of the price decline is to cause more x to be demanded. It is important to recognize that this person does not actually make a series of choices from x , y to B and then to x , y . We never observe point B; only the two optimal positions are reﬂected in observed behavior. However, the notion of income and substitution effects is analytically valuable because it shows that a price change affects the quantity of x that is demanded in two conceptually different ways. We will see how this separation offers major insights in the theory of demand.

Graphical analysis of an increase in price If the price of good x were to increase, a similar analysis would be used. In Figure 5.4, the budget line has been shifted inward because of an increase in the price of x from p1x to p2x . The movement from the initial point of utility maximization (x , y ) to the new point (x , y ) can be decomposed into two effects. First, even if this person could stay on the initial indifference curve (U2 ), there would still be an incentive to substitute y for x and move along U2 to point B. However, because purchasing power has been reduced by the rise in the price of x, he or she must move to a lower level of utility. This movement is again called the income effect. Notice in Figure 5.4 that both the income and substitution effects work in the same direction and cause the quantity of x demanded to be reduced in response to an increase in its price.

Effects of price changes for inferior goods So far we have shown that substitution and income effects tend to reinforce one another. For a price decline, both cause more of the good to be demanded, whereas for a price increase, both cause less to be demanded. Although this analysis is accurate for the case of normal (noninferior) goods, the possibility of inferior goods complicates the story. In this case, income and substitution effects work in opposite directions, and the combined result of a price change is indeterminate. A fall in price, for example, will always cause an individual to tend to consume more of a good because of the substitution effect. But if the good is inferior, the increase in purchasing power caused by the price decline may cause less of the good to be bought. The result is therefore indeterminate: the substitution effect tends to increase the quantity of the inferior good bought, whereas the (perverse) income effect tends to reduce this quantity. Unlike the situation for normal goods, it is not possible here to predict even the direction of the effect of a change in px on the quantity of x consumed.

Chapter 5 Income and Substitution Effects

FIGURE 5.4 Demonstration of the Income and Substitution Effects of an Increase in the Price of x When the price of x increases, the budget constraint shifts inward. The movement from the initial utility-maximizing point (x , y ) to the new point (x , y ) can be analyzed as two separate effects. The substitution effect would be depicted as a movement to point B on the initial indifference curve (U2 ). The price increase, however, would create a loss of purchasing power and a consequent movement to a lower indifference curve. This is the income effect. In the diagram, both the income and substitution effects cause the quantity of x to fall as a result of the increase in its price. Again, the point I =py is not affected by the change in the price of x. Quantity of y I py

U1

U2

B

y**

I = px2 x + pyy

y*

I = px1x + pyy

U2 U1 x**

xB

x*

Quantity of x

Income Substitution effect effect Total reduction in x

Giffen’s paradox If the income effect of a price change is strong enough, the change in price and the resulting change in the quantity demanded could actually move in the same direction. Legend has it that the English economist Robert Giffen observed this paradox in nineteenth-century Ireland: when the price of potatoes rose, people reportedly consumed more of them. This peculiar result can be explained by looking at the size of the income effect of a change in the price of potatoes. Potatoes were not only inferior goods, they also used up a large portion of the Irish people’s income. An increase in the price of potatoes therefore reduced real income substantially. The Irish were forced to cut back on other luxury food consumption in order to buy more potatoes. Even though this rendering of events is historically implausible, the

147

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possibility of an increase in the quantity demanded in response to an increase in the price of a good has come to be known as Giffen’s paradox.2 Later we will provide a mathematical analysis of how Giffen’s paradox can occur.

A summary Hence, our graphical analysis leads to the following conclusions. OPTIMIZATION PRINCIPLE

Substitution and income effects. The utility-maximization hypothesis suggests that, for normal goods, a fall in the price of a good leads to an increase in quantity purchased because: (1) the substitution effect causes more to be purchased as the individual moves along an indifference curve; and (2) the income effect causes more to be purchased because the price decline has increased purchasing power, thereby permitting movement to a higher indifference curve. When the price of a normal good rises, similar reasoning predicts a decline in the quantity purchased. For inferior goods, substitution and income effects work in opposite directions, and no deﬁnite predictions can be made.

THE INDIVIDUAL’S DEMAND CURVE Economists frequently wish to graph demand functions. It will come as no surprise to you that these graphs are called “demand curves.” Understanding how such widely used curves relate to underlying demand functions provides additional insights to even the most fundamental of economic arguments. To simplify the development, assume there are only two goods and that, as before, the demand function for good x is given by x ¼ xð p , p , I Þ. x

y

The demand curve_ derived from this function looks at the relationship between x and px while holding py , I , and preferences constant. That is, it shows the relationship _ _ x ¼ xð px , p y , I Þ, (5.8) where the bars over py and I indicate that these determinants of demand are being held constant. This construction is shown in Figure 5.5. The graph shows utility-maximizing choices of x and y as this individual is presented with successively lower prices of good x (while holding py and I constant). We assume that the quantities of x chosen increase from x 0 to x 00 to x 000 as that good’s price falls from px0 to px00 to px000 . Such an assumption is in accord with our general conclusion that, except in the unusual case of Giffen’s paradox, ∂x=∂px is negative. In Figure 5.5b, information about the utility-maximizing choices of good x is transferred to a demand curve with px on the vertical axis and sharing the same horizontal axis as Figure 5.5a. The negative slope of the curve again reﬂects the assumption that ∂x=∂px is negative. Hence, we may deﬁne an individual demand curve as follows. DEFINITION

Individual demand curve. An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual, assuming that all other determinants of demand are held constant.

2

A major problem with this explanation is that it disregards Marshall’s observation that both supply and demand factors must be taken into account when analyzing price changes. If potato prices increased because of the potato blight in Ireland, then supply should have become smaller, so how could more potatoes possibly have been consumed? Also, since many Irish people were potato farmers, the potato price increase should have increased real income for them. For a detailed discussion of these and other fascinating bits of potato lore, see G. P. Dwyer and C. M. Lindsey, “Robert Giffen and the Irish Potato,” American Economic Review (March 1984): 188–92.

FIGURE 5.5

Construction of an Individual’s Demand Curve

In (a), the individual’s utility-maximizing choices of x and y are shown for three different prices of x (px0 , px00 , and px000 ). In (b), this relationship between px and x is used to construct the demand curve for x. The demand curve is drawn on the assumption that py , I , and preferences remain constant as px varies. Quantity of y per period I /py I = p x′ x + p y y I = p x″ x + p y y I = p x″‴ x + p y y U3 U2 U1 x′

x″

x‴

Quantity of x per period

(a) Individual’s indifference curve map

px

p x′ p x″ p x‴

x( p x, p y, I)

x′ (b) Demand curve

x″

x‴

Quantity of x per period

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Part 2 Choice and Demand

The demand curve illustrated in Figure 5.5 stays in a ﬁxed position only so long as all other determinants of demand remain unchanged. If one of these other factors were to change then the curve might shift to a new position, as we now describe.

Shifts in the demand curve Three factors were held constant in deriving this demand curve: (1) income; (2) prices of other goods (say, py ); and (3) the individual’s preferences. If any of these were to change, the entire demand curve might shift to a new position. For example, if I were to increase, the curve would shift outward (provided that ∂x=∂I > 0, that is, provided the good is a “normal” good over this income range). More x would be demanded at each price. If another price (say, py ) were to change then the curve would shift inward or outward, depending precisely on how x and y are related. In the next chapter we will examine that relationship in detail. Finally, the curve would shift if the individual’s preferences for good x were to change. A sudden advertising blitz by the McDonald’s Corporation might shift the demand for hamburgers outward, for example. As this discussion makes clear, one must remember that the demand curve is only a twodimensional representation of the true demand function (Equation 5.8) and that it is stable only if other things do stay constant. It is important to keep clearly in mind the difference between a movement along a given demand curve caused by a change in px and a shift in the entire curve caused by a change in income, in one of the other prices, or in preferences. Traditionally, the term an increase in demand is reserved for an outward shift in the demand curve, whereas the term an increase in the quantity demanded refers to a movement along a given curve caused by a change in px .

EXAMPLE 5.2 Demand Functions and Demand Curves To be able to graph a demand curve from a given demand function, we must assume that the preferences that generated the function remain stable and that we know the values of income and other relevant prices. In the ﬁrst case studied in Example 5.1, we found that 0:3I (5.9) x¼ px and y¼

0:7I . py

If preferences do not change and if this individual’s income is $100, these functions become 30 , x¼ px (5.10) 70 y ¼ , py or px x ¼ 30, py y ¼ 70, which makes clear that the demand curves for these two goods are simple hyperbolas. A rise in income would shift both of the demand curves outward. Notice also, in this case, that the demand curve for x is not shifted by changes in py and vice versa.

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For the second case examined in Example 5.1, the analysis is more complex. For good x, we know that ! 1 I (5.11) x¼ ⋅ , 1 þ px =py px so to graph this in the px –x plane we must know both I and py . If we again assume I ¼ 100 and let py ¼ 1, then Equation 5.11 becomes 100 , (5.12) x¼ 2 p x þ px which, when graphed, would also show a general hyperbolic relationship between price and quantity consumed. In this case the curve would be relatively flatter because substitution effects are larger than in the Cobb-Douglas case. From Equation 5.11, we also know that ! ∂x 1 1 >0 (5.13) ¼ ⋅ ∂I 1 þ px =py px and ∂x I ¼ > 0, ∂py ð px þ py Þ2 so increases in I or py would shift the demand curve for good x outward. QUERY: How would the demand functions in Equations 5.10 change if this person spent half of his or her income on each good? Show that these demand functions predict the same x consumption at the point px ¼ 1, py ¼ 1, I ¼ 100 as does Equation 5.11. Use a numerical example to show that the CES demand function is more responsive to an increase in px than is the Cobb-Douglas demand function.

COMPENSATED DEMAND CURVES In Figure 5.5, the level of utility this person gets varies along the demand curve. As px falls, he or she is made increasingly better-off, as shown by the increase in utility from U1 to U2 to U3 . The reason this happens is that the demand curve is drawn on the assumption that nominal income and other prices are held constant; hence, a decline in px makes this person better off by increasing his or her real purchasing power. Although this is the most common way to impose the ceteris paribus assumption in developing a demand curve, it is not the only way. An alternative approach holds real income (or utility) constant while examining reactions to changes in px . The derivation is illustrated in Figure 5.6, where we hold utility constant (at U2 ) while successively reducing px . As px falls, the individual’s nominal income is effectively reduced, thus preventing any increase in utility. In other words, the effects of the price change on purchasing power are “compensated” so as to constrain the individual to remain on U2 . Reactions to changing prices include only substitution effects. If we were instead to examine effects of increases in px , income compensation would be positive: This individual’s income would have to be increased to permit him or her to stay on the U2 indifference curve in response to the price rises. We can summarize these results as follows. Compensated demand curve. A compensated demand curve shows the relationship beDEFINITION tween the price of a good and the quantity purchased on the assumption that other prices and utility are held constant. The curve (which is sometimes termed a “Hicksian” demand curve

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FIGURE 5.6

Construction of a Compensated Demand Curve The curve x c shows how the quantity of x demanded changes when px changes, holding py and utility constant. That is, the individual’s income is “compensated” so as to keep utility constant. Hence, x c reﬂects only substitution effects of changing prices.

Quantity of y Slope = –

p x′ py

Slope = –

p x″ py Slope = –

p x‴ py

U2 x*

x″

x‴

Quantity of x

(a) Individual’s indifference curve map px p x′ p x″ p x‴ x c ( p x ,p y,U)

x*

x″

x **

Quantity of x

(b) Compensated demand curve

after the British economist John Hicks) therefore illustrates only substitution effects. Mathematically, the curve is a two-dimensional representation of the compensated demand function x ¼ x c ð px , py , U Þ.

(5.14)

Relationship between compensated and uncompensated demand curves This relationship between the two demand curve concepts is illustrated in Figure 5.7. At px00 the curves intersect, because at that price the individual’s income is just sufﬁcient to attain

Chapter 5 Income and Substitution Effects

FIGURE 5.7

Comparison of Compensated and Uncompensated Demand Curves

The compensated (x c ) and uncompensated (x) demand curves intersect at px00 because x 00 is demanded under each concept. For prices above px00 , the individual’s income is increased with the compensated demand curve, so more x is demanded than with the uncompensated curve. For prices below px00 , income is reduced for the compensated curve, so less x is demanded than with the uncompensated curve. The standard demand curve is ﬂatter because it incorporates both substitution and income effects whereas the curve x c reﬂects only substitution effects. px

p x′ p x″ p x‴ x( p x ,p y,I) x c ( p x ,p y,U) x′

x*

x″

x**

x‴

Quantity of x

utility level U2 (compare Figures 5.5 and Figure 5.6). Hence, x 00 is demanded under either demand concept. For prices below px00 , however, the individual suffers a compensating reduction in income on the curve x c that prevents an increase in utility from the lower price. Hence, assuming x is a normal good, it follows that less x is demanded at px000 along x c than along the uncompensated curve x. Alternatively, for a price above px00 (such as px0 ), income compensation is positive because the individual needs some help to remain on U2 . Hence, again assuming x is a normal good, at px0 more x is demanded along x c than along x. In general, then, for a normal good the compensated demand curve is somewhat less responsive to price changes than is the uncompensated curve. This is because the latter reﬂects both substitution and income effects of price changes, whereas the compensated curve reﬂects only substitution effects. The choice between using compensated or uncompensated demand curves in economic analysis is largely a matter of convenience. In most empirical work, uncompensated curves (which are sometimes called “Marshallian demand curves”) are used because the data on prices and nominal incomes needed to estimate them are readily available. In the Extensions to Chapter 12 we will describe some of these estimates and show how they might be employed for practical policy purposes. For some theoretical purposes, however, compensated demand curves are a more appropriate concept because the ability to hold utility constant offers some advantages. Our discussion of “consumer surplus” later in this chapter offers one illustration of these advantages.

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EXAMPLE 5.3 Compensated Demand Functions In Example 3.1 we assumed that the utility function for hamburgers (y) and soft drinks (x) was given by utility ¼ U ðx, yÞ ¼ x 0:5 y 0:5 ,

(5.15)

and in Example 4.1 we showed that we can calculate the Marshallian demand functions for such utility functions as αI I ¼ , x ¼ px 2px (5.16) βI I y ¼ ¼ . py 2py Also, in Example 4.3 we calculated the indirect utility function by combining Equations 5.15 and 5.16 as I . (5.17) utility ¼ V ðI , px , py Þ ¼ 0:5 2p x p 0:5 y To obtain the compensated demand functions for x and y, we simply use Equation 5.17 to solve for I and then substitute this expression involving V into Equations 5.16. This permits us to interchange income and utility so we may hold the latter constant, as is required for the compensated demand concept. Making these substitutions yields x¼

Vp 0:5 y p 0:5 x

,

Vp 0:5 x y ¼ 0:5 . py

(5.18)

These are the compensated demand functions for x and y. Notice that now demand depends on utility (V ) rather than on income. Holding utility constant, it is clear that increases in px reduce the demand for x, and this now reflects only the substitution effect (see also Example 5.4). Although py did not enter into the uncompensated demand function for good x, it does enter into the compensated function: increases in py shift the compensated demand curve for x outward. The two demand concepts agree at the assumed initial point px ¼ 1, py ¼ 4, I ¼ 8, and V ¼ 2; Equations 5.16 predict x ¼ 4, y ¼ 1 at this point, as do Equations 5.18. For px > 1 or px < 1, the demands differ under the two concepts, however. If, say, px ¼ 4, then the uncompensated functions (Equations 5.16) predict x ¼ 1, y ¼ 1, whereas the compensated functions (Equations 5.18) predict x ¼ 2, y ¼ 2. The reduction in x resulting from the rise in its price is smaller with the compensated demand function than it is with the uncompensated function because the former concept adjusts for the negative effect on purchasing power that comes about from the price rise. This example makes clear the different ceteris paribus assumptions inherent in the two demand concepts. With uncompensated demand, expenditures are held constant at I ¼ 2 and so the rise in px from 1 to 4 results in a loss of utility; in this case, utility falls from 2 to 1. In the compensated demand case, utility is held constant at V ¼ 2. To keep utility constant, expenditures must rise to E ¼ 1ð2Þ þ 1ð2Þ ¼ 4 in order to offset the effects of the price rise (see Equation 5.17). QUERY: Are the compensated demand functions given in Equations 5.18 homogeneous of degree 0 in px and py if utility is held constant? Would you expect that to be true for all compensated demand functions?

Chapter 5 Income and Substitution Effects

A MATHEMATICAL DEVELOPMENT OF RESPONSE TO PRICE CHANGES Up to this point we have largely relied on graphical devices to describe how individuals respond to price changes. Additional insights are provided by a more mathematical approach. Our basic goal is to examine the partial derivative ∂x=∂px —that is, how a change in the price of a good affects its purchase, ceteris paribus. In the next chapter, we take up the question of how changes in the price of one commodity affect purchases of another commodity.

Direct approach Our goal is to use the utility-maximization model to learn something about how the demand for good x changes when px changes; that is, we wish to calculate ∂x=∂px . The direct approach to this problem makes use of the ﬁrst-order conditions for utility maximization (Equations 4.8). Differentiation of these n þ 1 equations yields a new system of n þ 1 equations, which eventually can be solved for the derivative we seek.3 Unfortunately, obtaining this solution is quite cumbersome and the steps required yield little in the way of economic insights. Hence, we will instead adopt an indirect approach that relies on the concept of duality. In the end, both approaches yield the same conclusion, but the indirect approach is much richer in terms of the economics it contains.

Indirect approach To begin our indirect approach,4 we will assume (as before) there are only two goods (x and y) and focus on the compensated demand function, x c ð px , py , U Þ, introduced in Equation 5.14. We now wish to illustrate the connection between this demand function and the ordinary demand function, xð px , py , I Þ. In Chapter 4 we introduced the expenditure function, which records the minimal expenditure necessary to attain a given utility level. If we denote this function by minimum expenditure ¼ Eðpx , py , U Þ

(5.19)

x c ð px , py , U Þ ¼ x½ px , py , Eðpx , py ,U Þ.

(5.20)

then, by definition, This conclusion was already introduced in connection with Figure 5.7, which showed that the quantity demanded is identical for the compensated and uncompensated demand functions when income is exactly what is needed to attain the required utility level. Equation 5.20 is obtained by inserting that expenditure level into the demand function, xðpx , py , I Þ. Now we can proceed by partially differentiating Equation 5.20 with respect to px and recognizing that this variable enters into the ordinary demand function in two places. Hence ∂x c ∂x ∂x ∂E ¼ þ , (5.21) ⋅ ∂px ∂px ∂E ∂px and rearranging terms yields ∂x ∂x c ∂x ∂E ¼ . ⋅ ∂px ∂px ∂E ∂px

(5.22)

3

See, for example, Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947), pp. 101–3. The following proof is adapted from Phillip J. Cook, “A ‘One Line’ Proof of the Slutsky Equation,” American Economic Review 62 (March 1972): 139.

4

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The substitution effect Consequently, the derivative we seek has two terms. Interpretation of the ﬁrst term is straightforward: It is the slope of the compensated demand curve. But that slope represents movement along a single indifference curve; it is, in fact, what we called the “substitution effect” earlier. The ﬁrst term on the right of Equation 5.22 is a mathematical representation of that effect.

The income effect The second term in Equation 5.22 reﬂects the way in which changes in px affect the demand for x through changes in necessary expenditure levels (that is, changes in purchasing power). This term therefore reﬂects the income effect. The negative sign in Equation 5.22 shows the direction of the effect. For example, an increase in px increases the expenditure level that would have been needed to keep utility constant (mathematically, ∂E=∂px > 0). But because nominal income is held constant in Marshallian demand, these extra expenditures are not available. Hence x (and y) must be reduced to meet this shortfall. The extent of the reduction in x is given by ∂x=∂E. On the other hand, if px falls, the expenditure level required to attain a given utility also falls. The decline in x that would normally accompany such a fall in expenditures is precisely the amount that must be added back through the income effect. Notice that in this case the income effect works to increase x.

The Slutsky equation The relationships embodied in Equation 5.22 were ﬁrst discovered by the Russian economist Eugen Slutsky in the late nineteenth century. A slight change in notation is required to state the result the way Slutsky did. First, we write the substitution effect as ∂x c ∂x ¼ (5.23) substitution effect ¼ ∂px ∂px U ¼constant to indicate movement along a single indifference curve. For the income effect, we have income effect ¼

∂x ∂E ∂x ∂E ¼ , ⋅ ⋅ ∂E ∂px ∂I ∂px

(5.24)

because changes in income or expenditures amount to the same thing in the function xð px , py , I ). The second term in the income effect can be studied most directly by using the envelope theorem. Remember that expenditure functions represent a minimization problem in which the expenditure required to reach a minimum level of utility is minimized. The Lagrangian _ expression for this minimization is ℒ ¼ px x þ py y þ λ½ U U ðx, yÞ. Applying the envelope theorem to this problem yields ∂E ∂ℒ ¼ ¼ x. (5.25) ∂px ∂px In words, the envelope theorem shows that partial differentiation of the expenditure function with respect to a good’s price yields the demand function for that good. Because utility is held constant in the expenditure function, this demand function will be a compensated one. This result, and a similar one in the theory of the firm, is usually called Shephard’s lemma after the economist who first studied this approach to demand theory in detail. The result is extremely useful in both theoretical and applied microeconomics; partial differentiation of maximized or minimized functions is often the easiest way to derive demand

Chapter 5 Income and Substitution Effects

157

functions.5 Notice also that the result makes intuitive sense. If we ask how much extra expenditure is necessary to compensate for a rise in the price of good x, a simple approximation would be given by the number of units of x currently being consumed. By combining Equations 5.23–5.25, we can arrive at the following complete statement of the response to a price change. Slutsky equation. The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by

or

∂x ¼ substitution effect þ income effect, ∂px

(5.26)

∂x ∂x ∂x ¼ x . ∂px ∂px U ¼constant ∂I

(5.27)

The Slutsky equation allows a more deﬁnitive treatment of the direction and size of substitution and income effects than was possible with a graphic analysis. First, the substitution effect ð∂x=∂px jU ¼constant Þ is always negative as long as the MRS is diminishing. A fall (rise) in px reduces (increases) px =py , and utility maximization requires that the MRS fall (rise) too. But this can occur along an indifference curve only if x increases (or, in the case of a rise in px , if x decreases). Hence, insofar as the substitution effect is concerned, price and quantity always move in opposite directions. Equivalently, the slope of the compensated demand curve must be negative.6 We will show this result in a somewhat different way in the ﬁnal section of this chapter. The sign of the income effect ( x∂x=∂I ) depends on the sign of ∂x=∂I . If x is a normal good, then ∂x=∂I is positive and the entire income effect, like the substitution effect, is negative. Thus, for normal goods, price and quantity always move in opposite directions. For example, a fall in px raises real income and, because x is a normal good, purchases of x rise. Similarly, a rise in px reduces real income and so purchases of x fall. Overall, then, as we described previously using a graphic analysis, substitution and income effects work in the same direction to yield a negatively sloped demand curve. In the case of an inferior good, ∂x=∂I < 0 and the two terms in Equation 5.27 would have different signs. It is at least theoretically possible that, in this case, the second term could dominate the ﬁrst, leading to Giffen’s paradox (∂x=∂px > 0). EXAMPLE 5.4 A Slutsky Decomposition The decomposition of a price effect that was ﬁrst discovered by Slutsky can be nicely illustrated with the Cobb-Douglas example studied previously. In Example 5.3, we found that the Marshallian demand function for good x was 0:5I (5.28) xðpx , py , I Þ ¼ px (continued)

5

For instance, in Example 4.4, for expenditure we found a simple Cobb-Douglas utility function of the form p0:5 Eð px , py , V Þ ¼ 2Vpx0:5 py0:5 . Hence, from Shephard’s lemma we know that x ¼ ∂E=∂px ¼ Vp0:5 x y , which is the same result we obtained in Example 5.3. 6 It is possible that substitution effects would be zero if indifference curves have an L-shape (implying that x and y are used in fixed proportions). Some examples are provided in the Chapter 5 problems.

OPTIMIZATION PRINCIPLE

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EXAMPLE 5.4 CONTINUED and that the Hicksian (compensated) demand function was x c ðpx , py , V Þ ¼

Vp 0:5 y p 0:5 x

.

(5.29)

The overall effect of a price change on the demand for good x can be found by differentiating the Marshallian demand function: ∂x 0:5I ¼ . (5.30) ∂px p 2x Now we wish to show that this effect is the sum of the two effects that Slutsky identified. As before, the substitution effect is found by differentiating the compensated demand function: ∂x c 0:5Vp y ¼ . ∂px p 1:5 x 0:5

substitution effect ¼

(5.31)

We can eliminate indirect utility, V , by substitution from Equation 5.17: substitution effect ¼

0:5ð0:5Ip 0:5 p y0:5 Þp 0:5 x y p 1:5 x

¼

0:25I . p 2x

(5.32)

Calculation of the income effect in this example is considerably easier. Applying the results from Equation 5.27, we have

∂x 0:5I 0:5 0:25I ¼ ¼ . (5.33) income effect ¼ x ⋅ ∂I px px p 2x A comparison of Equation 5.30 with Equations 5.32 and 5.33 shows that we have indeed decomposed the price derivative of this demand function into substitution and income components. Interestingly, the substitution and income effects are of precisely the same size. This, as we will see in later examples, is one of the reasons that the Cobb-Douglas is a very special case. The well-worn numerical example we have been using also demonstrates this decomposition. When the price of x rises from $1 to $4, the (uncompensated) demand for x falls from x ¼ 4 to x ¼ 1 but the compensated demand for x falls only from x ¼ 4 to x ¼ 2. That decline of 50 percent is the substitution effect. The further 50 percent fall from x ¼ 2 to x ¼ 1 represents reactions to the decline in purchasing power incorporated in the Marshallian demand function. This income effect does not occur when the compensated demand notion is used. QUERY: In this example, the individual spends half of his or her income on good x and half on good y. How would the relative sizes of the substitution and income effects be altered if the exponents of the Cobb-Douglas utility function were not equal?

DEMAND ELASTICITIES So far in this chapter we have been examining how individuals respond to changes in prices and income by looking at the derivatives of the demand function. For many analytical questions this is a good way to proceed because calculus methods can be directly applied. However, as we pointed out in Chapter 2, focusing on derivatives has one major disadvantage for empirical work: the sizes of derivatives depend directly on how variables are measured.

Chapter 5 Income and Substitution Effects

159

That can make comparisons among goods or across countries and time periods very difﬁcult. For this reason, most empirical work in microeconomics uses some form of elasticity measure. In this section we introduce the three most common types of demand elasticities and explore some of the mathematical relations among them. Again, for simplicity we will look at a situation where the individual chooses between only two goods, though these ideas can be easily generalized.

Marshallian demand elasticities Most of the commonly used demand elasticities are derived from the Marshallian demand function xðpx , py , I Þ. Speciﬁcally, the following deﬁnitions are used. 1. Price elasticity of demand ðex, px Þ. This measures the proportionate change in quantity DEFINITION demanded in response to a proportionate change in a good’s own price. Mathematically, ex, px ¼

∆x=x ∆x px ∂x px ¼ ¼ . ⋅ ⋅ ∆px =px ∆px x ∂px x

(5.34)

2. Income elasticity of demand ðex, I Þ. This measures the proportionate change in quantity demanded in response to a proportionate change in income. In mathematical terms, ex, I ¼

∆x=x ∆x I ∂x I ¼ ⋅ ¼ ⋅ . ∆I =I ∆I x ∂I x

(5.35)

3. Cross-price elasticity of demand ðex, py Þ. This measures the proportionate change in the quantity of x demanded in response to a proportionate change in the price of some other good (y): ∆x=x ∆x py ∂x py ¼ . (5.36) ¼ ex, py ¼ ⋅ ⋅ ∆py =py ∆py x ∂py x Notice that all of these deﬁnitions use partial derivatives, which signiﬁes that all other determinants of demand are to be held constant when examining the impact of a speciﬁc variable. In the remainder of this section we will explore the own-price elasticity deﬁnition in some detail. Examining the cross-price elasticity of demand is the primary topic of Chapter 6.

Price elasticity of demand The (own-) price elasticity of demand is probably the most important elasticity concept in all of microeconomics. Not only does it provide a convenient way of summarizing how people respond to price changes for a wide variety of economic goods, but it is also a central concept in the theory of how ﬁrms react to the demand curves facing them. As you probably already learned in earlier economics courses, a distinction is usually made between cases of elastic demand (where price affects quantity signiﬁcantly) and inelastic demand (where the effect of price is small). One mathematical complication in making these ideas precise is that the price elasticity of demand itself is negative7 because, except in the unlikely case of Giffen’s paradox, ∂x=∂px is negative. The dividing line between large and small responses is generally set

7

Sometimes economists use the absolute value of the price elasticity of demand in their discussions. Although this is mathematically incorrect, such usage is quite common. For example, a study that finds that ex, px ¼ 1:2 may sometimes report the price elasticity of demand as “1.2.” We will not do so here, however.

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at 1. If ex, px ¼ 1, changes in x and px are of the same proportionate size. That is, a 1 percent increase in price leads to a fall of 1 percent in quantity demanded. In this case, demand is said to be “unit-elastic.” Alternatively, if ex, px < 1, then quantity changes are proportionately larger than price changes and we say that demand is “elastic.” For example, if ex, px ¼ 3, each 1 percent rise in price leads to a fall of 3 percent in quantity demanded. Finally, if ex, px > 1 then demand is inelastic and quantity changes are proportionately smaller than price changes. A value of ex, px ¼ 0:3, for example, means that a 1 percent increase in price leads to a fall in quantity demanded of 0.3 percent. In Chapter 12 we will see how aggregate data are used to estimate the typical individual’s price elasticity of demand for a good and how such estimates are used in a variety of questions in applied microeconomics.

Price elasticity and total spending The price elasticity of demand determines how a change in price, ceteris paribus, affects total spending on a good. The connection is most easily shown with calculus: ∂ðpx ⋅ xÞ ∂x ¼ px ⋅ þ x ¼ xðex, px þ 1Þ. ∂px ∂px

(5.37)

So, the sign of this derivative depends on whether ex, px is larger or smaller than 1. If demand is inelastic (0 > ex, px > 1), the derivative is positive and price and total spending move in the same direction. Intuitively, if price does not affect quantity demanded very much, then quantity stays relatively constant as price changes and total spending reflects mainly those price movements. This is the case, for example, with the demand for most agricultural products. Weather-induced changes in price for specific crops usually cause total spending on those crops to move in the same direction. On the other hand, if demand is elastic (ex, px < 1), reactions to a price change are so large that the effect on total spending is reversed: a rise in price causes total spending to fall (because quantity falls a lot) and a fall in price causes total spending to rise (quantity increases significantly). For the unit-elastic case (ex, px ¼ 1), total spending is constant no matter how price changes.

Compensated price elasticities Because some microeconomic analyses focus on the compensated demand function, it is also useful to deﬁne elasticities based on that concept. Such deﬁnitions follow directly from their Marshallian counterparts. DEFINITION

Let the compensated demand function be given by x c ð px , py , U Þ. Then we have the following deﬁnitions. 1. Compensated own-price elasticity of demand (ex c , px ). This elasticity measures the proportionate compensated change in quantity demanded in response to a proportionate change in a good’s own price: ex c , px ¼

∆x c =x c ∆x c px ∂x c px ¼ ¼ . ⋅ ⋅ ∆px =px ∆px x c ∂px x c

(5.38)

2. Compensated cross-price elasticity of demand (ex c , px ). This measures the proportionate compensated change in quantity demanded in response to a proportionate change in the price of another good: ∆x c =x c ∆x c py ∂x c py ¼ . (5.39) e x c , py ¼ ⋅ c ¼ ⋅ ∆py =py ∆py x ∂py x c

Chapter 5 Income and Substitution Effects

Whether these price elasticities differ much from their Marshallian counterparts depends on the importance of income effects in the overall demand for good x. The precise connection between the two can be shown by multiplying the Slutsky result from Equation 5.27 by the factor px =x: px ∂x p ∂x c p ∂x ¼ ex, px ¼ x ⋅ x ⋅x ⋅ (5.40) ¼ ex c , px sx ex, I , ⋅ x ∂px x ∂px x ∂I where sx ¼ px x=I is the share of total income devoted to the purchase of good x. Equation 5.40 shows that compensated and uncompensated own-price elasticities of demand will be similar if either of two conditions hold: (1) The share of income devoted to good x ð sx Þ is small; or (2) the income elasticity of demand for good x ðex, I Þ is small. Either of these conditions serves to reduce the importance of the income compensation employed in the construction of the compensated demand function. If good x is unimportant in a person’s budget, then the amount of income compensation required to offset a price change will be small. Even if a good has a large budget share, if demand does not react strongly to changes in income then the results of either demand concept will be similar. Hence, there will be many circumstances where one can use the two price elasticity concepts more or less interchangeably. Put another way, there are many economic circumstances in which substitution effects constitute the most important component of price responses.

Relationships among demand elasticities There are a number of relationships among the elasticity concepts that have been developed in this section. All of these are derived from the underlying model of utility maximization. Here we look at three such relationships that provide further insight on the nature of individual demand. Homogeneity. The homogeneity of demand functions can also be expressed in elasticity terms. Because any proportional increase in all prices and income leaves quantity demanded unchanged, the net sum of all price elasticities together with the income elasticity for a particular good must sum to zero. A formal proof of this property relies on Euler’s theorem (see Chapter 2). Applying that theorem to the demand function xðpx , py , I Þ and remembering that this function is homogeneous of degree 0 yields ∂x ∂x ∂x þ py ⋅ þI⋅ 0 ¼ px ⋅ . (5.41) ∂px ∂py ∂I If we divide Equation 5.41 by x then we obtain 0 ¼ ex, px þ ex, py þ ex, I ,

(5.42)

as intuition suggests. This result shows that the elasticities of demand for any good cannot follow a completely flexible pattern. They must exhibit a sort of internal consistency that reflects the basic utility-maximizing approach on which the theory of demand is based. Engel aggregation. In the Extensions to Chapter 4 we discussed the empirical analysis of market shares and took special note of Engel’s law that the share of income devoted to food declines as income increases. From an elasticity perspective, Engel’s law is a statement of the empirical regularity that the income elasticity of demand for food is generally found to be considerably less than 1. Because of this, it must be the case that the income elasticity of all nonfood items must be greater than 1. If an individual experiences an increase in his or her income then we would expect food expenditures to increase by a smaller proportional

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amount, but the income must be spent somewhere. In the aggregate, these other expenditures must increase proportionally faster than income. A formal statement of this property of income elasticities can be derived by differentiating the individual’s budget constraint (I ¼ px x þ py y) with respect to income while treating the prices as constants: ∂x ∂y þ py ⋅ . (5.43) 1 ¼ px ⋅ ∂I ∂I A bit of algebraic manipulation of this expression yields ∂x xI ∂y yI 1 ¼ px ⋅ þ py ⋅ ¼ sx ex, I þ sy ey, I ; (5.44) ⋅ ⋅ ∂I xI ∂I yI here, as before, si represents the share of income spent on good i. Equation 5.44 shows that the weighted average on income elasticities for all goods that a person buys must be 1. If we knew, say, that a person spent a fourth of his or her income on food and the income elasticity of demand for food were 0.5, then the income elasticity of demand for everything else must be approximately 1:17 ½¼ ð1 0:25 ⋅ 0:5Þ=0:75. Because food is an important “necessity,” everything else is in some sense a “luxury.” Cournot aggregation. The eighteenth-century French economist Antoine Cournot provided one of the ﬁrst mathematical analyses of price changes using calculus. His most important discovery was the concept of marginal revenue, a concept central to the proﬁtmaximization hypothesis for ﬁrms. Cournot was also concerned with how the change in a single price might affect the demand for all goods. Our ﬁnal relationship shows that there are indeed connections among all of the reactions to the change in a single price. We begin by differentiating the budget constraint again, this time with respect to px : ∂I ∂x ∂y ¼ 0 ¼ px ⋅ þ x þ py ⋅ . ∂px ∂px ∂px Multiplication of this equation by px =I yields ∂x px x px ∂y px y þ py ⋅ 0 ¼ px ⋅ ⋅ ⋅ ⋅ þx⋅ ⋅ , I ∂px I x ∂px I y 0 ¼ sx ex, px þ sx þ sy ey, px ,

(5.45)

so the final Cournot result is sx ex, px þ sy ey, px ¼ sx .

(5.46)

This equation shows that the size of the cross-price effect of a change in the price of x on the quantity of y consumed is restricted because of the budget constraint. Direct, own-price effects cannot be totally overwhelmed by cross-price effects. This is the first of many connections among the demands for goods that we will study more intensively in the next chapter. Generalizations. Although we have shown these aggregation results only for the case of two goods, they are actually easily generalized to the case of many goods. You are asked to do just that in Problem 5.11. A more difﬁcult issue is whether these results should be expected to hold for typical economic data in which the demands of many people are combined. Often economists treat aggregate demand relationships as describing the behavior of a “typical person,” and these relationships should in fact hold for such a person. But the situation may not be quite that simple, as we will show when discussing aggregation later in the book.

Chapter 5 Income and Substitution Effects

EXAMPLE 5.5 Demand Elasticities: The Importance of Substitution Effects In this example we calculate the demand elasticities implied by three of the utility functions we have been using. Although the possibilities incorporated in these functions are too simple to reﬂect how economists actually study demand empirically, they do show how elasticities ultimately reﬂect people’s preferences. One especially important lesson is to show why most of the variation in demand elasticities among goods probably arises because of differences in the size of substitution effects. Case 1: Cobb-Douglas ðσ ¼ 1Þ. U ðx, yÞ ¼ x α y β , where α þ β ¼ 1. The demand functions derived from this utility function are αI , xð px , py , I Þ ¼ px βI ð1 αÞI yðpx , py , I Þ ¼ ¼ . py py Application of the elasticity definitions shows that ∂x px αI px ¼ 2 ⋅ ex, px ¼ ¼ 1, ⋅ αI =px ∂px x px py ∂x py ¼0⋅ ¼ 0, ex, py ¼ ⋅ x ∂py x ex, I ¼

(5.47)

∂x I α I ¼ 1. ⋅ ⋅ ¼ ∂I x px αI =px

The elasticities for good y take on analogous values. Hence, the elasticities associated with the Cobb-Douglas utility function are constant over all ranges of prices and income and take on especially simple values. That these obey the three relationships shown in the previous section can be easily demonstrated using the fact that here sx ¼ α and sy ¼ β: Homogeneity: ex, px þ ex, py þ ex, I ¼ 1 þ 0 þ 1 ¼ 0.

Engel aggregation: sx ex, I þ sy ey, I ¼ α ⋅ 1 þ β ⋅ 1 ¼ α þ β ¼ 1. Cournot aggregation: sx ex, px þ sy ey, px ¼ αð1Þ þ β ⋅ 0 ¼ α ¼ sx . We can also use the Slutsky equation in elasticity form (Equation 5.40) to derive the compensated price elasticity in this example: ex c , px ¼ ex, px þ sx ex, I ¼ 1 þ αð1Þ ¼ α 1 ¼ β.

(5.48)

Here, then, the compensated price elasticity for x depends on how important other goods (y) are in the utility function. Case 2: CES ðσ ¼ 2; δ ¼ 0:5Þ. U ðx, yÞ ¼ x 0:5 þ y 0:5 . In Example 4.2 we showed that the demand functions that can be derived from this utility function are I , xðpx , py , I Þ ¼ px ð1 þ px p 1 y Þ I . yð px , py , I Þ ¼ py ð1 þ p 1 x py Þ

(continued)

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EXAMPLE 5.5 CONTINUED As you might imagine, calculating elasticities directly from these functions can take some time. Here we focus only on the own-price elasticity and make use of the result (from Problem 5.6) that the “share elasticity” of any good is given by ∂s p esx , px ¼ x ⋅ x ¼ 1 þ ex, px . (5.49) ∂px sx In this case, sx ¼

px x 1 , ¼ I 1 þ px p 1 y

so the share elasticity is more easily calculated and is given by e s x , px ¼

p 1 px p 1 ∂sx px px y y ¼ ¼ . ⋅ ⋅ 2 1 1 Þ1 ∂px sx 1 þ p p ð1 þ p p ð1 þ px p 1 Þ y x x y y

(5.50)

Because the units in which goods are measured are rather arbitrary in utility theory, we might as well define them so that initially px ¼ py , in which case8 we get ex, px ¼ esx , px 1 ¼

1 1 ¼ 1.5. 1þ1

(5.51)

Hence, demand is more elastic in this case than in the Cobb-Douglas example. The reason for this is that the substitution effect is larger for this version of the CES utility function. This can be shown by again applying the Slutsky equation (and using the facts that ex, I ¼ 1 and sx ¼ 0:5): ex c , px ¼ ex, px þ sx ex, I ¼ 1.5 þ 0.5ð1Þ ¼ 1,

(5.52)

which is twice the size of the substitution effect for the Cobb-Douglas. Case 3. CES ðσ ¼ 0:5; δ ¼ 1Þ: U ðx, yÞ ¼ x 1 y 1 . Referring back to Example 4.2, we can see that the share of good x implied by this utility function is given by 1 , sx ¼ 0:5 1 þ p y p x0:5 so the share elasticity is given by e s x , px ¼

0:5 0:5p y0:5 p 1:5 0:5p 0:5 ∂sx px px x y px ¼ ¼ . ⋅ ⋅ 0:5 Þ1 ∂px sx 1 þ p y0:5 p x0:5 ð1 þ p y0:5 p 0:5 Þ2 ð1 þ p 0:5 y px x

(5.53).

If we again adopt the simplification of equal prices, we can compute the own-price elasticity as 0:5 1 ¼ 0:75 (5.54) ex, px ¼ esx , px 1 ¼ 2 and the compensated price elasticity as ex c , px ¼ ex, px þ sx ex, I ¼ 0:75 þ 0:5ð1Þ ¼ 0:25.

(5.55)

So, for this version of the CES utility function, the own-price elasticity is smaller than in Case 1 and Case 2 because the substitution effect is smaller. Hence, the main variation among the cases is indeed caused by differences in the size of the substitution effect. 8

Notice that this substitution must be made after differentiation because the definition of elasticity requires that we change only px while holding py constant.

Chapter 5 Income and Substitution Effects

If you never want to work out this kind of elasticity again, it may be helpful to make use of the quite general result that (5.56) ex c , px ¼ ð1 sx Þσ. You may wish to check out that this formula works in these three examples (with sx ¼ 0:5 and σ ¼ 1, 2, 0.5, respectively), and Problem 5.9 asks you to show that this result is generally true. Because all of these cases based on the CES utility function have a unitary income elasticity, the own-price elasticity can be computed from the compensated price elasticity by simply adding sx to the figure computed in Equation 5.56. QUERY: Why is it that the budget share for goods other than x ð1 sx Þ enters into the compensated own-price elasticities in this example?

CONSUMER SURPLUS An important problem in applied welfare economics is to devise a monetary measure of the gains and losses that individuals experience when prices change. One use for such a measure is to place a dollar value on the welfare loss that people experience when a market is monopolized with prices exceeding marginal costs. Another application concerns measuring the welfare gains that people experience when technical progress reduces the prices they pay for goods. Related applications occur in environmental economics (measuring the welfare costs of incorrectly priced resources), law and economics (evaluating the welfare costs of excess protections taken in fear of lawsuits), and public economics (measuring the excess burden of a tax). In order to make such calculations, economists use empirical data from studies of market demand in combination with the theory that underlies that demand. In this section we will examine the primary tools used in that process.

Consumer welfare and the expenditure function The expenditure function provides the ﬁrst component for the study of the price/welfare connection. Suppose that we wished to measure the change in welfare that an individual experiences if the price of good x rises from p0x to p1x . Initially this person requires expenditures of Eðp0x , py , U0 Þ to reach a utility of U0 . To achieve the same utility once the price of x rises, he or she would require spending of at least Eðp1x , py , U0 Þ. In order to compensate for the price rise, therefore, this person would require a compensation (formally called a compensating variation or CV) of CV ¼ Eð p 1x , py , U0 Þ Eð p 0x , py , U0 Þ.

(5.57)

This situation is shown graphically in the top panel of Figure 5.8. Initially, this person consumes the combination x0 , y0 and obtains utility of U0 . When the price of x rises, he or she would be forced to move to combination x2 , y2 and suffer a loss in utility. If he or she were compensated with extra purchasing power of amount CV, he or she could afford to remain on the U0 indifference curve despite the price rise by choosing combination x1 , y1 . The distance CV, therefore, provides a monetary measure of how much this person needs in order to be compensated for the price rise.

Using the compensated demand curve to show CV Unfortunately, individuals’ utility functions and their associated indifference curve maps are not directly observable. But we can make some headway on empirical measurement by determining how the CV amount can be shown on the compensated demand curve in the

165

FIGURE 5.8

Showing Compensating Variation If the price of x rises from p0x to p1x , this person needs extra expenditures of CV to remain on the U0 indifference curve. Integration shows that CV can also be represented by the shaded area below the compensated demand curve in panel (b). Quantity of y

E( px1, . . . ,U0) CV E( px1, . . . ,U0)

E( px0, . . . ,U0)

y1 y2 y0 U0 U1 x2

x1

x0

E( px0, . . . ,U0) Quantity of x

(a) Indifference curve map

Price

p x2

p x1

B

p x0

A xc( px , . . . ,U0)

x1

x0

(b) Compensated demand curve

Quantity of x

Chapter 5 Income and Substitution Effects

bottom panel of Figure 5.8. Shephard’s lemma shows that the compensated demand function for a good can be found directly from the expenditure function by differentiation: ∂Eðpx , py , U Þ . (5.58) x c ðpx , py , U Þ ¼ ∂px Hence, the compensation described in Equation 5.57 can be found by integrating across a sequence of small increments to price from p0x to p1x : p 1x

CV ¼

p 1x

∫ dE ¼ ∫ x ðp , p , U Þ dp c

x

p 0x

y

0

x

(5.59)

p 0x

while holding py and utility constant. The integral defined in Equation 5.59 has a geometric interpretation, which is shown in the lower panel of Figure 5.9: it is the shaded area to the left of the compensated demand curve and bounded by p0x and p1x . So the welfare cost of this price increase can also be illustrated using changes in the area below the compensated demand curve.

The consumer surplus concept There is another way to look at this issue. We can ask how much this person would be willing to pay for the right to consume all of this good that he or she wanted at the market price of p0x rather than doing without the good completely. The compensated demand curve in the bottom panel of Figure 5.8 shows that if the price of x rose to p2x , this person’s consumption would fall to zero and he or she would require an amount of compensation equal to area p2x Ap0x in order to accept the change voluntarily. The right to consume x0 at a price of p0x is therefore worth this amount to this individual. It is the extra beneﬁt that this person receives by being able to make market transactions at the prevailing market price. This value, given by the area below the compensated demand curve and above the market price, is termed consumer surplus. Looked at in this way, the welfare problem caused by a rise in the price of x can be described as a loss in consumer surplus. When the price rises from p0x to p1x the consumer surplus “triangle” decreases in size from p2x Ap0x to p2x Bp1x . As the ﬁgure makes clear, that is simply another way of describing the welfare loss represented in Equation 5.59.

Welfare changes and the Marshallian demand curve So far our analysis of the welfare effects of price changes has focused on the compensated demand curve. This is in some ways unfortunate because most empirical work on demand actually estimates ordinary (Marshallian) demand curves. In this section we will show that studying changes in the area below a Marshallian demand curve may in fact be quite a good way to measure welfare losses. Consider the Marshallian demand curve xð px , …Þ illustrated in Figure 5.9. Initially this consumer faces the price p0x and chooses to consume x0 . This consumption yields a utility level of U0 , and the initial compensated demand curve for x [that is, x c ðpx , py , U0 Þ] also passes through the point x0 , p0x (which we have labeled point A). When price rises to p1x , the Marshallian demand for good x falls to x1 (point C on the demand curve) and this person’s utility also falls to, say, U1 . There is another compensated demand curve associated with this lower level of utility, and it also is shown in Figure 5.9. Both the Marshallian demand curve and this new compensated demand curve pass through point C. The presence of a second compensated demand curve in Figure 5.9 raises an intriguing conceptual question. Should we measure the welfare loss from the price rise as we did in Figure 5.8 using the compensating variation (CV) associated with the initial compensated demand curve (area p1x BAp0x ) or should we, perhaps, use this new compensated demand curve

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FIGURE 5.9

Welfare Effects of Price Changes and the Marshallian Demand Curve The usual Marshallian (nominal income constant) demand curve for good x is xð px , …Þ. Further, x c ð…, U0 Þ and x c ð…, U1 Þ denote the compensated demand curves associated with the utility levels experienced when p0x and p1x , respectively, prevail. The area to the left of xð px , …Þ between p0x and p1x is bounded by the similar areas to the left of the compensated demand curves. Hence, for small changes in price, the area to the left of the Marshallian demand curve is a good measure of welfare loss. px

px1

C

B A

px0

D

x(px , . . . ) xc( . . . ,U0) xc( .

x1

x0

. . ,U1)

Quantity of x per period

and measure the welfare loss as area p1x CDp0x ? A potential rationale for using the area under the second curve would be to focus on the individual’s situation after the price rise (with utility level U1 ). We might ask how much he or she would now be willing to pay to see the price return to its old, lower levels.9 The answer to this would be given by area p1x CDp0x . The choice between which compensated demand curve to use therefore boils down to choosing which level of utility one regards as the appropriate target for the analysis. Luckily, the Marshallian demand curve provides a convenient compromise between these two measures. Because the size of the area between the two prices and below the Marshallian curve (area p1x CAp0x ) is smaller than that below the compensated demand curve based on U0 but larger than that below the curve based on U1 , it does seem an attractive middle ground. Hence, this is the measure of welfare losses we will primarily use throughout this book. DEFINITION

Consumer surplus. Consumer surplus is the area below the Marshallian demand curve and above market price. It shows what an individual would pay for the right to make voluntary transactions at this price. Changes in consumer surplus can be used to measure the welfare effects of price changes. We should point out that some economists use either CV or EV to compute the welfare effects of price changes. Indeed, economists are often not very clear about which measure of welfare change they are using. Our discussion in the previous section shows that if income effects are small, it really does not make much difference in any case. 9

This alternative measure of compensation is sometimes termed the “equivalent variation” (EV).

Chapter 5 Income and Substitution Effects

EXAMPLE 5.6 Welfare Loss from a Price Increase These ideas can be illustrated numerically by returning to our old hamburger/soft drink example. Let’s look at the welfare consequences of an unconscionable price rise for soft drinks (good x) from $1 to $4. In Example 5.3, we found that the compensated demand for good x was given by Vp y0:5 x c ðpx , py , V Þ ¼ 0:5 . (5.60) px Hence, the welfare cost of the price increase is given by 4

CV ¼

4

∫ x ðp , p , V Þ dp ¼ ∫ c

x

y

x

1

0:5 Vp 0:5 dpx y px

¼

px ¼4

0:5 2Vp 0:5 y px

1

.

(5.61)

px ¼1

If we use the values we have been assuming throughout this gastronomic feast (V ¼ 2, py ¼ 4), then CV ¼ 2 ⋅ 2 ⋅ 2 ⋅ ð4Þ0:5 2 ⋅ 2 ⋅ 2 ⋅ ð1Þ0:5 ¼ 8.

(5.62)

This figure would be cut in half (to 4) if we believed that the utility level after the price rise (V ¼ 1) were the more appropriate utility target for measuring compensation. If instead we had used the Marshallian demand function xð px , py , I Þ ¼ 0:5Ip 1 x , the loss would be calculated as 4

loss ¼

4

∫ xðp , p , I Þ dp ¼ ∫ x

1

y

x

1

4 0:5Ip 1 x dpx ¼ 0:5I ln px .

(5.63)

1

So, with I ¼ 8, this loss is loss ¼ 4 lnð4Þ 4 lnð1Þ ¼ 4 lnð4Þ ¼ 4ð1:39Þ ¼ 5:55,

(5.64)

which seems a reasonable compromise between the two alternative measures based on the compensated demand functions. QUERY: In this problem, none of the demand curves has a ﬁnite price at which demand goes to precisely zero. How does this affect the computation of total consumer surplus? Does this affect the types of welfare calculations made here?

REVEALED PREFERENCE AND THE SUBSTITUTION EFFECT The principal unambiguous prediction that can be derived from the utility-maximation model is that the slope (or price elasticity) of the compensated demand curve is negative. The proof of this assertion relies on the assumption of a diminishing MRS and the related observation that, with a diminishing MRS, the necessary conditions for a utility maximum are also sufﬁcient. To some economists, the reliance on a hypothesis about an unobservable utility function represented a weak foundation indeed on which to base a theory of demand. An alternative approach, which leads to the same result, was ﬁrst proposed by Paul Samuelson in the late 1940s.10 This approach, which Samuelson termed the theory of revealed preference, deﬁnes a principle of rationality that is based on observed behavior and 10

Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947).

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Part 2 Choice and Demand

then uses this principle to approximate an individual’s utility function. In this sense, a person who follows Samuelson’s principle of rationality behaves as if he or she were maximizing a proper utility function and exhibits a negative substitution effect. Because Samuelson’s approach provides additional insights into our model of consumer choice, we will brieﬂy examine it here.

Graphical approach The principle of rationality in the theory of revealed preference is as follows: Consider two bundles of goods, A and B. If, at some prices and income level, the individual can afford both A and B but chooses A, we say that A has been “revealed preferred” to B. The principle of rationality states that under any different price-income arrangement, B can never be revealed preferred to A. If B is in fact chosen at another price-income conﬁguration, it must be because the individual could not afford A. The principle is illustrated in Figure 5.10. Suppose that, when the budget constraint is given by I1 , point A is chosen even though B also could have been purchased. Then A has been revealed preferred to B. If, for some other budget constraint, B is in fact chosen, then it must be a case such as that represented by I2 , where A could not have been bought. If B were chosen when the budget constraint is I3 , this would be a violation of the principle of rationality because, with I3 , both A and B can be bought. With budget constraint I3 , it is likely that some point other than either A or B (say, C) will be bought. Notice how this principle uses observable reactions to alternative budget constraints to rank commodities rather than assuming the existence of a utility function itself. Also notice

FIGURE 5.10

Demonstration of the Principle of Rationality in the Theory of Revealed Preference With income I1 the individual can afford both points A and B. If A is selected then A is revealed preferred to B. It would be irrational for B to be revealed preferred to A in some other price-income conﬁguration.

Quantity of y

ya

A C

B

yb

I2 I3

xa

xb

I1

Quantity of x

Chapter 5 Income and Substitution Effects

how the principle offers a glimpse of why indifference curves are convex. Now we turn to a formal proof.

Negativity of the substitution effect Suppose that an individual is indifferent between two bundles, C (composed of xC and yC ) C and D (composed of xD and yD ). Let pC x , p y be the prices at which bundle C is chosen and D D px , py the prices at which bundle D is chosen. Because the individual is indifferent between C and D, it must be the case that when C was chosen, D cost at least as much as C: C C C pC x xC þ p y yC p x xD þ p y yD .

(5.65)

A similar statement holds when D is chosen: D D D pD x xD þ p y y D p x xC þ p y y C .

(5.66)

Rewriting these equations gives C pC x ðxC xD Þ þ p y ðyC yD Þ 0,

(5.67)

D pD x ðxD xC Þ þ p y ðyD yC Þ 0.

(5.68)

Adding these together yields D C D ðp C x p x ÞðxC xD Þ þ ð p y p y ÞðyC yD Þ 0.

(5.69)

D Now suppose that only the price of x changes; assume that pC y ¼ p y . Then D ð pC x p x ÞðxC xD Þ 0.

(5.70)

But Equation 5.70 says that price and quantity move in the opposite direction when utility is held constant (remember, bundles C and D are equally attractive). This is precisely a statement about the nonpositive nature of the substitution effect: ∂x c ðpx , py , V Þ ∂x ¼ 0. (5.71) ∂px ∂px U ¼constant We have arrived at the result by an approach that requires neither the existence of a utility function nor the assumption of a diminishing MRS.

Mathematical generalization Generalizing the revealed preference idea to n goods is straightforward. If at prices p0i , bundle x 0i is chosen instead of x 1i and if bundle x 1i is also affordable, then n n X X p 0i x 0i p 0i x 1i ; (5.72) i¼1

i¼1

that is, bundle 0 has been “revealed preferred” to bundle 1. Consequently, at the prices that prevail when bundle 1 is bought (say, p1i ), it must be the case that x 0i is more expensive: n X i¼1

p 1i x 0i >

n X

p 1i x 1i .

(5.73)

i¼1

Although this initial deﬁnition of revealed preference focuses on the relationship between two bundles of goods, the most often used version of the basic principle requires a degree of

171

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Part 2 Choice and Demand

transitivity for preferences among an arbitrarily large number of bundles. This is summarized by the following “strong” axiom. DEFINITION

Strong axiom of revealed preference. The strong axiom of revealed preference states that if commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is revealed preferred to bundle 2, and if bundle 2 is revealed preferred to bundle 3, … , and if bundle K 1 is revealed preferred to bundle K , then bundle K cannot be revealed preferred to bundle 0 (where K is any arbitrary number of commodity bundles). Most other properties that we have developed using the concept of utility can be proved using this revealed preference axiom instead. For example, it is an easy matter to show that demand functions are homogeneous of degree 0 in all prices and income. It therefore is apparent that the revealed preference axiom and the existence of “well-behaved” utility functions are somehow equivalent conditions. That this is in fact the case was ﬁrst shown by H. S. Houthakker in 1950. Houthakker showed that a set of indifference curves can always be derived for an individual who obeys the strong axiom of revealed preference.11 Hence, this axiom provides a quite general and believable foundation for utility theory based on simple comparisons among alternative budget constraints. This approach is widely used in the construction of price indices and for a variety of other applied purposes.

SUMMARY In this chapter, we used the utility-maximization model to study how the quantity of a good that an individual chooses responds to changes in income or to changes in that good’s price. The ﬁnal result of this examination is the derivation of the familiar downward-sloping demand curve. In arriving at that result, however, we have drawn a wide variety of insights from the general economic theory of choice. •

•

•

Proportional changes in all prices and income do not shift the individual’s budget constraint and therefore do not change the quantities of goods chosen. In formal terms, demand functions are homogeneous of degree 0 in all prices and income. When purchasing power changes (that is, when income increases with prices remaining unchanged), budget constraints shift and individuals will choose new commodity bundles. For normal goods, an increase in purchasing power causes more to be chosen. In the case of inferior goods, however, an increase in purchasing power causes less to be purchased. Hence the sign of ∂xi =∂I could be either positive or negative, although ∂xi =∂I 0 is the most common case. A fall in the price of a good causes substitution and income effects that, for a normal good, cause more of the good to be purchased. For inferior goods, however, substitution and income effects work in opposite directions and no unambiguous prediction is possible.

•

Similarly, a rise in price induces both substitution and income effects that, in the normal case, cause less to be demanded. For inferior goods the net result is again ambiguous.

•

The Marshallian demand curve summarizes the total quantity of a good demanded at each possible price. Changes in price induce both substitution and income effects that prompt movements along the curve. For a normal good, ∂xi =∂pi 0 along this curve. If income, prices of other goods, or preferences change, then the curve may shift to a new location.

•

Compensated demand curves illustrate movements along a given indifference curve for alternative prices. They are constructed by holding utility constant and exhibit only the substitution effects from a price change. Hence, their slope is unambiguously negative.

•

Demand elasticities are often used in empirical work to summarize how individuals react to changes in prices and income. The most important such elasticity is the (own-) price elasticity of demand, ex, px . This measures the proportionate change in quantity in response to a 1 percent change in price. A similar elasticity can be deﬁned for movements along the compensated demand curve.

•

There are many relationships among demand elasticities. Some of the more important ones are: (1) own-price

H. S. Houthakker, “Revealed Preference and the Utility Function,” Economica 17 (May 1950): 159–74.

11

Chapter 5 Income and Substitution Effects elasticities determine how a price change affects total spending on a good; (2) substitution and income effects can be summarized by the Slutsky equation in elasticity form; and (3) various aggregation relations hold among elasticities—these show how the demands for different goods are related. •

173

demand curves. Such changes affect the size of the consumer surplus that individuals receive from being able to make market transactions. •

Welfare effects of price changes can be measured by changing areas below either compensated or ordinary

The negativity of the substitution effect is the most basic conclusion from demand theory. This result can be shown using revealed preference theory and so does not require assuming the existence of a utility function.

PROBLEMS 5.1 Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 liter and 2 liter. Because the water itself is identical, he regards these two “goods” as perfect substitutes. a. Assuming Ed’s utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of 0.75L containers (x) and 2L containers (y). b. State Ed’s demand function for x in terms of px , py , and I . c. Graph the demand curve for x, holding I and py constant. d. How do changes in I and py shift the demand curve for x? e. What would the compensated demand curve for x look like in this situation?

5.2 David N. gets $3 per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at $0.05 per ounce) and jelly (at $0.10 per ounce). Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a. How much peanut butter and jelly will David buy with his $3 allowance in a week? b. Suppose the price of jelly were to rise to $0.15 an ounce. How much of each commodity would be bought? c. By how much should David’s allowance be increased to compensate for the rise in the price of jelly in part (b)? d. Graph your results in parts (a) to (c). e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity. f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.

5.3 As deﬁned in Chapter 3, a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The MRS depends on the ratio y=x. a. Prove that, in this case, ∂x=∂I is constant. b. Prove that if an individual’s tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions; that is, prove that Giffen’s paradox cannot occur.

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Part 2 Choice and Demand

5.4 As in Example 5.1, assume that utility is given by utility ¼ U ðx, yÞ ¼ x 0:3 y 0:7 . a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case. b. Use the expenditure function calculated in part (a) together with Shephard’s lemma to compute the compensated demand function for good x. c. Use the results from part (b) together with the uncompensated demand function for good x to show that the Slutsky equation holds for this case.

5.5 Suppose the utility function for goods x and y is given by utility ¼ U ðx, yÞ ¼ xy þ y. a. Calculate the uncompensated (Marshallian) demand functions for x and y and describe how the demand curves for x and y are shifted by changes in I or the price of the other good. b. Calculate the expenditure function for x and y. c. Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good.

5.6 Over a three-year period, an individual exhibits the following consumption behavior: px

py

x

y

Year 1

3

3

7

4

Year 2

4

2

6

6

Year 3

5

1

7

3

Is this behavior consistent with the strong axiom of revealed preference?

5.7 Suppose that a person regards ham and cheese as pure complements—he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free. a. If the price of ham is equal to the price of cheese, show that the own-price elasticity of demand for ham is 0.5 and that the cross-price elasticity of demand for ham with respect to the price of cheese is also 0.5. b. Explain why the results from part (a) reﬂect only income effects, not substitution effects. What are the compensated price elasticities in this problem? c. Use the results from part (b) to show how your answers to part (a) would change if a slice of ham cost twice the price of a slice of cheese. d. Explain how this problem could be solved intuitively by assuming this person consumes only one good—a ham-and-cheese sandwich.

Chapter 5 Income and Substitution Effects

5.8 Show that the share of income spent on a good x is sx ¼

d ln E , where E is total expenditure. d ln px

Analytical Problems 5.9 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good, x, the income share is deﬁned as sx ¼ px x=I . In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good’s budget share with respect to income ðesx , I ¼ ∂sx =∂I ⋅ I =sx Þ is equal to ex, I 1. Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a good’s budget share with respect to its own price ðesx, px ¼ ∂sx =∂px ⋅ px =sx Þ is equal to ex, px þ 1. Again, interpret this ﬁnding with a few numerical examples. c. Use your results from part (b) to show that the “expenditure elasticity” of good x with respect to its own price ½ex ⋅px , px ¼ ∂ð px ⋅ xÞ=∂px ⋅ 1=x is also equal to ex, px þ 1. d. Show that the elasticity of a good’s budget share with respect to a change in the price of some other good ðesx , py ¼ ∂sx =∂py ⋅ py =sx Þ is equal to ex, py . e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good x is given by sx ¼ 1=ð1 þ pky pk x Þ, where k ¼ δ=ðδ 1Þ ¼ 1 σ. Use this share equation to prove Equation 5.56: ex c , px ¼ ð1 sx Þσ. Hint: This problem can be simpliﬁed by assuming px ¼ py , in which case sx ¼ 0:5.

5.10 More on elasticities Part (e) of Problem 5.9 has a number of useful applications because it shows how price responses depend ultimately on the underlying parameters of the utility function. Speciﬁcally, use that result together with the Slutsky equation in elasticity terms to show: a. In the Cobb-Douglas case ðσ ¼ 1Þ, the following relationship holds between the own-price elasticities of x and y: ex, px þ ey, py ¼ 2. b. If σ > 1 then ex, px þ ey, py < 2, and if σ < 1 then ex, px þ ey, py > 2. Provide an intuitive explanation for this result. c. How would you generalize this result to cases of more than two goods? Discuss whether such a generalization would be especially meaningful.

5.11 Aggregation of elasticities for many goods The three aggregation relationships presented in this chapter can be generalized to any number of goods. This problem asks you to do so. We assume that there are n goods and that the share of income devoted to good i is denoted by si . We also deﬁne the following elasticities: ∂x I ei, I ¼ i ⋅ , ∂I xi ∂xi I ei, j ¼ ⋅ . ∂pj xi Use this notation to show: P a. Homogeneity: nj¼1 ei, j þ ei, I ¼ 0. b. Engel aggregation:

Pn

c. Cournot aggregation:

i¼1 si ei, I

Pn

¼ 1.

i¼1 si ei, j

¼ sj .

175

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Part 2 Choice and Demand

5.12 Quasi-linear utility (revisited) Consider a simple quasi-linear utility function of the form U ðx, yÞ ¼ x þ ln y. a. Calculate the income effect for each good. Also calculate the income elasticity of demand for each good. b. Calculate the substitution effect for each good. Also calculate the compensated own-price elasticity of demand for each good. c. Show that the Slutsky equation applies to this function. d. Show that the elasticity form of the Slutsky equation also applies to this function. Describe any special features you observe.

5.13 The almost ideal demand system The general form of the almost ideal demand system (AIDS) is given by ln Eð ! p , U Þ ¼ a0 þ

n X

αi ln pi þ

i¼1

k n X n 1X β γij ln pi ln pj þ U β0 pk k , 2 i¼1 j ¼1 i¼1

∏

where ! p is the vector of prices, E is the expenditure function, and U is the level of utility required. For analytical ease, assume that the following restrictions apply: n n n X X X γij ¼ γji , αi ¼ 1, and γij ¼ βk ¼ 0. j ¼1

i¼1

k¼1

a. Derive the AIDS functional form for a two-goods case. p, U Þ is homogeneous of degree 1 in all prices. b. Given the previous restrictions, show that Eð ! This, along with the fact that this function resembles closely the actual data, makes it an “ideal” function. d ln E (see Problem 5.8), calculate the income share of each of the two c. Using the fact that sx ¼ d ln px goods.

5.14 Price indifference curves Price indifference curves are iso-utility curves with the prices of two goods on the x- and y-axes, respectively. Thus, they have the following general form: ð p1 , p2 Þj vð p1 , p2 , I Þ ¼ v0 . a. Derive the formula for the price indifference curves for the Cobb-Douglas case with α ¼ β ¼ 0:5. Sketch one of them. b. What does the slope of the curve show? c. What is the direction of increasing utility in your graph?

SUGGESTIONS FOR FURTHER READING Cook, P. J. “A ‘One Line’ Proof of the Slutsky Equation.” American Economic Review 62 (March 1972): 139. Clever use of duality to derive the Slutsky equation; uses the same method as in Chapter 5 but with rather complex notation.

Fisher, F. M., and K. Shell. The Economic Theory of Price Indices. New York: Academic Press, 1972. Complete, technical discussion of the economic properties of various price indexes; describes “ideal” indexes based on utility-maximizing models in detail.

Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapter 3 covers much of the material in this chapter at a somewhat higher level. Section I on measurement of the welfare effects of price changes is especially recommended.

Samuelson, Paul A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947, Chap. 5. Provides a complete analysis of substitution and income effects. Also develops the revealed preference notion.

Chapter 5 Income and Substitution Effects Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGrawHill, 2001. Provides an extensive derivation of the Slutsky equation and a lengthy presentation of elasticity concepts.

Sydsaetter, K., A. Strom, and P. Berck. Economist’s Mathematical Manual. Berlin: Springer-Verlag, 2003. Provides a compact summary of elasticity concepts. The coverage of elasticity of substitution notions is especially complete.

177

Varian, H. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992. Formal development of preference notions. Extensive use of expenditure functions and their relationship to the Slutsky equation. Also contains a nice proof of Roy‘s identity.

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Part 2 Choice and Demand

EXTENSIONS Demand Concepts and the Evaluation of Price Indices In Chapters 4 and 5 we introduced a number of related demand concepts, all of which were derived from the underlying model of utility maximization. Relationships among these various concepts are summarized in Figure E5.1. We have already looked at most of the links in the table formally. We have not yet discussed the mathematical relationship between indirect utility functions and Marshallian demand functions (Roy’s identity), and we will do that below. All of the entries in the table make clear that there are many ways to learn something about the relationship between individuals’ welfare and the prices they face. In this extension we will explore some of these approaches. Speciﬁcally, we will look at how the concepts can shed light on the accuracy of the consumer price index (CPI), the primary measure of inﬂation in the United States. We will also look at a few other price index concepts. The CPI is a “market basket” index of the cost of living. Researchers measure the amounts that people

FIGURE E5.1

consume of a set of goods in some base period (in the two-good case these base-period consumption levels might be denoted by x0 and y0 ) and then use current price data to compute the changing price of this market basket. Using this procedure, the cost of the market basket initially would be I0 ¼ p0x x0 þ p0y y0 and the cost in period 1 would be I1 ¼ p1x x0 þ p1y y0 . The change in the cost of living between these two periods would then be measured by I1 =I0 . Although this procedure is an intuitively plausible way of measuring inﬂation and market basket price indices are widely used, such indices have many shortcomings.

E5.1 Expenditure functions and substitution bias Market-basket price indices suffer from “substitution bias.” Because the indices do not permit individuals to make substitutions in the market basket in response to changes in relative prices, they will tend to overstate

Relationships among Demand Concepts

Primal

Dual

Maximize U(x, y) s.t. I = Pxx + Pyy

Minimize E(x, y) s.t. U = U(x, y)

Indirect utility function U* = V(px, py, I)

Roy’s identity

Marshallian demand ∂V ∂px x(px, py, I) = – ∂V ∂I

Inverses

Expenditure function E* = E(px, py, U)

Shephard’s lemma

Compensated demand xc(px, py, U) =

∂E ∂px

Chapter 5 Income and Substitution Effects

the welfare losses that people incur from rising prices. This exaggeration is illustrated in Figure E5.2. To achieve the utility level U0 initially requires expenditures of E0 , resulting in a purchase of the basket x0 , y0 . If px =py falls, the initial utility level can now be obtained with expenditures of E1 by altering the consumption bundle to x1 , y1 . Computing the expenditure level needed to continue consuming x0 , y0 exaggerates how much extra purchasing power this person needs to restore his or her level of well-being. Economists have extensively studied the extent of this substitution bias. Aizcorbe and Jackman (1993), for example, ﬁnd that this difﬁculty with a market basket index may exaggerate the level of inﬂation shown by the CPI by about 0.2 percent per year.

FIGURE E5.2

E5.2 Roy’s identity and new goods bias When new goods are introduced, it takes some time for them to be integrated into the CPI. For example, Hausman (1999, 2003) states that it took more than 15 years for cell phones to appear in the index. The problem with this delay is that market basket indices will fail to reﬂect the welfare gains that people experience from using new goods. To measure these costs, Hausman sought to measure a “virtual” price (p ) at which the demand for, say, cell phones would be zero and then argued that the introduction of the good at its market price represented a change in consumer surplus that could be measured. Hence, the author

Substitution Bias in the CPI

Initially expenditures are given by E0 and this individual buys x0 , y0 . If px =py falls, utility level U0 can be reached most cheaply by consuming x1 , y1 and spending E1 . Purchasing x0 , y0 at the new prices would cost more than E1 . Hence, holding the consumption bundle constant imparts an upward bias to CPI-type computations.

Quantity of y

E0

y0 E1

U0

x0

x1

179

Quantity of x

180

Part 2 Choice and Demand

was faced with the problem of how to get from the Marshallian demand function for cell phones (which he estimated econometrically) to the expenditure function. To do so he used Roy’s identity (see Roy, 1942). Remember that the consumer’s utility-maximizing problem can be represented by the Lagrangian expression ℒ ¼ U ðx, yÞ þ λðI px x py yÞ. If we apply the envelope theorem to this expression, we know that ∂U ∂ℒ ¼ ¼ λxð px , py , I Þ, ∂px ∂px (i) ∂U ∂ℒ ¼ ¼ λ. ∂I ∂I Hence the Marshallian demand function is given by ∂U =∂px . (ii) xð px , py , I Þ ¼ ∂U =∂I Using his estimates of the Marshallian demand function, Hausman integrated Equation ii to obtain the implied indirect utility function and then calculated its inverse, the expenditure function (check Figure E5.1 to see the logic of the process). Though this certainly is a roundabout scheme, it did yield large estimates for the gain in consumer welfare from cell phones—a present value in 1999 of more than $100 billion. Delays in the inclusion of such goods into the CPI can therefore result in a misleading measure of consumer welfare.

E5.3 Other complaints about the CPI Researchers have found several other faults with the CPI as currently constructed. Most of these focus on the consequences of using incorrect prices to compute the index. For example, when the quality of a good improves, people are made better-off, though this may not show up in the good’s price. Throughout the 1970s and 1980s the reliability of color television sets improved dramatically, but the price of a set did not change very much. A market basket that included “one color television set” would miss this source of improved welfare. Similarly, the opening of “big box” retailers such as Costco and Home Depot during the 1990s undoubtedly reduced the prices that consumers paid for various goods. But including these new retail outlets into the sample scheme for the CPI took several years, so the index misrepresented what people were actually paying. Assessing the magnitude of error introduced by these cases where incorrect prices are used in the CPI can also be accomplished by using the

various demand concepts in Figure E5.1. For a summary of this research, see Moulton (1996).

E5.4 Exact price indices In principle, it is possible that some of the shortcomings of price indices such as the CPI might be ameliorated by more careful attention to demand theory. If the expenditure function for the representative consumer were known, for example, it would be possible to construct an “exact” index for changes in purchasing power that would take commodity substitution into account. To illustrate this, suppose there are only two goods and we wish to know how purchasing power has changed between period 1 and period 2. If the expenditure function is given by Eðpx , py , U Þ then the ratio _ Eð p 2x , p 2y , U Þ _ I1,2 ¼ (iii) Eð p 1x , p 1y , U Þ shows how the cost of attaining the target utility level _ U has changed between the two periods. If, for example, I1, 2 ¼ 1:04, then we would say that the cost of attaining the utility target had increased by 4 percent. Of course, this answer is only a conceptual one. Without knowing the representative person’s utility function, we would not know the specific form of the expenditure function. But in some cases Equation iii may suggest how to proceed in index construction. Suppose, for example, that the typical person’s preferences could be represented by the Cobb-Douglas utility function U ðx, yÞ ¼ x α y 1α . In this case it is easy to show that the expenditure function is a generalization of the one given in Example 4.4: U =αα ð1 αÞ1α ¼ kpαx py1α U . Eðpx , py , U Þ ¼ pαx p1α y Inserting this function into Equation iii yields _ kðp 2x Þα ðp 2y Þ1α U ð p 2x Þα ðp 2y Þ1α _ ¼ . (iv) I1,2 ¼ kðp 1x Þα ðp 1y Þ1α U ð p 1x Þα ðp 1y Þ1α So, in this case, the exact price index is a relatively simple function of the observed prices. The particularly useful feature of this example is that the utility target cancels out in the construction of the cost-of-living index (as it will anytime the expenditure function is homogeneous in utility). Notice also that the expenditure shares (α and 1 α) play an important role in the index—the larger a good’s share, the more important will changes be in that good’s price in the ﬁnal index.

Chapter 5 Income and Substitution Effects

E5.5 Development of exact price indices The Cobb-Douglas utility function is, of course, a very simple one. Much recent research on price indices has focused on more general types of utility functions and on the discovery of the exact price indices they imply. For example, Feenstra and Reinsdorf (2000) show that the almost ideal demand system described in the Extensions to Chapter 4 implies an exact price index (I ) that takes a “Divisia” form: n X wi ∆ ln pi (v) lnðI Þ ¼ i¼1

(here the wi are weights to be attached to the change in the logarithm of each good’s price). Often the weights in Equation v are taken to be the budget shares of the goods. Interestingly, this is precisely the price index implied by the Cobb-Douglas utility function in Equation iv, since lnðI1;2 Þ ¼ α ln p 2x þ ð1 αÞ ln p 2y α ln p 1x ð1 αÞ ln p 1y ¼ α∆ ln px þ ð1 − αÞ∆ ln py .

(vi)

In actual applications, the weights would change from period to period to reﬂect changing budget shares. Similarly, changes over several periods would be “chained” together from a number of single-period price change indices. Changing demands for food in China. China has one of the fastest growing economies in the world: its GDP per capita is currently growing at a rate of about 8 percent per year. Chinese consumers also spend a large fraction of their incomes on food—approximately 38 percent of total expenditures in recent survey data. One implication of the rapid growth in Chinese

181

incomes, however, is that patterns of food consumption are changing rapidly. Purchases of staples, such as rice or wheat, are declining in relative importance, whereas purchases of poultry, ﬁsh, and processed foods are growing rapidly. A recent paper by Gould and Villarreal (2006) studies these patterns in detail using the AIDS model. They identify a variety of substitution effects across speciﬁc food categories in response to changing relative prices. Such changing patterns imply that a ﬁxed market basket price index (such as the U.S. Consumer Price Index) would be particularly inappropriate for measuring changes in the cost of living in China and that some alternative approaches should be examined.

References Aizcorbe, Ana M., and Patrick C. Jackman. “The Commodity Substitution Effect in CPI Data, 1982–91.” Monthly Labor Review (December 1993): 25–33. Feenstra, Robert C., and Marshall B. Reinsdorf. “An Exact Price Index for the Almost Ideal Demand System.” Economics Letters (February 2000): 159–62. Gould, Brain W., and Hector J. Villarreal. “An Assessment of the Current Structure of Food Demand in Urban China.” Agricultural Economics (January 2006): 1–16. Hausman, Jerry. “Cellular Telephone, New Products, and the CPI.” Journal of Business and Economic Statistics (April 1999): 188–94. Hausman, Jerry. “Sources of Bias and Solutions to Bias in the Consumer Price Index.” Journal of Economic Perspectives (Winter 2003): 23–44. Moulton, Brent R. “Bias in the Consumer Price Index: What Is the Evidence?” Journal of Economic Perspectives (Fall 1996): 159–77. Roy, R. De l’utilité, contribution à la théorie des choix. Paris: Hermann, 1942.

CHAPTER

6 Demand Relationships among Goods In Chapter 5 we examined how changes in the price of a particular good (say, good x) affect the quantity of that good chosen. Throughout the discussion, we held the prices of all other goods constant. It should be clear, however, that a change in one of these other prices could also affect the quantity of x chosen. For example, if x were taken to represent the quantity of automobile miles that an individual drives, this quantity might be expected to decline when the price of gasoline rises or increase when air and bus fares rise. In this chapter we will use the utility-maximization model to study such relationships.

THE TWO-GOOD CASE We begin our study of the demand relationship among goods with the two-good case. Unfortunately, this case proves to be rather uninteresting because the types of relationships that can occur when there are only two goods are quite limited. Still, the two-good case is useful because it can be illustrated with two-dimensional graphs. Figure 6.1 starts our examination by showing two examples of how the quantity of x chosen might be affected by a change in the price of y. In both panels of the ﬁgure, py has fallen. This has the result of shifting the budget constraint outward from I0 to I1 . In both cases, the quantity of good y chosen has also increased from y0 to y1 as a result of the decline in py , as would be expected if y is a normal good. For good x, however, the results shown in the two panels differ. In (a) the indifference curves are nearly L-shaped, implying a fairly small substitution effect. A decline in py does not induce a very large move along U0 as y is substituted for x. That is, x drops relatively little as a result of the substitution. The income effect, however, reﬂects the greater purchasing power now available, and this causes the total quantity of x chosen to increase. Hence, ∂x=∂py is negative (x and py move in opposite directions). In Figure 6.1b this situation is reversed: ∂x=∂py is positive. The relatively ﬂat indifference curves in Figure 6.1b result in a large substitution effect from the fall in py . The quantity of x declines sharply as y is substituted for x along U0 . As in Figure 6.1a, the increased purchasing power from the decline in py causes more x to be bought, but now the substitution effect dominates and the quantity of x declines to x1 . In this case, then, x and py move in the same direction.

A mathematical treatment The ambiguity in the effect of changes in py can be further illustrated by a Slutsky-type equation. By using procedures similar to those in Chapter 5, it is fairly simple to show that ∂xð px , py , I Þ ¼ substitution effect þ income effect ∂py ∂x ∂x ¼ , y ⋅ (6.1) ∂py U ¼constant ∂I 182

Chapter 6 Demand Relationships among Goods

FIGURE 6.1

Differing Directions of Cross-Price Effects

In both panels, the price of y has fallen. In (a), substitution effects are small so the quantity of x consumed increases along with y. Because ∂x=∂py < 0, x and y are Quantity of y

gross complements. In (b), substitution effects are large so the quantity of x chosen falls. Because ∂x=∂py > 0, x and y would be termed gross substitutes.

Quantity of y

I1

I1

I0

I0

y1

y1 y0

U1 U0

y0

U1 U0

x 0 x1

Quantity of x

(a) Gross complements

x1 x 0

Quantity of x

(b) Gross substitutes

or, in elasticity terms, ex, py ¼ ex c , py sy ex, I .

(6.2)

Notice that the size of the income effect is determined by the share of good y in this person’s purchases. The impact of a change in py on purchasing power is determined by how important y is to this person. For the two-good case, the terms on the right side of Equations 6.1 and 6.2 have different signs. Assuming that indifference curves are convex, the substitution effect ∂x=∂py jU ¼constant is positive. If we conﬁne ourselves to moves along one indifference curve, increases in py increase x and decreases in py decrease the quantity of x chosen. But, assuming x is a normal good, the income effect ( y∂x=∂I or sy ex, I ) is clearly negative. Hence, the combined effect is ambiguous; ∂x=∂py could be either positive or negative. Even in the two-good case, the demand relationship between x and py is rather complex.

EXAMPLE 6.1 Another Slutsky Decomposition for Cross-Price Effects In Example 5.4 we examined the Slutsky decomposition for the effect of a change in the price of x. Now let’s look at the cross-price effect of a change in y prices on x purchases. Remember that the uncompensated and compensated demand functions for x are given by 0:5I (6.3) xð px , py , I Þ ¼ px and

183

0:5 x c ðpx , py , V Þ ¼ Vp 0:5 . y px

(6.4) (continued)

184

Part 2 Choice and Demand

EXAMPLE 6.1 CONTINUED As we have pointed out before, the Marshallian demand function in this case yields ∂x=∂py ¼ 0; that is, changes in the price of y do not affect x purchases. Now we show that this occurs because the substitution and income effects of a price change are precisely counterbalancing. The substitution effect in this case is given by ∂x ∂x c ¼ ¼ 0:5Vp 0:5 p 0:5 . (6.5) y x ∂p ∂p y U ¼ constant

y

Substituting for V from the indirect utility function (V ¼ 0:5Ipy0:5 px0:5 ) gives a final statement for the substitution effect: ∂x 1 ¼ 0:25Ip 1 (6.6) y px . ∂p y U ¼ constant

Returning to the Marshallian demand function for y ðy ¼ 0:5Ip1 y ) to calculate the income effect yields ∂x 1 1 1 (6.7) y ¼ ½0:5Ip 1 y ⋅ ½0:5p x ¼ 0:25Ip y p x , ∂I and combining Equations 6.6 and 6.7 gives the total effect of the change in the price of y as ∂x 1 1 ¼ 0:25Ip 1 0:25Ip 1 y px y p x ¼ 0. ∂py

(6.8)

This makes clear that the reason that changes in the price of y have no effect on x purchases in the Cobb-Douglas case is that the substitution and income effects from such a change are precisely offsetting; neither of the effects alone, however, is zero. Returning to our numerical example ( px ¼ 1, py ¼ 4, I ¼ 8, V ¼ 2), suppose now that py falls to 2. This should have no effect on the Marshallian demand for good x. The compensated demand function in Equation 6.4 shows thatp the ﬃﬃﬃ price change would cause the quantity of x demanded to decline from 4 to 2.83 (¼ 2 2) as y is substituted for x with utility unchanged. However, the increased purchasing power arising from the price decline precisely reverses this effect. QUERY: Why would it be incorrect to argue that if ∂x =∂py ¼ 0, then x and y have no substitution possibilities—that is, they must be consumed in ﬁxed proportions? Is there any case in which such a conclusion could be drawn?

SUBSTITUTES AND COMPLEMENTS With many goods, there is much more room for interesting relations among goods. It is relatively easy to generalize the Slutsky equation for any two goods xi , xj as ∂xi ð p1 , …, pn , I Þ ∂xi ∂x (6.9) ¼ xj i , ∂pj U ¼constant ∂I ∂pj and again this can be readily translated into an elasticity relation: ei, j ¼ ei,c j sj ei, I .

(6.10)

This says that the change in the price of any good (here, good j ) induces income and substitution effects that may change the quantity of every good demanded. Equations 6.9 and 6.10 can be used to discuss the idea of substitutes and complements. Intuitively, these ideas are

Chapter 6 Demand Relationships among Goods

185

rather simple. Two goods are substitutes if one good may, as a result of changed conditions, replace the other in use. Some examples are tea and coffee, hamburgers and hot dogs, and butter and margarine. Complements, on the other hand, are goods that “go together,” such as coffee and cream, fish and chips, or brandy and cigars. In some sense, “substitutes” substitute for one another in the utility function whereas “complements” complement each other. There are two different ways to make these intuitive ideas precise. One of these focuses on the “gross” effects of price changes by including both income and substitution effects; the other looks at substitution effects alone. Because both deﬁnitions are used, we will examine each in detail.

Gross substitutes and complements Whether two goods are substitutes or complements can be established by referring to observed price reactions as follows. Gross substitutes and complements. Two goods, xi and xj , are said to be gross substiDEFINITION tutes if ∂xi > 0 (6.11) ∂pj and gross complements if

∂xi < 0. ∂pj

(6.12)

That is, two goods are gross substitutes if a rise in the price of one good causes more of the other good to be bought. The goods are gross complements if a rise in the price of one good causes less of the other good to be purchased. For example, if the price of coffee rises, the demand for tea might be expected to increase (they are substitutes), whereas the demand for cream might decrease (coffee and cream are complements). Equation 6.9 makes it clear that this deﬁnition is a “gross” deﬁnition in that it includes both income and substitution effects that arise from price changes. Because these effects are in fact combined in any real-world observation we can make, it might be reasonable always to speak only of “gross” substitutes and “gross” complements.

Asymmetry of the gross definitions There are, however, several things that are undesirable about the gross deﬁnitions of substitutes and complements. The most important of these is that the deﬁnitions are not symmetric. It is possible, by the deﬁnitions, for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1 . The presence of income effects can produce paradoxical results. Let’s look at a speciﬁc example.

EXAMPLE 6.2 Asymmetry in Cross-Price Effects Suppose the utility function for two goods (x and y) has the quasi-linear form U ðx, yÞ ¼ ln x þ y.

(6.13)

Setting up the Lagrangian expression ℒ ¼ ln x þ y þ λðI px x py yÞ

(6.14) (continued)

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EXAMPLE 6.2 CONTINUED yields the following ﬁrst-order conditions: ∂ℒ 1 ¼ λpx ¼ 0, ∂x x ∂ℒ (6.15) ¼ 1 λpy ¼ 0, ∂y ∂ℒ ¼ I px x py y ¼ 0. ∂λ Moving the terms in λ to the right and dividing the first equation by the second yields 1 p ¼ x, (6.16) py x px x ¼ py .

(6.17)

Substitution into the budget constraint now permits us to solve for the Marshallian demand function for y: I ¼ px x þ py y ¼ py þ py y. Hence, y¼

I py py

.

(6.18)

(6.19)

This equation shows that an increase in py must decrease spending on good y (that is, py y). Therefore, since px and I are unchanged, spending on x must rise. So ∂x > 0, (6.20) ∂py and we would term x and y gross substitutes. On the other hand, Equation 6.19 shows that spending on y is independent of px . Consequently, ∂y ¼ 0 (6.21) ∂px and, looked at in this way, x and y would be said to be independent of each other; they are neither gross substitutes nor gross complements. Relying on gross responses to price changes to define the relationship between x and y would therefore run into ambiguity. QUERY: In Example 3.4, we showed that a utility function of the form given by Equation 6.13 is not homothetic: the MRS does not depend only on the ratio of x to y. Can asymmetry arise in the homothetic case?

NET SUBSTITUTES AND COMPLEMENTS Because of the possible asymmetries involved in the deﬁnition of gross substitutes and complements, an alternative deﬁnition that focuses only on substitution effects is often used. DEFINITION

Net substitutes and complements. Goods xi and xj are said to be net substitutes if ∂xi >0 (6.22) ∂pj U ¼constant

Chapter 6 Demand Relationships among Goods

and net complements if

∂xi < 0. ∂pj U ¼constant

(6.23)

These deﬁnitions,1 then, look only at the substitution terms to determine whether two goods are substitutes or complements. This deﬁnition is both intuitively appealing (because it looks only at the shape of an indifference curve) and theoretically desirable (because it is unambiguous). Once xi and xj have been discovered to be substitutes, they stay substitutes, no matter in which direction the deﬁnition is applied. As a matter of fact, the deﬁnitions are perfectly symmetric: ∂xj ∂xi ¼ . (6.24) ∂p ∂p j U ¼constant

i U ¼constant

The substitution effect of a change in pi on good xj is identical to the substitution effect of a change in pj on the quantity of xi chosen. This symmetry is important in both theoretical and empirical work.2 The differences between the two deﬁnitions of substitutes and complements are easily demonstrated in Figure 6.1a. In this ﬁgure, x and y are gross complements, but they are net substitutes. The derivative ∂x=∂py turns out to be negative (x and y are gross complements) because the (positive) substitution effect is outweighed by the (negative) income effect (a fall in the price of y causes real income to increase greatly, and, consequently, actual purchases of x increase). However, as the ﬁgure makes clear, if there are only two goods from which to choose, they must be net substitutes, although they may be either gross substitutes or gross complements. Because we have assumed a diminishing MRS, the own-price substitution effect must be negative and, consequently, the cross-price substitution effect must be positive.

SUBSTITUTABILITY WITH MANY GOODS Once the utility-maximizing model is extended to many goods, a wide variety of demand patterns become possible. Whether a particular pair of goods are net substitutes or net complements is basically a question of a person’s preferences, so one might observe all sorts of odd relationships. A major theoretical question that has concerned economists is whether substitutability or complementarity is more prevalent. In most discussions, we tend to regard goods as substitutes (a price rise in one market tends to increase demand in most other markets). It would be nice to know whether this intuition is justiﬁed.

These are sometimes called “Hicksian” substitutes and complements, named after the British economist John Hicks, who originally developed the definitions.

1

2

This symmetry is easily shown using Shephard’s lemma. Compensated demand functions can be calculated from expenditure functions by differentiation: x ci ð p1 , …, pn , V Þ ¼

Hence, the substitution effect is given by

∂Eð p1 , …, pn , V Þ . ∂pi

∂x ci ∂xi ∂2 E ¼ ¼ ¼ Eij . ∂pj U ¼constant ∂pj ∂pj ∂pi

But now we can apply Young’s theorem to the expenditure function: ∂x cj ∂xj Eij ¼ Eji ¼ ¼ , ∂pi ∂pi U ¼constant which proves the symmetry.

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The British economist John Hicks studied this issue in some detail about 50 years ago and reached the conclusion that “most” goods must be substitutes. The result is summarized in what has come to be called “Hicks’ second law of demand.”3 A modern proof starts with the compensated demand function for a particular good: x ci ð p1 , …, pn , V Þ. This function is homogeneous of degree 0 in all prices (if utility is held constant and prices double, quantities demanded do not change because the utility-maximizing tangencies do not change). Applying Euler’s theorem to the function yields ∂x ci ∂x ci … ∂x ci þ p2 ⋅ þ þ pn ⋅ ¼ 0. (6.25) p1 ⋅ ∂p1 ∂p2 ∂pn We can put this result into elasticity terms by dividing Equation 6.25 by xi : c c c þ ei2 þ … þ ein ¼ 0. ei1

(6.26)

But we know that 0 because of the negativity of the own-substitution effect. Hence it must be the case that X eijc 0. (6.27) eiic

j ≠i

In words, the sum of all the compensated cross-price elasticities for a particular good must be positive (or zero). This is the sense that “most” goods are substitutes. Empirical evidence seems generally consistent with this theoretical finding: instances of net complementarity between goods are encountered relatively infrequently in empirical studies of demand.

COMPOSITE COMMODITIES Our discussion in the previous section showed that the demand relationships among goods can be quite complicated. In the most general case, an individual who consumes n goods will have demand functions that reﬂect nðn þ 1Þ=2 different substitution effects.4 When n is very large (as it surely is for all the speciﬁc goods that individuals actually consume), this general case can be unmanageable. It is often far more convenient to group goods into larger aggregates such as food, clothing, shelter, and so forth. At the most extreme level of aggregates, we might wish to examine one speciﬁc good (say, gasoline, which we might call x) and its relationship to “all other goods,” which we might call y. This is the procedure we have been using in some of our two-dimensional graphs, and we will continue to do so at many other places in this book. In this section we show the conditions under which this procedure can be defended. In the Extensions to this chapter, we explore more general issues involved in aggregating goods into larger groupings.

Composite commodity theorem Suppose consumers choose among n goods but that we are only interested speciﬁcally in one of them—say, x1 . In general, the demand for x1 will depend on the individual prices of the other n 1 commodities. But if all these prices move together, it may make sense to 3 See John Hicks, Value and Capital (Oxford: Oxford University Press, 1939), mathematical appendices. There is some debate about whether this result should be called Hicks’ “second” or “third” law. In fact, two other laws that we have already seen are listed by Hicks: (1) ∂x ci =∂pi 0 (negativity of the own-substitution effect); and (2) ∂x ci =∂pj ¼ ∂x cj =∂pi (symmetry of cross-substitution effects). But he refers explicitly only to two “properties” in his written summary of his results. 4 To see this, notice that all substitution effects, sij , could be recorded in an n n matrix. However, symmetry of the effects (sij ¼ sji ) implies that only those terms on and below the principal diagonal of this matrix may be distinctly different from each other. This amounts to half the terms in the matrix (n 2 =2) plus the remaining half of the terms on the main diagonal of the matrix (n=2).

Chapter 6 Demand Relationships among Goods

189

lump them into a single “composite commodity,” y. Formally, if we let p02 , …, p0n represent the initial prices of these goods, then we assume that these prices can only vary together. They might all double, or all decline by 50 percent, but the relative prices of x2 , …, xn would not change. Now we deﬁne the composite commodity y to be total expenditures on x2 , …, xn , using the initial prices p02 , …, p0n : (6.28) y ¼ p 02 x2 þ p 03 x3 þ … þ p 0n xn . This person’s initial budget constraint is given by I ¼ p1 x1 þ p 02 x2 þ … þ p 0n xn ¼ p1 x1 þ y.

(6.29)

By assumption, all of the prices p2 , …, pn change in unison. Assume all of these prices change by a factor of t ðt > 0Þ. Now the budget constraint is (6.30) I ¼ p1 x1 þ tp 02 x2 þ … þ tp 0n xn ¼ p1 x1 þ ty. Consequently, the factor of proportionality, t , plays the same role in this person’s budget constraint as did the price of yðpy Þ in our earlier two-good analysis. Changes in p1 or t induce the same kinds of substitution effects we have been analyzing. So long as p2 , …, pn move together, we can therefore confine our examination of demand to choices between buying x1 or buying “everything else.”5 Simplified graphs that show these two goods on their axes can therefore be defended rigorously so long as the conditions of this “composite commodity theorem” (that all other prices move together) are satisfied. Notice, however, that the theorem makes no predictions about how choices of x2 , …, xn behave; they need not move in unison. The theorem focuses only on total spending on x2 , …, xn , not on how that spending is allocated among specific items (although this allocation is assumed to be done in a utility-maximizing way).

Generalizations and limitations The composite commodity theorem applies to any group of commodities whose relative prices all move together. It is possible to have more than one such commodity if there are several groupings that obey the theorem (i.e., expenditures on “food,” “clothing,” and so forth). Hence, we have developed the following deﬁnition. Composite commodity. A composite commodity is a group of goods for which all prices DEFINITION move together. These goods can be treated as a single “commodity” in that the individual behaves as if he or she were choosing between other goods and total spending on the entire composite group. This deﬁnition and the related theorem are very powerful results. They help simplify many problems that would otherwise be intractable. Still, one must be rather careful in applying the theorem to the real world because its conditions are stringent. Finding a set of commodities whose prices move together is rare. Slight departures from strict proportionality may negate the composite commodity theorem if cross-substitution effects are large. In the Extensions to this chapter, we look at ways to simplify situations where prices move independently.

5 The idea of a “composite commodity” was also introduced by J. R. Hicks in Value and Capital, 2nd ed. (Oxford: Oxford University Press, 1946), pp. 312–13. Proof of the theorem relies on the notion that to achieve maximum utility, the ratio of the marginal utilities for x2 , …, xn must remain unchanged when p2 , …, pn all move together. Hence, the n-good problem can be reduced to the two-dimensional problem of equating the ratio of the marginal utility from x to that from y to the “price ratio” p1 =t .

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EXAMPLE 6.3 Housing Costs as a Composite Commodity Suppose that an individual receives utility from three goods: food (x), housing services (y) measured in hundreds of square feet, and household operations (z) as measured by electricity use. If the individual’s utility is given by the three-good CES function 1 1 1 , (6.31) utility ¼ U ðx, y, zÞ ¼ x y z then the Lagrangian technique can be used to calculate Marshallian demand functions for these goods as I x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ , px þ px py þ px pz y ¼ z ¼

I pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ , py þ py px þ py pz pz þ

(6.32)

I pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ . pz px þ pz py

If initially I ¼ 100, px ¼ 1, py ¼ 4, and pz ¼ 1, then the demand functions predict x ¼ 25, y ¼ 12:5, z ¼ 25:

(6.33)

Hence, 25 is spent on food and a total of 75 is spent on housing-related needs. If we assume that housing service prices (py ) and household operation prices (pz ) always move together, then we can use their initial prices to define the “composite commodity” housing (h) as h ¼ 4y þ 1z.

(6.34)

Here, we also (arbitrarily) define the initial price of housing (ph ) to be 1. The initial quantity of housing is simply total dollars spent on h: h ¼ 4ð12:5Þ þ 1ð25Þ ¼ 75.

(6.35)

Furthermore, because py and pz always move together, ph will always be related to these prices by (6.36) ph ¼ pz ¼ 0:25py . Using this information, we can recalculate the demand function for x as a function of I , px , and ph : I pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ x ¼ px þ 4px ph þ px ph I (6.37) ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ . py þ 3 px ph As before, initially I ¼ 100, px ¼ 1, and ph ¼ 1, so x ¼ 25. Spending on housing can be most easily calculated from the budget constraint as h ¼ 75, because spending on housing represents “everything” other than food. An increase in housing costs. If the prices of y and z were to rise proportionally to py ¼ 16, pz ¼ 4 (with px remaining at 1), then ph would also rise to 4. Equation 6.37 now predicts that the demand for x would fall to 100 100 pﬃﬃﬃ ¼ (6.38) x ¼ 7 1þ3 4

Chapter 6 Demand Relationships among Goods

and that housing purchases would be given by ph h ¼ 100

100 600 ¼ , 7 7

(6.39)

or, because ph ¼ 4, 150 . (6.40) 7 Notice that this is precisely the level of housing purchases predicted by the original demand functions for three goods in Equation 6.32. With I ¼ 100, px ¼ 1, py ¼ 16, and pz ¼ 4, these equations can be solved as 100 , x ¼ 7 100 (6.41) y ¼ , 28 100 , z ¼ 14 and so the total amount of the composite good “housing” consumed (according to Equation 6.34) is given by 150 . (6.42) h ¼ 4y þ 1z ¼ 7 Hence, we obtained the same responses to price changes regardless of whether we chose to examine demands for the three goods x, y, and z or to look only at choices between x and the composite good h. h ¼

QUERY: How do we know that the demand function for x in Equation 6.37 continues to ensure utility maximization? Why is the Lagrangian constrained maximization problem unchanged by making the substitutions represented by Equation 6.36?

HOME PRODUCTION, ATTRIBUTES OF GOODS, AND IMPLICIT PRICES So far in this chapter we have focused on what economists can learn about the relationships among goods by observing individuals’ changing consumption of these goods in reaction to changes in market prices. In some ways this analysis skirts the central question of why coffee and cream go together or why ﬁsh and chicken may substitute for each other in a person’s diet. To develop a deeper understanding of such questions, economists have sought to explore activities within individuals’ households. That is, they have devised models of nonmarket types of activities such as parental child care, meal preparation, or do-it-yourself construction to understand how such activities ultimately result in demands for goods in the market. In this section we brieﬂy review some of these models. Our primary goal is to illustrate some of the implications of this approach for the traditional theory of choice.

Household production model The starting point for most models of household production is to assume that individuals do not receive utility directly from goods they purchase in the market (as we have been assuming so far). Instead, it is only when market goods are combined with time inputs by the individual that utility-providing outputs are produced. In this view, then, raw beef and uncooked

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potatoes yield no utility until they are cooked together to produce stew. Similarly, market purchases of beef and potatoes can be understood only by examining the individual’s preferences for stew and the underlying technology through which it is produced. In formal terms, assume as before that there are three goods that a person might purchase in the market: x, y, and z. Purchasing these goods provides no direct utility, but the goods can be combined by the individual to produce either of two home-produced goods: a1 or a2 . The technology of this household production can be represented by the production functions f1 and f2 (see Chapter 9 for a more complete discussion of the production function concept). Therefore, a1 ¼ f1 ðx, y, zÞ, (6.43) a2 ¼ f2 ðx, y, zÞ, and utility ¼ U ða1 , a2 Þ.

(6.44)

The individual’s goal is to choose x, y, z so as to maximize utility subject to the production constraints and to a financial budget constraint:6 px x þ py y þ pz z ¼ I . (6.45) Although we will not examine in detail the results that can be derived from this general model, two insights that can be drawn from it might be mentioned. First, the model may help clarify the nature of market relationships among goods. Because the production functions in Equations 6.43 are in principle measurable using detailed data on household operations, households can be treated as “multi-product” firms and studied using many of the techniques economists use to study production. A second insight provided by the household production approach is the notion of the “implicit” or “shadow” prices associated with the home-produced goods a1 and a2 . Because consuming more a1 , say, requires the use of more of the “ingredients” x, y, and z, this activity obviously has an opportunity cost in terms of the quantity of a2 that can be produced. To produce more bread, say, a person must not only divert some ﬂour, milk, and eggs from using them to make cupcakes but may also have to alter the relative quantities of these goods purchased because he or she is bound by an overall budget constraint. Hence, bread will have an implicit price in terms of the number of cupcakes that must be forgone in order to be able to consume one more loaf. That implicit price will reﬂect not only the market prices of bread ingredients but also the available household production technology and, in more complex models, the relative time inputs required to produce the two goods. As a starting point, however, the notion of implicit prices can be best illustrated with a very simple model.

The linear attributes model A particularly simple form of the household production model was ﬁrst developed by K. J. Lancaster to examine the underlying “attributes” of goods.7 In this model, it is the attributes of goods that provide utility to individuals, and each speciﬁc good contains a ﬁxed set of attributes. If, for example, we focus only on the calories (a1 ) and vitamins (a2 ) that various foods provide, Lancaster’s model assumes that utility is a function of these attributes and that individuals purchase various foods only for the purpose of obtaining the calories and vitamins they offer. In mathematical terms, the model assumes that the “production”

6 Often household production theory also focuses on the individual’s allocation of time to producing a1 and a2 or to working in the market. In Chapter 16 we look at a few simple models of this type. 7

See K. J. Lancaster, “A New Approach to Consumer Theory,” Journal of Political Economy 74 (April 1966): 132–57.

Chapter 6 Demand Relationships among Goods

equations have the simple form a1 ¼ a 1x x þ a 1y y þ a 1z z, a2 ¼ a 2x x þ a 2y y þ a 2z z,

(6.46)

where a 1x represents the number of calories per unit of food x, a 2x represents the number of vitamins per unit of food x, and so forth. In this form of the model, then, there is no actual “production” in the home. Rather, the decision problem is how to choose a diet that provides the optimal mix of calories and vitamins given the available food budget.

Illustrating the budget constraints To begin our examination of the theory of choice under the attributes model, we ﬁrst illustrate the budget constraint. In Figure 6.2, the ray 0x records the various combinations of a1 and a2 available from successively larger amounts of good x. Because of the linear production technology assumed in the attributes model, these combinations of a1 and a2 lie along such a straight line, though in more complex models of home production that might not be the case. Similarly, rays of 0y and 0z show the quantities of the attributes a1 and a2 provided by various amounts of goods y and z that might be purchased. If this person spends all of his or her income on good x, then the budget constraint (Equation 6.45) allows the purchase of I (6.47) x ¼ , px and that will yield a 1 ¼ a 1x x ¼

a 1x I , px

a2I a 2 ¼ a 2x x ¼ x . px

(6.48)

This point is recorded as point x on the 0x ray in Figure 6.2. Similarly, the points y and z represent the combinations of a1 and a2 that would be obtained if all income were spent on good y or good z, respectively. Bundles of a1 and a2 that are obtainable by purchasing both x and y (with a ﬁxed budget) are represented by the line joining x and y in Figure 6.2.8 Similarly, the line x z represents the combinations of a1 and a2 available from x and z, and the line y z shows combinations available from mixing y and z. All possible combinations from mixing the three market goods are represented by the shaded triangular area x y z .

Corner solutions One fact is immediately apparent from Figure 6.2: A utility-maximizing individual would never consume positive quantities of all three of these goods. Only the northeast perimeter of the x y z triangle represents the maximal amounts of a1 and a2 available to this person given his or her income and the prices of the market goods. Individuals with a preference toward a1 will have indifference curves similar to U0 and will maximize utility by choosing a point such as E. The combination of a1 and a2 speciﬁed by that point can be obtained by 8

Mathematically, suppose a fraction α of the budget is spent on x and (1 α) on y; then a1 ¼ αa 1x x þ ð1 αÞa 1y y , a ¼ αa 2 x þ ð1 αÞa 2 y . 2

x

y

The line x y is traced out by allowing α to vary between 0 and 1. The lines x z and y z are traced out in a similar way, as is the triangular area x y z .

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FIGURE 6.2

Utility Maximization in the Attributes Model The points x , y , and z show the amounts of attributes a1 and a2 that can be purchased by buying only x, y, or z, respectively. The shaded area shows all combinations that can be bought with mixed bundles. Some individuals may maximize utility at E, others at E 0 . a2 x U′0 a*2

y

x*

E′ y* z

E U0 z*

0

a*1

a1

consuming only goods y and z. Similarly, a person with preferences represented by the indifference curve U 00 will choose point E 0 and consume only goods x and y. The attributes model therefore predicts that corner solutions at which individuals consume zero amounts of some commodities will be relatively common, especially in cases where individuals attach value to fewer attributes (here, two) than there are market goods to choose from (three). If income, prices, or preferences change, then consumption patterns may also change abruptly. Goods that were previously consumed may cease to be bought and goods previously neglected may experience a signiﬁcant increase in purchases. This is a direct result of the linear assumptions inherent in the production functions assumed here. In household production models with greater substitutability assumptions, such discontinuous reactions are less likely.

SUMMARY In this chapter, we used the utility-maximizing model of choice to examine relationships among consumer goods. Although these relationships may be complex, the analysis presented here provided a number of ways of categorizing and simplifying them. •

When there are only two goods, the income and substitution effects from the change in the price of one good (say, py ) on the demand for another good (x) usually work in opposite directions. The sign of ∂x=∂py is therefore ambiguous: its substitution effect is positive but its income effect is negative.

•

In cases of more than two goods, demand relationships can be speciﬁed in two ways. Two goods (xi and xj ) are “gross substitutes” if ∂xi =∂pj > 0 and “gross complements” if ∂xi =∂pj < 0. Unfortunately, because these price effects include income effects, they need not be symmetric. That is, ∂xi =∂pj does not necessarily equal ∂xj =∂pi .

•

Focusing only on the substitution effects from price changes eliminates this ambiguity because substitution effects are symmetric; that is, ∂x ci =∂pj ¼ ∂x cj =∂pi . Now two goods are deﬁned as net (or Hicksian) substitutes if ∂x ci =∂pj > 0 and net complements if ∂x ci =∂pj < 0. Hicks’

Chapter 6 Demand Relationships among Goods “second law of demand” shows that net substitutes are more prevalent. •

•

If a group of goods has prices that always move in unison, then expenditures on these goods can be treated as a “composite commodity” whose “price” is given by the size of the proportional change in the composite goods’ prices.

195

An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes. This may provide additional insights into relationships among goods.

PROBLEMS 6.1 Heidi receives utility from two goods, goat’s milk (m) and strudel (s), according to the utility function U ðm, sÞ ¼ m ⋅ s. a. Show that increases in the price of goat’s milk will not affect the quantity of strudel Heidi buys; that is, show that ∂s=∂pm ¼ 0. b. Show also that ∂m=∂ps ¼ 0. c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives in parts (a) and (b) are identical. d. Prove part (c) explicitly using the Marshallian demand functions for m and s.

6.2 Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good that exhibits Giffen’s paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop an intuitive explanation to suggest why a rise in the price of rotgut must cause fewer jelly donuts to be bought. That is, the goods must also be gross complements.

6.3 Donald, a frugal graduate student, consumes only coffee (c) and buttered toast (bt ). He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast. a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter (pb ) and toast (pt )? b. Explain why ∂c=∂pbt ¼ 0. c. Is it also true here that ∂c=∂pb and ∂c=∂pt are equal to 0?

6.4 Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by utility ¼ b ⋅ t ⋅ p, where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus travel (pt =pb ) never changes. a. How might one deﬁne a composite commodity for ground transportation? b. Phrase Sarah’s optimization problem as one of choosing between ground ( g) and air ( p) transportation. c. What are Sarah’s demand functions for g and p? d. Once Sarah decides how much to spend on g, how will she allocate those expenditures between b and t ?

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6.5 Suppose that an individual consumes three goods, x1 , x2 , and x3 , and that x2 and x3 are similar commodities (i.e., cheap and expensive restaurant meals) with p2 ¼ kp3 , where k < 1—that is, the goods’ prices have a constant relationship to one another. a. Show that x2 and x3 can be treated as a composite commodity. b. Suppose both x2 and x3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this transaction cost affect the price of x2 relative to that of x3 ? How will this effect vary with the value of t ? c. Can you predict how an income-compensated increase in t will affect expenditures on the composite commodity x2 and x3 ? Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between x2 and x3 ?

6.6 Apply the results of Problem 6.5 to explain the following observations: a. It is difﬁcult to ﬁnd high-quality apples to buy in Washington State or good fresh oranges in Florida. b. People with signiﬁcant baby-sitting expenses are more likely to have meals out at expensive (rather than cheap) restaurants than are those without such expenses. c. Individuals with a high value of time are more likely to ﬂy the Concorde than those with a lower value of time. d. Individuals are more likely to search for bargains for expensive items than for cheap ones. Note: Observations (b) and (d) form the bases for perhaps the only two murder mysteries in which an economist solves the crime; see Marshall Jevons, Murder at the Margin and The Fatal Equilibrium.

6.7 In general, uncompensated cross-price effects are not equal. That is, ∂xi ∂xj 6¼ . ∂pj ∂pi Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem 6.1.)

6.8 Example 6.3 computes the demand functions implied by the three-good CES utility function 1 1 1 U ðx, y, zÞ ¼ − − − . x y z a. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements. b. How would you determine whether x and y or x and z are net substitutes or net complements?

Analytical Problems 6.9 Consumer surplus with many goods In Chapter 5, we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods.

Chapter 6 Demand Relationships among Goods a. Suppose that an individual consumes n goods and that the prices of two of those goods (say, p1 and p2 ) rise. How would you use the expenditure function to measure the compensating variation (CV) for this person of such a price rise? b. A way to show these welfare costs graphically would be to use the compensated demand curves for goods x1 and x2 by assuming that one price rose before the other. Illustrate this approach. c. In your answer to part (b), would it matter in which order you considered the price changes? Explain. d. In general, would you think that the CV for a price rise of these two goods would be greater if the goods were net substitutes or net complements? Or would the relationship between the goods have no bearing on the welfare costs?

6.10 Separable utility A utility function is called separable if it can be written as U ðx, yÞ ¼ U1 ðxÞ þ U2 ðyÞ, where Ui0 > 0, U 00i < 0, and U1 , U2 need not be the same function. a. What does separability assume about the cross-partial derivative Ux y ? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to conclude deﬁnitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

6.11 Graphing complements Graphing complements is complicated because a complementary relationship between goods (under the Hicks deﬁnition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by x1 , x2 , x3 . Now proceed as follows. a. Draw an indifference curve for x2 and x3 , holding the quantity of x1 constant at x 01 . This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for x2 , x3 , holding x1 constant at x 01 h. For this new indifference curve, show the amount of extra x2 that would compensate this person for the loss of x1 ; call this amount j . Similarly, show that amount of extra x3 that would compensate for the loss of x1 and call this amount k. c. Suppose now that an individual is given both amounts j and k, thereby permitting him or her to move to an even higher x2 =x3 indifference curve. Show this move on your graph and draw this new indifference curve. d. Samuelson now suggests the following deﬁnitions: •

If the new indifference curve corresponds to the indifference curve when x1 ¼ x 01 2h, goods 2 and 3 are independent.

•

If the new indifference curve provides more utility than when x1 ¼ x 01 2h, goods 2 and 3 are complements.

•

If the new indifference curve provides less utility than when x1 ¼ x 01 2h, goods 2 and 3 are substitutes.

Show that these graphical definitions are symmetric.

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Part 2 Choice and Demand e. Discuss how these graphical deﬁnitions correspond to Hicks’ more mathematical deﬁnitions given in the text. f. Looking at your ﬁnal graph, do you think that this approach fully explains the types of relationships that might exist between x2 and x3 ?

6.12 Shipping the good apples out Details of the analysis suggested in Problems 6.5 and 6.6 were originally worked out by Borcherding and Silberberg (see the Suggested Readings) based on a supposition ﬁrst proposed by Alchian and Allen. These authors look at how a transaction charge affects the relative demand for two closely substitutable items. Assume that goods x2 and x3 are close substitutes and are subject to a transaction charge of t per unit. Suppose also that good 2 is the more expensive of the two goods (i.e., “good apples” as opposed to “cooking apples”). Hence the transaction charge lowers the relative price of the more expensive good [that is, ð p2 þ t Þ=ð p3 þ t Þ falls as t increases]. This will increase the relative demand for the expensive good if ∂ðx c2 =x c3 Þ=∂t > 0 (where we use compensated demand functions in order to eliminate pesky income effects). Borcherding and Silberberg show this result will probably hold using the following steps. a. Use the derivative of a quotient rule to expand ∂ðx c2 =x c3 Þ=∂t . b. Use your result from part (a) together with the fact that, in this problem, ∂x ci =∂t ¼ ∂x ci =∂p2 þ ∂x ci =∂p3 for i ¼ 2, 3, to show that the derivative we seek can be written as

∂ðx c2 =x c3 Þ x c2 s22 s23 s32 s33 , þ ¼ c x 3 x2 x2 x3 x3 ∂t where sij ¼ ∂x ci =∂pj . c. Rewrite the result from part (b) in terms of compensated price elasticities: ∂x c pj eijc ¼ i ⋅ c . ∂pj xi d. Use Hicks’ third law (Equation 6.26) to show that the term in brackets in parts (b) and (c) can now be written as ½ðe22 e32 Þð1=p2 1=p3 Þ þ ðe21 e31 Þ=p3 . e. Develop an intuitive argument about why the expression in part (d) is likely to be positive under the conditions of this problem. Hints: Why is the ﬁrst product in the brackets positive? Why is the second term in brackets likely to be small? f. Return to Problem 6.6 and provide more complete explanations for these various ﬁndings.

Chapter 6 Demand Relationships among Goods

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SUGGESTIONS FOR FURTHER READING Borcherding, T. E., and E. Silberberg. “Shipping the Good Apples Out—The Alchian-Allen Theorem Reconsidered,” Journal of Political Economy (February 1978): 131–38. Good discussion of the relationships among three goods in demand theory. See also Problems 6.5 and 6.6.

Hicks, J. R. Value and Capital, 2nd ed. Oxford: Oxford University Press, 1946. See Chaps. I–III and related appendices. Proof of the composite commodity theorem. Also has one of the ﬁrst treatments of net substitutes and complements.

Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Explores the consequences of the symmetry of compensated crossprice effects for various aspects of demand theory.

Rosen, S. “Hedonic Prices and Implicit Markets.” Journal of Political Economy (January/February 1974): 34–55. Nice graphical and mathematical treatment of the attribute approach to consumer theory and of the concept of “markets” for attributes.

Samuelson, P. A. “Complementarity—An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory.” Journal of Economic Literature (December 1977): 1255–89. Reviews a number of deﬁnitions of complementarity and shows the connections among them. Contains an intuitive, graphical discussion and a detailed mathematical appendix.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGrawHill, 2001. Good discussion of expenditure functions and the use of indirect utility functions to illustrate the composite commodity theorem and other results.

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EXTENSIONS Simplifying Demand and Two-Stage Budgeting In Chapter 6 we saw that the theory of utility maximization in its full generality imposes rather few restrictions on what might happen. Other than the fact that net cross-substitution effects are symmetric, practically any type of relationship among goods is consistent with the underlying theory. This situation poses problems for economists who wish to study consumption behavior in the real world—theory just does not provide very much guidance when there are many thousands of goods potentially available for study. There are two general ways in which simpliﬁcations are made. The ﬁrst uses the composite commodity theorem from Chapter 6 to aggregate goods into categories within which relative prices move together. For situations where economists are speciﬁcally interested in changes in relative prices within a category of spending (such as changes in the relative prices of various forms of energy), this process will not do, however. An alternative is to assume that consumers engage in a two-stage process in their consumption decisions. First they allocate income to various broad groupings of goods (food, clothing, and so forth) and then, given these expenditure constraints, they maximize utility within each of the subcategories of goods using only information about those goods’ relative prices. In that way, decisions can be studied in a simpliﬁed setting by looking only at one category at a time. This process is called “two-stage” budgeting. In these extensions, we ﬁrst look at the general theory of two-stage budgeting and then turn to examine some empirical examples.

E6.1 Theory of two-stage budgeting The issue that arises in two-stage budgeting can be stated succinctly: Does there exist a partition of goods into m nonoverlapping groups (denoted by r ¼ 1, m) and a separate budget (lr ) devoted to each category such that the demand functions for the goods within any one category depend only on the prices of goods within the category and on the category’s budget allocation? That is, can we partition goods so that demand is given by xi ð p1 , …, pn , I Þ ¼ xi2r ð pi2r , Ir Þ

(i)

for r ¼ 1, m, ? That it might be possible to do this is suggested by comparing the following two-stage maximization problem,

V ð p1 ,…,pn ,I1 ,…,Im Þ i h X pi xi Ir ,r ¼ 1,m ¼ max U ðx1 ,…,xn Þ s:t: x1 ,…,xn

i2r

(ii) and max V

I1 , …, Im

s. t.

m X

Ir ¼ I ,

r¼1

to the utility-maximization problem we have been studying, n X max U ðx1 , …, xn Þ s. t. pi xi I . (iii) xi

i¼1

Without any further restrictions, these two maximization processes will yield the same result; that is, Equation ii is just a more complicated way of stating Equation iii. So, some restrictions have to be placed on the utility function to ensure that the demand functions that result from solving the two-stage process will be of the form specified in Equation i. Intuitively, it seems that such a categorization of goods should work providing that changes in the price of a good in one category do not affect the allocation of spending for goods in any category other than its own. In Problem 6.9 we showed a case where this is true for an “additively separable” utility function. Unfortunately, this proves to be a very special case. The more general mathematical restrictions that must be placed on the utility function to justify two-stage budgeting have been derived (see Blackorby, Primont, and Russell, 1978), but these are not especially intuitive. Of course, economists who wish to study decentralized decisions by consumers (or, perhaps more importantly, by firms that operate many divisions) must do something to simplify matters. Now we look at a few applied examples.

E6.2 Relation to the composition commodity theorem Unfortunately, neither of the two available theoretical approaches to demand simpliﬁcation is completely satisfying. The composite commodity theorem requires that the relative prices for goods within one group remain constant over time, an assumption that has been rejected during many different historical periods.

Chapter 6 Demand Relationships among Goods

On the other hand, the kind of separability and twostage budgeting indicated by the utility function in Equation i also requires very strong assumptions about how changes in prices for a good in one group affect spending on goods in any other group. These assumptions appear to be rejected by the data (see Diewert and Wales, 1995). Economists have tried to devise even more elaborate, hybrid methods of aggregation among goods. For example, Lewbel (1996) shows how the composite commodity theorem might be generalized to cases where within-group relative prices exhibit considerable variability. He uses this generalization for aggregating U.S. consumer expenditures into six large groups (food, clothing, household operation, medical care, transportation, and recreation). Using these aggregates, he concludes that his procedure is much more accurate than assuming two-stage budgeting among these expenditure categories.

E6.3 Homothetic functions and energy demand One way to simplify the study of demand when there are many commodities is to assume that utility for certain subcategories of goods is homothetic and may be separated from the demand for other commodities. This procedure was followed by Jorgenson, Slesnick, and Stoker (1997) in their study of energy

201

demand by U.S. consumers. By assuming that demand functions for speciﬁc types of energy are proportional to total spending on energy, the authors were able to concentrate their empirical study on the topic that is of most interest to them: estimating the price elasticities of demand for various types of energy. They conclude that most types of energy (that is, electricity, natural gas, gasoline, and so forth) have fairly elastic demand functions. Demand appears to be most responsive to price for electricity.

References Blackorby, Charles, Daniel Primont, and R. Robert Russell. Duality, Separability and Functional Structure: Theory and Economic Applications. New York: North Holland, 1978. Diewert, W. Erwin, and Terrence J. Wales. “Flexible Functional Forms and Tests of Homogeneous Separability.” Journal of Econometrics (June 1995): 259–302. Jorgenson, Dale W., Daniel T. Slesnick, and Thomas M. Stoker. “Two-Stage Budgeting and Consumer Demand for Energy.” In Dale W. Jorgenson, Ed., Welfare, vol. 1: Aggregate Consumer Behavior, pp. 475–510. Cambridge, MA: MIT Press, 1997. Lewbel, Arthur. “Aggregation without Separability: A Standardized Composite Commodity Theorem.” American Economic Review (June 1996): 524–43.

CHAPTER

7 Uncertainty and Information In this chapter we will explore some of the basic elements of the theory of individual behavior in uncertain situations. Our general goal is to show why individuals do not like risk and how they may adopt strategies to reduce it. More generally, the chapter is intended to provide a brief introduction to issues raised by the possibility that information may be imperfect when individuals make utility-maximizing decisions. Some of the themes developed here will recur throughout the remainder of the book.

MATHEMATICAL STATISTICS Many of the formal tools for modeling uncertainty in economic situations were originally developed in the ﬁeld of mathematical statistics. Some of these tools were reviewed in Chapter 2 and in this chapter we will be making a great deal of use of the concepts introduced there. Speciﬁcally, four statistical ideas will recur throughout this chapter. •

Random variable: A random variable is a variable that records, in numerical form, the possible outcomes from some random event.1

•

Probability density function (PDF): A function that shows the probabilities associated with the possible outcomes from a random variable. Expected value of a random variable: The outcome of a random variable that will occur “on average.” The expected valueP is denoted by EðxÞ. If x is a discrete random variable with n outcomes then EðxÞ ¼ ni¼1 xi ¼ f ðxi Þ, where f ðxÞ is the PDF for the random variable x. If x is a continuous random variable, then EðxÞ ¼ ∞ ∫þ ∞ xf ðxÞ dx. Variance and standard deviation of a random variable: These concepts measure the dispersion of Pa random variable about its expected value. In the discrete case, VarðxÞ ¼ σ2x ¼ ni¼1 ½xi EðxÞ2 f ðxi Þ; in the continuous case, VarðxÞ ¼ σ2x ¼ 2 ∞ ∫þ ∞ ½x EðxÞ f ðxÞ dx. The standard deviation is the square root of the variance.

•

•

As we shall see, all of these concepts will come into play when we begin looking at the decision-making process of a person faced with a number of uncertain outcomes that can be conceptually represented by a random variable.

When it is necessary to differentiate between random variables and nonrandom variables, we will use the notation ∼ x to denote the fact that the variable x is random in that it takes on a number of potential randomly determined outcomes. Often, however, it will not be necessary to make the distinction because randomness will be clear from the context of the problem. 1

202

Chapter 7 Uncertainty and Information

FAIR GAMES AND THE EXPECTED UTILITY HYPOTHESIS A “fair game” is a random game with a speciﬁed set of prizes and associated probabilities that has an expected value of zero. For example, if you ﬂip a coin with a friend for a dollar, the expected value of this game is zero because EðxÞ ¼ 0:5ðþ$1Þ þ 0:5ð$1Þ ¼ 0, (7.1) where wins are recorded with a plus sign and losses with a minus sign. Similarly, a game that promised to pay you $10 if a coin came up heads but would cost you only $1 if it came up tails would be “unfair” because EðxÞ ¼ 0:5ðþ$10Þ þ 0:5ð$1Þ ¼ $4:50. (7.2) This game can easily be converted into a fair game, however, simply by charging you an entry fee of $4.50 for the right to play.2 It has long been recognized that most people would prefer not to play fair games. Although people may sometimes willingly ﬂip a coin for a few dollars, they would generally balk at playing a similar game whose outcome was +$1 million or $1 million. One of the ﬁrst mathematicians to study the reasons for this unwillingness to engage in fair bets was Daniel Bernoulli in the eighteenth century.3 His examination of the famous St. Petersburg paradox provided the starting point for virtually all studies of the behavior of individuals in uncertain situations.

St. Petersburg paradox In the St. Petersburg paradox, the following game is proposed: A coin is ﬂipped until a head appears. If a head ﬁrst appears on the nth ﬂip, the player is paid $2n . This game has an inﬁnite number of outcomes (a coin might be ﬂipped from now until doomsday and never come up a head, although the likelihood of this is small), but the ﬁrst few can easily be written down. If xi represents the prize awarded when the ﬁrst head appears on the ith trial, then x1 ¼ $2, x2 ¼ $4, x3 ¼ $8, …, xn ¼ $2n . (7.3) 1 i The probability of getting a head for the first time on the ith trial is 2 ; it is the probability of getting (i 1) tails and then a head. Hence the probabilities of the prizes given in Equation 7.3 are 1 1 1 1 (7.4) π1 ¼ , π2 ¼ , π3 ¼ , …, πn ¼ n . 2 4 8 2 The expected value of the St. Petersburg paradox game is therefore infinite: ∞ ∞ X X EðxÞ ¼ πi xi ¼ 2i ð1=2i Þ i¼1

i¼1

(7.5) ¼ 1 þ 1 þ 1 þ … þ 1 þ … ¼ ∞. Some introspection, however, should convince anyone that no player would pay very much (much less than infinity) to play this game. If I charged $1 billion to play the game, I would surely have no takers, despite the fact that $1 billion is still considerably less than the expected value of the game. This, then, is the paradox: Bernoulli’s game is in some sense not worth its (infinite) expected dollar value.

2

The games discussed here are assumed to yield no utility in their play other than the prizes; hence, the observation that many individuals gamble at “unfair” odds is not necessarily a refutation of this statement. Rather, such individuals can reasonably be assumed to be deriving some utility from the circumstances associated with the play of the game. It is therefore possible to differentiate the consumption aspect of gambling from the pure risk aspect.

3 The original Bernoulli paper has been reprinted as D. Bernoulli, “Exposition of a New Theory on the Measurement of Risk,” Econometrica 22 (January 1954): 23–36.

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Part 2 Choice and Demand

Expected utility Bernoulli’s solution to this paradox was to argue that individuals do not care directly about the dollar prizes of a game; rather, they respond to the utility these dollars provide. If we assume that the marginal utility of wealth declines as wealth increases, the St. Petersburg game may converge to a ﬁnite expected utility value that players would be willing to pay for the right to play. Bernoulli termed this expected utility value the moral value of the game because it represents how much the game is worth to the individual. Because utility may rise less rapidly than the dollar value of the prizes, it is possible that a game’s moral value will fall short of its monetary expected value. Example 7.1 looks at some issues related to Bernoulli’s solution.

EXAMPLE 7.1 Bernoulli’s Solution to the Paradox and Its Shortcomings Suppose, as did Bernoulli, that the utility of each prize in the St. Petersburg paradox is given by (7.6) U ðxi Þ ¼ lnðxi Þ. This logarithmic utility function exhibits diminishing marginal utility (that is, U 0 > 0 but U 00 < 0), and the expected utility value of this game converges to a finite number: ∞ X expected utility ¼ πi U ðxÞi i¼1 ∞ X 1 ¼ lnð2i Þ. i 2 i¼1

(7.7)

Some manipulation of this expression yields4 the result that the expected utility value of this game is 1.39. An individual with this type of utility function might therefore be willing to invest resources that otherwise yield up to 1.39 units of utility (a certain wealth of about $4 provides this utility) in purchasing the right to play this game. Assuming that the very large prizes promised by the St. Petersburg paradox encounter diminishing marginal utility therefore permitted Bernoulli to offer a solution to the paradox. Unbounded utility. Bernoulli’s solution to the St. Petersburg paradox, unfortunately, does not completely solve the problem. So long as there is no upper bound to the utility function, the paradox can be regenerated by redeﬁning the game’s prizes. For example, with the i logarithmic utility function, prizes can be set as xi ¼ e 2 , in which case i

(7.8) U ðxi Þ ¼ ln½e 2 ¼ 2i and the expected utility value of the game would again be infinite. Of course, the prizes in this redefined game are very large. For example, if a head first appears on the fifth flip, a person 5 would win e 2 ¼ e 32 ¼ $7:9 ⋅ 1013 , though the probability of winning this would be only 5 1=2 ¼ 0:031. The idea that people would pay a great deal (say, billions of dollars) to play games with small probabilities of such large prizes seems, to many observers, to be unlikely. Hence, in many respects the St. Petersburg game remains a paradox.

4

Proof : expected utility ¼

∞ ∞ X X i i ln 2 ¼ ln 2 : i ⋅ i 2 i¼1 i¼1 2

But the value of this final infinite series can be shown to be 2.0. Hence, expected utility ¼ 2 ln 2 ¼ 1:39.

Chapter 7 Uncertainty and Information

QUERY: Here are two alternative solutions to the St. Petersburg paradox. For each, calculate the expected value of the original game. 1. Suppose individuals assume that any probability less than 0.01 is in fact zero. 2. Suppose that the utility from the St. Petersburg prizes is given by

if xi 1,000,000, xi U ðxi Þ ¼ 1,000,000 if xi > 1,000,000.

THE VON NEUMANN–MORGENSTERN THEOREM In their book The Theory of Games and Economic Behavior, John von Neumann and Oscar Morgenstern developed mathematical models for examining the economic behavior of individuals under conditions of uncertainty.5 To understand these interactions, it was necessary ﬁrst to investigate the motives of the participants in such “games.” Because the hypothesis that individuals make choices in uncertain situations based on expected utility seemed intuitively reasonable, the authors set out to show that this hypothesis could be derived from more basic axioms of “rational” behavior. The axioms represent an attempt by the authors to generalize the foundations of the theory of individual choice to cover uncertain situations. Although most of these axioms seem eminently reasonable at ﬁrst glance, many important questions about their tenability have been raised. We will not pursue these questions here, however.6

The von Neumann–Morgenstern utility index To begin, suppose that there are n possible prizes that an individual might win by participating in a lottery. Let these prizes be denoted by x1 , x2 , …, xn and assume that these have been arranged in order of ascending desirability. Therefore, x1 is the least preferred prize for the individual and xn is the most preferred prize. Now assign arbitrary utility numbers to these two extreme prizes. For example, it is convenient to assign U ðx1 Þ ¼ 0, (7.9) U ðxn Þ ¼ 1, but any other pair of numbers would do equally well.7 Using these two values of utility, the point of the von Neumann–Morgenstern theorem is to show that a reasonable way exists to assign specific utility numbers to the other prizes available. Suppose that we choose any other prize, say, xi . Consider the following experiment. Ask the individual to state the probability, say, πi , at which he or she would be indifferent between xi with certainty, and a gamble offering prizes of xn with probability πi and x1 with probability ð1 πi Þ. It seems reasonable (although this is the most problematic assumption in the von Neumann–Morgenstern approach) that such a probability will exist: The individual will always be indifferent between a gamble and a sure thing, provided that a high enough probability of winning the best prize is offered. It also seems likely that πi will be higher the more desirable xi is; the better xi is, the 5

J. von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944). The axioms of rationality in uncertain situations are discussed in the book’s appendix.

6

For a discussion of some of the issues raised in the debate over the von Neumann–Morgenstern axioms, especially the assumption of independence, see C. Gollier, The Economics of Risk and Time (Cambridge, MA: MIT Press, 2001), chap. 1.

7

Technically, a von Neumann–Morgenstern utility index is unique only up to a choice of scale and origin—that is, only up to a “linear transformation.” This requirement is more stringent than the requirement that a utility function be unique up to a monotonic transformation.

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better the chance of winning xn must be to get the individual to gamble. The probability πi therefore measures how desirable the prize xi is. In fact, the von Neumann–Morgenstern technique is to define the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi : U ðxi Þ ¼ πi ⋅ U ðxn Þ þ ð1 πi Þ ⋅ U ðx1 Þ. Because of our choice of scale in Equation 7.9, we have

(7.10)

U ðxi Þ ¼ πi ⋅ 1 þ ð1 πi Þ ⋅ 0 ¼ πi . (7.11) By judiciously choosing the utility numbers to be assigned to the best and worst prizes, we have been able to devise a scale under which the utility number attached to any other prize is simply the probability of winning the top prize in a gamble the individual regards as equivalent to the prize in question. This choice of utility numbers is arbitrary. Any other two numbers could have been used to construct this utility scale, but our initial choice (Equation 7.9) is a particularly convenient one.

Expected utility maximization In line with the choice of scale and origin represented by Equation 7.9, suppose that probability πi has been assigned to represent the utility of every prize xi . Notice in particular that π1 ¼ 0, πn ¼ 1, and that the other utility values range between these extremes. Using these utility numbers, we can show that a “rational” individual will choose among gambles based on their expected “utilities” (that is, based on the expected value of these von Neumann–Morgenstern utility index numbers). As an example, consider two gambles. One gamble offers x2 , with probability q, and x3 , with probability (1 q). The other offers x5 , with probability t , and x6 , with probability (1 t ). We want to show that this person will choose gamble 1 if and only if the expected utility of gamble 1 exceeds that of gamble 2. Now for the gambles: expected utility ð1Þ ¼ q ⋅ U ðx2 Þ þ ð1 qÞ ⋅ U ðx3 Þ, (7.12) expected utility ð2Þ ¼ t ⋅ U ðx5 Þ þ ð1 t Þ ⋅ U ðx6 Þ. Substituting the utility index numbers (that is, π2 is the “utility” of x2 , and so forth) gives expected utilityð1Þ ¼ q ⋅ π2 þ ð1 qÞ ⋅ π3 , (7.13) expected utilityð2Þ ¼ t ⋅ π5 þ ð1 t Þ ⋅ π6 . We wish to show that the individual will prefer gamble 1 to gamble 2 if and only if (7.14) q ⋅ π2 þ ð1 qÞ ⋅ π3 > t ⋅ π5 þ ð1 t Þ ⋅ π6 . To show this, recall the definitions of the utility index. The individual is indifferent between x2 and a gamble promising x1 with probability (1 π2 ) and xn with probability π2 . We can use this fact to substitute gambles involving only x1 and xn for all utilities in Equation 7.13 (even though the individual is indifferent between these, the assumption that this substitution can be made implicitly assumes that people can see through complex lottery combinations). After a bit of messy algebra, we can conclude that gamble 1 is equivalent to a gamble promising xn with probability qπ2 þ ð1 qÞπ3 , and gamble 2 is equivalent to a gamble promising xn with probability t π5 þ ð1 t Þπ6 . The individual will presumably prefer the gamble with the higher probability of winning the best prize. Consequently, he or she will choose gamble 1 if and only if (7.15) qπ2 þ ð1 qÞπ3 > t π5 þ ð1 t Þπ6 . But this is precisely what we wanted to show. Consequently, we have proved that an individual will choose the gamble that provides the highest level of expected (von Neumann– Morgenstern) utility. We now make considerable use of this result, which can be summarized as follows.

Chapter 7 Uncertainty and Information

Expected utility maximization. If individuals obey the von Neumann–Morgenstern axioms of behavior in uncertain situations, they will act as if they choose the option that maximizes the expected value of their von Neumann–Morgenstern utility index.

RISK AVERSION Two lotteries may have the same expected monetary value but may differ in their riskiness. For example, ﬂipping a coin for $1 and ﬂipping a coin for $1,000 are both fair games, and both have the same expected value (0). However, the latter is in some sense more “risky” than the former, and fewer people would participate in the game where the prize was winning or losing $1,000. The purpose of this section is to discuss the meaning of the term risky and explain the widespread aversion to risk. The term risk refers to the variability of the outcomes of some uncertain activity.8 If variability is low, the activity may be approximately a sure thing. With no more precise notion of variability than this, it is possible to show why individuals, when faced with a choice between two gambles with the same expected value, will usually choose the one with a smaller variability of return. Intuitively, the reason behind this is that we usually assume that the marginal utility from extra dollars of prize money (that is, wealth) declines as the prizes get larger. A ﬂip of a coin for $1,000 promises a relatively small gain of utility if you win but a large loss of utility if you lose. A bet of only $1 is “inconsequential,” and the gain in utility from a win approximately counterbalances the decline in utility from a loss.9

Risk aversion and fair bets This argument is illustrated in Figure 7.1. Here W represents an individual’s current wealth and U ðW Þ is a von Neumann–Morgenstern utility index that reﬂects how he or she feels about various levels of wealth.10 In the ﬁgure, U ðW Þ is drawn as a concave function of W to reﬂect the assumption of a diminishing marginal utility. It is assumed that obtaining an extra dollar adds less to enjoyment as total wealth increases. Now suppose this person is offered two fair gambles: a 50–50 chance of winning or losing $h or a 50–50 chance of winning or losing $2h. The utility of present wealth is U ðW Þ: The expected utility if he or she participates in gamble 1 is given by U h ðW Þ: 1 1 U h ðW Þ ¼ U ðW þ hÞ þ U ðW hÞ, (7.16) 2 2 2h and the expected utility of gamble 2 is given by U ðW Þ: 1 1 (7.17) U 2h ðW Þ ¼ U ðW þ 2hÞ þ U ðW 2hÞ. 2 2 11 It is geometrically clear from the figure that U ðW Þ > U h ðW Þ > U 2h ðW Þ. (7.18)

8

Often the statistical concepts of variance and standard deviation are used to measure risk. We will do so at several places later in this chapter.

9

Technically, this result is a direct consequence of Jensen’s inequality in mathematical statistics. The inequality states that if x is a random variable and f ðxÞ is a concave function of that variable, then E½ f ðxÞ f ½EðxÞ. In the utility context, this means that if utility is concave in a random variable measuring wealth (i.e., if U 0 ðW Þ > 0 and U 00 ðW Þ < 0Þ, then the expected utility of wealth will be less than the utility associated with the expected value of W . 10 Technically, U ðW Þ is an indirect utility function because it is the consumption allowed by wealth that provides direct utility. In Chapter 17 we will take up the relationship between consumption-based utility functions and their implied indirect utility of wealth functions. 11

To see why the expected utilities for bet h and bet 2h are those shown, notice that these expected utilities are the average of the utilities from a favorable and an unfavorable outcome. Because W is halfway between W þ h and W h, U is also halfway between U ðW þ hÞ and U ðW hÞ.

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OPTIMIZATION PRINCIPLE

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Part 2 Choice and Demand

FIGURE 7.1

Utility of Wealth from Two Fair Bets of Differing Variability If the utility-of-wealth function is concave (i.e., exhibits a diminishing marginal utility of wealth), then this person will refuse fair bets. A 50–50 bet of winning or losing h dollars, for example, yields less utility ½U h ðW Þ than does refusing the bet. The reason for this is that winning h dollars means less to this individual than does losing h dollars. Utility U(W) U(W*) Uh(W*) U 2h(W*)

W* − 2h

W* − h

W W*

W* + h

W* + 2h

Wealth (W)

This person therefore will prefer his or her current wealth to that wealth combined with a fair gamble and will prefer a small gamble to a large one. The reason for this is that winning a fair bet adds to enjoyment less than losing hurts. Although in this case the prizes are equal, winning provides less than losing costs in utility terms.

Risk aversion and insurance As a matter of fact, this person might be willing to pay some amount to avoid participating in any gamble at all. Notice that a certain wealth of W provides the same utility as does participating in gamble 1. This person would be willing to pay up to W W in order to avoid participating in the gamble. This explains why people buy insurance. They are giving up a small, certain amount (the insurance premium) to avoid the risky outcome they are being insured against. The premium a person pays for automobile collision insurance, for example, provides a policy that agrees to repair his or her car should an accident occur. The widespread use of insurance would seem to imply that aversion to risk is quite prevalent. Hence, we introduce the following deﬁnition. DEFINITION

Risk aversion. An individual who always refuses fair bets is said to be risk averse. If individuals exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will be willing to pay something to avoid taking fair bets.

EXAMPLE 7.2 Willingness to Pay for Insurance To illustrate the connection between risk aversion and insurance, consider a person with a current wealth of $100,000 who faces the prospect of a 25 percent chance of losing his or her $20,000 automobile through theft during the next year. Suppose also that this person’s von Neumann–Morgenstern utility index is logarithmic; that is, U ðW Þ ¼ lnðW Þ:

Chapter 7 Uncertainty and Information

If this person faces next year without insurance, expected utility will be expected utility ¼ 0.75U ð100,000Þ þ 0.25U ð80,000Þ ¼ 0.75 ln 100,000 þ 0.25 ln 80,000 ¼ 11.45714. (7.19) In this situation, a fair insurance premium would be $5,000 (25 percent of $20,000, assuming that the insurance company has only claim costs and that administrative costs are $0). Consequently, if this person completely insures the car, his or her wealth will be $95,000 regardless of whether the car is stolen. In this case, then, expected utility ¼ U ð95,000Þ ¼ lnð95,000Þ ¼ 11.46163. (7.20) This person is made better-off by purchasing fair insurance. Indeed, we can determine the maximum amount that might be paid for this insurance protection (x) by setting expected utility ¼ U ð100,000 xÞ ¼ lnð100,000 xÞ ¼ 11.45714.

(7.21)

Solving this equation for x yields 100,000 x ¼ e 11.45714 . Therefore, the maximum premium is

(7.22)

x ¼ 5,426. (7.23) This person would be willing to pay up to $426 in administrative costs to an insurance company (in addition to the $5,000 premium to cover the expected value of the loss). Even when these costs are paid, this person is as well-off as he or she would be when facing the world uninsured. QUERY: Suppose utility had been linear in wealth. Would this person be willing to pay anything more than the actuarially fair amount for insurance? How about the case where utility is a convex function of wealth?

MEASURING RISK AVERSION In the study of economic choices in risky situations, it is sometimes convenient to have a quantitative measure of how averse to risk a person is. The most commonly used measure of risk aversion was initially developed by J. W. Pratt in the 1960s.12 This risk aversion measure, rðW Þ, is deﬁned as U 00 ðW Þ . rðW Þ ¼ 0 U ðW Þ

(7.24)

Because the distinguishing feature of risk-averse individuals is a diminishing marginal utility of wealth ½U 00 ðW Þ < 0, Pratt’s measure is positive in such cases. The measure is invariant with respect to linear transformations of the utility function, and therefore not affected by which particular von Neumann–Morgenstern ordering is used.

J. W. Pratt, “Risk Aversion in the Small and in the Large,” Econometrica (January/April 1964): 122–36.

12

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Part 2 Choice and Demand

Risk aversion and insurance premiums A useful feature of the Pratt measure of risk aversion is that it is proportional to the amount an individual will pay for insurance against taking a fair bet. Suppose the winnings from such a fair bet are denoted by the random variable h (this variable may be either positive or negative). Because the bet is fair, EðhÞ ¼ 0. Now let p be the size of the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble: E½U ðW þ hÞ ¼ U ðW pÞ, (7.25) where W is the individual’s current wealth. We now expand both sides of Equation 7.25 using Taylor’s series.13 Because p is a fixed amount, a linear approximation to the right-hand side of the equation will suffice: (7.26) U ðW pÞ ¼ U ðW Þ pU 0 ðW Þ þ higher-order terms. For the left-hand side, we need a quadratic approximation to allow for the variability in the gamble, h: h2 E½U ðW þ hÞ ¼ E U ðW Þ þ hU 0 ðW Þþ U 00 ðW Þ 2

þ higher-order terms (7.27) ¼ U ðW Þ þ EðhÞU 0 ðW Þ þ

Eðh 2 Þ 00 U ðW Þ 2

þ higher-order terms. (7.28) If we recall that EðhÞ ¼ 0 and then drop the higher-order terms and use the constant k to represent Eðh 2 Þ=2, we can equate Equations 7.26 and 7.28 as U ðW Þ pU 0 ðW Þ ≅ U ðW Þ þ kU 00 ðW Þ

(7.29)

or kU 00 ðW Þ ¼ krðW Þ. (7.30) U 0 ðW Þ That is, the amount that a risk-averse individual is willing to pay to avoid a fair bet is approximately proportional to Pratt’s risk aversion measure.14 Because insurance premiums paid are observable in the real world, these are often used to estimate individuals’ risk aversion coefficients or to compare such coefficients among groups of individuals. It is therefore possible to use market information to learn quite a bit about attitudes toward risky situations. p ≅

Risk aversion and wealth An important question is whether risk aversion increases or decreases with wealth. Intuitively, one might think that the willingness to pay to avoid a given fair bet would decline as wealth increases, because diminishing marginal utility would make potential losses less serious for high-wealth individuals. This intuitive answer is not necessarily correct, however, because diminishing marginal utility also makes the gains from winning gambles less attractive. So the 13 Taylor’s series provides a way of approximating any differentiable function around some point. If f ðxÞ has derivatives of all orders, it can be shown that

f ðx þ hÞ ¼ f ðxÞ þ hf 0 ðxÞ þ ðh 2 =2Þf 00 ðxÞ þ higher-order terms. The point-slope formula in algebra is a simple example of Taylor’s series. In this case, the factor of proportionality is also proportional to the variance of h because VarðhÞ ¼ E½h EðhÞ2 ¼ Eðh 2 Þ. For an illustration where this equation fits exactly, see Example 7.3.

14

Chapter 7 Uncertainty and Information

net result is indeterminate; it all depends on the precise shape of the utility function. Indeed, if utility is quadratic in wealth, U ðW Þ ¼ a þ bW þ cW 2 , where b > 0 and c < 0, then Pratt’s risk aversion measure is U 00 ðW Þ 2c ¼ , U 0 ðW Þ b þ 2cW which, contrary to intuition, increases as wealth increases. On the other hand, if utility is logarithmic in wealth, U ðW Þ ¼ lnðW Þ ðW > 0Þ, then we have rðW Þ ¼

U 00 ðW Þ 1 ¼ , U 0 ðW Þ W which does indeed decrease as wealth increases. The exponential utility function rðW Þ ¼

(7.31)

(7.32)

(7.33)

(7.34)

(7.35) U ðW Þ ¼ e AW ¼ expðAW Þ (where A is a positive constant) exhibits constant absolute risk aversion over all ranges of wealth, because now rðW Þ ¼

U 00 ðW Þ A 2 e AW ¼ ¼ A. Ae AW U 0 ðW Þ

(7.36)

This feature of the exponential utility function15 can be used to provide some numerical estimates of the willingness to pay to avoid gambles, as the next example shows.

EXAMPLE 7.3 Constant Risk Aversion Suppose an individual whose initial wealth is W0 and whose utility function exhibits constant absolute risk aversion is facing a 50–50 chance of winning or losing $1,000. How much (f ) would he or she pay to avoid the risk? To ﬁnd this value, we set the utility of W0 f equal to the expected utility from the gamble: exp½AðW0 f Þ ¼ 0.5 exp½AðW0 þ 1,000Þ (7.37) 0.5 exp½AðW 0 1,000Þ. Because the factor expðAW0 Þ is contained in all of the terms in Equation 7.37, this may be divided out, thereby showing that (for the exponential utility function) the willingness to pay to avoid a given gamble is independent of initial wealth. The remaining terms expðAf Þ ¼ 0.5 expð1,000AÞ þ 0.5 expð1,000AÞ (7.38) can now be used to solve for f for various values of A. If A ¼ 0:0001, then f ¼ 49:9; a person with this degree of risk aversion would pay about $50 to avoid a fair bet of $1,000. Alternatively, if A ¼ 0:0003, this more risk-averse person would pay f ¼ 147:8 to avoid the gamble. Because intuition suggests that these values are not unreasonable, values of the risk aversion parameter A in these ranges are sometimes used for empirical investigations. (continued)

15

Because the exponential utility function exhibits constant (absolute) risk aversion, it is sometimes abbreviated by the term CARA utility.

211

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Part 2 Choice and Demand

EXAMPLE 7.3 CONTINUED A normally distributed risk. The constant risk aversion utility function can be combined with the assumption that a person faces a random threat to his or her wealth that follows a normal distribution (see Chapter 2) to arrive at a particularly simple result. Speciﬁcally, if and σ2W , then a person’s risky wealth follows a normal distribution with mean μW p ﬃﬃﬃﬃﬃﬃvariance z 2 =2 , where z ¼ the probability density function for wealth is given by f ðW Þ ¼ ð1= 2πÞe ½ðW μW Þ=σW : If this person has a utility function for wealth given by U ðW Þ ¼ e AW , then expected utility from his or her risky wealth is given by ∞

E½U ðW Þ ¼

ﬃ e ∫ U ðW Þf ðW Þ dW ¼ p1ﬃﬃﬃﬃﬃﬃ 2π ∫

AW ½ðW μW Þ=σW 2 =2

e

dW .

(7.39)

∞

Perhaps surprisingly, this integration is not too difficult to accomplish, though it does take patience. Performing this integration and taking a variety of monotonic transformations of the resulting expression yields the final result that A 2 (7.40) E½U ðW Þ ≅ μW ⋅ σW . 2 Hence, expected utility is a linear function of the two parameters of the wealth probability density function, and the individual’s risk aversion parameter (A) determines the size of the negative effect of variability on expected utility. For example, suppose a person has invested his or her funds so that wealth has an expected value of $100,000 but a standard deviation ðσW Þ of $10,000. With the Normal distribution, he or she might therefore expect wealth to decline below $83,500 about 5 percent of the time and rise above $116,500 a similar fraction of the time. With these parameters, expected utility is given by E½U ðW Þ ¼ 100,000 ðA=2Þð10,000Þ2 : If A ¼ 0:0001 ¼ 104 , expected utility is given by 100,0000 0:5 ⋅ 104 ⋅ ð104 Þ2 ¼ 95, 000: Hence, this person receives the same utility from his or her risky wealth as would be obtained from a certain wealth of $95,000. A more risk-averse person might have A ¼ 0:0003 and in this case the “certainty equivalent” of his or her wealth would be $85,000. QUERY: Suppose this person had two ways to invest his or her wealth: Allocation 1, μW ¼ 107,000 and σW ¼ 10,000; Allocation 2, μW ¼ 102,000 and σW ¼ 2,000: How would this person’s attitude toward risk affect his or her choice between these allocations?16

Relative risk aversion It seems unlikely that the willingness to pay to avoid a given gamble is independent of a person’s wealth. A more appealing assumption may be that such willingness to pay is inversely proportional to wealth and that the expression U 00 ðW Þ (7.41) U 0 ðW Þ might be approximately constant. Following the terminology proposed by J. W. Pratt,17 the rrðW ) function defined in Equation 7.41 is a measure of relative risk aversion. The power utility function rrðW Þ ¼ WrðW Þ ¼ W

16 This numerical example (very roughly) approximates historical data on real returns of stocks and bonds, respectively, though the calculations are illustrative only.

Pratt, “Risk Aversion.”

17

Chapter 7 Uncertainty and Information

U ðW Þ ¼

WR R

ðR < 1, R 6¼ 0Þ

(7.42)

and U ðW Þ ¼ lnW exhibits diminishing absolute risk aversion,

ðR ¼ 0Þ

U 00 ðW Þ ðR 1ÞW R2 ðR 1Þ ¼ ¼ , 0 R1 W U ðW Þ W but constant relative risk aversion: rðW Þ ¼

(7.43)

rrðW Þ ¼ WrðW Þ ¼ ðR 1Þ ¼ 1 R. (7.44) Empirical evidence is generally consistent with values of R in the range of –3 to –1. Hence, individuals seem to be somewhat more risk averse than is implied by the logarithmic utility function, though in many applications that function provides a reasonable approximation. It is useful to note that the constant relative risk aversion utility function in Equation 7.42 has the same form as the general CES utility function we first described in Chapter 3. This provides some geometric intuition about the nature of risk aversion that we will explore later in this chapter. 18

EXAMPLE 7.4 Constant Relative Risk Aversion An individual whose behavior is characterized by a constant relative risk aversion utility function will be concerned about proportional gains or loss of wealth. We can therefore ask what fraction of initial wealth ( f ) such a person would be willing to give up to avoid a fair gamble of, say, 10 percent of initial wealth. First, we assume R ¼ 0, so the logarithmic utility function is appropriate. Setting the utility of this individual’s certain remaining wealth equal to the expected utility of the 10 percent gamble yields (7.45) ln½ð1 f ÞW0 ¼ 0:5 lnð1:1W0 Þ þ 0:5 lnð0:9W0 Þ. Because each term contains ln W0 , initial wealth can be eliminated from this expression: lnð1 f Þ ¼ 0:5½lnð1:1Þ þ lnð0:9Þ ¼ lnð0:99Þ0:5 ; hence ð1 f Þ ¼ ð0:99Þ0:5 ¼ 0:995 and f ¼ 0:005. (7.46) This person will thus sacrifice up to 0.5 percent of wealth to avoid the 10 percent gamble. A similar calculation can be used for the case R ¼ 2 to yield f ¼ 0:015. (7.47) Hence this more risk-averse person would be willing to give up 1.5 percent of his or her initial wealth to avoid a 10 percent gamble. QUERY: With the constant relative risk aversion function, how does this person’s willingness to pay to avoid a given absolute gamble (say, of 1,000) depend on his or her initial wealth?

Some authors write the utility function in Equation 7.42 as U ðW Þ ¼ W 1a =ð1 aÞ and seek to measure a ¼ 1 R. In this case, a is the relative risk aversion measure. The constant relative risk aversion function is sometimes abbreviated as CRRA.

18

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Part 2 Choice and Demand

THE PORTFOLIO PROBLEM One of the classic problems in the theory of behavior under uncertainty is the issue of how much of his or her wealth a risk-averse investor should invest in a risky asset. Intuitively, it seems that the fraction invested in risky assets should be smaller for more risk-averse investors, and one goal of our analysis will be to show that formally. To get started, assume that an investor has a certain amount of wealth, W0 , to invest in one of two assets. The ﬁrst asset yields a certain return of rf , whereas the second asset’s return is a random variable, re. If we let the amount invested in the risky asset be denoted by k, then this person’s wealth at the end of one period will be W ¼ ðW0 kÞð1 þ rf Þ þ kð1 þ reÞ ¼ W0 ð1 þ rf Þ þ kðre rf Þ.

(7.48)

Notice three things about this end-of-period wealth. First, W is a random variable because its value depends on re. Second, k can be either positive or negative here depending on whether this person buys the risky asset or sells it short. As we shall see, however, in the usual case Eðre rf Þ > 0 and this will imply k 0. Finally, notice also that Equation 7.48 allows for a solution in which k >W0 . In this case, this investor would leverage his or her investment in the risky asset by borrowing at the risk-free rate rf . If we let U ðW Þ represent this investor’s utility function, then the von Neumann–Morgenstern theorem states that he or she will choose k to maximize E½U ðW Þ. The ﬁrst-order condition for such a maximum is19 ∂E½U ðW Þ ∂E½U ðW0 ð1 þ rf Þ þ kðre rf ÞÞ ¼ E½U 0 ⋅ ðre rf Þ ¼ 0. ¼ ∂k ∂k

(7.49)

Because this first-order condition lies at the heart of many problems in the theory of uncertainty, it may be worthwhile spending some time to understand it intuitively. Equation 7.49 is looking at the expected value of the product of marginal utility and the term re rf . Both of these terms are random. Whether re rf is positive or negative will depend on how well the risky assets perform over the next period. But the return on this risky asset will also affect this investor’s end-of-period wealth and thus will affect his or her marginal utility. If the investment does well, W will be large and marginal utility will be relatively low (because of diminishing marginal utility). If the investment does poorly, wealth will be relatively low and marginal utility will be relatively high. Hence, in the expected value calculation in Equation 7.49, negative outcomes for re rf will be weighted more heavily than positive outcomes to take the utility consequences of these outcomes into account. If the expected value in Equation 7.49 were positive, a person could increase his or her expected utility by investing more in the risky asset. If the expected value were negative, he or she could increase expected utility by reducing the amount of the risky asset held. Only when the first-order condition holds will this person have an optimal portfolio. Two other conclusions can be drawn from the optimality condition in Equation 7.49. First, so long as Eðre rf Þ > 0, an investor will choose positive amounts of the risky asset. To see why, notice that meeting Equation 7.49 will require that fairly large values of U 0 be attached to situations where re rf turns out to be negative. That can only happen if the investor owns positive amounts of the risky asset so that end-of-period wealth is low in such situations. A second conclusion from the ﬁrst-order condition in Equation 7.49 is that investors who are more risk averse will hold smaller amounts of the risky asset than will investors who are more tolerant of risk. Again, the reason relates to the shape of the U 0 function. For very risk-averse investors, marginal utility rises rapidly as wealth falls. Hence, they need relatively little exposure to potential negative outcomes from holding the risky asset to satisfy 19 In calculating this first-order condition, we can differentiate through the expected value operator. See Chapter 2 for a discussion of differentiating integrals.

Chapter 7 Uncertainty and Information

Equation 7.49. Investors who are more tolerant of risk will ﬁnd that U 0 rises less rapidly when the risky asset performs poorly, so they will be willing to hold more of it. In summary, then, a formal study of the portfolio problem conﬁrms simple intuitions about how people choose to invest. To make further progress on the question requires that we make some speciﬁc assumptions about the investor’s utility function. In Example 7.5, we look at a two examples. EXAMPLE 7.5 The Portfolio Problem with Specific Utility Functions In this problem we show the implications of assuming either CARA or CRRA utility for the solution to the portfolio allocation problem. 1. CARA Utility. If U ðW Þ ¼ expðAW Þ then the marginal utility function is given by U 0 ðW Þ ¼ A expðAW Þ; substituting for end-of-period wealth, we have U 0 ðW Þ ¼ A exp½AðW0 ð1 þ rf Þ þ kðre rf ÞÞ ¼ A exp½AW0 ð1 þ rf Þ exp½Akðe r rf Þ.

(7.50)

That is, the marginal utility function can be separated into a random part and a nonrandom part (both initial wealth and the risk-free rate are nonrandom). Hence, the optimality condition from Equation 7.49 can be written as E½U 0 ⋅ ðre rf Þ ¼ A exp½AW0 ð1 þ rf Þ E½expðAkðre rf ÞÞ ⋅ ðre rf Þ ¼ 0:

(7.51)

Now we can divide by the exponential function of initial wealth, leaving an optimality condition that involves only terms in k,A, and re rf . Solving this condition for the optimal level of k can in general be quite difficult (but see Problem 7.14). Regardless of the specific solution, however, Equation 7.51 shows that this optimal investment amount will be a constant regardless of the level of initial wealth. Hence, the CARA function implies that the fraction of wealth that an investor holds in risky assets should decline as wealth increases—a conclusion that seems precisely contrary to empirical data, which tend to show the fraction of wealth held in risky assets rising with wealth. 2. CRRA Utility. If U ðW Þ ¼ W R =R then the marginal utility function is given by U 0 ðW Þ ¼ W R1 . Substituting the expression for ﬁnal wealth into this equation yields U 0 ðW Þ ¼ ½W0 ð1 þ rf Þ þ kðe r rf ÞR1

k ¼ ½W0 ð1 þ rf ÞR1 1 þ ⋅ ðre rf Þ . W0 ð1 þ rf Þ

(7.52)

Inserting this expression into the optimality condition in Equation 7.49 shows that the term ½W0 ð1 þ rf ÞR1 can be canceled out, implying that the optimal solution will not involve the absolute level of initial wealth but only the ratio k=W0 ð1 þ rf Þ. In words, the CRRA utility function implies that all individuals with the same risk tolerance will hold the same fraction of wealth in risky assets, regardless of their absolute levels of wealth. Though this conclusion is slightly more in accord with the facts than is the conclusion from the CARA function, it still falls short of explaining why the fraction of wealth held in risky assets tends to rise with wealth. QUERY: Can you suggest a reason why investors might increase the proportion of their portfolios invested in risky assets as wealth increases even though their preferences are characterized by the CRRA utility function?

215

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Part 2 Choice and Demand

THE STATE-PREFERENCE APPROACH TO CHOICE UNDER UNCERTAINTY Although our analysis in this chapter has offered insights on a number of issues, it seems rather different from the approach we took in other chapters. The basic model of utility maximization subject to a budget constraint seems to have been lost. In order to make further progress in the study of behavior under uncertainty, we will therefore develop some new techniques that will permit us to bring the discussion of such behavior back into the standard choice-theoretic framework.

States of the world and contingent commodities We start by assuming that the outcomes of any random event can be categorized into a number of states of the world. We cannot predict exactly what will happen, say, tomorrow, but we assume that it is possible to categorize all of the possible things that might happen into a ﬁxed number of well-deﬁned states. For example, we might make the very crude approximation of saying that the world will be in only one of two possible states tomorrow: It will be either “good times” or “bad times.” One could make a much ﬁner gradation of states of the world (involving even millions of possible states), but most of the essentials of the theory can be developed using only two states. A conceptual idea that can be developed concurrently with the notion of states of the world is that of contingent commodities. These are goods delivered only if a particular state of the world occurs. As an example, “$1 in good times” is a contingent commodity that promises the individual $1 in good times but nothing should tomorrow turn out to be bad times. It is even possible, by stretching one’s intuitive ability somewhat, to conceive of being able to purchase this commodity: I might be able to buy from someone the promise of $1 if tomorrow turns out to be good times. Because tomorrow could be bad, this good will probably sell for less than $1. If someone were also willing to sell me the contingent commodity “$1 in bad times,” then I could assure myself of having $1 tomorrow by buying the two contingent commodities “$1 in good times” and “$1 in bad times.”

Utility analysis Examining utility-maximizing choices among contingent commodities proceeds formally in much the same way we analyzed choices previously. The principal difference is that, after the fact, a person will have obtained only one contingent good (depending on whether it turns out to be good or bad times). Before the uncertainty is resolved, however, the individual has two contingent goods from which to choose and will probably buy some of each because he or she does not know which state will occur. We denote these two contingent goods by Wg (wealth in good times) and Wb (wealth in bad times). Assuming that utility is independent of which state occurs20 and that this individual believes that good times will occur with probability π, the expected utility associated with these two contingent goods is V ðWg , Wb Þ ¼ πU ðWg Þ þ ð1 πÞU ðWb Þ.

(7.53)

This is the magnitude this individual seeks to maximize given his or her initial wealth, W . 20

This assumption is untenable in circumstances where utility of wealth depends on the state of the world. For example, the utility provided by a given level of wealth may differ depending on whether an individual is “sick” or “healthy.” We will not pursue such complications here, however. For most of our analysis, utility is assumed to be concave in wealth: U 0 ðW Þ > 0, U 00 ðW Þ < 0.

Chapter 7 Uncertainty and Information

Prices of contingent commodities Assuming that this person can purchase a dollar of wealth in good times for pg and a dollar of wealth in bad times for pb , his or her budget constraint is then W ¼ pg Wg þ pb Wb .

(7.54)

The price ratio pg =pb shows how this person can trade dollars of wealth in good times for dollars in bad times. If, for example, pg ¼ 0:80 and pb ¼ 0:20, the sacrifice of $1 of wealth in good times would permit this person to buy contingent claims yielding $4 of wealth should times turn out to be bad. Whether such a trade would improve utility will, of course, depend on the specifics of the situation. But looking at problems involving uncertainty as situations in which various contingent claims are traded is the key insight offered by the statepreference model.

Fair markets for contingent goods If markets for contingent wealth claims are well developed and there is general agreement about the likelihood of good times (π), then prices for these claims will be actuarially fair— that is, they will equal the underlying probabilities: pg ¼ π, (7.55) pb ¼ ð1 πÞ: Hence, the price ratio pg =pb will simply reflect the odds in favor of good times: pg π ¼ . (7.56) pb 1 π In our previous example, if pg ¼ π ¼ 0:8 and pb ¼ ð1 πÞ ¼ 0:2 then π=ð1 πÞ ¼ 4. In this case the odds in favor of good times would be stated as “4-to-1.” Fair markets for contingent claims (such as insurance markets) will also reflect these odds. An analogy is provided by the “odds” quoted in horse races. These odds are “fair” when they reflect the true probabilities that various horses will win.

Risk aversion We are now in a position to show how risk aversion is manifested in the state-preference model. Speciﬁcally, we can show that, if contingent claims markets are fair, then a utilitymaximizing individual will opt for a situation in which Wg ¼ Wb ; that is, he or she will arrange matters so that the wealth ultimately obtained is the same no matter what state occurs. As in previous chapters, maximization of utility subject to a budget constraint requires that this individual set the MRS of Wg for Wb equal to the ratio of these “goods” prices: ∂V =∂Wg

πU 0 ðWg Þ

pg ¼ . (7.57) ∂V =∂Wb ð1 πÞU 0 ðWb Þ pb In view of the assumption that markets for contingent claims are fair (Equation 7.56), this first-order condition reduces to U 0 ðWg Þ ¼1 U 0 ðWb Þ or21 MRS ¼

¼

Wg ¼ Wb .

This step requires that utility be state independent and that U 0 ðW Þ > 0.

21

(7.58)

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FIGURE 7.2

Risk Aversions in the State-Preference Model The line I represents the individual’s budget constraint for contingent wealth claims: W ¼ pg Wg þ pb Wb . If the market for contingent claims is actuarially fair ½ pg =pb ¼ π=ð1 − πÞ, then utility maximization will occur on the certainty line where Wg ¼ Wb ¼ W . If prices are not actuarially fair, the budget constraint may resemble I 0 and utility maximization will occur at a point where Wg > Wb . Wb

Certainty line

W*

l W*

l′

U1 Wg

Hence this individual, when faced with fair markets in contingent claims on wealth, will be risk averse and will choose to ensure that he or she has the same level of wealth regardless of which state occurs.

A graphic analysis Figure 7.2 illustrates risk aversion with a graph. This individual’s budget constraint (I ) is shown to be tangent to the U1 indifference curve where Wg ¼ Wb —a point on the “certainty line” where wealth ðW Þ is independent of which state of the world occurs. At W the slope of the indifference curve ½π=ð1 πÞ is precisely equal to the price ratio pg =pb . If the market for contingent wealth claims were not fair, utility maximization might not occur on the certainty line. Suppose, for example, that π=ð1 πÞ ¼ 4 but that pg =pb ¼ 2 because ensuring wealth in bad times proves quite costly. In this case the budget constraint would resemble line I 0 in Figure 7.2 and utility maximization would occur below the certainty line.22 In this case this individual would gamble a bit by opting for Wg > Wb , because claims on Wb are relatively costly. Example 7.6 shows the usefulness of this approach in evaluating some of the alternatives that might be available.

Because (as Equation 7.58 shows) the MRS on the certainty line is always π=ð1 − πÞ, tangencies with a flatter slope than this must occur below the line.

22

Chapter 7 Uncertainty and Information

EXAMPLE 7.6 Insurance in the State-Preference Model We can illustrate the state-preference approach by recasting the auto insurance illustration from Example 7.2 as a problem involving the two contingent commodities “wealth with no theft” ðWg Þ and “wealth with a theft” ðWb Þ. If, as before, we assume logarithmic utility and that the probability of a theft (that is, 1 π) is 0.25, then expected utility ¼ 0.75U ðWg Þ þ 0:25U ðWb Þ ¼ 0.75 ln Wg þ 0:25 ln Wb .

(7.59)

If the individual takes no action then utility is determined by the initial wealth endowment, W g ¼ 100,000 and W b ¼ 80,000, so expected utility ¼ 0:75 ln 100,000 þ 0.25 ln 80,000 (7.60) ¼ 11.45714. To study trades away from these initial endowments, we write the budget constraint in terms of the prices of the contingent commodities, pg and pb : pg W g þ pb W b ¼ pg Wg þ pb Wb . (7.61) Assuming that these prices equal the probabilities of the two states ðpg ¼ 0:75, pb ¼ 0:25Þ, this constraint can be written 0:75ð100,000Þ þ 0:25ð80,000Þ ¼ 95,000 ¼ 0:75Wg þ 0:25Wb ;

(7.62)

that is, the expected value of wealth is $95,000, and this person can allocate this amount between Wg and Wb . Now maximization of utility with respect to this budget constraint yields Wg ¼ Wb ¼ 95,000. Consequently, the individual will move to the certainty line and receive an expected utility of expected utility ¼ ln 95,000 ¼ 11.46163, (7.63) a clear improvement over doing nothing. To obtain this improvement, this person must be able to transfer $5,000 of wealth in good times (no theft) into $15,000 of extra wealth in bad times (theft). A fair insurance contract would allow this because it would cost $5,000 but return $20,000 should a theft occur (but nothing should no theft occur). Notice here that the wealth changes promised by insurance—dWb =dWg ¼ 15,000= 5,000 ¼ 3— exactly equal the negative of the odds ratio π=ð1 πÞ ¼ 0:75=0:25 ¼ 3. A policy with a deductible provision. A number of other insurance contracts might be utility improving in this situation, though not all of them would lead to choices that lie on the certainty line. For example, a policy that cost $5,200 and returned $20,000 in case of a theft would permit this person to reach the certainty line with Wg ¼ Wb ¼ 94,800 and expected utility ¼ ln 94, 800 ¼ 11.45953, (7.64) which also exceeds the utility obtainable from the initial endowment. A policy that costs $4,900 and requires the individual to incur the first $1,000 of a loss from theft would yield Wg ¼ 100,000 4,900 ¼ 95,100, Wb ¼ 80,000 4,900 þ 19,000 ¼ 94,100;

(7.65)

then expected utility ¼ 0:75 ln 95,100 þ 0:25 ln 94,100 ¼ 11:46004: (7.66) Although this policy does not permit this person to reach the certainty line, it is utility improving. Insurance need not be complete in order to offer the promise of higher utility. (continued)

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EXAMPLE 7.6 CONTINUED QUERY: What is the maximum amount an individual would be willing to pay for an insurance policy under which he or she had to absorb the ﬁrst $1,000 of loss?

Risk aversion and risk premiums The state-preference model is also especially useful for analyzing the relationship between risk aversion and individuals’ willingness to pay for risk. Consider two people, each of whom starts with a certain wealth, W . Each person seeks to maximize an expected utility function of the form WR g

WR b . (7.67) R R Here the utility function exhibits constant relative risk aversion (see Example 7.4). Notice also that the function closely resembles the CES utility function we examined in Chapter 3 and elsewhere. The parameter R determines both the degree of risk aversion and the degree of curvature of indifference curves implied by the function. A very risk-averse individual will have a large negative value for R and have sharply curved indifference curves, such as U1 shown in Figure 7.3. A person with more tolerance for risk will have a higher value of R and flatter indifference curves (such as U2 ).23 V ðWg , Wb Þ ¼ π

þ ð1 πÞ

Tangency of U1 and U2 at W is ensured, because the MRS along the certainty line is given by π=ð1 πÞ regardless of the value of R.

23

FIGURE 7.3

Risk Aversion and Risk Premiums Indifference curve U1 represents the preferences of a very risk-averse person, whereas the person with preferences represented by U2 is willing to assume more risk. When faced with the risk of losing h in bad times, person 2 will require compensation of W2 − W in good times whereas person 1 will require a larger amount given by W1 − W .

Wb

Certainty line

W* W* − h U1

U2 W*

W2 W 1

Wg

Chapter 7 Uncertainty and Information

Suppose now these individuals are faced with the prospect of losing h dollars of wealth in bad times. Such a risk would be acceptable to individual 2 if wealth in good times were to increase from W to W2 . For the very risk-averse individual 1, however, wealth would have to increase to W1 to make the risk acceptable. The difference between W1 and W2 therefore indicates the effect of risk aversion on willingness to assume risk. Some of the problems in this chapter make use of this graphic device for showing the connection between preferences (as reﬂected by the utility function in Equation 7.67) and behavior in risky situations.

THE ECONOMICS OF INFORMATION Information is a valuable economic resource. People who know where to buy high-quality goods cheaply can make their budgets stretch further than those who don’t; farmers with access to better weather forecasting may be able to avoid costly mistakes; and government environmental regulation can be more efﬁcient if it is based on good scientiﬁc knowledge. Although these observations about the value of information have long been recognized, formal economic modeling of information acquisition and its implications for resource allocation are fairly recent.24 Despite its late start, the study of information economics has become one of the major areas in current research. In this chapter we brieﬂy survey some of the issues raised by this research. Far more detail on the economics of information is provided in Chapter 18.

PROPERTIES OF INFORMATION One difﬁculty encountered by economists who wish to study the economics of information is that “information” itself is not easy to deﬁne. Unlike the economic goods we have been studying so far, the “quantity” of information obtainable from various actions is not well deﬁned, and what information is obtained is not homogeneous among its users. The forms of economically useful information are simply too varied to permit the kinds of price-quantity characterizations we have been using for basic consumer goods. Instead, economists who wish to study information must take some care to specify what the informational environment is in a particular decision problem (this is sometimes called the information set) and how that environment might be changed through individual actions. As might be expected, this approach has resulted in a vast number of models of speciﬁc situations with little overall commonality among them. A second complication involved in the study of information concerns some technical properties of information itself. Most information is durable and retains value after it has been used. Unlike a hot dog, which is consumed only once, knowledge of a special sale can be used not only by the person who discovers it but also by any friends with whom the information is shared. The friends then may gain from this information even though they don’t have to spend anything to obtain it. Indeed, in a special case of this situation, information has the characteristic of a pure public good (see Chapter 19). That is, the information is both nonrival in that others may use it at zero cost and nonexclusive in that no individual can prevent others from using the information. The classic example of these properties is a new scientiﬁc discovery. When some prehistoric people invented the wheel, others could use it without detracting from the value of the discovery, and everyone who saw the wheel could copy it freely. These technical properties of information imply that market mechanisms may often operate imperfectly in allocating resources to information provision and acquisition. Standard 24 The formal modeling of information is sometimes dated from the path-breaking article by G. J. Stigler, “The Economics of Information,” Journal of Political Economy (June 1961): 213–25.

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models of supply and demand may therefore be of relatively limited use in understanding such activities. At a minimum, models have to be developed that accurately reﬂect the properties being assumed about the informational environment. Throughout the latter portions of this book, we will describe some of the situations in which such models are called for. Here, however, we will pay relatively little attention to supply-demand equilibria and will instead focus primarily on information issues that arise in the theory of individual choice.

THE VALUE OF INFORMATION Developing models of information acquisition necessarily requires using tools from our study of uncertainty earlier in this chapter. Lack of information clearly represents a problem involving uncertainty for a decision maker. In the absence of perfect information, he or she may not be able to know exactly what the consequences of a particular action will be. Better information can reduce that uncertainty and therefore lead to better decisions that provide increased utility.

Information and subjective possibilities This relationship between uncertainty and information acquisition can be illustrated using the state-preference model. Earlier we assumed that an individual forms subjective opinions about the probabilities of the two states of the world, “good times” and “bad times.” In this model, information is valuable because it allows the individual to revise his or her estimates of these probabilities and to take advantage of these revisions. For example, information that foretold that tomorrow would deﬁnitely be “good times” would cause this person to revise his or her probabilities to πg ¼ 1, πb ¼ 0 and to change his or her purchases accordingly. When the information received is less deﬁnitive, the probabilities may be changed only slightly, but even small revisions may be quite valuable. If you ask some friends about their experiences with a few brands of DVD players you are thinking of buying, you may not want their opinions to dictate your choice. The prices of the players and other types of information (say, obtained from consulting Consumer Reports) will also affect your views. Ultimately, however, you must process all of these factors into a decision that reﬂects your assessment of the probabilities of various “states of the world” (in this case, the quality obtained from buying different brands).

A formal model To illustrate why information has value, assume that an individual faces an uncertain situation involving “good” and “bad” times and that he or she can invest in a “message” that will yield some information about the probabilities of these outcomes. This message can take on two potential values, 1 or 2, with probabilities p and ð1 pÞ, respectively. If the message takes the value 1, then this person will believe that the probability of good times is given by π1g [and the probability of bad times by π1b ¼ ð1 π1g Þ]. If the message takes the value 2, on the other hand, the probabilities are π2g and ð1 π2g Þ. Once the message is received, this person has the opportunity to maximize expected utility on the basis of these probabilities. In general, it would be expected that he or she will make different decisions depending on what the message is. Let V1 be the (indirect) maximal expected utility when the message takes the value 1 and V2 be this maximal utility when the message takes the value 2. Hence, when this person is considering purchasing the message (that is, before it is actually received), expected utility is given by: (7.68) Ewith m ¼ pV1 þ ð1 pÞV2 . Now let’s consider the situation of this person when he or she decides not to purchase the message. In this case, a single decision must be made that is based on the probabilities of

Chapter 7 Uncertainty and Information

good and bad times, π0g and ð1 π0g Þ. Because the individual knows the various probabilities involved, consistency requires that π0g ¼ pπ1g þ ð1 pÞπ2g . Now let V0 represent the maximal expected utility this person can obtain with these probabilities. Hence, we can write expected utility without the message as (7.69) Ewithout m ¼ V0 ¼ pV0 þ ð1 pÞV0 . A comparison of Equations 7.68 and 7.69 shows that this person can always achieve Ewithout m when he or she has the information provided by the message. That is, he or she can just choose to disregard what the message says. But if he or she chooses to make new, different decisions based on the information in the message, it must be the case that this information has value. That is: (7.70) Ewith m Ewithout m . Presumably, then, this person will be willing to pay something for the message because of the better decision-making opportunities it provides.25 Example 7.7 provides a simple illustration. EXAMPLE 7.7 The Value of Information on Prices To illustrate how new information may affect utility maximization, let’s return to one of the ﬁrst models we used in Chapter 4. There we showed that if an individual consumes two goods and utility is given by U ðx, yÞ ¼ x 0:5 y 0:5 , then the indirect utility function is I . (7.71) V ðpx , py , I Þ ¼ 2p x0:5 p 0:5 y As a numerical example, we considered the case px ¼ 1, py ¼ 4, I ¼ 8, and calculated that V ¼ I =2 ⋅ 1 ⋅ 2 ¼ 2. Now suppose that good y represents, say, a can of brand-name tennis balls, and this consumer knows that these can be bought at a price of either $3 or $5 from two stores but does not know which store charges which price. Because it is equally likely that either store has the lower price, the expected value of the price is $4. But, because the indirect utility function is convex in price, this person receives an expected value of greater than V ¼ 2 from shopping because he or she can buy more if the low-priced store is encountered. Before shopping, expected utility is E½V ðpx , py , I Þ ¼ 0:5 ⋅ V ð1, 3, 8Þ þ 0:5 ⋅ V ð1, 5, 8Þ ¼ 1:155 þ 0:894 ¼ 2:049. (7.72) If the consumer knew which store offered the lower price, utility would be even greater. If this person could buy at py ¼ 3 with certainty, then indirect utility would be V ¼ 2:309 and we can use this result to calculate what the value of this information is. That is, we can ask what level of income, I , would yield the same utility when py ¼ 3, as is obtained when this person must choose which store to patronize by chance. Hence we need to solve the equation I I ¼ ¼ 2:049. (7.73) V ð px , py , I Þ ¼ 2p x0:5 p y0:5 2 ⋅ 1 ⋅ 30:5 Solving this yields a value of I ¼ 7:098. Hence, this person would be willing to pay up to 0.902 ð¼ 8 7.098Þ for the information. Notice that availability of the price information (continued) 25

A more general way to state this result is to consider the properties of the individual’s indirect expected utility function (V ) as dependent on the probabilities in the problem. That is, V ðπg Þ ¼ max½πg U ðWg Þ þ ð1 πg ÞU ðWb Þ. Comparing Equations 7.68 and 7.69 amounts to comparing pV ðπ1g Þ þ ð1 pÞV ðπ2g Þ to V ðπ0g Þ ¼ V ½ pπ1g þ ð1 pÞπ2g . Because the V function is convex in πg , the inequality in Equation 7.70 necessarily holds.

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EXAMPLE 7.7 CONTINUED helps this person in two ways: (1) it increases the probability he or she will patronize the low-price store from 0.5 to 1.0; and (2) it permits this person to take advantage of the lower price offered by buying more. QUERY: It seems odd in this problem that expected utility with price uncertainty (V ¼ 2.049) is greater than utility when price takes its expected value (V ¼ 2). Does this violate the assumption of risk aversion?

FLEXIBILITY AND OPTION VALUE The availability of new information allows individuals to make better decisions in situations involving uncertainty. It may therefore be beneﬁcial to try to postpone making decisions until the information arrives. Of course, ﬂexibility may sometimes involve costs of its own, so the decision-making process can become complex. For example, someone planning a trip to the Caribbean would obviously like to know whether he or she will have good weather. A vacationer who could wait until the last minute in deciding when to go could use the latest weather forecast to make that decision. But waiting may be costly (perhaps because lastminute airfares are much higher), so the choice can be a difﬁcult one. Clearly the option to delay the decision is valuable, but whether this “option value” exceeds the costs involved in delay is the crucial question. Modeling the importance of ﬂexibility in decision making has become a major topic in the study of uncertainty and information. “Real option theory” has come to be an important component of ﬁnancial and management theory. Other applications are beginning to emerge in such diverse ﬁelds as development economics, natural resource economics, and law and economics. Because this book focuses on general theory, however, we cannot pursue these interesting innovations here. Rather, our brief treatment will focus on how questions of ﬂexibility might be incorporated into some of the models we have already examined, followed by a few concluding remarks.

Flexibility in the portfolio model Some of the basic principles of real option theory can be illustrated by combining the portfolio choice model that we introduced earlier in this chapter with the idea of information messages introduced in the previous section. Suppose that an investor is considering putting some portion of his or her wealth (k) into a risky asset. The return on the asset is random and its characteristics will depend on whether there are “good times” or “bad times.” The returns under these two situations are designated by re1 and re2 , respectively. First, consider a situation where this person will get a message telling him or her whether it is good or bad times, but the message will arrive after the investment decision is made. The probability that the message will indicate good times is given by p. In this case, this person can be viewed as investing in a risky asset whose return is given by re0 ¼ pre1 þ ð1 pÞre2 . Following the procedure outlined earlier, associated with this asset will be an optimal investment, k0 , and the expected utility associated with this portfolio will be U0 . Suppose, alternatively, that this person has the ﬂexibility to wait until after the message is received to decide on how his or her portfolio will be allocated. If the message reveals good times, then he or she will choose to invest k1 in the risky asset and expected utility will be U1 .

Chapter 7 Uncertainty and Information

On the other hand, if the message reveals bad times, then he or she will choose to invest k2 in the risky asset and expected utility will be U2 . Hence, the expected utility provided by the option of waiting before choosing k will be (7.74) U ¼ pU þ ð1 pÞU . 1

2

As before, it is clear that U U0 . The investor could always choose to invest k0 no matter what the message says, but if he or she chooses differing k’s depending on the information in the message, it must be because that strategy provides more expected utility. When U > U0 , the option to wait has real value and this person will be willing to pay something (say, in forgone interest receipts) for that possibility.

Financial options In some cases option values can be observed in actual markets. For example, ﬁnancial options provide a buyer the right, but not the obligation, to conduct an economic transaction (typically buying or selling a stock) at speciﬁed terms at a certain date in the future. An option on Microsoft Corporation shares, for instance, might give the buyer the right (but not the obligation) to buy the stock in six months at a price of $30 per share. Or a foreign exchange option might provide the buyer with the right to buy euros at a price of $1.30 per euro in three months. All such options have value because they permit the owner to either make or decline the speciﬁed transaction depending on what new information becomes available over the option’s duration. Such built-in ﬂexibility is useful in a wide variety of investment strategies.

Options embedded in other transactions Many other types of economic transactions have options embedded in them. For example, the purchase of a good that comes with a “money-back guarantee” gives the buyer an option to reverse the transaction should his or her experience with the good be unfavorable. Similarly, many mortgages provide the homeowner with the option to pay off the loan without penalty should conditions change. All such options are clearly valuable. A car buyer is not required to return his or her purchase if the car runs well and the homeowner need not pay off the mortgage if interest rates rise. Hence, embedding a buyer’s option in a transaction can only increase the value of that transaction to the buyer. Contracts with such options would be expected to have higher prices. On the other hand, transactions with embedded seller options (for example, the right to repurchase a house at a stated price) will have lower prices. Examining price differences can therefore be one way to infer the value of some embedded options.

ASYMMETRY OF INFORMATION One obvious implication of the study of information acquisition is that the level of information that an individual buys will depend on the per-unit price of information messages. Unlike the market price for most goods (which we usually assume to be the same for everyone), there are many reasons to believe that information costs may differ signiﬁcantly among individuals. Some individuals may possess speciﬁc skills relevant to information acquisition (they may be trained mechanics, for example) whereas others may not possess such skills. Some individuals may have other types of experience that yield valuable information, whereas others may lack that experience. For example, the seller of a product will usually know more about its limitations than will a buyer, because the seller will know precisely how the good was made and where possible problems might arise. Similarly, large-scale repeat buyers of a good may have greater access to information about it than would ﬁrst-time buyers. Finally, some

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individuals may have invested in some types of information services (for example, by having a computer link to a brokerage ﬁrm or by subscribing to Consumer Reports) that make the marginal cost of obtaining additional information lower than for someone without such an investment. All of these factors suggest that the level of information will sometimes differ among the participants in market transactions. Of course, in many instances, information costs may be low and such differences may be minor. Most people can appraise the quality of fresh vegetables fairly well just by looking at them, for example. But when information costs are high and variable across individuals, we would expect them to ﬁnd it advantageous to acquire different amounts of information. We will postpone a detailed study of such situations until Chapter 18.

SUMMARY The goal of this chapter was to provide some basic material for the study of individual behavior in uncertain situations. The key concepts covered may be listed as follows. •

The most common way to model behavior under uncertainty is to assume that individuals seek to maximize the expected utility of their actions.

•

Individuals who exhibit a diminishing marginal utility of wealth are risk averse. That is, they generally refuse fair bets.

•

Risk-averse individuals will wish to insure themselves completely against uncertain events if insurance premiums are actuarially fair. They may be willing to pay more than actuarially fair premiums in order to avoid taking risks.

•

Two utility functions have been extensively used in the study of behavior under uncertainty: the constant absolute risk aversion (CARA) function and the con-

stant relative risk aversion (CRRA) function. Neither is completely satisfactory on theoretical grounds. •

One of the most extensively studied issues in the economics of uncertainty is the “portfolio problem,” which asks how an investor will split his or her wealth between risky and risk-free assets. In some cases it is possible to obtain precise solutions to this problem, depending on the nature of the risky assets that are available.

•

The state-preference approach allows decision making under uncertainty to be approached in a familiar choicetheoretic framework. The approach is especially useful for looking at issues that arise in the economics of information.

•

Information is valuable because it permits individuals to make better decisions in uncertain situations. Information can be most valuable when individuals have some ﬂexibility in their decision making.

PROBLEMS 7.1 George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utility-of-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win?

7.2 Show that if an individual’s utility-of-wealth function is convex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

7.3 An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip.

Chapter 7 Uncertainty and Information a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?

7.4 Suppose there is a 50–50 chance that a risk-averse individual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000. a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 7.1) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) a fair policy covering the complete loss; and (2) a fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

7.5 Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it ðY Þ, given by U ðY Þ ¼ ln Y . a. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip’s expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the $1,000 (say, by purchasing traveler’s checks) at an “actuarially fair” premium of $250. Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the $1,000 without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1,000?

7.6 In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is p and that the ﬁne for receiving the ticket is f . Suppose that all individuals are risk averse (that is, U 00 ðW Þ < 0, where W is the individual’s wealth). Will a proportional increase in the probability of being caught or a proportional increase in the ﬁne be a more effective deterrent to illegal parking? Hint: Use the Taylor series approximation U ðW f Þ ¼ U ðW Þ f U 0 ðW Þ þ ð f 2 =2ÞU 00 ðW Þ.

7.7 A farmer believes there is a 50–50 chance that the next growing season will be abnormally rainy. His expected utility function has the form 1 1 expected utility ¼ ln YNR þ ln YR , 2 2 where YNR and YR represent the farmer’s income in the states of “normal rain” and “rainy,” respectively.

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Part 2 Choice and Demand a. Suppose the farmer must choose between two crops that promise the following income prospects: Crop YNR YR Wheat Corn

$28,000 19,000

$10,000 15,000

Which of the crops will he plant? b. Suppose the farmer can plant half his ﬁeld with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance—which is available to farmers who grow only wheat and which costs $4,000 and pays off $8,000 in the event of a rainy growing season—cause this farmer to change what he plants?

7.8 In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble (h) is given by p ¼ 0:5Eðh 2 ÞrðW Þ, where rðW Þ is the measure of absolute risk aversion at this person’s initial level of wealth. In this problem we look at the size of this payment as a function of the size of the risk faced and this person’s level of wealth. a. Consider a fair gamble (v) of winning or losing $1. For this gamble, what is Eðv2 Þ? b. Now consider varying the gamble in part (a) by multiplying each prize by a positive constant k. Let h ¼ kv. What is the value of Eðh 2 Þ? c. Suppose this person has a logarithmic utility function U ðW Þ ¼ ln W . What is a general expression for rðW Þ? d. Compute the risk premium (p) for k ¼ 0:5, 1, and 2 and for W ¼ 10 and 100. What do you conclude by comparing the six values?

Analytical Problems 7.9 HARA Utility The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute risk aversion (HARA) functions. The general form for this function is U ðW Þ ¼ θðμ þ W =γÞ1γ , where the various parameters obey the following restrictions:

• •

γ 1, μ þ W =γ > 0,

•

θ½ð1 γÞ=γ > 0.

The reasons for the first two restrictions are obvious; the third is required so that U 0 > 0. a. Calculate rðW Þ for this function. Show that the reciprocal of this expression is linear in W . This is the origin of the term “harmonic” in the function’s name. b. Show that, when μ ¼ 0 and θ ¼ ½ð1 γÞ=γγ1 , this function reduces to the CRRA function given in Chapter 7 (see footnote 17). c. Use your result from part (a) to show that if γ ! ∞ then rðW Þ is a constant for this function. d. Let the constant found in part (c) be represented by A. Show that the implied form for the utility function in this case is the CARA function given in Equation 7.35. e. Finally, show that a quadratic utility function can be generated from the HARA function simply by setting γ ¼ 1. f. Despite the seeming generality of the HARA function, it still exhibits several limitations for the study of behavior in uncertain situations. Describe some of these shortcomings.

Chapter 7 Uncertainty and Information

7.10 The resolution of uncertainty In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today (c1 ) but that tomorrow’s consumption (c2 ) will be either 10 or 2.5, depending on whether a coin comes up heads or tails. Suppose also that the individual’s utility function has the simple Cobb-Douglas form pﬃﬃﬃﬃﬃﬃﬃﬃﬃ U ðc1 , c2 Þ ¼ c1 c2 . a. If an individual cares only about the expected value of utility, will it matter whether the coin is ﬂipped just before day 1 or just before day 2? Explain. b. More generally, suppose that the individual’s expected utility depends on the timing of the coin ﬂip. Speciﬁcally, assume that expected utility ¼ E1 ½fE2 ½U ðc1 , c2 Þgα , where E1 represents expectations taken at the start of day 1, E2 represents expectations at the start of day 2, and α represents a parameter that indicates timing preferences. Show that if α ¼ 1, the individual is indifferent about when the coin is flipped. c. Show that if α ¼ 2, the individual will prefer early resolution of the uncertainty—that is, ﬂipping the coin at the start of day 1. d. Show that if α ¼ 0.5, the individual will prefer later resolution of the uncertainty (ﬂipping at the start of day 2). e. Explain your results intuitively and indicate their relevance for information theory. Note: This problem is an illustration of “resolution seeking” and “resolution averse” behavior; see D. M. Kreps and E. L. Porteus, “Temporal Resolution of Uncertainty and Dynamic Choice Theory,” Econometrica (January 1978): 185–200.

7.11 More on the CRRA function For the constant relative risk aversion utility function (Equation 7.42), we showed that the degree of risk aversion is measured by ð1 RÞ. In Chapter 3 we showed that the elasticity of substitution for the same function is given by 1=ð1 RÞ. Hence, the measures are reciprocals of each other. Using this result, discuss the following questions. a. Why is risk aversion related to an individual’s willingness to substitute wealth between states of the world? What phenomenon is being captured by both concepts? b. How would you interpret the polar cases R ¼ 1 and R ¼ ∞ in both the risk-aversion and substitution frameworks? c. A rise in the price of contingent claims in “bad” times ðPb Þ will induce substitution and income effects into the demands for Wg and Wb . If the individual has a ﬁxed budget to devote to these two goods, how will choices among them be affected? Why might Wg rise or fall depending on the degree of risk aversion exhibited by the individual? d. Suppose that empirical data suggest an individual requires an average return of 0.5 percent before being tempted to invest in an investment that has a 50–50 chance of gaining or losing 5 percent. That is, this person gets the same utility from W0 as from an even bet on 1.055W0 and 0.955W0 .

(1) What value of R is consistent with this behavior? (2) How much average return would this person require to accept a 50–50 chance of gaining or losing 10 percent? Note: This part requires solving nonlinear equations, so approximate solutions will sufﬁce. The comparison of the risk-reward trade-off illustrates what is called the “equity premium puzzle” in that risky investments seem actually to earn much more than is consistent with the degree of risk aversion suggested by other data. See N. R. Kocherlakota, “The Equity Premium: It’s Still a Puzzle,” Journal of Economic Literature (March 1996): 42–71.

229

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Part 2 Choice and Demand

7.12 Graphing risky investments Investment in risky assets can be examined in the state-preference framework by assuming that W dollars invested in an asset with a certain return r will yield W ð1 þ rÞ in both states of the world, whereas investment in a risky asset will yield W ð1 þ rg Þ in good times and W ð1 þ rb Þ in bad times (where rg > r > rb ). a. Graph the outcomes from the two investments. b. Show how a “mixed portfolio” containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals’ attitudes toward risk will determine the mix of risk-free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual’s utility takes the constant relative risk aversion form (Equation 7.42), explain why this person will not change the fraction of risky assets held as his or her wealth increases.26

7.13 Taxing risks assets Suppose the asset returns in Problem 7.12 are subject to taxation. a. Show, under the conditions of Problem 7.12, why a proportional tax on wealth will not affect the fraction of wealth allocated to risky assets. b. Suppose that only the returns from the safe asset were subject to a proportional income tax. How would this affect the fraction of wealth held in risky assets? Which investors would be most affected by such a tax? c. How would your answer to part (b) change if all asset returns were subject to a proportional income tax? Note: This problem asks you to compute the pre-tax allocation of wealth that will result in post-tax utility maximization.

7.14 The portfolio problem with a Normally distributed risky asset In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form E½U ðW Þ ¼ μW ðA=2Þσ2W , where μW is the expected value of wealth and σ2W is its variance. Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest k of his or her wealth in a Normally distributed risky asset whose expected return is μr and variance in return is σ2r (your answer should depend on A). Explain your results intuitively.

This problem and the next are taken from J. E. Stiglitz, “The Effects of Income, Wealth, and Capital Gains Taxation in Risk Taking,” Quarterly Journal of Economics (May 1969): 263–83.

26

Chapter 7 Uncertainty and Information

231

SUGGESTIONS FOR FURTHER READING Arrow, K. J. “The Role of Securities in the Optimal Allocation of Risk Bearing.” Review of Economic Studies 31 (1963): 91–96.

Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995, chap. 6.

Introduces the state-preference concept and interprets securities as claims on contingent commodities.

Provides a good summary of the foundations of expected utility theory. Also examines the “state independence” assumption in detail and shows that some notions of risk aversion carry over into cases of state dependence.

———. “Uncertainty and the Welfare Economics of Medical Care.” American Economic Review 53 (1963): 941–73. Excellent discussion of the welfare implications of insurance. Has a clear, concise, mathematical appendix. Should be read in conjunction with Pauly’s article on moral hazard (see Chapter 18).

Bernoulli, D. “Exposition of a New Theory on the Measurement of Risk.” Econometrica 22 (1954): 23–36. Reprint of the classic analysis of the St. Petersburg paradox.

Dixit, A. K., and R. S. Pindyck. Investment under Uncertainty. Princeton: Princeton University Press, 1994. Focuses mainly on the investment decision by ﬁrms but has a good coverage of option concepts.

Friedman, M., and L. J. Savage. “The Utility Analysis of Choice.” Journal of Political Economy 56 (1948): 279–304. Analyzes why individuals may both gamble and buy insurance. Very readable.

Gollier, Christian. The Economics of Risk and Time. Cambridge, MA: MIT Press, 2001. Contains a complete treatment of many of the issues discussed in this chapter. Especially good on the relationship between allocation under uncertainty and allocation over time.

Pratt, J. W. “Risk Aversion in the Small and in the Large.” Econometrica 32 (1964): 122–36. Theoretical development of risk-aversion measures. Fairly technical treatment but readable.

Rothschild, M., and J. E. Stiglitz. “Increasing Risk: 1. A Definition.” Journal of Economic Theory 2 (1970): 225–43. Develops an economic deﬁnition of what it means for one gamble to be “riskier” than another. A sequel article in the Journal of Economic Theory provides economic illustrations.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGrawHill, 2001. Chapter 13 provides a nice introduction to the relationship between statistical concepts and expected utility maximization. Also shows in detail the integration mentioned in Example 7.3.

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Part 2 Choice and Demand

EXTENSIONS Portfolios of Many Risky Assets The portfolio problem we studied in Chapter 7 looked at an investor’s decision to invest a portion of his or her wealth in a single risky asset. In these Extensions we will see how this model can be generalized to consider portfolios with many such assets. Throughout our discussion we will assume that there are n risky assets. The return on each asset is a random variable denoted by ri ði ¼ 1, nÞ. The expected values and variances of these assets’ returns are denoted by Eðri Þ ¼ μi and Varðri Þ ¼ σ2i , respectively. An investor who invests a portion of his or her wealth in a portfolio of these assets will obtain a random return ðrP Þ given by n X rP ¼ αi ri , (i) i¼1

where αi ð 0Þ is the fraction Pn of the risky portfolio held in asset i and where i¼1 αi ¼ 1. In this situation, the expected return on this portfolio will be n X EðrP Þ ¼ μP ¼ αi μi . (ii)

invests, the expected return on the portfolio will be the same: μP ¼ α1 μ1 þ ð1 α1 Þμ2 ¼ μ1 ¼ μ2 . (iv) But the variance will depend on the allocation between the two assets: σ2P ¼ α21 σ21 þ ð1 α1 Þ2 σ22 ¼ ð1 2α1 þ 2α21 Þσ21 . (v) Choosing α1 to minimize this expression yields α1 ¼ 0:5 and (vi) σ2P ¼ 0:5σ21 . Hence, holding half of one’s portfolio in each asset yields the same expected return as holding only one asset, but it promises a variance of return that is only half as large. As we showed earlier in Chapter 7, this is the primary benefit of diversification.

E7.2 Efficient portfolios

i¼1

If the returns of each asset are independent, then the variance of the portfolio’s return will be n X α2i σ2i . (iii) VarðrP Þ ¼ σ2P ¼ i¼1

If the returns are not independent, Equation iii would have to be modified to take covariances among the returns into account. Using this general notation, we now proceed to look at some aspects of this portfolio allocation problem.

E7.1 Diversification with two risky assets Equation iii provides the basic rationale for holding many assets in a portfolio: so that diversiﬁcation can reduce risk. Suppose, for example, that there are only two independent assets and that the expected returns and variances of those returns for each of the assets are identical. That is, assume μ1 ¼ μ2 and σ21 ¼ σ22 . A person who invests his or her risky portfolio in only one of these seemingly identical assets will obtain μP ¼ μ1 ¼ μ2 and σ2P ¼ σ21 ¼ σ22 . By mixing the assets, however, this investor can do better in the sense that he or she can get the same expected yield with lower variance. Notice that, no matter how this person

With many assets, the optimal diversiﬁcation problem is to choose asset weightings (the α’s) so as to minimize the variance (or standard deviation) of the portfolio for each potential expected return. The solution to this problem yields an “efﬁciency frontier” for risky asset portfolios such as that represented by the line EE in Figure E7.1. Portfolios that lie below this frontier are inferior to those on the frontier because they offer lower expected returns for any degree of risk. Portfolio returns above the frontier are unattainable. Sharpe (1970) discusses the mathematics associated with constructing the EE frontier. Mutual funds The notion of portfolio efﬁciency has been widely applied to the study of mutual funds. In general, mutual funds are a good answer to small investors’ diversiﬁcation needs. Because such funds pool the funds of many individuals, they are able to achieve economies of scale in transactions and management costs. This permits fund owners to share in the fortunes of a much wider variety of equities than would be possible if each acted alone. But mutual fund managers have incentives of their own, so the portfolios they hold may not always be perfect representations of the risk attitudes of their clients. For example, Scharfstein and Stein (1990) develop a model that shows why mutual fund

Chapter 7 Uncertainty and Information

FIGURE E7.1

233

Efficient Portfolios

The frontier EE represents optimal mixtures of risky assets that minimize the standard deviation of the portfolio, σP , for each expected return, μP . A risk-free asset with return μf offers investors the opportunity to hold mixed portfolios along PP that mix this risk-free asset with the market portfolio, M .

P

P E M

M

P f

E

M

managers have incentives to “follow the herd” in their investment picks. Other studies, such as the classic investigation by Jensen (1968), ﬁnd that mutual fund managers are seldom able to attain extra returns large enough to offset the expenses they charge investors. In recent years this has led many mutual fund buyers to favor “index” funds that seek simply to duplicate the market average (as represented, say, by the Standard and Poor’s 500 stock index). Such funds have very low expenses and therefore permit investors to achieve diversiﬁcation at minimal cost.

E7.3 Portfolio separation If there exists a risk-free asset with expected return μf and σf ¼ 0, then optimal portfolios will consist of mixtures of this asset with risky ones. All such portfolios will lie along the line PP in Figure 7.1, because this shows the maximum return attainable for each value of σ for various portfolio allocations. These allocations will contain only one speciﬁc set of risky assets: the set represented by point M . In equilibrium this will be the “market portfolio” consisting of all capital assets held in proportion to their market valuations. This market portfolio will provide an expected return of μM and a

P

standard deviation of that return of σM . The equation for the line PP that represents any mixed portfolio is given by the linear equation μP ¼ μf þ

μM μf σM

⋅ σP .

(vii)

This shows that the market line PP permits individual investors to “purchase” returns in excess of the riskfree return ðμM μf Þ by taking on proportionally more risk ðσP =σM Þ. For choices on PP to the left of the market point M , σP =σM < 1 and μf < μP < μM . High-risk points to the right of M —which can be obtained by borrowing to produce a leveraged portfolio—will have σP =σM > 1 and will promise an expected return in excess of what is provided by the market portfolio ðμP > μM Þ. Tobin (1958) was one of the ﬁrst economists to recognize the role that risk-free assets play in identifying the market portfolio and in setting the terms on which investors can obtain returns above risk-free levels.

E7.4 Individual choices Figure E7.2 illustrates the portfolio choices of various investors facing the options offered by the line PP .

234

Part 2 Choice and Demand

FIGURE E7.2

Investor Behavior and Risk Aversion Given the market options PP , investors can choose how much risk they wish to assume. Very riskaverse investors (UI ) will hold mainly risk-free assets, whereas risk takers (UIII ) will opt for leveraged portfolios.

P

UIII P UII

UI

f

M

P

P

This ﬁgure illustrates the type of portfolio choice model previously described in this chapter. Individuals with low tolerance for risk (I ) will opt for portfolios that are heavily weighted toward the risk-free asset. Investors willing to assume a modest degree of risk (II ) will opt for portfolios close to the market portfolio. High-risk investors (III ) may opt for leveraged portfolios. Notice that all investors face the same “price” of risk ðμM μf Þ with their expected returns being determined by how much relative risk ðσP =σM Þ they are willing to incur. Notice also that the risk associated with an investor’s portfolio depends only on the fraction of the portfolio invested in the market portfolio ðαÞ, since σ2P ¼ α2 σ2M þ ð1 αÞ2 ⋅ 0. Hence, σP =σM ¼ α and so the investor’s choice of portfolio is equivalent to his or her choice of risk.

E7.5 Capital asset pricing model Although the analysis of E7.4 shows how a portfolio that mixes a risk-free asset with the market portfolio

will be priced, it does not describe the risk-return tradeoff for a single asset. Because (assuming transactions are costless) an investor can always avoid risk unrelated to the overall market by choosing to diversify with a “market portfolio,” such “unsystematic” risk will not warrant any excess return. An asset will, however, earn an excess return to the extent that it contributes to overall market risk. An asset that does not yield such extra returns would not be held in the market portfolio, so it would not be held at all. This is the fundamental insight of the capital asset pricing model (CAPM). To examine these results formally, consider a portfolio that combines a small amount ðαÞ of an asset with a random return of x with the market portfolio (which has a random return of M ). The return on this portfolio (z) would be given by z ¼ αx þ ð1 αÞM . (viii) The expected return is (ix) μz ¼ αμx þ ð1 αÞμM

Chapter 7 Uncertainty and Information

with variance σ2z ¼ α2 σ2x þ ð1 αÞ2 σ2M

þ 2αð1 αÞσx,M ,

(x)

where σx,M is the covariance between the return on x and the return on the market. But our previous analysis shows σ (xi) μz ¼ μf þ ðμM μf Þ ⋅ z . σM Setting Equation ix equal to xi and differentiating with respect to α yields μM μf ∂σz ∂μz ¼ μx μM ¼ . (xii) ∂α σM ∂α By calculating ∂σz =∂α from Equation x and taking the limit as α approaches zero, we get ! μM μf σx, M σ2M , (xiii) μx μM ¼ σM σM or, rearranging terms, μx ¼ μf þ ðμM μf Þ ⋅

σx, M

. (xiv) σ2M Again, risk has a reward of μM μf , but now the quantity of risk is measured by σx, M =σ2M . This ratio of the covariance between the return x and the market to the variance of the market return is referred to as the beta coefficient for the asset. Estimated beta coefficients for financial assets are reported in many publications. Studies of the CAPM This version of the capital asset pricing model carries strong implications about the determinants of any

235

asset’s expected rate of return. Because of this simplicity, the model has been subject to a large number of empirical tests. In general these ﬁnd that the model’s measure of systemic risk (beta) is indeed correlated with expected returns, while simpler measures of risk (for example, the standard deviation of past returns) are not. Perhaps the most inﬂuential early empirical test that reached such a conclusion was Fama and MacBeth (1973). But the CAPM itself explains only a small fraction of differences in the returns of various assets. And, contrary to the CAPM, a number of authors have found that many other economic factors signiﬁcantly affect expected returns. Indeed, a prominent challenge to the CAPM comes from one of its original founders—see Fama and French (1992).

References Fama, E. F., and K. R. French. “The Cross Section of Expected Stock Returns.” Journal of Finance 47 (1992): 427–66. Fama, E. F., and J. MacBeth. “Risk, Return, and Equilibrium.” Journal of Political Economy 8 (1973): 607–36. Jensen, M. “The Performance of Mutual Funds in the Period 1945–1964.” Journal of Finance (May 1968): 386–416. Scharfstein, D. S., and J. Stein. “Herd Behavior and Investment.” American Economic Review (June 1990): 465–89. Sharpe, W. F. Portfolio Theory and Capital Markets. New York: McGraw-Hill, 1970. Tobin, J. “Liquidity Preference as Behavior Towards Risk.” Review of Economic Studies (February 1958): 65–86.

CHAPTER

8 Strategy and Game Theory This chapter provides an introduction to noncooperative game theory, a tool used to understand the strategic interactions among two or more agents. The range of applications of game theory has been growing constantly, including all areas of economics (from labor economics to macroeconomics) and other ﬁelds such as political science and biology. Game theory is particularly useful in understanding the interaction between ﬁrms in an oligopoly, so the concepts learned here will be used extensively in Chapter 15. We begin with the central concept of Nash equilibrium and study its application in simple games. We then go on to study reﬁnements of Nash equilibrium that are used in games with more complicated timing and information structures.

BASIC CONCEPTS So far in Part II of this text, we have studied individual decisions made in isolation. In this chapter we study decision making in a more complicated, strategic setting. In a strategic setting, a person may no longer have an obvious choice that is best for him or her. What is best for one decision maker may depend on what the other is doing and vice versa. For example, consider the strategic interaction between drivers and the police. Whether drivers prefer to speed may depend on whether the police set up speed traps. Whether the police ﬁnd speed traps valuable depends on how much drivers speed. This confusing circularity would seem to make it difﬁcult to make much headway in analyzing strategic behavior. In fact, the tools of game theory will allow us to push the analysis nearly as far, for example, as our analysis of consumer utility maximization in Chapter 4. There are two major tasks involved when using game theory to analyze an economic situation. The ﬁrst is to distill the situation into a simple game. Because the analysis involved in strategic settings quickly grows more complicated than in simple decision problems, it is important to simplify the setting as much as possible by retaining only a few essential elements. There is a certain art to distilling games from situations that is hard to teach. The examples in the text and problems in this chapter can serve as models that may help in approaching new situations. The second task is to “solve” the given game, which results in a prediction about what will happen. To solve a game, one takes an equilibrium concept (Nash equilibrium, for example) and runs through the calculations required to apply it to the given game. Much of the chapter will be devoted to learning the most widely used equilibrium concepts (including Nash equilibrium) and to practicing the calculations necessary to apply them to particular games. A game is an abstract model of a strategic situation. Even the most basic games have three essential elements: players, strategies, and payoffs. In complicated settings, it is sometimes also necessary to specify additional elements such as the sequence of moves and the information that players have when they move (who knows what when) to describe the game fully. 236

Chapter 8

Strategy and Game Theory

Players Each decision maker in a game is called a player. These players may be individuals (as in poker games), ﬁrms (as in markets with few ﬁrms), or entire nations (as in military conﬂicts). A player is characterized as having the ability to choose from among a set of possible actions. Usually, the number of players is ﬁxed throughout the “play” of the game. Games are sometimes characterized by the number of players involved (two-player, three-player, or n-player games). As does much of the economic literature, this chapter often focuses on two-player games because this is the simplest strategic setting. We will label the players with numbers, so in a two-player game we will have players 1 and 2. In an n-player game we will have players 1, 2, ..., n, with the generic player labeled i.

Strategies Each course of action open to a player during the game is called a strategy. Depending on the game being examined, a strategy may be a simple action (drive over the speed limit or not) or a complex plan of action that may be contingent on earlier play in the game (say, speeding only if the driver has observed speed traps less than a quarter of the time in past drives). Many aspects of game theory can be illustrated in games in which players choose between just two possible actions. Let S1 denote the set of strategies open to player 1, S2 the set open to player 2, and (more generally) Si the set open to player i. Let s1 2 S1 be a particular strategy chosen by player 1 from the set of possibilities, s2 2 S2 the particular strategy chosen by player 2, and si 2 Si for player i. A strategy proﬁle will refer to a listing of particular strategies chosen by each of a group of players.

Payoffs The ﬁnal returns to the players at the conclusion of a game are called payoffs. Payoffs are measured in levels of utility obtained by the players. For simplicity, monetary payoffs (say, proﬁts for ﬁrms) are often used. More generally, payoffs can incorporate nonmonetary outcomes such as prestige, emotion, risk preferences, and so forth. Players are assumed to prefer higher payoffs than lower payoffs. In a two-player game, u1 ðs1 , s2 Þ denotes player 1’s payoff given that he or she chooses s1 and the other player chooses s2 and similarly u2 ðs2 , s1 Þ denotes player 2’s payoff. The fact player 1’s payoff may depend on 2’s strategy (and vice versa) is where the strategic interdependence shows up. In an n-player game, we can write the payoff of a generic player i as ui ðsi , si Þ, which depends on player i’s own strategy si and the proﬁle si ¼ ðs1 , …, si1 , siþ1 , …, sn Þ of the strategies of all players other than i.

PRISONERS’ DILEMMA The Prisoners’ Dilemma, introduced by A. W. Tucker in the 1940s, is one of the most famous games studied in game theory and will serve here as a nice example to illustrate all the notation just introduced. The title stems from the following situation. Two suspects are arrested for a crime. The district attorney has little evidence in the case and is eager to extract a confession. She separates the suspects and tells each: “If you ﬁnk on your companion but your companion doesn’t ﬁnk on you, I can promise you a reduced (one-year) sentence, whereas your companion will get four years. If you both ﬁnk on each other, you will each get a three-year sentence.” Each suspect also knows that if neither of them ﬁnks then the lack of evidence will result in being tried for a lesser crime for which the punishment is a two-year sentence.

237

238

Part 2 Choice and Demand

Boiled down to its essence, the Prisoners’ Dilemma has two strategic players, the suspects, labeled 1 and 2. (There is also a district attorney, but since her actions have already been fully speciﬁed, there is no reason to complicate the game and include her in the speciﬁcation.) Each player has two possible strategies open to him: ﬁnk or remain silent. We therefore write their strategy sets as S1 ¼ S2 ffink, silentg. To avoid negative numbers we will specify payoffs as the years of freedom over the next four years. For example, if suspect 1 ﬁnks and 2 does not, suspect 1 will enjoy three years of freedom and 2 none, that is, u1 ðfink, silentÞ ¼ 3 and u2 ðsilent, finkÞ ¼ 0.

Extensive form There are 22 ¼ 4 combinations of strategies and two payoffs to specify for each combination. So instead of listing all the payoffs, it will be clearer to organize them in a game tree or a matrix. The game tree, also called the extensive form, is shown in Figure 8.1. The action proceeds from left to right. Each node (shown as a dot on the tree) represents a decision point for the player indicated there. The ﬁrst move in this game belongs to player 1; he must choose whether to ﬁnk or be silent. Then player 2 makes his decision. The dotted oval drawn around the nodes at which player 2 moves indicates that the two nodes are in the same information set, that is, player 2 does not know what player 1 has chosen when 2 moves. We put the two nodes in the same information set because the district attorney approaches each suspect separately and does not reveal what the other has chosen. We will later look at games in which the second mover does have information about the ﬁrst mover’s choice and so the two nodes are in separate information sets. Payoffs are given at the end of the tree. The convention is for player 1’s payoff to be listed ﬁrst, then player 2’s. FIGURE 8.1

Extensive Form for the Prisoners’ Dilemma In this game, player 1 chooses to ﬁnk or be silent, and player 2 has the same choice. The oval surrounding 2’s nodes indicates that they share the same (lack of ) information: 2 does not know what strategy 1 has chosen because the district attorney approaches each player in secret. Payoffs are listed at the right. u1 = 1, u2 = 1 Fink 2

Fink

Silent

Silent

Fink

u1 = 3, u2 = 0

1

u1 = 0, u2 = 3 2

Silent

u1 = 2, u2 = 2

Chapter 8

TABLE 8.1

Strategy and Game Theory

Normal Form for the Prisoners’ Dilemma

Suspect 1

Suspect 2 Fink

Silent

Fink

u1 ¼ 1, u2 ¼ 1

u1 ¼ 3, u2 ¼ 0

Silent

u1 ¼ 0, u2 ¼ 3

u1 ¼ 2, u2 ¼ 2

Normal form Although the extensive form in Figure 8.1 offers a useful visual presentation of the complete structure of the game, sometimes it is more convenient to represent games in matrix form, called the normal form of the game; this is shown for the Prisoners’ Dilemma in Table 8.1. Player 1 is the row player, and 2 is the column player. Each entry in the matrix lists the payoffs ﬁrst for player 1 and then for 2.

Thinking strategically about the Prisoners’ Dilemma Although we have not discussed how to solve games yet, it is worth thinking about what we might predict will happen in the Prisoners’ Dilemma. Studying Table 8.1, on ﬁrst thought one might predict that both will be silent. This gives the most total years of freedom for both (four) compared to any other outcome. Thinking a bit deeper, this may not be the best prediction in the game. Imagine ourselves in player 1’s position for a moment. We don’t know what player 2 will do yet since we haven’t solved out the game, so let’s investigate each possibility. Suppose 2 chose to ﬁnk. By ﬁnking ourselves we would earn one year of freedom versus none if we remained silent, so ﬁnking is better for us. Suppose 2 chose to remain silent. Finking is still better for us than remaining silent since we get three rather than two years of freedom. Regardless of what the other player does, ﬁnking is better for us than being silent since it results in an extra year of freedom. Since players are symmetric, the same reasoning holds if we imagine ourselves in player 2’s position. Therefore, the best prediction in the Prisoners’ Dilemma is that both will ﬁnk. When we formally introduce the main solution concept—Nash equilibrium—we will indeed ﬁnd that both ﬁnking is a Nash equilibrium. The prediction has a paradoxical property: by both ﬁnking, the suspects only enjoy one year of freedom, but if they were both silent they would both do better, enjoying two years of freedom. The paradox should not be taken to imply that players are stupid or that our prediction is wrong. Rather, it reveals a central insight from game theory that pitting players against each other in strategic situations sometimes leads to outcomes that are inefﬁcient for the players. (When we say the outcome is inefﬁcient, we are focusing just on the suspects’ utilities; if the focus were shifted to society at large, then both ﬁnking might be quite a good outcome for the criminal justice system—presumably the motivation behind the district attorney’s offer.) The suspects might try to avoid the extra prison time by coming to an agreement beforehand to remain silent, perhaps reinforced by threats to retaliate afterwards if one or the other ﬁnks. Introducing agreements and threats leads to a game that differs from the basic Prisoners’ Dilemma, a game that should be analyzed on its own terms using the tools we will develop shortly. Solving the Prisoners’ Dilemma was easy because there were only two players and two strategies and because the strategic calculations involved were fairly straightforward. It would be useful to have a systematic way of solving this as well as more complicated games. Nash equilibrium provides us with such a systematic solution.

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NASH EQUILIBRIUM In the economic theory of markets, the concept of equilibrium is developed to indicate a situation in which both suppliers and demanders are content with the market outcome. Given the equilibrium price and quantity, no market participant has an incentive to change his or her behavior. In the strategic setting of game theory, we will adopt a related notion of equilibrium, formalized by John Nash in the 1950s, called Nash equilibrium.1 Nash equilibrium involves strategic choices that, once made, provide no incentives for the players to alter their behavior further. A Nash equilibrium is a strategy for each player that is the best choice for each player given the others’ equilibrium strategies. Nash equilibrium can be deﬁned very simply in terms of best responses. In an n-player game, strategy si is a best response to rivals’ strategies si if player i cannot obtain a strictly higher payoff with any other possible strategy si0 2 Si given that rivals are playing si . DEFINITION

Best response. si is a best response for player i to rivals’ strategies si , denoted si 2 BRi ðsi Þ, if ui ðsi , si Þ ui ðsi0 , si Þ

for all

si0 2 Si .

(8.1)

A technicality embedded in the deﬁnition is that there may be a set of best responses rather than a unique one; that is why we used the set inclusion notation si 2 BRi ðsi Þ. There may be a tie for the best response, in which case the set BRi ðsi Þ will contain more than one element. If there isn’t a tie, then there will be a single best response si and we can simply write si ¼ BRi ðsi Þ. We can now deﬁne a Nash equilibrium in an n-player game as follows. DEFINITION

Nash equilibrium. A Nash equilibrium is a strategy proﬁle ðs 1 , s 2 , …, s n Þ such that, for each . That player i ¼ 1, 2, …, n, s i is a best response to the other players’ equilibrium strategies si is, s i 2 BRi ðsi Þ. These deﬁnitions involve a lot of notation. The notation is a bit simpler in a two-player game. In a two-player game, ðs 1 , s 2 Þ is a Nash equilibrium if s 1 and s 2 are mutual best responses against each other: u1 ðs 1 , s 2 Þ u1 ðs1 , s 2 Þ

for all

s1 2 S1

(8.2)

u2 ðs 2 , s 1 Þ u2 ðs2 , s 1 Þ

for all

s2 2 S2 .

(8.3)

and

A Nash equilibrium is stable in that, even if all players revealed their strategies to each other, no player would have an incentive to deviate from his or her equilibrium strategy and choose something else. Nonequilibrium strategies are not stable in this way. If an outcome is not a Nash equilibrium, then at least one player must benefit from deviating. Hyperrational players could be expected to solve the inference problem and deduce that all would play a Nash equilibrium (especially if there is a unique Nash equilibrium). Even if players are not hyperrational, over the long run we can expect their play to converge to a Nash equilibrium as they abandon strategies that are not mutual best responses.

1 John Nash, “Equilibrium Points in n-Person Games,” Proceedings of the National Academy of Sciences 36 (1950): 48–49. Nash is the principal figure in the 2001 film A Beautiful Mind (see Problem 8.7 for a game-theory example from the film) and co-winner of the 1994 Nobel Prize in economics.

Chapter 8

Strategy and Game Theory

Besides this stability property, another reason Nash equilibrium is used so widely in economics is that it is guaranteed to exist for all games we will study (allowing for mixed strategies, to be deﬁned below; Nash equilibria in pure strategies do not have to exist). Nash equilibrium has some drawbacks. There may be multiple Nash equilibria, making it hard to come up with a unique prediction. Also, the deﬁnition of Nash equilibrium leaves unclear how a player can choose a best-response strategy before knowing how rivals will play.

Nash equilibrium in the Prisoners’ Dilemma Let’s apply the concepts of best response and Nash equilibrium to the example of the Prisoners’ Dilemma. Our educated guess was that both players will end up ﬁnking. We will show that both ﬁnking is a Nash equilibrium of the game. To do this, we need to show that ﬁnking is a best response to the other players’ ﬁnking. Refer to the payoff matrix in Table 8.1. If player 2 ﬁnks, we are in the ﬁrst column of the matrix. If player 1 also ﬁnks, his payoff is 1; if he is silent, his payoff is 0. Since he earns the most from ﬁnking given player 2 ﬁnks, ﬁnking is player 1’s best response to player 2’s ﬁnking. Since players are symmetric, the same logic implies that player 2’s ﬁnking is a best response to player 1’s ﬁnking. Therefore, both ﬁnking is indeed a Nash equilibrium. We can show more: that both ﬁnking is the only Nash equilibrium. To do so, we need to rule out the other three outcomes. Consider the outcome in which player 1 ﬁnks and 2 is silent, abbreviated (ﬁnk, silent), the upper right corner of the matrix. This is not a Nash equilibrium. Given that player 1 ﬁnks, as we have already said, player 2’s best response is to ﬁnk, not to be silent. Symmetrically, the outcome in which player 1 is silent and 2 ﬁnks in the lower left corner of the matrix is not a Nash equilibrium. That leaves the outcome in which both are silent. Given that player 2 is silent, we focus our attention on the second column of the matrix: the two rows in that column show that player 1’s payoff is 2 from being silent and 3 from ﬁnking. Therefore, silent is not a best response to ﬁnk and so both being silent cannot be a Nash equilibrium. To rule out a Nash equilibrium, it is enough to ﬁnd just one player who is not playing a best response and so would want to deviate to some other strategy. Considering the outcome (ﬁnk, silent), although player 1 would not deviate from this outcome (he earns 3, which is the most possible), player 2 would prefer to deviate from silent to ﬁnk. Symmetrically, considering the outcome (silent, ﬁnk), although player 2 does not want to deviate, player 1 prefers to deviate from silent to ﬁnk, so this is not a Nash equilibrium. Considering the outcome (silent, silent), both players prefer to deviate to another strategy. Having two players prefer to deviate is more than enough to rule out a Nash equilibrium.

Underlining best-response payoffs A quick way to ﬁnd the Nash equilibria of a game is to underline best-response payoffs in the matrix. The underlining procedure is demonstrated for the Prisoners’ Dilemma in Table 8.2. The ﬁrst step is to underline the payoffs corresponding to player 1’s best responses. Player 1’s

Underlining Procedure in the Prisoners’ Dilemma Suspect 2 Suspect 1

TABLE 8.2

Fink

Silent

Fink

u1 ¼ 1, u2 ¼ 1

u1 ¼ 3, u2 ¼ 0

Silent

u1 ¼ 0, u2 ¼ 3

u1 ¼ 2, u2 ¼ 2

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Part 2 Choice and Demand

best response is to ﬁnk if player 2 ﬁnks, so we underline u1 ¼ 1 in the upper left box, and to ﬁnk if player 2 is silent, so we underline u1 ¼ 3 in the upper left box. Next, we move to underlining the payoffs corresponding to player 2’s best responses. Player 2’s best response is to ﬁnk if player 1 ﬁnks, so we underline u2 ¼ 1 in the upper left box, and to ﬁnk if player 1 is silent, so we underline u2 ¼ 3 in the lower left box. Now that the best-response payoffs have been underlined, we look for boxes in which every player’s payoff is underlined. These boxes correspond to Nash equilibria. (There may be additional Nash equilibria involving mixed strategies, deﬁned later in the chapter.) In Table 8.2, only in the upper left box are both payoffs underlined, verifying that (ﬁnk, ﬁnk)— and none of the other outcomes—is a Nash equilibrium.

Dominant Strategies (Fink, ﬁnk) is a Nash equilibrium in the Prisoners’ Dilemma because ﬁnking is a best response to the other player’s ﬁnking. We can say more: ﬁnking is the best response to all of the other player’s strategies, ﬁnk and silent. (This can be seen, among other ways, from the underlining procedure shown in Table 8.2: all player 1’s payoffs are underlined in the row in which he plays ﬁnk, and all player 2’s payoffs are underlined in the column in which he plays ﬁnk.) A strategy that is a best response to any strategy the other players might choose is called a dominant strategy. Players do not always have dominant strategies, but when they do there is strong reason to believe they will play that way. Complicated strategic considerations do not matter when a player has a dominant strategy because what is best for that player is independent of what others are doing. DEFINITION

Dominant strategy. A dominant strategy is a strategy s i for player i that is a best response to all strategy proﬁles of other players. That is, s i 2 BRi ðsi Þ for all si . Note the difference between a Nash equilibrium strategy and a dominant strategy. A strategy that is part of a Nash equilibrium need only be a best response to one strategy proﬁle of other players—namely, their equilibrium strategies. A dominant strategy must be a best response not just to the Nash equilibrium strategies of other players but to all the strategies of those players. If all players in a game have a dominant strategy, then we say the game has a dominant strategy equilibrium. As well as being the Nash equilibrium of the Prisoners’ Dilemma, (ﬁnk, ﬁnk) is a dominant strategy equilibrium. As is clear from the deﬁnitions, in any game with a dominant strategy equilibrium, the dominant strategy equilibrium is a Nash equilibrium. Problem 8.4 will show that when a dominant strategy exists, it is the unique Nash equilibrium.

Battle of the Sexes The famous Battle of the Sexes game is another example that illustrates the concepts of best response and Nash equilibrium. The story goes that a wife (player 1) and husband (player 2) would like to meet each other for an evening out. They can go either to the ballet or to a boxing match. Both prefer to spend time together than apart. Conditional on being together, the wife prefers to go to the ballet and the husband to boxing. The extensive form of the game is presented in Figure 8.2 and the normal form in Table 8.3. For brevity we dispense with the u1 and u2 labels on the payoffs and simply re-emphasize the convention that the ﬁrst payoff is player 1’s and the second player 2’s. We will work with the normal form, examining each of the four boxes in Table 8.3 and determining which are Nash equilibria and which are not. Start with the outcome in which both players choose ballet, written (ballet, ballet), the upper left corner of the payoff matrix. Given that the husband plays ballet, the wife’s best response is to play ballet (this gives her her

Chapter 8

FIGURE 8.2

Strategy and Game Theory

Extensive Form for the Battle of the Sexes

In this game, player 1 (wife) and player 2 (husband) choose to attend the ballet or a boxing match. They prefer to coordinate but disagree on which event to coordinate. Because they choose simultaneously, the husband does not know the wife’s choice when he moves, so his decision nodes are connected in the same information set.

2, 1 Ballet 2

Boxing

Ballet

0, 0 1

Ballet

Boxing

0, 0

2

Boxing 1, 2

TABLE 8.3

Normal Form for the Battle of the Sexes

Player 1 ðWifeÞ

Player 2 (Husband) Ballet

Boxing

Ballet

2, 1

0, 0

Boxing

0, 0

1, 2

highest payoff in the matrix of 2). Using notation, ballet ¼ BR1 (ballet). [We don’t need the fancy set-inclusion symbol as in “ballet 2 BR1 ðballetÞ” because the husband has only one best response to the wife’s choosing ballet.] Given that the wife plays ballet, the husband’s best response is to play ballet. If he deviated to boxing then he would earn 0 rather than 1, since they would end up not coordinating. Using notation, ballet ¼ BR2 (ballet). So (ballet, ballet) is indeed a Nash equilibrium. Symmetrically, (boxing, boxing) is a Nash equilibrium. Consider the outcome (ballet, boxing) in the upper left corner of the matrix. Given the husband chooses boxing, the wife earns 0 from choosing ballet but 1 from choosing boxing, so ballet is not a best response for the wife to the husband’s choosing boxing. In notation, ballet 62 BR1 ðboxingÞ. Hence (ballet, boxing) cannot be a Nash equilibrium. [The husband’s strategy of boxing is not a best response to the wife’s playing ballet either, so in fact both players would prefer to deviate from (ballet, boxing), although we only need to ﬁnd one player who would want to deviate to rule out an outcome as a Nash equilibrium.] Symmetrically, (boxing, ballet) is not a Nash equilibrium, either.

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TABLE 8.4

Underlining Procedure in the Battle of the Sexes

Player 1 ðWifeÞ

Player 2 (Husband) Ballet

Boxing

Ballet

2, 1

0, 0

Boxing

0, 0

1, 2

The Battle of the Sexes is an example of a game with more than one Nash equilibrium (in fact, it has three—a third in mixed strategies, as we will see). It is hard to say which of the two we have found so far is more plausible, since they are symmetric. It is therefore difﬁcult to make a ﬁrm prediction in this game. The Battle of the Sexes is also an example of a game with no dominant strategies. A player prefers to play ballet if the other plays ballet and boxing if the other plays boxing. Table 8.4 applies the underlining procedure, used to ﬁnd Nash equilibria quickly, to the Battle of the Sexes. The procedure veriﬁes that the two outcomes in which the players succeed in coordinating are Nash equilibria and the two outcomes in which they don’t coordinate are not. Examples 8.1, 8.2, and 8.3 provide additional practice in ﬁnding Nash equilibria in more complicated settings (a game that has many ties for best responses in Example 8.1, a game with three strategies for each player in Example 8.2, and a game with three players in Example 8.3). EXAMPLE 8.1 The Prisoners’ Dilemma Redux In this variation on the Prisoners’ Dilemma, a suspect is convicted and receives a sentence of four years if he is ﬁnked on and goes free if not. The district attorney does not reward ﬁnking. Table 8.5 presents the normal form for the game before and after applying the procedure for underlining best responses. Payoffs are again restated in terms of years of freedom. Ties for best responses are rife. For example, given player 2 ﬁnks, player 1’s payoff is 0 whether he ﬁnks or is silent. So there is a tie for 1’s best response to 2’s ﬁnking. This is an example of the set of best responses containing more than one element: BR1 ðfinkÞ ¼ ffink, silentg. TABLE 8.5

The Prisoners’ Dilemma Redux

(a) Normal form

Suspect 1

Suspect 2 Fink

Silent

Fink

0, 0

1, 0

Silent

0, 1

1, 1

(b) Underlining procedure

Suspect 1

Suspect 2 Fink

Silent

Fink

0, 0

1, 0

Silent

0, 1

1, 1

Chapter 8

Strategy and Game Theory

The underlining procedure shows that there is a Nash equilibrium in each of the four boxes. Given that suspects receive no personal reward or penalty for ﬁnking, they are both indifferent between ﬁnking and being silent; thus any outcome can be a Nash equilibrium. QUERY: Does any player have a dominant strategy? Can you draw the extensive form for the game?

EXAMPLE 8.2 Rock, Paper, Scissors Rock, Paper, Scissors is a children’s game in which the two players simultaneously display one of three hand symbols. Table 8.6 presents the normal form. The zero payoffs along the diagonal show that if players adopt the same strategy then no payments are made. In other cases, the payoffs indicate a $1 payment from loser to winner under the usual hierarchy (rock breaks scissors, scissors cut paper, paper covers rock). TABLE 8.6

Rock, Paper, Scissors

(a) Normal form

Player 1

Player 2 Rock

Paper

Scissors

Rock

0, 0

1, 1

1, 1

Paper

1, 1

0, 0

1, 1

Scissors

1, 1

1, 1

0, 0

(b) Underlying procedure

Player 1

Player 2 Rock

Paper

Scissors

Rock

0, 0

1, 1

1, 1

Paper

1, 1

0, 0

1, 1

Scissors

1, 1

1, 1

0, 0

As anyone who has played this game knows, and as the underlining procedure reveals, none of the nine boxes represents a Nash equilibrium. Any strategy pair is unstable because it offers at least one of the players an incentive to deviate. For example, (scissors, scissors) provides an incentive for either player 1 or 2 to choose rock; (paper, rock) provides an incentive for 2 to choose scissors. The game does have a Nash equilibrium—not any of the nine boxes in Table 8.6 but in mixed strategies, deﬁned in the next section. QUERY: Does any player have a dominant strategy? Why isn’t (paper, scissors) a Nash equilibrium?

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Part 2 Choice and Demand

EXAMPLE 8.3 Three’s Company Three’s Company is a three-player version of the Battle of the Sexes based on a 1970s sitcom of the same name about the misadventures of a man (Jack) and two women (Janet and Chrissy) who shared an apartment to save rent. Modify the payoffs from the Battle of the Sexes as follows. Players get one “util” from attending their favorite event (Jack’s is boxing and Janet and Chrissy’s is ballet). Players get an additional util for each of the other players who shows up at the event with them. Table 8.7 presents the normal form. For each of player 3’s strategies, there is a separate payoff matrix with all combinations of player 1 and 2’s strategies. Each box lists the three players’ payoffs in order. TABLE 8.7

Three’s Company

(a) Normal form Player 3 (Jack) plays Boxing

Player 3 (Jack) plays Ballet

Player 2 (Chrissy)

Ballet

Boxing

Ballet

3, 3, 2

2, 0, 1

Boxing

0, 2, 1

1, 1, 0

Player 1 ðJanetÞ

Player 1 ðJanetÞ

Player 2 (Chrissy)

Ballet

Boxing

Ballet

2, 2, 1

1, 1, 2

Boxing

1, 1, 2

2, 2, 3

(b) Underlining Procedure Player 3 (Jack) plays Boxing

Player 3 (Jack) plays Ballet

Player 2 (Chrissy)

Ballet

Boxing

Ballet

3, 3, 2

2, 0, 1

Boxing

0, 2, 1

1, 1, 0

Player 1 ðJanetÞ

Player 1 ðJanetÞ

Player 2 (Chrissy)

Ballet

Boxing

Ballet

2, 2, 1

1, 1, 2

Boxing

1, 1, 2

2, 2, 3

For players 1 and 2, the underlining procedure is the same as in a two-player game except that it must be repeated for the two payoff matrices. To underline player 3’s best-response payoffs, compare the two boxes in the same position in the two different matrices. For example, given Janet and Chrissy both play ballet, we compare the third payoff in the upper-left box in both matrices: Jack’s payoff is 2 in the ﬁrst matrix (in which he plays ballet) and 1 in the second (in which he plays boxing). So we underline the 2. As in the Battle of the Sexes, Three’s Company has two Nash equilibria, one in which all go to ballet and one in which all go to boxing. QUERY: What payoffs might make Three’s Company even closer in spirit to the Battle of the Sexes? What would the normal form look like for Four’s Company? (Four’s Company is similar to Three’s Company except with two men and two women.)

Chapter 8

Strategy and Game Theory

MIXED STRATEGIES Players’ strategies can be more complicated than simply choosing an action with certainty. In this section we study mixed strategies, which have the player randomly select from several possible actions. By contrast, the strategies considered in the examples so far have a player choose one action or another with certainty; these are called pure strategies. For example, in the Battle of the Sexes, we have considered the pure strategies of choosing either ballet or boxing for sure. A possible mixed strategy in this game would be to ﬂip a coin and then attend the ballet if and only if the coin comes up heads, yielding a 50–50 chance of showing up at either event. Although at ﬁrst glance it may seem bizarre to have players ﬂipping coins to determine how they will play, there are good reasons for studying mixed strategies. First, some games (such as Rock, Paper, Scissors) have no Nash equilibria in pure strategies. As we will see in the section on existence, such games will always have a Nash equilibrium in mixed strategies, so allowing for mixed strategies will enable us to make predictions in such games where it was impossible to do so otherwise. Second, strategies involving randomization are quite natural and familiar in certain settings. Students are familiar with the setting of class exams. Class time is usually too limited for the professor to examine students on every topic taught in class, but it may be sufﬁcient to test students on a subset of topics to induce them to study all of the material. If students knew which topics were on the test then they might be inclined to study only those and not the others, so the professor must choose the topics at random in order to get the students to study everything. Random strategies are also familiar in sports (the same soccer player sometimes shoots to the right of the net and sometimes to the left on penalty kicks) and in card games (the poker player sometimes folds and sometimes bluffs with a similarly poor hand at different times). Third, it is possible to “purify” mixed strategies by specifying a more complicated game in which one or the other action is better for the player for privately known reasons and where that action is played with certainty.2 For example, a history professor might decide to ask an exam question about World War I because, unbeknownst to the students, she recently read an interesting journal article about it. To be more formal, suppose that player i has a set of M possible actions Ai ¼ fa 1i , …, m a i , …, a M i g, where the subscript refers to the player and the superscript to the different choices. A mixed strategy is a probability distribution over the M actions, si ¼ ðσ1i , …, M m σm i , …, σi Þ, where σi is a number between 0 and 1 that indicates the probability of player m … þ σM ¼ 1. i playing action a i . The probabilities in si must sum to unity: σ1i þ … þ σm i þ i In the Battle of the Sexes, for example, both players have the same two actions of ballet and boxing, so we can write A1 ¼ A2 ¼ fballet, boxingg. We can write a mixed strategy as a pair of probabilities ðσ, 1 σÞ, where σ is the probability that the player chooses ballet. The probabilities must sum to unity and so, with two actions, once the probability of one action is speciﬁed, the probability of the other is determined. Mixed strategy (1=3, 2=3) means that the player plays ballet with probability 1=3 and boxing with probability 2=3; (1=2, 1=2) means that the player is equally likely to play ballet or boxing; (1, 0) means that the player chooses ballet with certainty; and (0, 1) means that the player chooses boxing with certainty. In our terminology, a mixed strategy is a general category that includes the special case of a pure strategy. A pure strategy is the special case in which only one action is played with

2 John Harsanyi, “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed-Strategy Equilibrium Points,” International Journal of Game Theory 2 (1973): 1–23. Harsanyi was a co-winner (along with Nash) of the 1994 Nobel Prize in economics.

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Part 2 Choice and Demand

positive probability. Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies. Returning to the examples from the previous paragraph of mixed strategies in the Battle of the Sexes, all four strategies (1=3, 2=3), (1=2, 1=2), (1, 0), and (0, 1) are mixed strategies. The ﬁrst two are strictly mixed and the second two are pure strategies. With this notation for actions and mixed strategies behind us, we do not need new deﬁnitions for best response, Nash equilibrium, and dominant strategy. The deﬁnitions introduced when si was taken to be a pure strategy also apply to the case in which si is taken to be a mixed strategy. The only change is that the payoff function ui ðsi , si Þ, rather than being a certain payoff, must be reinterpreted as the expected value of a random payoff, with probabilities given by the strategies si and si . Example 8.4 provides some practice in computing expected payoffs in the Battle of the Sexes. EXAMPLE 8.4 Expected Payoffs in the Battle of the Sexes Let’s compute players’ expected payoffs if the wife chooses the mixed strategy (1=9, 8=9) and the husband (4=5, 1=5) in the Battle of the Sexes. The wife’s expected payoff is 1 8 4 1 1 4 1 1 , , , ¼ U1 ðballet, balletÞ þ U1 ðballet, boxingÞ U1 9 9 5 5 9 5 9 5 8 4 8 1 U1 ðboxing, balletÞ þ U1 ðboxing, boxingÞ þ 9 5 9 5 1 4 1 1 8 4 8 1 ð2Þ þ ð0Þ þ ð0Þ þ ð1Þ ¼ 9 5 9 5 9 5 9 5 ¼

16 . 45

(8.4)

To understand Equation 8.4, it is helpful to review the concept of expected value from Chapter 2. Equation (2.176) indicates that an expected value of a random variable equals the sum over all outcomes of the probability of the outcome multiplied by the value of the random variable in that outcome. In the Battle of the Sexes, there are four outcomes, corresponding to the four boxes in Table 8.3. Since players randomize independently, the probability of reaching a particular box equals the product of the probabilities that each player plays the strategy leading to that box. So, for example, the probability of (boxing, ballet)—that is, the wife plays boxing and the husband plays ballet—equals ð8=9Þ ð4=5Þ. The probabilities of the four outcomes are multiplied by the value of the relevant random variable (in this case, player 1’s payoff) in each outcome. Next we compute the wife’s expected payoff if she plays the pure strategy of going to ballet [the same as the mixed strategy (1, 0)] and the husband continues to play the mixed strategy ð4=5, 1=5Þ. Now there are only two relevant outcomes, given by the two boxes in the row in which the wife plays ballet. The probabilities of the two outcomes are given by the probabilities in the husband’s mixed strategy. Therefore, 4 1 4 1 ¼ U1 ðballet, balletÞ þ U1 ðballet, boxingÞ , U1 ballet, 5 5 5 5 4 1 8 ð2Þ þ ð0Þ ¼ . ¼ (8.5) 5 5 5 Finally, we will compute the general expression for the wife’s expected payoff when she plays mixed strategy ðw, 1 wÞ and the husband plays ðh, 1 hÞ: if the wife plays ballet with probability w and the husband with probability h, then

Chapter 8

Strategy and Game Theory

u1 ððw, 1 wÞ, ðh, 1 h Þ Þ ¼ ðwÞðhÞU1 ðballet, balletÞ þ ðwÞð1 hÞU1 ðballet, boxingÞ þ ð1 wÞðhÞU1 ðboxing, balletÞ þ ð1 wÞð1 hÞU1 ðboxing, boxingÞ ¼ ðwÞðhÞð2Þ þ ðwÞð1 hÞð0Þ þ ð1 wÞðhÞð0Þ þ ð1 wÞð1 hÞð1Þ ¼ 1 h w þ 3hw.

(8.6)

QUERY: What is the husband’s expected payoff in each case? Show that his expected payoff is 2 2h 2w þ 3hw in the general case. Given the husband plays the mixed strategy ð4=5, 1=5Þ, what strategy provides the wife with the highest payoff?

Computing Nash equilibrium of a game when strictly mixed strategies are involved is quite a bit more complicated than when pure strategies are involved. Before wading in, we can save a lot of work by asking whether the game even has a Nash equilibrium in strictly mixed strategies. If it does not then, having found all the pure-strategy Nash equilibria, one has ﬁnished analyzing the game. The key to guessing whether a game has a Nash equilibrium in strictly mixed strategies is the surprising result that almost all games have an odd number of Nash equilibria.3 Let’s apply this insight to some of the examples considered so far. We found an odd number (one) of pure-strategy Nash equilibria in the Prisoners’ Dilemma, suggesting we need not look further for one in strictly mixed strategies. In the Battle of the Sexes, we found an even number (two) of pure-strategy Nash equilibria, suggesting the existence of a third one in strictly mixed strategies. Example 8.2—Rock, Paper, Scissors—has no pure-strategy Nash equilibria. To arrive at an odd number of Nash equilibria, we would expect to ﬁnd one Nash equilibrium in strictly mixed strategies. EXAMPLE 8.5 Mixed-Strategy Nash Equilibrium in the Battle of the Sexes A general mixed strategy for the wife in the Battle of the Sexes is ðw, 1 wÞ and for the husband is ðh, 1 hÞ; where w and h are the probabilities of playing ballet for the wife and husband, respectively. We will compute values of w and h that make up Nash equilibria. Both players have a continuum of possible strategies between 0 and 1. Therefore, we cannot write these strategies in the rows and columns of a matrix and underline best-response payoffs to ﬁnd the Nash equilibria. Instead, we will use graphical methods to solve for the Nash equilibria. Given players’ general mixed strategies, we saw in Example 8.4 that the wife’s expected payoff is u1 ððw, 1 wÞ, ðh, 1 hÞÞ ¼ 1 h w þ 3hw.

(8.7)

As Equation 8.7 shows, the wife’s best response depends on h. If h < 1=3, she wants to set w as low as possible: w ¼ 0. If h > 1=3, her best response is to set w as high as possible: w ¼ 1. When h ¼ 1=3, her expected payoff equals 2=3 regardless of what w she chooses. In this case there is a tie for the best response, including any w from 0 to 1. (continued) 3 John Harsanyi, “Oddness of the Number of Equilibrium Points: A New Proof,” International Journal of Game Theory 2 (1973): 235–50. Games in which there are ties between payoffs may have an even or infinite number of Nash equilibria. Example 8.1, the Prisoners’ Dilemma Redux, has several payoff ties. The game has four pure-strategy Nash equilibria and an infinite number of different mixed strategy equilibria.

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EXAMPLE 8.5 CONTINUED FIGURE 8.3

Nash Equilibria in Mixed Strategies in the Battle of the Sexes Ballet is chosen by the wife with probability w and by the husband with probability h. Players’ best responses are graphed on the same set of axes. The three intersection points E1 , E2 , and E3 are Nash equilibria. The Nash equilibrium in strictly mixed strategies, E3 , is w ¼ 2=3 and h ¼ 1=3. h E2

1

Husband’s best response, BR2

2/3

E3

1/3

Wife’s best response, BR1

E1 0

w 1/3

2/3

1

In Example 8.4, we stated that the husband’s expected payoff is U2 ððh, 1 hÞ, ðw, 1 wÞÞ ¼ 2 2h 2w þ 3hw.

(8.8)

When w < 2=3, his expected payoff is maximized by h ¼ 0; when w > 2=3, his expected payoff is maximized by h ¼ 1; and when w ¼ 2=3, he is indifferent among all values of h, obtaining an expected payoff of 2=3 regardless. The best responses are graphed in Figure 8.3. The Nash equilibria are given by the intersection points between the best responses. At these intersection points, both players are best responding to each other, which is what is required for the outcome to be a Nash equilibrium. There are three Nash equilibria. The points E1 and E2 are the pure-strategy Nash equilibria we found before, with E1 corresponding to the pure-strategy Nash equilibrium in which both play boxing and E2 to that in which both play ballet. Point E3 is the strictly mixed-strategy Nash equilibrium, which can can be spelled out as “the wife plays ballet with probability 2=3 and boxing with probability 1=3 and the husband plays ballet with probability 1=3 and boxing with probability 2=3.” More succinctly, having deﬁned w and h, we may write the equilibruim as “w ¼ 2=3 and h ¼ 1=3.” QUERY: What is a player’s expected payoff in the Nash equilibrium in strictly mixed strategies? How does this payoff compare to those in the pure-strategy Nash equilibria? What arguments might be offered that one or another of the three Nash equilibria might be the best prediction in this game?

Chapter 8

Strategy and Game Theory

Example 8.5 runs through the lengthy calculations involved in ﬁnding all the Nash equilibria of the Battle of the Sexes, those in pure strategies and those in strictly mixed strategies. The steps involve ﬁnding players’ expected payoffs as functions of general mixed strategies, using these to ﬁnd players’ best responses, and then graphing players’ best responses to see where they intersect. A shortcut to ﬁnding the Nash equilibrium in strictly mixed strategies is based on the insight that a player will be willing to randomize between two actions in equilibrium only if he or she gets the same expected payoff from playing either action or, in other words, is indifferent between the two actions in equilibrium. Otherwise, one of the two actions would provide a higher expected payoff, and the player would prefer to play that action with certainty. Suppose the husband is playing mixed strategy ðh, 1 hÞ; that is, playing ballet with probability h and boxing with probability 1 h. The wife’s expected payoff from playing ballet is U1 ðballet, ðh, 1 hÞÞ ¼ ðhÞð2Þ þ ð1 hÞð0Þ ¼ 2h.

(8.9)

Her expected payoff from playing boxing is U1 ðboxing, ðh, 1 hÞÞ ¼ ðhÞð0Þ þ ð1 hÞð1Þ ¼ 1 h.

(8.10)

For the wife to be indifferent between ballet and boxing in equilibrium, Equations 8.9 and 8.10 must be equal: 2h ¼ 1 h, implying h ¼ 1=3. Similar calculations based on the husband’s indifference between playing ballet and boxing in equilibrium show that the wife’s probability of playing ballet in the strictly mixed strategy Nash equilibrium is w ¼ 2=3. (Work through these calculations as an exercise.) Notice that the wife’s indifference condition does not “pin down” her equilibrium mixed strategy. The wife’s indifference condition cannot pin down her own equilibrium mixed strategy because, given that she is indifferent between the two actions in equilibrium, her overall expected payoff is the same no matter what probability distribution she plays over the two actions. Rather, the wife’s indifference condition pins down the other player’s—the husband’s—mixed strategy. There is a unique probability distribution he can use to play ballet and boxing that makes her indifferent between the two actions and thus makes her willing to randomize. Given any probability of his playing ballet and boxing other than ð1=3, 2=3Þ, it would not be a stable outcome for her to randomize. Thus, two principles should be kept in mind when seeking Nash equilibria in strictly mixed strategies. One is that a player randomizes over only those actions among which he or she is indifferent, given other players’ equilibrium mixed strategies. The second is that one player’s indifference condition pins down the other player’s mixed strategy.

EXISTENCE One of the reasons Nash equilibrium is so widely used is that a Nash equilibrium is guaranteed to exist in a wide class of games. This is not true for some other equilibrium concepts. Consider the dominant strategy equilibrium concept. The Prisoners’ Dilemma has a dominant strategy equilibrium (both suspects ﬁnk), but most games do not. Indeed, there are many games—including, for example, the Battle of the Sexes—in which no player has a dominant strategy, let alone all the players. In such games, we can’t make predictions using dominant strategy equilibrium but we can using Nash equilibrium. The Extensions section at the end of this chapter will provide the technical details behind John Nash’s proof of the existence of his equilibrium in all ﬁnite games (games with a ﬁnite number of players and a ﬁnite number of actions). The existence theorem does not guarantee the existence of a pure-strategy Nash equilibrium. We already saw an example: Rock, Paper,

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Scissors in Example 8.2. However, if a ﬁnite game does not have a pure-strategy Nash equilibrium, the theorem guarantees that it will have a mixed-strategy Nash equilibrium. The proof of Nash’s theorem is similar to the proof in Chapter 13 of the existence of prices leading to a general competitive equilibrium. The Extensions section includes an existence proof for games with a continuum of actions, as studied in the next section.

CONTINUUM OF ACTIONS Most of the insight from economic situations can often be gained by distilling the situation down to a few or even two actions, as with all the games studied so far. Other times, additional insight can be gained by allowing a continuum of actions. To be clear, we have already encountered a continuum of strategies—in our discussion of mixed strategies—but still the probability distributions in mixed strategies were over a ﬁnite number of actions. In this section we focus on continuum of actions. Some settings are more realistically modeled via a continuous range of actions. In Chapter 15, for example, we will study competition between strategic ﬁrms. In one model (Bertrand), ﬁrms set prices; in another (Cournot), ﬁrms set quantities. It is natural to allow ﬁrms to choose any nonnegative price or quantity rather than artiﬁcially restricting them to just two prices (say, $2 or $5) or two quantities (say, 100 or 1,000 units). Continuous actions have several other advantages. With continuous actions, the familiar methods from calculus can often be used to solve for Nash equilibria. It is also possible to analyze how the equilibrium actions vary with changes in underlying parameters. With the Cournot model, for example, we might want to know how equilibrium quantities change with a small increase in a ﬁrm’s marginal costs or a demand parameter.

Tragedy of the Commons Example 8.6 illustrates how to solve for the Nash equilibrium when the game (in this case, the Tragedy of the Commons) involves a continuum of actions. The ﬁrst step is to write down the payoff for each player as a function of all players’ actions. The next step is to compute the ﬁrst-order condition associated with each player’s payoff maximum. This will give an equation that can be rearranged into the best response of each player as a function of all other players’ actions. There will be one equation for each player. With n players, the system of n equations for the n unknown equilibrium actions can be solved simultaneously by either algebraic or graphical methods. EXAMPLE 8.6 Tragedy of the Commons The term “Tragedy of the Commons” has come to signify environmental problems of overuse that arise when scarce resources are treated as common property.4 A game-theoretic illustration of this issue can be developed by assuming that two herders decide how many sheep to graze on the village commons. The problem is that the commons is quite small and can rapidly succumb to overgrazing. In order to add some mathematical structure to the problem, let qi be the number of sheep that herder i ¼ 1, 2 grazes on the commons, and suppose that the per-sheep value of grazing on the commons (in terms of wool and sheep-milk cheese) is

4

This term was popularized by G. Hardin, “The Tragedy of the Commons,” Science 162 (1968): 1243–48.

Chapter 8

Strategy and Game Theory

vðq1 , q2 Þ ¼ 120 ðq1 þ q2 Þ:

(8.11)

This function implies that the value of grazing a given number of sheep is lower the more sheep are around competing for grass. We cannot use a matrix to represent the normal form of this game of continuous actions. Instead, the normal form is simply a listing of the herders’ payoff functions u1 ðq1 , q2 Þ ¼ q1 vðq1 , q2 Þ ¼ q1 ð120 q1 q2 Þ,

(8.12)

u2 ðq1 , q2 Þ ¼ q2 vðq1 , q2 Þ ¼ q2 ð120 q1 q2 Þ:

To find the Nash equilibrium, we solve herder 1’s value-maximization problem: maxfq1 ð120 q1 q2 Þg:

(8.13)

q1

The first-order condition for a maximum is 120 2q1 q2 ¼ 0

(8.14)

or, rearranging, q2 ¼ BR1 ðq2 Þ: 2 Similar steps show that herder 2’s best response is q1 ¼ 60

(8.15)

q1 ¼ BR2 ðq1 Þ: (8.16) 2 The Nash equilibrium is given by the pair ðq , q Þ that satisfies Equations 8.15 and 8.16 q2 ¼ 60

1

2

simultaneously. Taking an algebraic approach to the simultaneous solution, Equation 8.16 can be substituted into Equation 8.15, which yields 1 q 60 1 ; q1 ¼ 60 (8.17) 2 2 upon rearranging, this implies q 1 ¼ 40. Substituting q 1 ¼ 40 into Equation 8.17 implies q 2 ¼ 40 as well. Thus, each herder will graze 40 sheep on the common. Each earns a payoff of 1,600, as can be seen by substituting q ¼ q ¼ 40 into the payoff function in Equation 8.13. 1

2

Equations 8.15 and 8.16 can also be solved simultaneously using graphical methods. Figure 8.4 plots the two best responses on a graph with player 1’s action on the horizontal axis and player 2’s on the vertical axis. These best responses are simply lines and so are easy to graph in this example. (To be consistent with the axis labels, the inverse of Equation 8.15 is actually what is graphed.) The two best responses intersect at the Nash equilibrium E1 . The graphical method is useful for showing how the Nash equilibrium shifts with changes in the parameters of the problem. Suppose the per-sheep value of grazing increases for the ﬁrst herder while the second remains as in Equation 8.11, perhaps because the ﬁrst herder starts raising merino sheep with more valuable wool. This change would shift the best response out for herder 1 while leaving 2’s the same. The new intersection point (E2 in Figure 8.4), which is the new Nash equilibrium, involves more sheep for 1 and fewer for 2. The Nash equilibrium is not the best use of the commons. In the original problem, both herders’ per-sheep value of grazing is given by Equation 8.11. If both grazed only 30 sheep then each would earn a payoff of 1,800, as can be seen by substituting q1 ¼ q2 ¼ 30 into Equation 8.13. Indeed, the “joint payoff maximization” problem maxfðq1 þ q2 Þvðq1 , q2 Þg ¼ maxfðq1 þ q2 Þð120 q1 q2 Þg q1

(8.18)

q1

is solved by q1 ¼ q2 ¼ 30 or, more generally, by any q1 and q2 that sum to 60. (continued)

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EXAMPLE 8.6 CONTINUED FIGURE 8.4

Best-Response Diagram for the Tragedy of the Commons The intersection, E1 , between the two herders’ best responses is the Nash equilibrium. An increase in the per-sheep value of grazing in the Tragedy of the Commons shifts out herder 1’s best response, resulting in a Nash equilibrium E2 in which herder 1 grazes more sheep (and herder 2, fewer sheep) than in the original Nash equilibrium. q2

120 BR1(q2)

60 E1

40

E2 BR2(q1) q1 0

40

60

120

QUERY: How would the Nash equilibrium shift if both herders’ beneﬁts increased by the same amount? What about a decrease in (only) herder 2’s beneﬁt from grazing?

As Example 8.6 shows, graphical methods are particularly convenient for quickly determining how the equilibrium shifts with changes in the underlying parameters. The example shifted the beneﬁt of grazing to one of herders. This exercise nicely illustrates the nature of strategic interaction. Herder 2’s payoff function hasn’t changed (only herder 1’s has), yet his equilibrium action changes. The second herder observes the ﬁrst’s higher beneﬁt, anticipates that the ﬁrst will increase the number of sheep he grazes, and reduces his own grazing in response. The Tragedy of the Commons shares with the Prisoners’ Dilemma the feature that the Nash equilibrium is less efﬁcient for all players than some other outcome. In the Prisoners’ Dilemma, both ﬁnk in equilibrium when it would be more efﬁcient for both to be silent. In the Tragedy of the Commons, the herders graze more sheep in equilibrium than is efﬁcient. This insight may explain why ocean ﬁshing grounds and other common resources can end up being overused even to the point of exhaustion if their use is left unregulated. More detail on such problems—involving what we will call negative externalities—is provided in Chapter 19.

Chapter 8

Strategy and Game Theory

SEQUENTIAL GAMES In some games, the order of moves matters. For example, in a bicycle race with a staggered start, it may help to go last and thus know the time to beat. On the other hand, competition to establish a new high-deﬁnition video format may be won by the ﬁrst ﬁrm to market its technology, thereby capturing an installed base of consumers. Sequential games differ from the simultaneous games we have considered so far in that a player that moves later in the game can observe how others have played up to that moment. The player can use this information to form more sophisticated strategies than simply choosing an action; the player’s strategy can be a contingent plan with the action played depending on what the other players have done. To illustrate the new concepts raised by sequential games—and, in particular, to make a stark contrast between sequential and simultaneous games—we take a simultaneous game we have discussed already, the Battle of the Sexes, and turn it into a sequential game.

Sequential Battle of the Sexes Consider the Battle of the Sexes game analyzed previously with all the same actions and payoffs, but now change the timing of moves. Rather than the wife and husband making a simultaneous choice, the wife moves ﬁrst, choosing ballet or boxing; the husband observes this choice (say, the wife calls him from her chosen location) and then the husband makes his choice. The wife’s possible strategies have not changed: she can choose the simple actions ballet or boxing (or perhaps a mixed strategy involving both actions, although this will not be a relevant consideration in the sequential game). The husband’s set of possible strategies has expanded. For each of the wife’s two actions, he can choose one of two actions, so he has four possible strategies, which are listed in Table 8.8. The vertical bar in the husband’s strategies means “conditional on” and so, for example, “boxing | ballet” should be read as “the husband chooses boxing conditional on the wife’s choosing ballet”. Given that the husband has four pure strategies rather than just two, the normal form (given in Table 8.9) must now be expanded to eight boxes. Roughly speaking, the normal form is twice as complicated as that for the simultaneous version of the game in Table 8.3. By contrast, the extensive form, given in Figure 8.5, is no more complicated than the extensive form for the simultaneous version of the game in Figure 8.2. The only difference between the TABLE 8.8

TABLE 8.9

Husband's Contingent Strategies Contingent strategy Always go to the ballet

Written in conditional format (ballet | ballet, ballet | boxing)

Follow his wife

(ballet | ballet, boxing | boxing)

Do the opposite

(boxing | ballet, ballet | boxing)

Always go to boxing

(boxing | ballet, boxing | boxing)

Normal Form for the Sequential Battle of the Sexes

Wife

Husband

Ballet Boxing

(Ballet | Ballet Ballet | Boxing) 2, 1 0, 0

(Ballet | Ballet Boxing | Boxing) 2, 1 1, 2

(Boxing | Ballet (Boxing | Ballet Ballet | Boxing) Boxing | Boxing) 0, 0 0, 0 0, 0

1, 2

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FIGURE 8.5

Extensive Form for the Sequential Battle of the Sexes In the sequential version of the Battle of the Sexes, the husband moves second after observing his wife’s move. The husband’s decision nodes are not gathered in the same information set.

2, 1 Ballet 2

Ballet

Boxing

Boxing

Ballet

0, 0

1 0, 0 2

Boxing

1, 2

extensive forms is that the oval around the husband’s decision nodes has been removed. In the sequential version of the game, the husband’s decision nodes are not gathered together in the same information set because the husband observes his wife’s action and so knows which node he is on before moving. We can begin to see why the extensive form becomes more useful than the normal form for sequential games, especially in games with many rounds of moves. To solve for the Nash equilibria, consider the normal form in Table 8.9. Applying the method of underlining best-response payoffs—being careful to underline both payoffs in cases of ties for the best response—reveals three pure-strategy Nash equilibria: 1. wife plays ballet, husband plays (ballet | ballet, ballet | boxing); 2. wife plays ballet, husband plays (ballet | ballet, boxing | boxing); 3. wife plays boxing, husband plays (boxing | ballet, boxing | boxing). As with the simultaneous version of the Battle of the Sexes, here again we have multiple equilibria. Yet now game theory offers a good way to select among the equilibria. Consider the third Nash equilibrium. The husband’s strategy (boxing | ballet, boxing | boxing) involves the implicit threat that he will choose boxing even if his wife chooses ballet. This threat is sufﬁcient to deter her from choosing ballet. Given that she chooses boxing in equilibrium, his strategy earns him 2, which is the best he can do in any outcome. So the outcome is a Nash equilibrium. But the husband’s threat is not credible— that is, it is an empty threat. If the wife really were to choose ballet ﬁrst, then he would be giving up a payoff of 1 by choosing boxing rather than ballet. It is clear why he would want to threaten to choose boxing, but it is not clear that such a threat should be believed. Similarly, the husband’s strategy (ballet | ballet, ballet | boxing) in the ﬁrst Nash equilibrium also involves an empty threat: that he will choose ballet if his wife chooses boxing. (This is an odd threat to make since he does not gain from making it, but it is an empty threat nonetheless.) Another way to understand empty versus credible threats is by using the concept of the equilibrium path, the connected path through the game tree implied by equilibrium strategies.

Chapter 8

FIGURE 8.6

Strategy and Game Theory

Equilibrium Path

In the third of the Nash equilibria listed for the sequential Battle of the Sexes, the wife plays boxing and the husband plays (boxing | ballet, boxing | boxing), tracing out the branches indicated with thick lines (both solid and dashed). The dashed line is the equilibrium path; the rest of the tree is referred to as being “off the equilibrium path.” 2, 1 Ballet 2

Ballet

Boxing

Boxing

Ballet

0, 0

1 0, 0 2

Boxing

1, 2

Figure 8.6 uses a dashed line to illustrate the equilibrium path for the third of the listed Nash equilibria in the sequential Battle of the Sexes. The third outcome is a Nash equilibrium because the strategies are rational along the equilibrium path. However, following the wife’s choosing ballet—an event that is off the equilibrium path—the husband’s strategy is irrational. The concept of subgame-perfect equilibrium in the next section will rule out irrational play both on and off the equilibrium path.

Subgame-perfect equilibrium Game theory offers a formal way of selecting the reasonable Nash equilibria in sequential games using the concept of subgame-perfect equilibrium. Subgame-perfect equilibrium is a reﬁnement that rules out empty threats by requiring strategies to be rational even for contingencies that do not arise in equilibrium. Before deﬁning subgame-perfect equilibrium formally, we need a few preliminary deﬁnitions. A subgame is a part of the extensive form beginning with a decision node and including everything that branches out to the right of it. A proper subgame is a subgame that starts at a decision node not connected to another in an information set. Conceptually, this means that the player who moves ﬁrst in a proper subgame knows the actions played by others that have led up to that point. It is easier to see what a proper subgame is than to deﬁne it in words. Figure 8.7 shows the extensive forms from the simultaneous and sequential versions of the Battle of the Sexes with boxes drawn around the proper subgames in each. In the simultaneous Battle of the Sexes, there is only one decision node—the topmost mode—that is not connected to another in the same information set; hence there is only one proper subgame, the game itself. In the sequential Battle of the Sexes, there are three proper subgames: the game itself and two lower subgames starting with decision nodes where the husband gets to move.

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FIGURE 8.7

Proper Subgames in the Battle of the Sexes The simultaneous Battle of the Sexes in (a) has only one proper subgame: the whole game itself, labeled A. The sequential Battle of the Sexes in (b) has three proper subgames, labeled B, C, and D.

A

2, 1

B

C

Ballet 2

2

Boxing

Ballet

2, 1 Ballet

Boxing

Ballet 0, 0

1

0, 0 1

Ballet

Boxing

0, 0

2

D Ballet

Boxing 2

Boxing

Boxing 1, 2

(a) Simultaneous

DEFINITION

0, 0

1, 2

(b) Sequential

Subgame-perfect equilibrium. A subgame-perfect equilibrium is a strategy proﬁle ðs 1 , s 2 , …, s n Þ that constitutes a Nash equilibrium for every proper subgame. A subgame-perfect equilibrium is always a Nash equilibrium. This is true because the whole game is a proper subgame of itself and so a subgame-perfect equilibrium must be a Nash equilibrium for the whole game. In the simultaneous version of the Battle of the Sexes, there is nothing more to say because there are no subgames other than the whole game itself. In the sequential version of the Battle of the Sexes, subgame-perfect equilibrium has more bite. Strategies must not only form a Nash equilibrium on the whole game itself, they must also form Nash equilibria on the two proper subgames starting with the decision points at which the husband moves. These subgames are simple decision problems, so it is easy to compute the corresponding Nash equilibria. For subgame C, beginning with the husband’s decision node following his wife’s choosing ballet, he has a simple decision between ballet (which earns him a payoff of 1) and boxing (which earns him a payoff of 0). The Nash equilibrium in this simple decision subgame is for the husband to choose ballet. For the other subgame, D, he has a simple decision between ballet, which earns him 0, and boxing, which earns him 2. The Nash equilibrium in this simple decision subgame is for him to choose boxing. The husband therefore has only one strategy that can be part of a subgame-perfect equilibrium: (ballet | ballet, boxing | boxing). Any other strategy has him playing something that is not a Nash equilibrium for some proper subgame. Returning to the three enumerated Nash equilibria, only the second is subgame perfect; the ﬁrst and the third are not. For example, the third equilibrium, in which the husband always goes to boxing, is ruled out as a subgame-perfect equilibrium because the

Chapter 8

FIGURE 8.8

Strategy and Game Theory

Applying Backward Induction

The last subgames (where player 2 moves) are replaced by the Nash equilibria on these subgames. The simple game that results at right can be solved for player 1’s equilibrium action.

2, 1 Ballet 2 plays ballet | ballet payoff 2, 1

2

Boxing

Ballet

Ballet 0, 0

1

1

Ballet

Boxing

0, 0 Boxing

2

2 plays boxing | boxing payoff 1, 2 Boxing 1, 2

husband’s strategy (boxing | boxing) is not a Nash equilibrium in proper subgame C: Subgameperfect equilibrium thus rules out the empty threat (of always going to boxing) that we were uncomfortable with earlier. More generally, subgame-perfect equilibrium rules out any sort of empty threat in a sequential game. In effect, Nash equilibrium requires behavior to be rational only on the equilibrium path. Players can choose potentially irrational actions on other parts of the game tree. In particular, one player can threaten to damage both in order to scare the other from choosing certain actions. Subgame-perfect equilibrium requires rational behavior both on and off the equilibrium path. Threats to play irrationally—that is, threats to choose something other than one’s best response—are ruled out as being empty. Subgame-perfect equilibrium is not a useful reﬁnement for a simultaneous game. This is because a simultaneous game has no proper subgames besides the game itself and so subgame-perfect equilibrium would not reduce the set of Nash equilibria.

Backward induction Our approach to solving for the equilibrium in the sequential Battle of the Sexes was to ﬁnd all the Nash equilibria using the normal form and then to seek among those for the subgame-perfect equilibrium. A shortcut for ﬁnding the subgame-perfect equilibrium directly is to use backward induction, the process of solving for equilibrium by working backwards from the end of the game to the beginning. Backward induction works as follows. Identify all of the subgames at the bottom of the extensive form. Find the Nash equilibria on these subgames. Replace the (potentially complicated) subgames with the actions and payoffs resulting from Nash equilibrium play on these subgames. Then move up to the next level of subgames and repeat the procedure. Figure 8.8 illustrates the use of backward induction to solve for the subgame-perfect equilibrium of the sequential Battle of the Sexes. First, we compute the Nash equilibria of

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the bottommost subgames at the husband’s decision nodes. In the subgame following his wife’s choosing ballet, he would choose ballet, giving payoffs 2 for her and 1 for him. In the subgame following his wife’s choosing boxing, he would choose boxing, giving payoffs 1 for her and 2 for him. Next, substitute the husband’s equilibrium strategies for the subgames themselves. The resulting game is a simple decision problem for the wife (drawn in the lower panel of the ﬁgure): a choice between ballet, which would give her a payoff of 2, and boxing, which would give her a payoff of 1. The Nash equilibrium of this game is for her to choose the action with the higher payoff, ballet. In sum, backward induction allows us to jump straight to the subgame-perfect equilibrium in which the wife chooses ballet and the husband chooses (ballet | ballet, boxing | boxing), bypassing the other Nash equilibria. Backward induction is particularly useful in games that feature multiple rounds of sequential play. As rounds are added, it quickly becomes too hard to solve for all the Nash equilibria and then to sort through which are subgame-perfect. With backward induction, an additional round is simply accommodated by adding another iteration of the procedure.

REPEATED GAMES In the games examined so far, each player makes one choice and the game ends. In many real-world settings, players play the same game over and over again. For example, the players in the Prisoners’ Dilemma may anticipate committing future crimes and thus playing future Prisoners’ Dilemmas together. Gasoline stations located across the street from each other, when they set their prices each morning, effectively play a new pricing game every day. The simple constituent game (e.g., the Prisoners’ Dilemma or the gasoline-pricing game) that is played repeatedly is called the stage game. As we saw with the Prisoners’ Dilemma, the equilibrium in one play of the stage game may be worse for all players than some other, more cooperative, outcome. Repeated play of the stage game opens up the possibility of cooperation in equilibrium. Players can adopt trigger strategies, whereby they continue to cooperate as long as all have cooperated up to that point but revert to playing the Nash equilibrium if anyone deviates from cooperation. We will investigate the conditions under which trigger strategies work to increase players’ payoffs. As is standard in game theory, we will focus on subgame-perfect equilibria of the repeated games.

Finitely repeated games For many stage games, repeating them a known, ﬁnite number of times does not increase the possibility for cooperation. To see this point concretely, suppose the Prisoners’ Dilemma were repeated for T periods. Use backward induction to solve for the subgame-perfect equilibrium. The lowest subgame is the Prisoners’ Dilemma stage game played in period T : Regardless of what happened before, the Nash equilibrium on this subgame is for both to ﬁnk. Folding the game back to period T 1, trigger strategies that condition period-T play on what happens in period T 1 are ruled out. Although a player might like to promise to play cooperatively in period T and so reward the other for playing cooperatively in period T 1, we have just seen that nothing that happens in period T 1 affects what happens subsequently because players both ﬁnk in period T regardless. It is as if period T 1 were the last, and the Nash equilibrium of this subgame is again for both to ﬁnk. Working backward in this way, we see that players will ﬁnk each period; that is, players will simply repeat the Nash equilibrium of the stage game T times. Reinhard Selten, winner of the Nobel Prize in economics for his contributions to game theory, showed that the same logic applies more generally to any stage game with a unique Nash equilibrium.5 This result is called Selten’s theorem: If the stage game has a unique Nash equilibrium, then the unique subgame-perfect equilibrium of the ﬁnitely repeated game is to play the Nash equilibrium every period.

Chapter 8

Strategy and Game Theory

If the stage game has multiple Nash equilibria, it may be possible to achieve some cooperation in a ﬁnitely repeated game. Players can use trigger strategies, sustaining cooperation in early periods on an outcome that is not an equilibrium of the stage game, by threatening to play in later periods the Nash equilibrium that yields a worse outcome for the player who deviates from cooperation. Example 8.7 illustrates how such trigger strategies work to sustain cooperation. EXAMPLE 8.7 Cooperation in a Finitely Repeated Game The stage game given in normal form in Table 8.10 has two pure-strategy Nash equilibria. In the “bad” pure-strategy equilibrium, each plays B and earns a payoff of 1; in the “good” equilibrium, each plays C and earns a payoff of 3. Players would earn still more (i.e., 4) if both played A, but this is not a Nash equilibrium. If one plays A, then the other would prefer to deviate to B and earn 5. There is a third, mixed-strategy Nash equilibrium in which each plays B with probability 3=4 and C with probability 1=4. The payoffs are graphed as solid circles in Figure 8.9. TABLE 8.10

Stage Game for Example 8.7

Player 1

Player 2 A

B

C

A

4, 4

0, 5

0, 0

B

5, 0

1, 1

0, 0

C

0, 0

0, 0

3, 3

If the stage game is repeated twice, a wealth of new possibilities arise in subgame-perfect equilibria. The same per-period payoffs (1 or 3) from the stage game can be obtained simply by repeating the pure-strategy Nash equilibria from the stage game twice. Per-period average payoffs of 2.5 can be obtained by alternating between the good and the bad stage-game equilibria. A more cooperative outcome can be sustained with the following strategy: begin by playing A in the ﬁrst period; if no one deviates from A, play C in the second period; if a player deviates from A, then play B in the second period. Backward induction can be used to show that these strategies form a subgame-perfect equilibrium. The strategies form a Nash equilibrium in second-period subgames by construction. It remains to check whether the strategies form a Nash equilibrium on the game as a whole. In equilibrium with these strategies, players earn 4 þ 3 ¼ 7 in total across the two periods. By deviating to B in the ﬁrst period, a player can increase his or her ﬁrst-period payoff from 4 to 5, but this leads to both playing B in the second period, reducing the second-period payoff from 3 to 1. The total payoff across the two periods from this deviation is 5 þ 1 ¼ 6, less than the 7 earned in the proposed equilibrium. The average per-period payoff in this subgame-perfect equilibrium is 7=2 ¼ 3.5 for each player. Asymmetric equilibria are also possible. In one, player 1 begins by playing B and player 2 by playing A; if no one deviates then both play the good stage-game Nash equilibrium (both play C), and if someone deviates then both play the bad equilibrium (both play B). Player 2 (continued) R. Selten, “A Simple Model of Imperfect Competition, Where 4 Are Few and 6 Are Many,” International Journal of Game Theory 2 (1973): 141–201.

5

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EXAMPLE 8.7 CONTINUED FIGURE 8.9

Per-Period Average Payoffs in Example 8.7 Solid circles indicate payoffs in Nash equilibria of the stage game. Squares (in addition to circles) indicate per-period average payoffs in subgame-perfect equilibria for T ¼ 2 repetitions of the stage game. Triangles (in addition to circles and squares) indicate per-period average payoffs for T ¼ 3.

u2 5

4

3

2

1

0

1

2

3

4

5

u1

does not want to deviate to playing B in the ﬁrst period because he or she earns 1 from this deviation in the ﬁrst period and 1 in the second when they play the bad equilibrium for a total of 1 þ 1 ¼ 2, whereas he or she earns more, 0 þ 3 ¼ 3; in equilibrium. The average perperiod payoff in this subgame-perfect equilibrium is ð5 þ 3Þ=2 ¼ 4 for player 1 and 3=2 ¼ 1.5 for player 2. The reverse payoffs can be obtained by reversing the strategies. The average perperiod payoffs from the additional subgame-perfect equilibria we computed for the twicerepeated game are graphed as squares in Figure 8.9. If the game is repeated three times (T ¼ 3), then additional payoff combinations are possible in subgame-perfect equilibria. Players can cooperate on playing A for two periods and C in the last, a strategy that is sustained by the threat of immediately moving to the bad equilibrium (both play B) if anyone deviates in the ﬁrst two periods. This subgame-perfect equilibrium gives each a per-period average payoff of ð4 þ 4 þ 3Þ=2 3:7, more than the 3.5 that was the most both could earn in the T ¼ 2 game. Asymmetric equilibria in the T ¼ 3 game include the possibility that 1 plays B and 2 plays A for the ﬁrst two periods and then both play C, with the threat of immediately moving to the bad equilibrium if anyone deviates. Player 1’s per-period average payoff in this subgame-perfect equilibrium is ð5 þ 5 þ 3Þ=3 4:3, and player 2’s payoff is ð0 þ 0 þ 3Þ=3 ¼ 1: The reverse strategies and payoffs also constitute a possible subgame-perfect equilibrium. The payoffs from the additional subgame-perfect equilibria of the T ¼ 3 game are graphed as triangles in Figure 8.9.

Chapter 8

Strategy and Game Theory

QUERY: There are many other subgame-perfect equilibrium payoffs for the repeated game than are shown in Figure 8.9. For the T ¼ 2 game, can you ﬁnd at least two other combinations of average per-period payoffs that can be attained in a subgame-perfect equilibrium?

For cooperation to be sustained in a subgame-perfect equilibrium, the stage game must be repeated often enough that the punishment for deviation (repeatedly playing the lesspreferred Nash equilibrium) is severe enough to deter deviation. The more repetitions of the stage game T , the more severe the possible punishment and thus the greater the level of cooperation and the higher the payoffs that can be sustained in a subgame-perfect equilibrium. In Example 8.7, the most both players can earn in a subgame-perfect equilibrium increases from 3 to 3.5 to about 3.7 as T increases from 1 to 2 to 3. Example 8.7 suggests that the range of sustinable payoffs in a subgame-perfect equilibrium expands as the number of repetitions T increases. In fact, the associated Figure 8.9 understates the expansion because it does not graph all subgame-perfect equilibrium payoffs for T ¼ 2 and T ¼ 3 (the Query in Example 8.7 asks you to ﬁnd two more, for example). We are left to wonder how much the set of possibilities might expand for yet higher T : Jean Pierre Benoit and Vijay Krishna answer this question with their folk theorem for ﬁnitely repeated games :6 Suppose that the stage game has multiple Nash equilibria and no player earns a constant payoff across all equilibria. Any feasible payoff in the stage game greater than the player’s pure-strategy minmax value can be approached arbitrarily closely by the player’s per-period average payoff in some subgame-perfect equilibrium of the ﬁnitely repeated game for large enough T :7

We will encounter other folk theorems in later sections of this chapter. Generally speaking, a folk theorem is a result that “anything is possible” in the limit with repeated games. Such results are called “folk” theorems because they were understood informally and thus were part of the “folk wisdom” of game theory well before anyone wrote down formal proofs. To understand the folk theorem fully, we need to understand what feasible payoffs and minmax values are. A feasible payoff is one that can be achieved by some mixed-strategy proﬁle in the stage game. Graphically, the feasible payoff set appears as the convex hull of the purestrategy stage-game payoffs. The convex hull of a set of points is the border and interior of the largest polygon that can be formed by connecting the points with line segments. For example, Figure 8.10 graphs the feasible payoff set for the stage game from Example 8.7 as the upwardhatched region. To derive this set, one ﬁrst graphs the pure-strategy payoffs from the stage game. Referring to the normal form in Table 8.10, the distinct pure-strategy payoffs are (4, 4), (0, 5), (0, 0), (5, 0), (1, 1), and (3, 3). The convex hull is the polygon formed by line segments going from (0, 0) to (0, 5) to (4, 4) to (5, 0), and back to (0, 0). Each point in the convex hull corresponds to the expected payoffs from some combination of mixed strategies for players 1 and 2 over actions A, B, and C: For example, the point (3, 0) on the boundary of the convex hull corresponds to players’ expected payoffs if 1 plays the mixed strategy (0, 3=5, 2=5) and 2 plays A: A minmax value is the least that player i can be forced to earn.

6 7

J. P. Benoit and V. Krishna, “Finitely Repeated Games,” Econometrica 53 (1985): 890–904.

An additional, technical condition is that the dimension of the feasible set of payoffs must equal the number of players. In the two-player game in Example 8.7, this condition would require the feasible payoff set to be a region (which is the case, as shown in Figure 8.12) rather than a line or a point.

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FIGURE 8.10

Folk Theorem for Finitely Repeated Games in Example 8.7 The feasible payoffs for the stage game in Example 8.7 are in the upward-hatched region; payoffs greater than each player’s minmax values are in the downward-hatched region. Their intersection (the cross-hatched region) constitutes the per-period average payoffs that can be approached by some subgame-perfect equilibrium of the repeated game, according to the folk theorem for ﬁnitely repeated games. Regions are superimposed on the equilibrium payoffs from Figure 8.9.

u2 5

4

3

2

1

0

DEFINITION

1

2

3

4

5

Minmax value. The minmax value is the following payoff for player i: i h min max ui ðsi , si Þ , si

si

u1

(8.19)

that is, the lowest payoff player i can be held to if all other players work against him or her but player i is allowed to choose a best response to them. In Example 8.7, if 2 plays the mixed strategy (0, 3=4, 1=4) then the most player 1 can earn in the stage game is 3=4 (by playing any mixed strategy involving only actions B and C). A little work shows that 3=4 is indeed player 1’s minmax value: any other strategy for 2 besides (0, 3=4, 1=4) would allow 1 to earn a higher payoff than 3=4. The folk theorem for ﬁnitely repeated games involves the pure-strategy minmax value—that is, the minmax value when players are restricted to using only pure strategies. The pure-strategy minmax value is easier to compute than the general minmax value. The lowest that player 2 can hold 1 to in Example 8.7 is a payoff of 1; player 2 does this by playing B and then 1 responds by playing B: Figure 8.10 graphs the payoffs exceeding both players’ pure-strategy minmax values as the downward-hatched region. The folk theorem for ﬁnitely repeated games assures us that any payoffs in the crosshatched region of Figure 8.10—payoffs that are feasible and above both players’ pure strategy

Chapter 8

Strategy and Game Theory

minmax values—can be approached as the per-period average payoffs in a subgame-perfect equilibrium if the stage game in Example 8.7 is repeated often enough. Payoffs (4, 4) can be approached by having players cooperate on playing A for hundreds of periods and then playing C in the last period (threatening the bad equilibrium in which both play B if anyone deviates from cooperation). The average of hundreds of payoffs of 4 with one payoff of 3 comes arbitrarily close to 4. Therefore, a considerable amount of cooperation is possible if the game is repeated often enough. Figure 8.10 also shows that many outcomes other than full cooperation are possible if the number of repetitions, T , is large. Although subgame-perfect equilibrium was selective in the sequential version of the Battle of the Sexes, allowing us to select one of three Nash equilibria, we see that subgame perfection may not be selective in repeated games. The folk theorem states that if the stage game has multiple Nash equilibria then almost anything can happen in the repeated game for T large enough.8

Infinitely repeated games With ﬁnitely repeated games, the folk theorem applies only if the stage game has multiple equilibria. If, like the Prisoners’ Dilemma, the stage game has only one Nash equilibrium, then Selten’s theorem tells us that the ﬁnitely repeated game has only one subgame-perfect equilibrium: repeating the stage-game Nash equilibrium each period. Backward induction starting from the last period T unravels any other outcomes. With inﬁnitely repeated games, however, there is no deﬁnite ending period T from which to start backward induction. A folk theorem will apply to inﬁnitely repeated games even if the underlying stage game has only one Nash equilibrium. Therefore, while both players ﬁnk every period in the unique subgame-perfect equilibrium of the ﬁnitely repeated Prisoners’ Dilemma, players may end up cooperating (being silent) in the inﬁnitely repeated version. One difﬁculty with inﬁnitely repeated games involves adding up payoffs across periods. With ﬁnitely repeated games, we could focus on average payoffs. With inﬁnitely repeated games, the average is not well-deﬁned because it involves an inﬁnite sum of payoffs divided by an inﬁnite number of periods. We will circumvent this problem with the aid of discounting. Let δ be the discount factor (discussed in the Chapter 17 Appendix) measuring how much a payoff unit is worth if received one period in the future rather than today. In Chapter 17 we show that δ is inversely related to the interest rate. If the interest rate is high then a person would much rather receive payment today than next period because investing today’s payment would provide a return of principal plus a large interest payment next period. Besides the interest rate, δ can also incorporate uncertainty about whether the game continues in future periods. The higher the probability that the game ends after the current period, the lower the expected return from stage games that might not actually be played. Factoring in a probability that the repeated game ends after each period makes the setting of an inﬁnitely repeated game more believable. The crucial issue with an inﬁnitely repeated game is not that it goes on forever but that its end is indeterminate. Interpreted in this way, there is a sense in which inﬁnitely repeated games are more realistic than ﬁnitely repeated games with large T : Suppose we expect two neighboring gasoline stations to play a pricing game each day until electric cars replace gasoline-powered ones. It is unlikely the gasoline stations would know that electric cars were coming in exactly T ¼ 2,000 days. More realistically, the gasoline stations will be uncertain about the end of gasoline-powered cars and so the end of their pricing game is indeterminate.

8

The folk theorem for finitely repeated games does not necessarily capture all subgame-perfect equilibria. In Figure 8.12, the point (3=4, 3=4) lies outside the cross-hatched region; nonetheless, it can be achieved in a subgame-perfect equilibrium in which, each period, both players play the Nash equilibrium of the stage game in strictly mixed strategies. Payoffs (3=4, 3=4) are in a “gray area” between player’s pure-strategy and mixed-strategy minmax values.

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Players can sustain cooperation in inﬁnitely repeated games by using trigger strategies : players continue cooperating unless someone has deviated from cooperation, and this deviation triggers some sort of punishment. In order for trigger strategies to form an equilibrium, the punishment must be severe enough to deter deviation. Suppose both players use the following trigger strategy in the Prisoners’ Dilemma: continue being silent if no one has deviated by playing ﬁnk; ﬁnk forever afterward if anyone has deviated to ﬁnk in the past. To show that this trigger strategy forms a subgame-perfect equilibrium, we need to check that a player cannot gain from deviating. Along the equilibrium path, both players are silent every period; this provides each with a payoff of 2 every period for a present discounted value of V eq ¼ 2 þ 2δ þ 2δ2 þ 2δ3 þ … þ 2ð1 þ δ þ δ2 þ δ3 þ …Þ ¼

2 : 1δ

(8.20)

A player who deviates by finking earns 3 in that period, but then both players fink every period from then on—each earning 1 per period for a total presented discounted payoff of V dev ¼ 3 þ ð1ÞðδÞ þ ð1Þðδ2 Þ þ ð1Þðδ3 Þ þ … þ 3 þ δð1 þ δ þ δ2 þ …Þ δ : (8.20) 1δ The trigger strategies form a subgame-perfect equilibrium if V eq V dev ; implying that ¼3þ

2 δ 3þ ; (8.22) 1δ 1δ after multiplying through by 1 δ and rearranging, we obtain δ 1=2: In other words, players will find continued cooperative play desirable provided they do not discount future gains from such cooperation too highly. If δ < 1=2, then no cooperation is possible in the infinitely repeated Prisoners’ Dilemma; the only subgame-perfect equilibrium involves finking every period. The trigger strategy we considered has players revert to the stage-game Nash equilibrium of ﬁnking each period forever. This strategy, which involves the harshest possible punishment for deviation, is called the grim strategy. Less harsh punishments include the so-called tit-fortat strategy, which involves only one round of punishment for cheating. Since it involves the harshest punishment possible, the grim strategy elicits cooperation for the largest range of cases (the lowest value of δ ) of any strategy. Harsh punishments work well because, if players succeed in cooperating, they never experience the losses from the punishment in equilibrium.9 The discount factor δ is crucial in determining whether trigger strategies can sustain cooperation in the Prisoners’ Dilemma or, indeed, in any stage game. As δ approaches 1, grim-strategy punishments become inﬁnitely harsh because they involve an unending stream of undiscounted losses. Inﬁnite punishments can be used to sustain a wide range of possible outcomes. This is the logic behind the folk theorem for inﬁnitely repeated games :10 9

Nobel Prize–winning economist Gary Becker introduced a related point, the maximal punishment principle for crime. The principle says that even minor crimes should receive draconian punishments, which can deter crime with minimal expenditure on policing. The punishments are costless to society because no crimes are committed in equilibrum, so punishments never have to be carried out. See G. Becker, “Crime and Punishment: An Economic Approach,” Journal of Political Economy 76 (1968): 169–217. Less harsh punishments may be suitable in settings involving uncertainty. For example, citizens may not be certain about the penal code; police may not be certain they have arrested the guilty party.

10 This folk theorem is due to D. Fudenberg and E. Maskin, “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information,” Econometrica 54 (1986): 533–56.

Chapter 8

FIGURE 8.11

Strategy and Game Theory

Folk-Theorem Payoffs in the Infinitely Repeated Prisoners' Dilemma

Feasible payoffs are in the upward-hatched region; payoffs greater than each player’s minmax values are in the downward-hatched region. Their intersection (the cross-hatched region) constitutes the achievable payoffs according to the folk theorem for inﬁnitely repeated games. u2 3

2

1

0

1

2

3

u1

Any feasible payoff in the stage game greater than the player’s minmax value can be obtained as the player’s normalized payoff (normalized by multiplying by 1 δ:) in some subgame-perfect equilibrium of the inﬁnitely repeated game for δ close enough to 1.11

A few differences with the folk theorem for ﬁnitely repeated games are worth emphasizing. First, the limit involves increases in δ rather than in the number of periods T : The two limits are related. Interpreting δ as capturing the probability that the game continues into the next period, an increase in δ increases the expected number of periods the game is played in total— similar to an increase in T with the difference that now the end of the game is indeﬁnite. Another difference between the two folk theorems is that the one for inﬁnitely repeated games holds even if the stage game has just a single Nash equilibrium whereas the theorem for ﬁnitely repeated games requires the stage game to have multiple Nash equilibria. A ﬁnal technicality is that comparing stage-game payoffs with the present discounted value of a stream of payoffs from the inﬁnitely repeated game is like comparing apples with oranges. To make the two comparable, we “normalize” the payoff from the inﬁnitely repeated game via multiplying by 1 δ: This normalization allows us to think of all payoffs in per-period terms for easy comparison.12 11

As in footnote 9, an additional technical condition on the dimension of the feasible payoff set is also required.

12

For example, suppose a player earns $1 at the beginning of each period. The present discounted value of the stream of these $1 payoffs for an infinite number of periods is $1 : $1 þ $1 δ þ $1 δ2 þ $1 δ3 þ … ¼ 1δ

Multiplying through by 1 δ converts this stream of payments back into the per-period payoff of $1. The Chapter 17 Appendix provides more detail on the calculation of present discounted values of annuity streams (though beware the

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Figure 8.11 illustrates the folk theorem for inﬁnitely repeated games in the case of the Prisoners’ Dilemma. The ﬁgure shows the range of normalized payoffs that are possible in some subgame-perfect equilibrium of the inﬁnitely repeated Prisoners’ Dilemma. Again we see that subgame perfection may not be particularly selective in certain repeated games.

INCOMPLETE INFORMATION In the games studied so far, players knew everything there was to know about the setup of the game, including each others’ strategy sets and payoffs. Matters become more complicated, and potentially more interesting, if some players have information about the game that others do not. Poker would be quite different if all hands were played face up. The fun of playing poker comes from knowing what is in your hand but not others’. Incomplete information arises in many other real-world contexts besides parlor games. A sports team may try to hide the injury of a star player from future opponents to prevent them from exploiting this weakness. Firms’ production technologies may be trade secrets, and thus ﬁrms may not know whether they face efﬁcient or weak competitors. This section (and the next two) will introduce the tools needed to analyze games of incomplete information. The analysis integrates the material on game theory developed so far in this chapter with the material on uncertainty and information from the previous chapter. Games of incomplete information can quickly become very complicated. Players that lack full information about the game will try to use what they do know to make inferences about what they do not. The inference process can be quite involved. In poker, for example, knowing what is in your hand can tell you something about what is in others’. A player that holds two aces knows that others are less likely to hold aces because two of the four aces are not available. Information on others’ hands can also come from the size of their bets or from their facial expressions (of course, a big bet may be a bluff and a facial expression may be faked). Probability theory provides a formula, called Bayes’ rule, for making inferences about hidden information. We will encounter Bayes’ rule in a later section. The relevance of Bayes’ rule in games of incomplete information has led them to be called Bayesian games. To limit the complexity of the analysis, we will focus on the simplest possible setting throughout. We will focus on two-player games in which one of the players (player 1) has private information and the other (player 2) does not. The analysis of games of incomplete information is divided into two sections. The next section begins with the simple case in which the players move simultaneously. The subsequent section then analyzes games in which the informed player 1 moves ﬁrst. Such games, called signaling games, are more complicated than simultaneous games because player 1’s action may signal something about his private information to the uninformed player 2. We will introduce Bayes’ rule at that point to help analyze player 2’s inference about player 1’s hidden information based on observations of player 1’s action.

SIMULTANEOUS BAYESIAN GAMES In this section we study a two-player, simultaneous-move game in which player 1 has private information but player 2 does not. (We will use “he” for player 1 and “she” for player 2 in order to facilitate the exposition.) We begin by studying how to model private information.

subtle difference that in Chapter 17 the annuity payments come at the end of each period rather than at the beginning as assumed here).

Chapter 8

TABLE 8.11

Strategy and Game Theory

Simple Game of Incomplete Information

Player 1

Player 2 L

R

U

t, 2

0, 0

D

2, 0

2, 4

Note: t ¼ 6 with probability 1=2 and t ¼ 0 with probability 1=2.

Player types and beliefs John Harsanyi, who received the Nobel Prize in economics for his work on games with incomplete information, provided a simple way to model private information by introducing player characteristics or types.13 Player 1 can be one of a number of possible such types, denoted t : Player 1 knows his own type. Player 2 is uncertain about t and must decide on her strategy based on beliefs about t : Formally, the game begins at an initial node, called a chance node, at which a particular value tk is randomly drawn for player 1’s type t from a set of possible types T ¼ ft1 , …, tk , …, tK g: Let Prðtk Þ be the probability of drawing the particular type tk : Player 1 sees which type is drawn. Player 2 does not see the draw and only knows the probabilities, using them to form her beliefs about player 1’s type. Thus the probability that player 2 places on player 1’s being of type tk is Prðtk Þ: Since player 1 observes his type t before moving, his strategy can be conditioned on t : Conditioning on this information may be a big beneﬁt to a player. In poker, for example, the stronger a player’s hand, the more likely the player is to win the pot and the more aggressively the player may want to bid. Let s1 ðt Þ be 1’s strategy contingent on his type. Since player 2 does not observe t , her strategy is simply the unconditional one, s2 : As with games of complete information, players’ payoffs depend on strategies. In Bayesian games, payoffs may also depend on types. We therefore write player 1’s payoff as u1 ðs1 ðt Þ, s2 , t Þ and 2’s as u2 ðs2 , s1 ðt Þ, t Þ: Note that t appears in two places in 2’s payoff function. Player 1’s type may have a direct effect on 2’s payoffs. Player 1’s type also has an indirect effect through its effect on 1’s strategy s1 ðt Þ, which in turn affects 2’s payoffs. Since 2’s payoffs depend on t in these two ways, her beliefs about t will be crucial in the calculation of her optimal strategy. Table 8.11 provides a simple example of a simultaneous Bayesian game. Each player chooses one of two actions. All payoffs are known except for 1’s payoff when 1 chooses U and 2 chooses L: Player 1’s payoff in outcome ðU , LÞ is identiﬁed as his type, t : There are two possible values for player 1’s type, t ¼ 6 and t ¼ 0; each occurring with equal probability. Player 1 knows his type before moving. Player 2’s beliefs are that each type has probability 1=2. The extensive form is drawn in Figure 8.12.

Bayesian-Nash equilibrium Extending Nash equilibrium to Bayesian games requires two small matters of interpretation. First, recall that player 1 may play a different action for each of his types. Equilibrium requires that 1’s strategy be a best response for each and every one of his types. Second, recall that player 2 is uncertain about player 1’s type. Equilibrium requires that 2’s strategy maximize an expected payoff, where the expectation is taken with respect to her beliefs about 1’s type. We encountered expected payoffs in our discussion of mixed strategies. The calculations involved in computing the best response to the pure strategies of different types of rivals in a game of 13 J. Harsanyi, “Games with Incomplete Information Played by Bayesian Players,” Management Science 14 (1967∕68): 159–82, 320–34, 486–502.

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Part 2 Choice and Demand

FIGURE 8.12

Extensive Form for Simple Game of Incomplete Information This ﬁgure translates Table 8.11 into an extensive-form game. The initial chance node is indicated by an open circle. Player 2’s decision nodes are in the same information set because she does not observe 1’s type or action prior to moving.

L

6, 2

R

0, 0

L

2, 0

R

2, 4

L

0, 2

R

0, 0

L

2, 0

R

2, 4

2 U 1 t=6 Pr = 1/2

D

2

2

t=0 Pr = 1/2

U 1

D

2

incomplete information are similar to the calculations involved in computing the best response to a rival’s mixed strategy in a game of complete information. Interpreted in this way, Nash equilibrium in the setting of a Bayesian game is called Bayesian-Nash equilibrium. DEFINITION

Bayesian-Nash equilibrium. In a two-player, simultaneous-move game in which player 1 has private information, a Bayesian-Nash equilibrium is a strategy proﬁle ðs 1 ðt Þ, s 2 Þ such that s 1 ðt Þ is a best response to s 2 for each type t 2 T of player 1, U1 ðs 1 ðt Þ, s 2 , t Þ U1 ðs10 , s 2 , t Þ for all s10 2 S1 , (8.23) and such that s 2 is a best response to s 1 ðt Þ given player 2’s beliefs Prðtk Þ about player 1’s types: X X Prðtk ÞU2 ðs 2 , s 1 ðtk Þ, tk Þ Prðtk ÞU2 ðs20 , s 1 ðtk Þ, tk Þ for all s20 2 S2 . (8.24) tk 2T

tk 2T

Since the difference between Nash equilibrium and Bayesian-Nash equilibrium is only a matter of interpretation, all our previous results for Nash equilibrium (including the existence proof ) apply to Bayesian-Nash equilibrium as well.

Chapter 8

Strategy and Game Theory

EXAMPLE 8.8 Bayesian-Nash Equilibrium of Game in Figure 8.12 To solve for the Bayesian-Nash equilibrium of the game in Figure 8.12, ﬁrst solve for the informed player’s (player 1’s) best responses for each of his types. If player 1 is of type t ¼ 0 then he would choose D rather than U because he earns 0 by playing U and 2 by playing D regardless of what 2 does. If player 1 is of type t ¼ 6, then his best response is U to 2’s playing L and D to her playing R. This leaves only two possible candidates for an equilibrium in pure strategies: 1 plays ðU jt ¼ 6, Djt ¼ 0Þ and 2 plays L; 1 plays ðDjt ¼ 6, Djt ¼ 0Þ and 2 plays R. The ﬁrst candidate cannot be an equilibrium because, given that 1 plays ðU jt ¼ 6, Djt ¼ 0Þ, 2 earns an expected payoff of 1 from playing L: Player 2 would gain by deviating to R, earning an expected payoff of 2. The second candidate is a Bayesian-Nash equilibrium. Given that 2 plays R, 1’s best response is to play D, providing a payoff of 2 rather than 0 regardless of his type. Given that both types of player 1 play D, player 2’s best response is to play R, providing a payoff of 4 rather than 0. QUERY: If the probability that player 1 is of type t ¼ 6 is high enough, can the ﬁrst candidate be a Bayesian-Nash equilibrium? If so, compute the threshold probability.

EXAMPLE 8.9 Tragedy of the Commons as a Bayesian Game For an example of a Bayesian game with continuous actions, consider the Tragedy of the Commons in Example 8.6 but now suppose that herder 1 has private information regarding his value of grazing per sheep: v1 ðq1 , q2 , t Þ ¼ t ðq1 þ q2 Þ,

(8.25)

where 1’s type is t ¼ 130 (the “high” type) with probability 2=3 and t ¼ 100 (the “low” type) with probability 1=3. Herder 2’s value remains the same as in Equation 8.11. To solve for the Bayesian-Nash equilibrium, we ﬁrst solve for the informed player’s (herder 1’s) best responses for each of his types. For any type t and rival’s strategy q2 , herder 1’s value-maximization problem is maxfq1 v1 ðq1 , q2 , t Þg ¼ maxfq1 ðt q1 q2 Þg. q1

q1

(8.26)

The ﬁrst-order condition for a maximum is t 2q1 q2 ¼ 0.

(8.27)

Rearranging and then substituting the values t ¼ 130 and t ¼ 100, we obtain q2 q and q1L ¼ 50 2 , (8.28) 2 2 where q1H is the quantity for the “high” type of herder 1 (that is, the t ¼ 130 type) and q1L for the “low” type (the t ¼ 130 type). Next we solve for 2’s best response. Herder 2’s expected payoff is q1H ¼ 65

_ 2 1 ½q ð120 q1H q2 Þ þ ½q2 ð120 q1L q2 Þ ¼ q2 ð120 q 1 q2 Þ, 3 2 3

(8.29) (continued)

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EXAMPLE 8.9 CONTINUED FIGURE 8.13

Equilibrium of the Bayesian Tragedy of the Commons Best responses for herder 2 and both types of herder 1 are drawn as thick solid lines; the expected best response as perceived by 2 is drawn as the thick dashed line. The Bayesian-Nash equilibrium of the incomplete-information game is given by points A and C; Nash equilibria of the corresponding fullinformation games are given by points A 0 and C 0 .

q2 High type’s best response Low type’s best response

40

A′ A

B C C′ 2’s best response

0

where

q1

30 40 45

_ 2 1 q 1 ¼ q1H þ q1L . 3 3

(8.30)

Rearranging the ﬁrst-order condition from the maximization of Equation 8.29 with respect to q2 gives _ q1 . (8.31) q2 ¼ 60 2 Substituting for q1H and q1L_from Equation 8.28 into Equation 8.30 and then substituting the resulting expression for q 1 into Equation 8.31 yields q2 ¼ 30 þ

q2 , 4

(8.32)

¼ 45 and implying that q 2 ¼ 40: Substituting q 2 ¼ 40 back into Equation 8.28 implies q1H q1L ¼ 30: Figure 8.13 depicts the Bayesian-Nash equilibrium graphically. Herder 2 imagines playing against an average type of herder 1, whose average best response is given by the thick dashed line. The intersection of this best response and herder 2’s at point B determines 2’s equilibrium quantity, q 2 ¼ 40: The best response of the low (resp. high) type of herder 1 to q 2 ¼ 40 is given by point A (resp. point C). For comparison, the full-information Nash equilibria are drawn when herder 1 is known to be the low type (point A 0 ) or the high type (point C 0 ). QUERY: Suppose herder 1 is the high type. How does the number of sheep each herder grazes change as the game moves from incomplete to full information (moving from point C 0

Chapter 8

Strategy and Game Theory

to C)? What if herder 1 is the low type? Which type prefers full information and thus would like to signal its type? Which type prefers incomplete information and thus would like to hide its type? We will study the possibility player 1 can signal his type in the next section.

SIGNALING GAMES In this section we move from simultaneous-move games of private information to sequential games in which the informed player, 1, takes an action that is observable to 2 before 2 moves. Player 1’s action provides information, a signal, that 2 can use to update her beliefs about 1’s type, perhaps altering the way 2 would play in the absence of such information. In poker, for instance, player 2 may take a big raise by player 1 as a signal that he has a good hand, perhaps leading 2 to fold. A ﬁrm considering whether to enter a market may take the incumbent ﬁrm’s low price as a signal that the incumbent is a low-cost producer and thus a tough competitor, perhaps keeping the entrant out of the market. A prestigious college degree may signal that a job applicant is highly skilled. The analysis of signaling games is more complicated than simultaneous games because we need to model how player 2 processes the information in 1’s signal and then updates her beliefs about 1’s type. To ﬁx ideas, we will focus on a concrete application: a version of Michael Spence’s model of job-market signaling, for which he won the 2001 Nobel Prize in economics.14

Job-market signaling Player 1 is a worker who can be one of two types, high-skilled ðt ¼ H Þ or low-skilled ðt ¼ LÞ: Player 2 is a ﬁrm that considers hiring the applicant. A low-skilled worker is completely unproductive and generates no revenue for the ﬁrm; a high-skilled worker generates revenue π: If the applicant is hired, the ﬁrm must pay the worker w (think of this wage as being ﬁxed by government regulation). Assume π > w > 0: Therefore, the ﬁrm wishes to hire the applicant if and only if he or she is high-skilled. But the ﬁrm cannot observe the applicant’s skill; it can observe only the applicant’s prior education. Let cH be the high type’s cost of obtaining an education and cL the low type’s. Assume cH < cL , implying that education requires less effort for the high-skilled applicant than the low-skilled one. We make the extreme assumption that education does not increase the worker’s productivity directly. The applicant may still decide to obtain an education because of its value as a signal of ability to future employers. Figure 8.14 shows the extensive form. Player 1 observes his or her type at the start; player 2 observes only 1’s action (education signal) before moving. Let PrðH Þ and PrðLÞ be 2’s beliefs prior to observing 1’s education signal that 1 is high- or low-skilled, respectively. These are called 1’s prior beliefs. Observing 1’s action will lead 2 to revise its beliefs to form what are called posterior beliefs. For example, the probability that the worker is high-skilled is, conditional on the worker’s having obtained an education, PrðH jEÞ and, conditional on no education, PrðH jNEÞ: Player 2’s posterior beliefs are used to compute its best response to 1’s education decision. Suppose 2 sees 1 choose E: Then 2’s expected payoff from playing J is PrðH jEÞðπ wÞ þ PrðLjEÞðwÞ ¼ PrðH jEÞπ w,

(8.33)

where the left-hand side of this equation follows from the fact that, since L and H are the only types, PrðLjEÞ ¼ 1 PrðH jEÞ: Player 2’s payoff from playing NJ is 0. To determine its M. Spence, “Job-Market Signaling,” Quarterly Journal of Economics 87 (1973): 355–74.

14

273

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Part 2 Choice and Demand

FIGURE 8.14

Job-Market Signaling Player 1 (worker) observes his or her own type. Then 1 chooses to become educated (E) or not (NE). After observing 1’s action, player 2 (ﬁrm) decides to make him or her a job offer ( J ) or not (NJ ). The nodes in 2’s information sets are labeled n1 , …, n4 for reference.

J

w – cH, π – w

2 n1 E

NJ

1

J

Pr(H)

–cH, 0

w – cL , – w

2 NE

n2 NJ

J

w, π – w

2

E

Pr(L)

–cL, 0

n3 NJ

1

J

NE

0, 0

w, –w

2 n4 NJ

0, 0

best response to E, player 2 compares the expected payoff in Equation 8.33 to 0. Player 2’s best response is J if and only if PrðH jEÞ w=π: The question remains of how to compute posterior beliefs such as PrðH jEÞ: Rational players use a statistical formula, called Bayes’ rule, to revise their prior beliefs to form posterior beliefs based on the observation of a signal.

Bayes’ rule Bayes’ rule gives the following formula for computing player 2’s posterior belief PrðH jEÞ:15 15 Equation 8.34 can be derived from the definition of conditional probability in footnote 24 of Chapter 2. (Equation 8.35 can be derived similarly.) By definition,

PrðH jEÞ ¼

PrðH and EÞ . PrðEÞ

Reversing the order of the two events in the conditional probability yields PrðEjH Þ ¼

PrðH and EÞ PrðH Þ

or, after rearranging, PrðH and EÞ ¼ PrðEjH Þ PrðH Þ. Substituting the preceding equation into the first displayed equation of this footnote gives the numerator of Equation 8.34.

Chapter 8

FIGURE 8.15

Strategy and Game Theory

Bayes’ Rule as a Black Box

Bayes’ rule is a formula for computing player 2’s posterior beliefs from other pieces of information in the game.

Inputs Player 2’s prior beliefs

Output Bayes’ rule

Player 2’s posterior beliefs

Player 1’s strategy

PrðH jEÞ ¼

PrðEjH Þ PrðH Þ . PrðEjH Þ PrðH Þ þ PrðEjLÞ PrðLÞ

(8.34)

Similarly, PrðH jEÞ is given by PrðH jNEÞ ¼

PrðNEjH Þ PrðH Þ . PrðNEjH Þ PrðH Þ þ PrðNEjLÞ PrðLÞ

(8.35)

Two sorts of probabilities appear on the left-hand side of Equations 8.34 and 8.35: •

the prior beliefs PrðH Þ and PrðLÞ;

•

the conditional probabilities PrðEjH Þ, PrðNEjLÞ, and so forth.

The prior beliefs are given in the speciﬁcation of the game by the probabilities of the different branches from the initial chance node. The conditional probabilities PrðEjH Þ, PrðNEjLÞ, and so forth are given by player 1’s equilibrium strategy. For example, PrðEjH Þ is the probability that 1 plays E if he or she is of type H , PrðNEjLÞ is the probability that 1 plays NE if he or she is of type L, and so forth. As the schematic diagram in Figure 8.15 summarizes, Bayes’ rule can be thought of as a “black box” that takes prior beliefs and strategies as inputs and gives as outputs the beliefs we must know in order to solve for an equilibrium of the game: player 2’s posterior beliefs. When 1 plays a pure strategy, Bayes’ rule often gives a simple result. Suppose, for example, that PrðEjH Þ ¼ 1 and PrðEjLÞ ¼ 0 or, in other words, that player 1 obtains an education if and only if he or she is high-skilled. Then Equation 8.34 implies PrðH jEÞ ¼

1 ⋅ PrðH Þ ¼ 1. 1 ⋅ PrðH Þ þ 0 ⋅ PrðLÞ

(8.36)

That is, player 2 believes that 1 must be high-skilled if it sees 1 choose E: On the other hand, suppose that PrðEjH Þ ¼ PrðEjLÞ ¼ 1—that is, suppose player 1 obtains an education regardless of his or her type. Then Equation 8.34 implies

The denominator follows because the events of player 1’s being of type H or L are mutually exclusive and jointly exhaustive, so PrðEÞ ¼ PrðE and H Þ þ PrðE and LÞ ¼ PrðEjH Þ PrðH Þ þ PrðEjLÞ PrðLÞ.

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Part 2 Choice and Demand

PrðH jEÞ ¼

1 ⋅ PrðH Þ ¼ PrðH Þ, 1 ⋅ PrðH Þ þ 1 ⋅ PrðLÞ

(8.37)

since PrðH Þ þ PrðLÞ ¼ 1: That is, seeing 1 play E provides no information about 1’s type, so 2’s posterior belief is the same as its prior. More generally, if 2 plays the mixed strategy PrðEjH Þ ¼ p and PrðEjLÞ ¼ q, then Bayes’ rule implies that PrðH jEÞ ¼

p PrðH Þ . p PrðH Þ þ q PrðLÞ

(8.38)

Perfect Bayesian equilibrium With games of complete information, we moved from Nash equilibrium to the reﬁnement of subgame-perfect equilibrium in order to rule out noncredible threats in sequential games. For the same reason, with games of incomplete information we move from Bayesian-Nash equilibrium to the reﬁnement of perfect Bayesian equilibrium. DEFINITION

Perfect Bayesian equilibrium. A perfect Bayesian equilibrium consists of a strategy proﬁle and a set of beliefs such that • •

at each information set, the strategy of the player moving there maximizes his or her expected payoff, where the expectation is taken with respect to his or her beliefs; and at each information set, where possible, the beliefs of the player moving there are formed using Bayes’ rule (based on prior beliefs and other players’ strategies).

The requirement that players play rationally at each information set is similar to the requirement from subgame-perfect equilibrium that play on every subgame form a Nash equilibrium. The requirement that players use Bayes’ rule to update beliefs ensures that players incorporate the information from observing others’ play in a rational way. The remaining wrinkle in the deﬁnition of perfect Bayesian equilibrium is that Bayes’ rule need only be used “where possible.” Bayes’ rule is useless following a completely unexpected event—in the context of a signaling model, an action that is not played in equilibrium by any type of player 1. For example, if neither H nor L type chooses E in the job-market signaling game, then the denominators of Equations 8.34 and 8.35 equal zero and the fraction is undeﬁned. If Bayes’ rule gives an undeﬁned answer, then perfect Bayesian equilibrium puts no restrictions on player 2’s posterior beliefs and so we can assume any beliefs we like. As we saw with games of complete information, signaling games may have multiple equilibria. The freedom to specify any beliefs when Bayes’ rule gives an undeﬁned answer may support additional perfect Bayesian equilibria. A systematic analysis of multiple equilibria starts by dividing the equilibria into three classes—separating, pooling, and hybrid. Then we look for perfect Bayesian equilibria within each class. In a separating equilibrium, each type of player 1 chooses a different action. Therefore, player 2 learns 1’s type with certainty after observing 1’s action. The posterior beliefs that come from Bayes’ rule are all zeros and ones. In a pooling equilibrium, different types of player 1 choose the same action. Observing 1’s action provides 2 with no information about 1’s type. Pooling equilibria arise when one of player 1’s types chooses an action that would otherwise be suboptimal in order to hide his or her private information. In a hybrid equilibrium, one type of player 1 plays a strictly mixed strategy; it is called a hybrid equilibrium because the mixed strategy sometimes results in the types being separated and sometimes pooled. Player 2 learns a little about 1’s type (Bayes’ rule reﬁnes 2’s beliefs a bit) but doesn’t learn 1’s type with certainty. Player 2 may respond to the uncertainty by playing a mixed strategy itself. The next three examples solve for the three different classes of equilibrium in the job-market signaling game.

Chapter 8

Strategy and Game Theory

EXAMPLE 8.10 Separating Equilibrium in the Job-Market Signaling Game A good guess for a separating equilibrium is that the high-skilled worker signals his or her type by getting an education and the low-skilled worker does not. Given these strategies, player 2’s beliefs must be PrðH jEÞ ¼ PrðLjNEÞ ¼ 1 and PrðH jNEÞ ¼ PrðLjEÞ ¼ 0 according to Bayes’ rule. Conditional on these beliefs, if player 2 observes that player 1 obtains an education then 2 knows it must be at node n1 rather than n2 in Figure 8.14. Its best response is to offer a job ( J ), given the payoff of π w > 0: If player 2 observes that player 1 does not obtain an eduation then 2 knows it must be at node n4 rather than n3 , and its best response is not to offer a job (NJ ) because 0 > w: The last step is to go back and check that player 1 would not want to deviate from the separating strategy ðEjH , NEjLÞ given that 2 plays ðJ jE, NJ jNEÞ: Type H of player 1 earns w cH by obtaining an education in equilibrium. If type H deviates and does not obtain an education, then he or she earns 0 because player 2 believes that 1 is type L and does not offer a job. For type H not to prefer to deviate, it must be that w cH 0: Next turn to type L of player 1. Type L earns 0 by not obtaining an education in equilibrium. If type L deviates and obtains an education, then he or she earns w cL because player 2 believes that 1 is type H and offers a job. For type L not to prefer to deviate, we must have w cL 0: Putting these conditions together, there is separating equilibrium in which the worker obtains an education if and only if he or she is high-skilled and in which the ﬁrm offers a job only to applicants with an education if and only if cH w cL : Another possible separating equilibrium is for player 1 to obtain an education if and only if he or she is low-skilled. This is a bizarre outcome—since we expect education to be a signal of high rather than low skill—and fortunately we can rule it out as a perfect Bayesian equilibrium. Player 2’s best response would be to offer a job if and only if 1 did not obtain an education. Type L would earn cL from playing E and w from playing NE, so it would deviate to NE: QUERY: Why does the worker sometimes obtain an education even though it does not raise his or her skill level? Would the separating equilibrium exist if a low-skilled worker could obtain an education more easily than a high-skilled one?

EXAMPLE 8.11 Pooling Equilibria in the Job-Market Signaling Game Let’s investigate a possible pooling equilibrium in which both types of player 1 choose E: For player 1 not to deviate from choosing E, player 2’s strategy must be to offer a job if and only if the worker is educated—that is, ð J jE, NJ jNEÞ: If 2 doesn’t offer jobs to educated workers, then 1 might as well save the cost of obtaining an education and choose NE: If 2 offers jobs to uneducated workers, then 1 will again choose NE because he or she saves the cost of obtaining an education and still earns the wage from the job offer. Next, we investigate when ðJ jE, NJ jNEÞ is a best response for 2. Player 2’s posterior beliefs after seeing E are the same as its prior beliefs in this pooling equilibrium. Player 2’s expected payoff from choosing J is PrðH jEÞðπ wÞ þ PrðLjEÞðwÞ ¼ PrðH Þðπ wÞ þ PrðLÞðwÞ ¼ PrðH Þπ w.

(8.39)

For J to be a best response to E, Equation 8.39 must exceed 2’s zero payoff from choosing NJ , which upon rearranging implies that PrðH Þ w=π: Player 2’s posterior beliefs at nodes n3 and n4 are not pinned down by Bayes’ rule, because NE is never played in (continued)

277

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Part 2 Choice and Demand

EXAMPLE 8.11 CONTINUED equilibrium and so seeing 1 play NE is a completely unexpected event. Perfect Bayesian equilibrium allows us to specify any probability distribution we like for the posterior beliefs PrðH jNEÞ at node n3 and PrðLjNEÞ at node n4 : Player 2’s payoff from choosing NJ is 0. For NJ to be a best response to NE, 0 must exceed 2’s expected payoff from playing J : 0 > PrðH jNEÞðπ wÞ þ PrðLjNEÞð wÞ ¼ PrðH jNEÞπ w,

(8.40)

where the right-hand side follows because PrðH jNEÞ þ PrðLjNEÞ ¼ 1: Rearranging yields PrðH jNEÞ w=π: In sum, in order for there to be a pooling equilibrium in which both types of player 1 obtain an education, we need PrðH jNEÞ w=π PrðH Þ: The ﬁrm has to be optimistic about the proportion of skilled workers in the population—PrðH Þ must be sufﬁciently high— and pessimistic about the skill level of uneducated workers—PrðH jNEÞ must be sufﬁciently low. In this equilibrium, type L pools with type H in order to prevent player 2 from learning anything about the worker’s skill from the education signal. The other possibility for a pooling equilibrium is for both types of player 1 to choose NE: There are a number of such equilibria depending on what is assumed about player 2’s posterior beliefs out of equilibrium (that is, 2’s beliefs after it observes 1 choosing E). Perfect Bayesian equilibrium does not place any restrictions on these posterior beliefs. Problem 8.12 asks you to search for various of these equilibria and introduces a further reﬁnement of perfect Bayesian equilibrium (the intuitive criterion) that helps rule out unreasonable out-ofequilibrium beliefs and thus implausible equilibria. QUERY: Return to the pooling outcome in which both types of player 1 obtain an education. Consider 2’s posterior beliefs following the unexpected event that a worker shows up with no education. Perfect Bayesian equilibrium leaves us free to assume anything we want about these posterior beliefs. Suppose we assume that the ﬁrm obtains no information from the “no education” signal and so maintains its prior beliefs. Is the proposed pooling outcome an equilibrium? What if we assume that the ﬁrm takes “no education” as a bad signal of skill, believing that 1’s type is L for certain?

EXAMPLE 8.12 Hybrid Equilibria in the Job-Market Signaling Game One possible hybrid equilibrium is for type H always to obtain an education and for type L to randomize, sometimes pretending to be a high type by obtaining an education. Type L randomizes between playing E and NE with probabilities e and 1 e: Player 2’s strategy is to offer a job to an educated applicant with probability j and not to offer a job to an uneducated applicant. We need to solve for the equilibrium values of the mixed strategies e and j and the posterior beliefs PrðH jEÞ and PrðH jNEÞ that are consistent with perfect Bayesian equilibrium. The posterior beliefs are computed using Bayes’ rule: PrðH jEÞ ¼

PrðH Þ PrðH Þ ¼ PrðH Þ þ e PrðLÞ PrðH Þ þ e½1 PrðH Þ

(8.41)

and PrðH jNEÞ ¼ 0: For type L of player 1 to be willing to play a strictly mixed strategy, he or she must get the same expected payoff from playing E—which equals jw cL , given 2’s mixed strategy—as from playing NE—which equals 0 given that player 2 does not offer a job to uneducated applicants. Hence jw cL ¼ 0 or, solving for j , j ¼ cL =w:

Chapter 8

Strategy and Game Theory

Player 2 will play a strictly mixed strategy (conditional on observing E) only if it gets the same expected payoff from playing J , which equals PrðH jEÞðπ wÞ þ PrðLjEÞðwÞ ¼ PrðH jEÞπ w,

(8.42)

as from playing NJ , which equals 0. Setting Equation 8.42 equal to 0, substituting for PrðH jEÞ from Equation 8.41, and then solving for e gives e ¼

ðπ wÞPrðH Þ . w½1 PrðH Þ

(8.43)

QUERY: To complete our analysis: in this equilibrium, type H of player 1 cannot prefer to deviate from E: Is this true? If so, can you show it? How does the probability of type L trying to “pool” with the high type by obtaining an education vary with player 2’s prior belief that player 1 is the high type?

Cheap Talk Education is nothing more than a costly display in the job-market signaling game. The display must be costly—indeed, it must be more costly to the low-skilled worker—or else the skill levels could not be separated in equilibrium. While we do see some information communicated through costly displays in the real world, most information is communicated simply by having one party talk to another at low or no cost (“cheap talk”). Game theory can help explain why cheap talk is prevalent but also why cheap talk sometimes fails, forcing parties to resort to costly displays. We will model cheap talk as a two-player signaling game in which player 1’s strategy space consists of messages sent costlessly to player 2. The timing is otherwise the same as before: player 1 ﬁrst learns his type (“state of the world” might be a better label than “type” here because player 1’s private information will enter both players’ payoff functions directly), player 1 communicates to 2, and 2 takes some action affecting both players’ payoffs. The space of messages is potentially limitless: player 1 can use a more or less sophisticated vocabulary, can write a more or less detailed message, can speak in any of the thousands of languages in the world, and so forth. So the set of equilibria is even larger than would normally be the case in signaling games. We will analyze the range of possible equilibria from the least to the most informative perfect Bayesian equilibrium. The maximum amount of information that can be contained in player 1’s message will depend on how well-aligned the players’ payoff functions are. Player 2 would like to know the state of the world because she might have different actions that are suitable in different situations. If player 1 has the same preferences as 2 over which of 2’s actions are best in each state of the world, then 1 has every incentive to tell 2 precisely what the state of the world is, and 2 has every reason to believe 1’s report. On the other hand, if their preferences diverge, then 1 would have an incentive to lie about the state of the world to induce 2 to take the action that 1 prefers. Of course, 2 would anticipate 1’s lying and would refuse to believe the report. As preferences diverge, messages become less and less informative. In the limit, 1’s messages are completely uninformative (“babble”); to communicate real information, player 1 would have to resort to costly displays. In the job-market signaling game, for example, the preferences of the worker and ﬁrm diverge when the worker is low-skilled. The worker would like to be hired and the ﬁrm would like not to hire the worker. The high-skilled worker must resort to the costly display (education) in order to signal his or her type. The reason we see relatively more cheap talk than costly displays in the real world is probably because people try to associate with others with whom they share common interests and avoid those with whom they don’t. Members of a family, players on a team, or coworkers within a ﬁrm tend to have the same goals and usually have little reason to lie to each

279

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Part 2 Choice and Demand

other. Even in these examples, players’ interests may not be completely aligned and so cheap talk may not be completely informative (think about teenagers talking to parents). EXAMPLE 8.13 Simple Cheap Talk Game Consider a game with three states of the world: A, B, and C: First player 1 privately observes the state, then 1 sends a message to player 2, and then 2 chooses an action, L or R: The interests of players 1 and 2 are aligned in states A and B: both agree that 2 should play L in state A and R in state B: Their interests diverge in state C: 1 prefers 2 to play L and 2 prefers to play R: Assume that states A and B are equally likely. Let d be the probability of state C: Here, d measures the divergence between players’ preferences. Instead of the extensive form, which is complicated by having three states and an ill-deﬁned message space for player 1, the game is represented schematically by the matrices in Table 8.12. TABLE 8.12

Simple Cheap Talk Game Player 2 Player 1

L

R

State A

1, 1

0, 0

PrðAÞ ¼ ð1 dÞ=2

Player 2 Player 1

L

R

State B

0, 0

1, 1

PrðBÞ ¼ ð1 dÞ=2

Player 2 Player 1

L

R

0, 1

1, 0

State C PrðCÞ ¼ d

If d ¼ 0 then players’ incentives are completely aligned. The most informative equilibrium results in perfect communication: 1 announces the state truthfully; 2 plays L if 1 announces “A” and R if 1 announces “B”.16 For d > 0; there cannot be perfect communication. If communication were perfect, then whatever message 1 sends when the state is A perfectly reveals the state and so leads 2 to play L: But then 1 would have an incentive to lie when the true state is C and would thus send the same message as when the state is A: Player 1’s messages can be no more reﬁned than issuing one of the two messages “the state is either A or C” or “the state is B”; any attempt to distinguish between A and C would not be believed. If there is not too much divergence between players’ interests—in particular, if d 1=3— then there is an equilibrium with imperfect but still informative communication. In this equilibrium, player 1 sends one of two truthful messages: “A or C” or “B.” Then player 2 plays L conditional on the message “A or C” and R conditional on “B.” Player 2’s expected payoff from playing L following the message “A or C” equals (8.44) PrðAj“A or C ”Þð1Þ þ PrðC j “A or C ”Þð0Þ ¼ PrðAj “A or C ”Þ. By Bayes’ rule, Prð“A or C ” jAÞ PrðAÞ 1d ¼ . PrðAj “A or C ”Þ ¼ Prð“A or C ” jAÞ PrðAÞ þ Prð“A or C ” jC Þ PrðC Þ 1þd (8.45) 16 At the other extreme, for d ¼ 0 and indeed for all parameters, there is always an uninformative “babbling” equilibrium in which 1’s messages contain no information and 2 pays no attention to what 1 says.

Chapter 8

Strategy and Game Theory

The second equality in Equation 8.45 holds upon substituting Prð“A or C” jAÞ ¼ Prð“A or C” jCÞ ¼ 1 (if the state is A or C, player 1’s strategy is to announce “A or C” with certainty) and substituting the values of PrðAÞ and PrðCÞ in terms of d from Table 8.12. Player 2’s expected payoff from deviating to U can be shown (using calculations similar to Equations 8.44 and 8.45) to equal PrðC j “A or C ”Þ ¼

2d . 1þd

(8.46)

In equilibrium, Equation 8.45 must exceed Equation 8.46, implying that d 1=3: If players’ interests are yet more divergent—in particular, if d > 1=3—then there are only uninformative “babbling” equilibria. QUERY: Are players better-off in more informative equilibria? What difference would it make if player 1 announced “purple” instead of “A or C” and “yellow” instead of “B”? What features of a language would make it more or less efﬁcient in a cheap-talk setting?

EXPERIMENTAL GAMES Experimental economics is a recent branch of research that explores how well economic theory matches the behavior of experimental subjects in laboratory settings. The methods are similar to those used in experimental psychology—often conducted on campus using undergraduates as subjects—although experiments in economics tend to involve incentives in the form of explicit monetary payments paid to subjects. The importance of experimental economics was highlighted in 2002, when Vernon Smith received the Nobel Prize in economics for his pioneering work in the ﬁeld. An important area in this ﬁeld is the use of experimental methods to test game theory.

Experiments with the Prisoners’ Dilemma There have been hundreds of tests of whether players ﬁnk in the Prisoners’ Dilemma as predicted by Nash equilibrium or whether they play the cooperative outcome of Silent. In one experiment, subjects played the game 20 times with each player being matched with a different, anonymous opponent to avoid repeated-game effects. Play converged to the Nash equilibrium as subjects gained experience with the game. Players played the cooperative action 43 percent of the time in the ﬁrst ﬁve rounds, falling to only 20 percent of the time in the last ﬁve rounds.17 As is typical with experiments, subjects’ behavior tended to be noisy. Although 80 percent of the decisions were consistent with Nash-equilibrium play by the end of the experiment, still 20 percent of them were anomalous. Even when experimental play is roughly consistent with the predictions of theory, it is rarely entirely consistent.

Experiments with the Ultimatum Game Experimental economics has also tested to see whether subgame-perfect equilibrium is a good predictor of behavior in sequential games. In one widely studied sequential game, the Ultimatum Game, the experimenter provides a pot of money to two players. The ﬁrst mover (Proposer) proposes a split of this pot to the second mover. The second mover (Responder) then decides whether to accept the offer, in which case players are given the amount of money indicated, or reject the offer, in which case both players get nothing. In the subgame-perfect R. Cooper, D. V. DeJong, R. Forsythe, and T. W. Ross, “Cooperation Without Reputation: Experimental Evidence from Prisoner’s Dilemma Games,” Games and Economic Behavior (February 1996): 187–218.

17

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equilibrium, the Proposer offers a minimal share of the pot and this is accepted by the Responder. One can see this by applying backward induction: the Responder should accept any positive division no matter how small; knowing this, the Proposer should offer the Responder only a minimal share. In experiments, the division tends to be much more even than in the subgame-perfect equilibrium.18 The most common offer is a 50–50 split. Responders tend to reject offers giving them less than 30 percent of the pot. This result is observed even when the pot is as high as $100, so that rejecting a 30 percent offer means turning down $30. Some economists have suggested that the money players receive may not be a true measure of their payoffs. They may care about other factors such as fairness and so obtain a beneﬁt from a more equal division of the pot. Even if a Proposer does not care directly about fairness, the fear that the Responder may care about fairness and thus might reject an uneven offer out of spite may lead the Proposer to propose an even split. The departure of experimental behavior from the predictions of game theory was too systematic in the Ultimatum Game to be attributed to noisy play, leading some game theorists to rethink the theory and add an explicit consideration for fairness.19

Experiments with the Dictator Game To test whether players care directly about fairness or act out of fear of the other player’s spite, researchers experimented with a related game, the Dictator Game. In the Dictator Game, the Proposer chooses a split of the pot, and this split is implemented without input from the Responder. Proposers tend to offer a less even split than in the Ultimatum Game but still offer the Responder some of the pot, suggesting that Responders have some residual concern for fairness. The details of the experimental design are crucial, however, as one ingenious experiment showed.20 The experiment was designed so that the experimenter would never learn which Proposers had made which offers. With this element of anonymity, Proposers almost never gave an equal split to Responders and indeed took the whole pot for themselves two thirds of the time. Proposers seem to care more about appearing fair to the experimenter than truly being fair.

EVOLUTIONARY GAMES AND LEARNING The frontier of game-theory research regards whether and how players come to play a Nash equilibrium. Hyperrational players may deduce each others’ strategies and instantly settle upon the Nash equilibrium. How can they instantly coordinate on a single outcome when there are multiple Nash equilibria? What outcome would real-world players, for whom hyperrational deductions may be too complex, settle on? Game theorists have tried to model the dynamic process by which an equilibrium emerges over the long run from the play of a large population of agents who meet others at random and play a pairwise game. Game theorists analyze whether play converges to Nash equilibrium or some other outcome, which Nash equilibrium (if any) is converged to if there are multiple equilibria, and how long such convergence takes. Two models, which make varying assumptions about the level of players’ rationality, have been most widely studied: an evolutionary model and a learning model. 18 For a review of Ultimatum Game experiments and a textbook treatment of experimental economics more generally, see D. D. Davis and C. A. Holt, Experimental Economics (Princeton, NJ: Princeton University Press, 1993). 19 See, for example, M. Rabin, “Incorporating Fairness into Game Theory and Economics,” American Economic Review (December 1993): 1281–1302. 20 E. Hoffman, K. McCabe, K. Shachat, and V. Smith, “Preferences, Property Rights, and Anonymity in Bargaining Games,” Games and Economic Behavior (November 1994): 346–80.

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In the evolutionary model, players do not make rational decisions; instead, they play the way they are genetically programmed. The more successful a player’s strategy in the population, the more ﬁt is the player and the more likely will the player survive to pass its genes on to future generations and so the more likely the strategy spreads in the population. Evolutionary models were initially developed by John Maynard Smith and other biologists to explain the evolution of such animal behavior as how hard a lion ﬁghts to win a mate or an ant ﬁghts to defend its colony. While it may be more of a stretch to apply evolutionary models to humans, evolutionary models provide a convenient way of analyzing population dynamics and may have some direct bearing on how social conventions are passed down, perhaps through culture. In a learning model, players are again matched at random with others from a large population. Players use their experiences of payoffs from past play to teach them how others are playing and how they themselves can best respond. Players usually are assumed to have a degree of rationality in that they can choose a static best response given their beliefs, may do some experimenting, and will update their beliefs according to some reasonable rule. Players are not fully rational in that they do not distort their strategies in order to affect others’ learning and thus future play. Game theorists have investigated whether more-or less-sophisticated learning strategies converge more or less quickly to a Nash equilibrium. Current research seeks to integrate theory with experimental study, trying to identify the speciﬁc algorithms that real-world subjects use when they learn to play games.

SUMMARY This chapter provided a structured way to think about strategic situations. We focused on the most important solution concept used in game theory, Nash equilibrium. We then progressed to several more-reﬁned solution concepts that are in standard use in game theory in more complicated settings (with sequential moves and incomplete information). Some of the principal results are as follows. • All games have the same basic components: players, strategies, payoffs, and an information structure. • Games can be written down in normal form (providing a payoff matrix or payoff functions) or extensive form (providing a game tree). • Strategies can be simple actions, more complicated plans contingent on others’ actions, or even probability distributions over simple actions (mixed strategies). • A Nash equilibrium is a set of strategies, one for each player, that are mutual best responses. In other words, a player’s strategy in a Nash equilibrium is optimal given that all others play their equilibrium strategies. • A Nash equilibrium always exists in ﬁnite games (in mixed if not pure strategies).

• Subgame-perfect equilibrium is a reﬁnement of Nash equilibrium that helps to rule out equilibria in sequential games involving noncredible threats. • Repeating a stage game a large number of times introduces the possibility of using punishment strategies to attain higher payoffs than if the stage game is played once. If a ﬁnite game with multiple stages is repeated often enough or if players are sufﬁciently patient in an inﬁnitely repeated game, then a folk theorem holds implying that essentially any payoffs are possible in the repeated game. • In games of private information, one player knows more about his or her “type” than another. Players maximize their expected payoffs given knowledge of their own type and beliefs about the others’. • In a perfect Bayesian equilibrium of a signaling game, the second mover uses Bayes’ rule to update his or her beliefs about the ﬁrst mover’s type after observing the ﬁrst mover’s action. • The frontier of game-theory research combines theory with experiments to determine whether players who may not be hyperrational come to play a Nash equilibrium, which particular equilibrium (if there are more than one), and what path leads to the equilibrium.

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PROBLEMS 8.1 Consider the following game:

Player 1

Player 2 D

E

F

A

7, 6

5, 8

0, 0

B

5, 8

7, 6

1, 1

C

0, 0

1, 1

4, 4

a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the ﬁrst two actions. c. Compute players’ expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.

8.2 The mixed-strategy Nash equilibrium in the Battle of the Sexes in Table 8.3 may depend on the numerical values for the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by

Player 1 ðWifeÞ

Player 2 (Husband) Ballet

Boxing

Ballet

K, 1

0, 0

Boxing

0, 0

1, K

where K 1: Show how the mixed-strategy Nash equilibrium depends on the value of K :

8.3 The game of Chicken is played by two macho teens who speed toward each other on a single-lane road. The ﬁrst to veer off is branded the chicken, whereas the one who doesn’t veer gains peer-group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game are provided in the following table.

Teen 1

Teen 2 Veer

Don’t veer

Veer

2, 2

1, 3

Don’t veer

3, 1

0, 0

Chapter 8

Strategy and Game Theory

a. Draw the extensive form. b. Find the pure-strategy Nash equilibrium or equilibria. c. Compute the mixed-strategy Nash equilibrium. As part of your answer, draw the best-response function diagram for the mixed strategies. d. Suppose the game is played sequentially, with teen A moving ﬁrst and committing to this action by throwing away the steering wheel. What are teen B’s contingent strategies? Write down the normal and extensive forms for the sequential version of the game. e. Using the normal form for the sequential version of the game, solve for the Nash equilibria. f. Identify the proper subgames in the extensive form for the sequential version of the game. Use backward induction to solve for the subgame-perfect equilibrium. Explain why the other Nash equilibria of the sequential game are “unreasonable.”

8.4 Two neighboring homeowners, i ¼ 1, 2, simultaneously choose how many hours li to spend maintaining a beautiful lawn. The average beneﬁt per hour is 10 li þ

lj 2

,

and the (opportunity) cost per hour for each is 4. Homeowner i ’s average beneﬁt is increasing in the hours neighbor j spends on his own lawn, since the appearance of one’s property depends in part on the beauty of the surrounding neighborhood. a. Compute the Nash equilibrium. b. Graph the best-response functions and indicate the Nash equilibrium on the graph. c. On the graph, show how the equilibrium would change if the intercept of one of the neighbor’s average beneﬁt functions fell from 6 to some smaller number.

8.5 The Academy Award–winning movie A Beautiful Mind about the life of John Nash dramatizes Nash’s scholarly contribution in a single scene: his equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest brunette, and agree that the blond is more desirable than the brunettes. The Nash character views the situation as a game among the male graduate students, along the following lines. Suppose there are n males who simultaneously approach either the blond or one of the brunettes. If male i alone approaches the blond, then he is successful in getting a date with her and earns payoff a: If one or more other males approach the blond along with i, the competition causes them all to lose her, and i (as well as the others who approached her) earns a payoff of zero. On the other hand, male i earns a payoff of b > 0 from approaching a brunette, since there are more brunettes than males, so i is certain to get a date with a brunette. The desirability of the blond implies a > b: a. Argue that this game does not have a symmetric pure-strategy Nash equilibrium. b. Solve for the symmetric mixed-strategy equilibrium. That is, letting p be the probability that a male approaches the blond, ﬁnd p . c. Show that the more males there are, the less likely it is in the equilibrium from part (b) that the blond is approached by at least one of them. Note: This paradoxical result was noted by S. Anderson and M. Engers in “Participation Games: Market Entry, Coordination, and the Beautiful Blond,” Journal of Economic Behavior & Organization 63 (2007): 120–37.

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8.6 Consider the following stage game.

Player 2 Player 1

286

A

B

C

A

10, 10

1, 15

1, 12

B

15, 1

0, 0

1, 1

C

12, 1

1, 1

8, 8

a. Compute a player’s minmax value if the rival is restricted to pure strategies. Is this minmax value different than if the rival is allowed to use mixed strategies? b. Suppose the stage game is played twice. Characterize the subgame-perfect equilibrium providing the highest total payoffs. c. Draw a graph of the set of feasible per-period payoffs in the limit in a ﬁnitely repeated game according to the folk theorem.

8.7 Return to the game with two neighbors in Problem 8.5. Continue to suppose that player i ’s average beneﬁt per hour of work on landscaping is 10 li þ

lj 2

.

Continue to suppose that player 2’s opportunity cost of an hour of landscaping work is 4. Suppose that 1’s opportunity cost is either 3 or 5 with equal probability and that this cost is 1’s private information. a. Solve for the Bayesian-Nash equilibrium. b. Indicate the Bayesian-Nash equilibrium on a best-response function diagram. c. Which type of player 1 would like to send a truthful signal to 2 if it could? Which type would like to hide its private information?

8.8 In Blind Texan Poker, player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it. Player 1 moves ﬁrst, deciding whether to stay or fold. If player 1 folds, he must pay player 2 $50. If player 1 stays, the action goes to player 2. Player 2 can fold or call. If player 2 folds, she must pay player 1 $50. If 2 calls, the card is examined. If it is a low card (2 through 8), player 2 pays player 1 $100. If it is a high card (9, 10, jack, queen, king, or ace), player 1 pays player 2 $100. a. Draw the extensive form for the game. b. Solve for the hybrid equilibrium. c. Compute the players’ expected payoffs.

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Analytical Problems 8.9 Dominant strategies Prove that an equilibrium in dominant strategies is the unique Nash equilibrium.

8.10 Rotten Kid Theorem In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child’s parent (player 2). The child moves ﬁrst, choosing an action r that affects his own income Y1 ðrÞ ½Y10 ðrÞ > 0 and the income of the parent Y2 ðrÞ ½Y20 ðrÞ < 0: Later, the parent moves, leaving a monetary bequest L to the child. The child cares only for his own utility, U1 ðY1 þ LÞ, but the parent maximizes U2 ðY2 LÞ þ αU1 , where α > 0 reﬂects the parent’s altruism toward the child. Prove that, in a subgame-perfect equilibrium, the child will opt for the value of r that maximizes Y1 þ Y2 even though he has no altruistic intentions. Hint: Apply backward induction to the parent’s problem ﬁrst, which will give a ﬁrst-order condition that implicitly determines L ; although an explicit solution for L cannot be found, the derivative of L with respect to r—required in the child’s ﬁrststage optimization problem—can be found using the implicit function rule.

8.11 Alternatives to Grim Strategy Suppose that the Prisoners’ Dilemma stage game (see Table 8.1) is repeated for inﬁnitely many periods. a. Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash equilibrium for just one period and then returning to cooperation? Are two periods of punishment enough? b. Suppose players use strategies that punish deviation from cooperation by reverting to the stagegame Nash equilibrium for ten periods before returning to cooperation. Compute the threshold discount factor above which cooperation is possible on the outcome that maximizes the joint payoffs.

8.12 Refinements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.11. a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the ﬁrm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the ﬁrm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to E and so an educated worker must be highskilled? Cho and Kreps’s intuitive criterion is one of a series of complicated reﬁnements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identiﬁed in this part; see I. K. Cho and D. M. Kreps, “Signalling Games and Stable Equilibria,” Quarterly Journal of Economics 102 (1987): 179–221.

SUGGESTIONS FOR FURTHER READING Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991. A comprehensive survey of game theory at the graduate-student level, though selected sections are accessible to advanced undergraduates.

Holt, C. A. Markets, Games, & Strategic Behavior. Boston: Pearson, 2007. An undergraduate text with emphasis on experimental games.

Rasmusen, E. Games and Information, 4th ed. Malden, MA: Blackwell, 2007. An advanced undergraduate text with many real-world applications.

Watson, Joel. Strategy: An Introduction to Game Theory. New York: Norton, 2002. An undergraduate text that balances rigor with simple examples (often 2 2 games). Emphasis on bargaining and contracting examples.

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EXTENSIONS Existence of Nash Equilibrium This section will sketch John Nash’s original proof that all ﬁnite games have at least one Nash equilibrium (in mixed if not in pure strategies). We will provide some of the details of the proof here; the original proof is in Nash (1950), and a clear textbook presentation of the full proof is provided in Fudenberg and Tirole (1991). The section concludes by mentioning a related existence theorem for games with continuous actions. Nash’s proof is similar to the proof of the existence of a general competitive equilibrium in Chapter 13. Both proofs rely on a ﬁxed point theorem. The proof of the existence of Nash equilibrium requires a slightly more powerful theorem. Instead of Brouwer’s ﬁxed FIGURE E8.1

point theorem, which applies to functions, Nash’s proof relies on Kakutani’s ﬁxed point theorem, which applies to correspondences—more general mappings than functions.

E8.1 Correspondences versus functions A function maps each point in a ﬁrst set to a single point in a second set. A correspondence maps a single point in the ﬁrst set to possibly many points in the second set. Figure E8.1 illustrates the difference.

Comparision of Functions and Correspondences The function graphed in (a) looks like a familiar curve. Each value of x is mapped into a single value of y. With the correspondence graphed in (b), each value of x may be mapped into many values of y. Correspondences can thus have bulges as shown by the gray regions in (b). y

x (a) Function y

x (b) Correspondence

Chapter 8

An example of a correspondence that we have already seen is the best response, BRi ðsi Þ: The best response need not map other players’ strategies si into a single strategy that is a best response for player i: There may be ties among several best responses. As shown in Figure 8.3, in the Battle of the Sexes, the husband’s best response to the wife’s playing the mixed strategy of going to ballet with probability 2=3 and boxing with probability 1=3 (or just w ¼ 2=3 for short) is not just a single point but the whole interval of possible mixed strategies. Both the husband’s and the wife’s best responses in this ﬁgure are correspondences, not functions. The reason Nash needed a ﬁxed point theorem involving correspondences rather than just functions is precisely because his proof works with players’ best responses to prove existence.

E8.2 Kakutani’s fixed point theorem Here is the statement of Kakutani’s ﬁxed point theorem: Any convex, upper-semicontinuous corrrespondence ½ f ðxÞ from a closed, bounded, convex set into itself has at least one ﬁxed point ðx Þ such that x 2 f ðx Þ:

Comparing the statement of Kakutani’s ﬁxed point theorem with Brouwer’s in Chapter 13, they are similar except for the substitution of “correspondence” for “function” and for the conditions on the correspondence. Brouwer’s theorem requires the function to be continuous; Kakutani’s theorem requires the correspondence to be convex and upper semicontinuous. These properties, which are related to continuity, are less familiar and worth spending a moment to understand. Figure E8.2 provides examples of correspondences violating (a) convexity and (b) upper semicontinuity. The ﬁgure shows why the two properties are needed to guarantee a ﬁxed point. Without both properties, the correspondence can “jump” across the 45° line and so fail to have a ﬁxed point—that is, a point for which x ¼ f ðxÞ:

E8.3 Nash’s proof We use RðsÞ to denote the correspondence that underlies Nash’s existence proof. This correspondence takes any proﬁle of players’ strategies s ¼ ðs1 , s2 , …, sn Þ (possibly mixed) and maps it into another mixed strategy

Strategy and Game Theory

289

proﬁle, the proﬁle of best responses: RðsÞ ¼ ðBR1 ðs1 Þ, BR2 ðs2 Þ, …, BRn ðsn ÞÞ.

(i)

A ﬁxed point of the correspondence is a strategy for which s 2 Rðs Þ; this is a Nash equilibrium because each player’s strategy is a best response to others’ strategies. The proof checks that all the conditions involved in Kakutani’s ﬁxed point theorem are satisﬁed by the best-response correspondence RðsÞ: First, we need to show that the set of mixed-strategy proﬁles is closed, bounded, and convex. Since a strategy proﬁle is just a list of individual strategies, the set of strategy proﬁles will be closed, bounded, and convex if each player’s strategy set Si has these properties individually. As Figure E8.3 shows for the case of two and three actions, the set of mixed strategies over actions has a simple shape.1 The set is closed (contains its boundary), bounded (does not go off to inﬁnity in any direction), and convex (the segment between any two points in the set is also in the set). We then need to check that the best-response correspondence RðsÞ is convex. Individual best responses cannot look like (a) in Figure E8.2, because if any two mixed strategies such as A and B are best responses to others’ strategies then mixed strategies between them must also be best responses. For example, in the Battle of the Sexes, if (1=3, 2=3) and (2=3, 1=3) are best responses for the husband against his wife’s playing (2=3, 1=3) (where, in each pair, the ﬁrst number is the probability of playing ballet and the second of playing boxing), then mixed strategies between the two such as (1=2, 1=2) must also be best responses for him. Figure 8.3 showed that in fact all possible mixed strategies for the husband are best responses to the wife’s playing (2=3, 1=3). Finally, we need to check that RðsÞ is upper semicontinuous. Individual best responses cannot look like (b) in Figure E8.2. They cannot have holes like point D punched out of them because payoff functions ui ðsi , si Þ are continuous. Recall that payoffs, when written as functions of mixed strategies, are actually expected values with probabilities given by the strategies si and si : As Equation 2.176 showed, expected values are linear functions of the underlying probabilities. Linear functions are of course continuous.

1 Mathematicians study them so frequently that they have a special name for such a set: a simplex.

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Part 2 Choice and Demand

FIGURE E8.2

Kakutani’s Conditions on Correspondences The correspondence in (a) is not convex because the dashed vertical segment between A and B is not inside the correspondence. The correspondence in (b) is not upper semicontinuous because there is a path (C) inside the correspondence leading to a point (D) that, as indicated by the open circle, is not inside the correspondence. Both (a) and (b) fail to have ﬁxed points. f(x)

1 45° A

B x 1 (a) Correspondence that is not convex f(x)

1 45°

D

C

x (b) Correspondence that is not upper semicontinuous

Chapter 8

FIGURE E8.3

Strategy and Game Theory

291

Set of Mixed Strategies for an Individual

Player 1’s set of possible mixed strategies over two actions is given by the diagonal line segment in (a). The set for three actions is given by the shaded triangle on the three-dimensional graph in (b).

p12

1

0

p11

1 (a) Two actions

p13 1

0 1

p12

1 p11 (b) Three actions

E8.4 Games with continuous actions Nash’s existence theorem applies to ﬁnite games— that is, games with a ﬁnite number of players and actions per player. Nash’s theorem does not apply to games, such as the Tragedy of the Commons in Example 8.6, that feature continuous actions. Is a Nash equilibrium guaranteed to exist for these games, too? Glicksberg (1952) proved that the answer is “yes” as long as payoff functions are continuous.

References Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991, sec. 1.3. Glicksberg, I. L. “A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points.” Proceedings of the National Academy of Sciences 38 (1952): 170–74. Nash, John. “Equilibrium Points in n-Person Games.” Proceedings of the National Academy of Sciences 36 (1950): 48–49.

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P A R T

3

Production and Supply CHAPTER 9

Production Functions

CHAPTER 10 Cost Functions CHAPTER 11 Proﬁt Maximization

In this part we examine the production and supply of economic goods. Institutions that coordinate the transformation of inputs into outputs are called ﬁrms. They may be large institutions (such as Microsoft, Sony, or the U.S. Department of Defense) or small ones (such as “Mom and Pop” stores or self-employed individuals). Although they may pursue different goals (Microsoft may seek maximum proﬁts, whereas an Israeli kibbutz may try to make members of the kibbutz as well off as possible), all ﬁrms must make certain basic choices in the production process. The purpose of Part 3 is to develop some tools for analyzing those choices. In Chapter 9 we examine ways of modeling the physical relationship between inputs and outputs. We introduce the concept of a production function, a useful abstraction from the complexities of real-world production processes. Two measurable aspects of the production function are stressed: its returns to scale (that is, how output expands when all inputs are increased) and its elasticity of substitution (that is, how easily one input may be replaced by another while maintaining the same level of output). We also brieﬂy describe how technical improvements are reﬂected in production functions. The production function concept is then used in Chapter 10 to discuss costs of production. We assume that all ﬁrms seek to produce their output at the lowest possible cost, an assumption that permits the development of cost functions for the ﬁrm. Chapter 10 also focuses on how costs may differ between the short run and the long run. In Chapter 11 we investigate the ﬁrm’s supply decision. To do so, we assume that the ﬁrm’s manager will make input and output choices so as to maximize proﬁts. The chapter concludes with the fundamental model of supply behavior by proﬁt-maximizing ﬁrms that we will use in many subsequent chapters.

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CHAPTER

9 Production Functions The principal activity of any ﬁrm is to turn inputs into outputs. Because economists are interested in the choices the ﬁrm makes in accomplishing this goal, but wish to avoid discussing many of the engineering intricacies involved, they have chosen to construct an abstract model of production. In this model the relationship between inputs and outputs is formalized by a production function of the form

q ¼ f ðk, l, m, …Þ,

(9.1)

where q represents the ﬁrm’s output of a particular good during a period,1 k represents the machine (that is, capital) usage during the period, l represents hours of labor input, m represents raw materials used,2 and the notation indicates the possibility of other variables aﬀecting the production process. Equation 9.1 is assumed to provide, for any conceivable set of inputs, the engineer’s solution to the problem of how best to combine those inputs to get output.

MARGINAL PRODUCTIVITY In this section we look at the change in output brought about by a change in one of the productive inputs. For the purposes of this examination (and indeed for most of the purposes of this book), it will be more convenient to use a simpliﬁed production function deﬁned as follows. Production function. The ﬁrm’s production function for a particular good, q, q ¼ f ðk, lÞ,

DEFINITION

(9.2)

shows the maximum amount of the good that can be produced using alternative combinations of capital ðkÞ and labor ðlÞ. Of course, most of our analysis will hold for any two inputs to the production process we might wish to examine. The terms capital and labor are used only for convenience. Similarly, it would be a simple matter to generalize our discussion to cases involving more than two inputs; occasionally, we will do so. For the most part, however, limiting the discussion to two inputs will be quite helpful because we can show these inputs on two-dimensional graphs.

Marginal physical product To study variation in a single input, we deﬁne marginal physical product as follows. Here we use a lowercase q to represent one ﬁrm’s output. We reserve the uppercase Q to represent total output in a market. Generally, we assume that a ﬁrm produces only one output. Issues that arise in multiproduct ﬁrms are discussed in a few footnotes and problems.

1

2

In empirical work raw material inputs often are disregarded and output, q, is measured in terms of “value added.”

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Part 3 Production and Supply

DEFINITION

Marginal physical product. The marginal physical product of an input is the additional output that can be produced by employing one more unit of that input while holding all other inputs constant. Mathematically, ∂q ¼ fk , marginal physical product of capital ¼ MPk ¼ ∂k (9.3) ∂q ¼ fl . marginal physical product of labor ¼ MPl ¼ ∂l Notice that the mathematical deﬁnitions of marginal product use partial derivatives, thereby properly reﬂecting the fact that all other input usage is held constant while the input of interest is being varied. For an example, consider a farmer hiring one more laborer to harvest the crop but holding all other inputs constant. The extra output this laborer produces is that farmhand’s marginal physical product, measured in physical quantities, such as bushels of wheat, crates of oranges, or heads of lettuce. We might observe, for example, that 50 workers on a farm are able to produce 100 bushels of wheat per year, whereas 51 workers, with the same land and equipment, can produce 102 bushels. The marginal physical product of the 51st worker is then 2 bushels per year.

Diminishing marginal productivity We might expect that the marginal physical product of an input depends on how much of that input is used. Labor, for example, cannot be added indeﬁnitely to a given ﬁeld (while keeping the amount of equipment, fertilizer, and so forth ﬁxed) without eventually exhibiting some deterioration in its productivity. Mathematically, the assumption of diminishing marginal physical productivity is an assumption about the second-order partial derivatives of the production function: ∂MPk ∂2 f ¼ 2 ¼ fkk ¼ f11 < 0, ∂k ∂k ∂MPl ∂2 f ¼ 2 ¼ fll ¼ f22 < 0: ∂l ∂l

(9.4)

The assumption of diminishing marginal productivity was originally proposed by the nineteenth-century economist Thomas Malthus, who worried that rapid increases in population would result in lower labor productivity. His gloomy predictions for the future of humanity led economics to be called the “dismal science.” But the mathematics of the production function suggests that such gloom may be misplaced. Changes in the marginal productivity of labor over time depend not only on how labor input is growing, but also on changes in other inputs, such as capital. That is, we must also be concerned with ∂MPl =∂k ¼ flk . In most cases, flk > 0, so declining labor productivity as both l and k increase is not a foregone conclusion. Indeed, it appears that labor productivity has risen signiﬁcantly since Malthus’ time, primarily because increases in capital inputs (along with technical improvements) have offset the impact of diminishing marginal productivity alone.

Average physical productivity In common usage, the term labor productivity often means average productivity. When it is said that a certain industry has experienced productivity increases, this is taken to mean that output per unit of labor input has increased. Although the concept of average productivity is not nearly as important in theoretical economic discussions as marginal productivity is, it receives a great deal of attention in empirical discussions. Because average productivity is

Chapter 9 Production Functions

easily measured (say, as so many bushels of wheat per labor-hour input), it is often used as a measure of efﬁciency. We deﬁne the average product of labor (APl ) to be output q f ðk, lÞ ¼ ¼ . (9.5) APl ¼ labor input l l Notice that APl also depends on the level of capital employed. This observation will prove to be quite important when we examine the measurement of technical change at the end of this chapter.

EXAMPLE 9.1 A Two-Input Production Function Suppose the production function for ﬂyswatters during a particular period can be represented by (9.6) q ¼ f ðk, lÞ ¼ 600k2 l 2 k3 l 3 . To construct the marginal and average productivity functions of labor (l) for this function, we must assume a particular value for the other input, capital (k). Suppose k ¼ 10. Then the production function is given by (9.7) q ¼ 60,000l 2 1,000l 3 . Marginal product. The marginal productivity function (when k ¼ 10) is given by MPl ¼

∂q ¼ 120,000l 3,000l 2 , ∂l

(9.8)

which diminishes as l increases, eventually becoming negative. This implies that q reaches a maximum value. Setting MPl equal to 0, 120,000l 3,000l 2 ¼ 0

(9.9)

40l ¼ l 2

(9.10)

l ¼ 40

(9.11)

yields or as the point at which q reaches its maximum value. Labor input beyond 40 units per period actually reduces total output. For example, when l ¼ 40, Equation 9.7 shows that q ¼ 32 million ﬂyswatters, whereas when l ¼ 50, production of ﬂyswatters amounts to only 25 million. Average product. To ﬁnd the average productivity of labor in ﬂyswatter production, we divide q by l, still holding k ¼ 10: q (9.12) APl ¼ ¼ 60,000l 1,000l 2 . l Again, this is an inverted parabola that reaches its maximum value when ∂APl ¼ 60,000 2,000l ¼ 0, ∂l

(9.13)

which occurs when l ¼ 30. At this value for labor input, Equation 9.12 shows that APl ¼ 900,000, and Equation 9.8 shows that MPl is also 900,000. When APl is at a maximum, average and marginal productivities of labor are equal.3 (continued) 3

This result is quite general. Because ∂APl l ⋅ MPl q ¼ , ∂l l2

at a maximum l MPl ¼ q or MPl ¼ APl .

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EXAMPLE 9.1 CONTINUED Notice the relationship between total output and average productivity that is illustrated by this example. Even though total production of ﬂyswatters is greater with 40 workers (32 million) than with 30 workers (27 million), output per worker is higher in the second case. With 40 workers, each worker produces 800,000 ﬂyswatters per period, whereas with 30 workers each worker produces 900,000. Because capital input (ﬂyswatter presses) is held constant in this deﬁnition of productivity, the diminishing marginal productivity of labor eventually results in a declining level of output per worker. QUERY: How would an increase in k from 10 to 11 affect the MPl and APl functions here? Explain your results intuitively.

ISOQUANT MAPS AND THE RATE OF TECHNICAL SUBSTITUTION To illustrate possible substitution of one input for another in a production function, we use its isoquant map. Again, we study a production function of the form q ¼ f ðk, lÞ, with the understanding that “capital” and “labor” are simply convenient examples of any two inputs that might happen to be of interest. An isoquant (from iso, meaning “equal”) records those combinations of k and l that are able to produce a given quantity of output. For example, all those combinations of k and l that fall on the curve labeled “q ¼ 10” in Figure 9.1 are capable of producing 10 units of output per period. This isoquant then records the fact that there are many alternative ways of producing 10 units of output. One way might be represented by point A: We would use lA and kA to produce 10 units of output. Alternatively, we might prefer FIGURE 9.1

An Isoquant Map Isoquants record the alternative combinations of inputs that can be used to produce a given level of output. The slope of these curves shows the rate at which l can be substituted for k while keeping output constant. The negative of this slope is called the (marginal) rate of technical substitution (RTS). In the ﬁgure, the RTS is positive and diminishing for increasing inputs of labor. k per period

kA

A

q = 30 q = 20

kB

B

lA

lB

q = 10

l per period

Chapter 9 Production Functions

299

to use relatively less capital and more labor and therefore would choose a point such as B. Hence, we may deﬁne an isoquant as follows. Isoquant. An isoquant shows those combinations of k and l that can produce a given level of DEFINITION output (say, q0 ). Mathematically, an isoquant records the set of k and l that satisﬁes f ðk, lÞ ¼ q0 .

(9.14)

As was the case for indifference curves, there are inﬁnitely many isoquants in the k–l plane. Each isoquant represents a different level of output. Isoquants record successively higher levels of output as we move in a northeasterly direction. Presumably, using more of each of the inputs will permit output to increase. Two other isoquants (for q ¼ 20 and q ¼ 30) are shown in Figure 9.1. You will notice the similarity between an isoquant map and the individual’s indifference curve map discussed in Part 2. They are indeed similar concepts, because both represent “contour” maps of a particular function. For isoquants, however, the labeling of the curves is measurable—an output of 10 units per period has a quantiﬁable meaning. Economists are therefore more interested in studying the shape of production functions than in examining the exact shape of utility functions.

The marginal rate of technical substitution (RTS) The slope of an isoquant shows how one input can be traded for another while holding output constant. Examining the slope provides information about the technical possibility of substituting labor for capital. A formal deﬁnition follows. Marginal rate of technical substitution. The marginal rate of technical substitution (RTS) DEFINITION shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. In mathematical terms, dk . (9.15) RTS ðl for kÞ ¼ dl q¼q0 In this deﬁnition, the notation is intended as a reminder that output is to be held constant as l is substituted for k. The particular value of this trade-off rate will depend not only on the level of output but also on the quantities of capital and labor being used. Its value depends on the point on the isoquant map at which the slope is to be measured.

RTS and marginal productivities To examine the shape of production function isoquants, it is useful to prove the following result: the RTS (of l for k) is equal to the ratio of the marginal physical productivity of labor (MPl ) to the marginal physical productivity of capital (MPk ). We begin by setting up the total differential of the production function: dq ¼

∂f ∂f ⋅ dl þ ⋅ dk ¼ MPl ⋅ dl þ MPk ⋅ dk, ∂l ∂k

(9.16)

which records how small changes in l and k affect output. Along an isoquant, dq ¼ 0 (output is constant), so (9.17) MPl ⋅ dl ¼ MPk ⋅ dk. This says that along an isoquant, the gain in output from increasing l slightly is exactly balanced by the loss in output from suitably decreasing k. Rearranging terms a bit gives dk MPl ¼ RTS ðl for kÞ ¼ . (9.18) MPk dl q¼q0 Hence the RTS is given by the ratio of the inputs’ marginal productivities.

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Equation 9.18 shows that those isoquants that we actually observe must be negatively sloped. Because both MPl and MPk will be nonnegative (no ﬁrm would choose to use a costly input that reduced output), the RTS also will be positive (or perhaps zero). Because the slope of an isoquant is the negative of the RTS, any ﬁrm we observe will not be operating on the positively sloped portion of an isoquant. Although it is mathematically possible to devise production functions whose isoquants have positive slopes at some points, it would not make economic sense for a ﬁrm to opt for such input choices.

Reasons for a diminishing RTS The isoquants in Figure 9.1 are drawn not only with a negative slope (as they should be) but also as convex curves. Along any one of the curves, the RTS is diminishing. For high ratios of k to l, the RTS is a large positive number, indicating that a great deal of capital can be given up if one more unit of labor becomes available. On the other hand, when a lot of labor is already being used, the RTS is low, signifying that only a small amount of capital can be traded for an additional unit of labor if output is to be held constant. This assumption would seem to have some relationship to the assumption of diminishing marginal productivity. A hasty use of Equation 9.18 might lead one to conclude that a rise in l accompanied by a fall in k would result in a fall in MPl , a rise in MPk , and, therefore, a fall in the RTS. The problem with this quick “proof” is that the marginal productivity of an input depends on the level of both inputs—changes in l affect MPk and vice versa. It is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone. To see why this is so mathematically, assume that q ¼ f ðk, lÞ and that fk and fl are positive (that is, the marginal productivities are positive). Assume also that fkk < 0 and fll < 0 (that the marginal productivities are diminishing). To show that isoquants are convex, we would like to show that dðRTSÞ=dl < 0. Since RTS ¼ fl =fk , we have dRTS dð fl =fk Þ ¼ . dl dl

(9.19)

Because fl and fk are functions of both k and l, we must be careful in taking the derivative of this expression: dRTS f ð f þ flk ⋅ dk=dlÞ fl ð fkl þ fkk ⋅ dk=dlÞ . ¼ k ll dl ð fk Þ2

(9.20)

Using the fact that dk=dl ¼ fl =fk along an isoquant and Young’s theorem (fkl ¼ flk ), we have f 2 f 2fk fl fkl þ f 2l fkk dRTS ¼ k ll . dl ð f k Þ3

(9.21)

Because we have assumed fk > 0, the denominator of this function is positive. Hence the whole fraction will be negative if the numerator is negative. Because fll and fkk are both assumed to be negative, the numerator deﬁnitely will be negative if fkl is positive. If we can assume this, we have shown that dRTS=dl < 0 (that the isoquants are convex)4.

Importance of cross-productivity eﬀects Intuitively, it seems reasonable that the cross-partial derivative fkl ¼ flk should be positive. If workers had more capital, they would have higher marginal productivities. But, although this is probably the most prevalent case, it does not necessarily have to be so. Some production functions have fkl < 0, at least for a range of input values. When we assume a diminishing 4

As we pointed out in Chapter 2, functions for which the numerator in Equation 9.21 is negative are called (strictly) quasiconcave functions.

Chapter 9 Production Functions

RTS (as we will throughout most of our discussion), we are therefore making a stronger assumption than simply diminishing marginal productivities for each input—speciﬁcally, we are assuming that marginal productivities diminish “rapidly enough” to compensate for any possible negative cross-productivity effects. Of course, as we shall see later, with three or more inputs, things become even more complicated. EXAMPLE 9.2 A Diminishing RTS In Example 9.1, the production function for ﬂyswatters was given by q ¼ f ðk, lÞ ¼ 600k2 l 2 k3 l 3 .

(9.22)

General marginal productivity functions for this production function are ∂q ¼ 1,200k 2 l 3k 3 l 2 , ∂l ∂q MPk ¼ fk ¼ ¼ 1,200kl 2 3k2 l 3 . ∂k MPl ¼ fl ¼

(9.23)

Notice that each of these depends on the values of both inputs. Simple factoring shows that these marginal productivities will be positive for values of k and l for which kl < 400. Because fll ¼ 1,200k2 6k 3 l and fkk ¼ 1,200l 2 6kl 3 ,

(9.24)

it is clear that this function exhibits diminishing marginal productivities for sufﬁciently large values of k and l. Indeed, again by factoring each expression, it is easy to show that fll , fkk < 0 if kl > 200. However, even within the range 200 < kl < 400 where the marginal productivity relations for this function behave “normally,” this production function may not necessarily have a diminishing RTS. Cross-differentiation of either of the marginal productivity functions (Equation 9.23) yields fkl ¼ flk ¼ 2,400kl 9k2 l 2 ,

(9.25)

which is positive only for kl < 266. The numerator of Equation 9.21 will therefore deﬁnitely be negative for 200 < kl < 266, but for larger-scale ﬂyswatter factories the case is not so clear, because fkl is negative. When fkl is negative, increases in labor input reduce the marginal productivity of capital. Hence, the intuitive argument that the assumption of diminishing marginal productivities yields an unambiguous prediction about what will happen to the RTS ð¼ fl =fk Þ as l increases and k falls is incorrect. It all depends on the relative effects on marginal productivities of diminishing marginal productivities (which tend to reduce fl and increase fk ) and the contrary effects of cross-marginal productivities (which tend to increase fl and reduce fk ). Still, for this ﬂyswatter case, it is true that the RTS is diminishing throughout the range of k and l, where marginal productivities are positive. For cases where 266 < kl < 400, the diminishing marginal productivities exhibited by the function are sufﬁcient to overcome the inﬂuence of a negative value for fkl on the convexity of isoquants. QUERY: For cases where k ¼ l, what can be said about the marginal productivities of this production function? How would this simplify the numerator for Equation 9.21? How does this permit you to more easily evaluate this expression for some larger values of k and l?

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RETURNS TO SCALE We now proceed to characterize production functions. A ﬁrst question that might be asked about them is how output responds to increases in all inputs together. For example, suppose that all inputs were doubled: Would output double or would the relationship not be quite so simple? This is a question of the returns to scale exhibited by the production function that has been of interest to economists ever since Adam Smith intensively studied the production of pins. Smith identiﬁed two forces that came into operation when the conceptual experiment of doubling all inputs was performed. First, a doubling of scale permits a greater division of labor and specialization of function. Hence, there is some presumption that efﬁciency might increase—production might more than double. Second, doubling of the inputs also entails some loss in efﬁciency because managerial overseeing may become more difﬁcult given the larger scale of the ﬁrm. Which of these two tendencies will have a greater effect is an important empirical question. Presenting a technical deﬁnition of these concepts is misleadingly simple. DEFINITION

Returns to scale. If the production function is given by q ¼ f ðk, lÞ and if all inputs are multiplied by the same positive constant t (where t > 1), then we classify the returns to scale of the production function by Effect on Output

Returns to Scale

I. f ðtk, tlÞ ¼ tf ðk, lÞ ¼ tq

Constant

II. f ðtk, tlÞ < tf ðk, lÞ ¼ tq

Decreasing

III. f ðtk, tlÞ > tf ðk, l Þ ¼ tq

Increasing

In intuitive terms, if a proportionate increase in inputs increases output by the same proportion, the production function exhibits constant returns to scale. If output increases less than proportionately, the function exhibits diminishing returns to scale. And if output increases more than proportionately, there are increasing returns to scale. As we shall see, it is theoretically possible for a function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels.5 Often, however, economists refer to the degree of returns to scale of a production function with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered.

Constant returns to scale There are economic reasons why a ﬁrm’s production function might exhibit constant returns to scale. If the ﬁrm operates many identical plants, it may increase or decrease production simply by varying the number of them in current operation. That is, the ﬁrm can double output by doubling the number of plants it operates, and that will require it to employ precisely twice as many inputs. Alternatively, if one were modeling the behavior of an entire industry composed of many ﬁrms, the constant returns-to-scale assumption might 5

A local measure of returns to scale is provided by the scale elasticity, deﬁned as eq, t ¼

∂f ðtk, tlÞ t , ⋅ ∂t f ðtk, tlÞ

where this expression is to be evaluated at t ¼ 1. This parameter can, in principle, take on different values depending on the level of input usage. For some examples using this concept, see Problem 9.9.

Chapter 9 Production Functions

make sense because the industry can expand or contract by adding or dropping an arbitrary number of identical ﬁrms (see Chapter 12). Finally, studies of the entire U.S. economy have found that constant returns to scale is a reasonably good approximation to use for an “aggregate” production function. For all of these reasons, then, the constant returns-toscale case seems worth examining in somewhat more detail. When a production function exhibits constant returns to scale, it meets the deﬁnition of “homogeneity” that we introduced in Chapter 2. That is, the production is homogeneous of degree 1 in its inputs because f ðtk, tlÞ ¼ t 1 f ðk, lÞ ¼ tq.

(9.26)

In Chapter 2 we showed that, if a function is homogeneous of degree k, its derivatives are homogeneous of degree k 1. In this context this implies that the marginal productivity functions derived from a constant returns-to-scale production function are homogeneous of degree 0. That is, ∂f ðk, lÞ ∂f ðtk, tlÞ ¼ , ∂k ∂k ∂f ðk, lÞ ∂f ðtk, tlÞ ¼ MPl ¼ ∂l ∂l

MPk ¼

(9.27)

for any t > 0. In particular, we can let t ¼ 1=l in Equations 9.27 and get ∂f ðk=l, 1Þ , ∂k ∂f ðk=l, 1Þ . MPl ¼ ∂l

MPk ¼

(9.28)

That is, the marginal productivity of any input depends only on the ratio of capital to labor input, not on the absolute levels of these inputs. This fact is especially important, for example, in explaining differences in productivity among industries or across countries.

Homothetic production functions One consequence of Equations 9.28 is that the RTS ð¼ MPl =MPk Þ for any constant returnsto-scale production function will depend only on the ratio of the inputs, not on their absolute levels. That is, such a function will be homothetic (see Chapter 2)—its isoquants will be radial expansions of one another. This situation is shown in Figure 9.2. Along any ray through the origin (where the ratio k=l does not change), the slopes of successively higher isoquants are identical. This property of the isoquant map will be very useful to us on several occasions. A simple numerical example may provide some intuition about this result. Suppose a roof can be installed in one day by three workers with one hammer each or by two workers with two hammers each (these workers are ambidextrous). The RTS of hammers for workers is therefore one for one—one extra hammer can be substituted for one worker. If this production process exhibits constant returns to scale, two roofs can be installed in one day either by six workers with six hammers or by four workers with eight hammers. In the latter case, two hammers are substituted for two workers, so again the RTS is one for one. In constant returns-to-scale cases, expanding the level of production does not alter trade-offs among inputs, so production functions are homothetic. A production function can have a homothetic indifference curve map even if it does not exhibit constant returns to scale. As we showed in Chapter 2, this property of homotheticity is retained by any monotonic transformation of a homogeneous function. Hence, increasing or decreasing returns to scale can be incorporated into a constant returns-to-scale function

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FIGURE 9.2

Isoquant Map for a Constant Returns-to-Scale Production Function For a constant returns-to-scale production function, the RTS depends only on the ratio of k to l, not on the scale of production. Consequently, each isoquant will be a radial blowup of the unit isoquant. Along any ray through the origin (a ray of constant k=l), the RTS will be the same on all isoquants.

k per period

q=3 q=2 q=1

l per period

through an appropriate transformation. Perhaps the most common such transformation is exponential. So, if f ðk, lÞ is a constant returns-to-scale producton function, we can let F ðk, lÞ ¼ ½ f ðk, lÞγ ,

(9.29)

where γ is any positive exponent. If γ > 1 then F ðtk, tlÞ ¼ ½ f ðtk, tlÞγ ¼ ½tf ðk, lÞγ ¼ t γ ½ f ðk, lÞγ ¼ t γ F ðk, lÞ > tF ðk, lÞ

(9.30)

for any t > 1. Hence, this transformed production function exhibits increasing returns to scale. An identical proof shows that the function F exhibits decreasing returns to scale for γ < 1 . Because this function remains homothetic through all such transformations, we have shown that there are important cases where the issue of returns to scale can be separated from issues involving the shape of an isoquant. In the next section, we will look at how shapes of isoquants can be described.

The n-input case The deﬁnition of returns to scale can be easily generalized to a production function with n inputs. If that production function is given by q ¼ f ðx1 , x2 , …, xn Þ

(9.31)

and if all inputs are multiplied by t > 1, we have f ðtx1 , tx2 , …, txn Þ ¼ t k f ðx1 , x2 , …, xn Þ ¼ t k q

(9.32)

for some constant k. If k ¼ 1, the production function exhibits constant returns to scale. Diminishing and increasing returns to scale correspond to the cases k < 1 and k > 1, respectively. The crucial part of this mathematical deﬁnition is the requirement that all inputs be increased by the same proportion, t . In many real-world production processes, this provision may make little economic sense. For example, a ﬁrm may have only one “boss,” and that

Chapter 9 Production Functions

305

number would not necessarily be doubled even if all other inputs were. Or the output of a farm may depend on the fertility of the soil. It may not be literally possible to double the acres planted while maintaining fertility, because the new land may not be as good as that already under cultivation. Hence, some inputs may have to be ﬁxed (or at least imperfectly variable) for most practical purposes. In such cases, some degree of diminishing productivity (a result of increasing employment of variable inputs) seems likely, although this cannot properly be called “diminishing returns to scale” because of the presence of inputs that are held ﬁxed.

THE ELASTICITY OF SUBSTITUTION Another important characteristic of the production function is how “easy” it is to substitute one input for another. This is a question about the shape of a single isoquant rather than about the whole isoquant map. Along one isoquant, the rate of technical substitution will decrease as the capital-labor ratio decreases (that is, as k=l decreases); now we wish to deﬁne some parameter that measures this degree of responsiveness. If the RTS does not change at all for changes in k=l, we might say that substitution is easy because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the RTS changes rapidly for small changes in k=l, we would say that substitution is difﬁcult because minor variations in the input mix will have a substantial effect on the inputs’ relative productivities. A scale-free measure of this responsiveness is provided by the elasticity of substitution, a concept we encountered in Part 2. Now we can provide a formal deﬁnition. Elasticity of substitution. For the production function q ¼ f ðk, lÞ, the elasticity of substitution ðσÞ measures the proportionate change in k=l relative to the proportionate change in D E F I N I T I O N the RTS along an isoquant. That is, σ¼

percent ∆ðk=lÞ dðk=lÞ RTS ∂ ln k=l ∂ ln k=l . ¼ ¼ ¼ ⋅ percent ∆RTS dRTS k=l ∂ ln RTS ∂ ln fl =fk

(9.33)

Because along an isoquant, k=l and RTS move in the same direction, the value of σ is always positive. Graphically, this concept is illustrated in Figure 9.3 as a movement from point A to point B on an isoquant. In this movement, both the RTS and the ratio k=l will change; we are interested in the relative magnitude of these changes. If σ is high, then the RTS will not change much relative to k=l and the isoquant will be relatively ﬂat. On the other hand, a low value of σ implies a rather sharply curved isoquant; the RTS will change by a substantial amount as k=l changes. In general, it is possible that the elasticity of substitution will vary as one moves along an isoquant and as the scale of production changes. Often, however, it is convenient to assume that σ is constant along an isoquant. If the production function is also homothetic, then— because all the isoquants are merely radial blowups—σ will be the same along all isoquants. We will encounter such functions later in this chapter and in many of its problems.6

The n-input case Generalizing the elasticity of substitution to the many-input case raises several complications. One approach is to adopt a deﬁnition analogous to Equation 9.33; that is, to deﬁne the elasticity of substitution between two inputs to be the proportionate change in the ratio of 6

The elasticity of substitution can be phrased directly in terms of the production function and its derivatives in the constant returns-to-scale case as f ⋅f σ¼ k l . f ⋅ fk, l

But this form is quite cumbersome. Hence usually the logarithmic deﬁnition in Equation 9.33 is easiest to apply. For a compact summary, see P. Berck and K. Sydsaeter, Economist’s Mathematical Manual (Berlin: Springer-Verlag, 1999), chap. 5.

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FIGURE 9.3

Graphic Description of the Elasticity of Substitution In moving from point A to point B on the q ¼ q0 isoquant, both the capital-labor ratio (k=l) and the RTS will change. The elasticity of substitution (σ) is deﬁned to be the ratio of these proportional changes; it is a measure of how curved the isoquant is. k per period

A

RTSA RTSB

B q = q0 (k /l ) A (k /l ) B l per period

the two inputs to the proportionate change in the RTS between them while holding output constant.7 To make this deﬁnition complete, it is necessary to require that all inputs other than the two being examined be held constant. However, this latter requirement (which is not relevant when there are only two inputs) restricts the value of this potential deﬁnition. In real-world production processes, it is likely that any change in the ratio of two inputs will also be accompanied by changes in the levels of other inputs. Some of these other inputs may be complementary with the ones being changed, whereas others may be substitutes, and to hold them constant creates a rather artiﬁcial restriction. For this reason, an alternative deﬁnition of the elasticity of substitution that permits such complementarity and substitutability in the ﬁrm’s cost function is generally used in the n-good case. Because this concept is usually measured using cost functions, we will describe it in the next chapter.

FOUR SIMPLE PRODUCTION FUNCTIONS In this section we illustrate four simple production functions, each characterized by a different elasticity of substitution. These are shown only for the case of two inputs, but generalization to many inputs is easily accomplished (see the Extensions for this chapter). 7

That is, the elasticity of substitution between input i and input j might be deﬁned as σij ¼

∂ lnðxi =xj Þ ∂ lnð fj =fi Þ

for movements along f ðx1 , x2 , …, xn Þ ¼ c. Notice that the use of partial derivatives in this deﬁnition effectively requires that all inputs other than i and j be held constant when considering movements along the c isoquant.

Chapter 9 Production Functions

Case 1: Linear (σ ¼ ∞) Suppose that the production function is given by q ¼ f ðk, lÞ ¼ ak þ bl.

(9.34)

It is easy to show that this production function exhibits constant returns to scale: For any t > 1, f ðtk, tlÞ ¼ atk þ btl ¼ t ðak þ blÞ ¼ tf ðk, lÞ. (9.35) All isoquants for this production function are parallel straight lines with slope b=a. Such an isoquant map is pictured in panel (a) of Figure 9.4. Because the RTS is constant along any straight-line isoquant, the denominator in the deﬁnition of σ (Equation 9.33) is equal to 0 and hence σ is inﬁnite. Although this linear production function is a useful example, it is FIGURE 9.4

Isoquant Maps for Simple Production Functions with Various Values for σ

Three possible values for the elasticity of substitution are illustrated in these ﬁgures. In (a), capital and labor are perfect substitutes. In this case, the RTS will not change as the capital-labor ratio changes. In (b), the ﬁxed-proportions case, no substitution is possible. The capital-labor ratio is ﬁxed at b=a. A case of limited substitutability is illustrated in (c).

k per period

k per period σ=∞

σ=0

–b Slope = __ a q __3 a q1

q2

q3 q2

q3

q1 l per period (a)

(b)

q 3 __ b

k per period σ=1

q3 q2 q1 l per period (c)

l per period

307

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Part 3 Production and Supply

rarely encountered in practice because few production processes are characterized by such ease of substitution. Indeed, in this case, capital and labor can be thought of as perfect substitutes for each other. An industry characterized by such a production function could use only capital or only labor, depending on these inputs’ prices. It is hard to envision such a production process: Every machine needs someone to press its buttons, and every laborer requires some capital equipment, however modest.

Case 2: Fixed proportions (σ ¼ 0) The production function characterized by σ ¼ 0 is the important case of a ﬁxed-proportions production function. Capital and labor must always be used in a ﬁxed ratio. The isoquants for this production function are L-shaped and are pictured in panel (b) of Figure 9.4. A ﬁrm characterized by this production function will always operate along the ray where the ratio k=l is constant. To operate at some point other than at the vertex of the isoquants would be inefﬁcient, because the same output could be produced with fewer inputs by moving along the isoquant toward the vertex. Because k=l is a constant, it is easy to see from the deﬁnition of the elasticity of substitution that σ must equal 0. The mathematical form of the ﬁxed-proportions production function is given by q ¼ minðak, blÞ,

a, b > 0,

(9.36)

where the operator “min” means that q is given by the smaller of the two values in parentheses. For example, suppose that ak < bl; then q ¼ ak, and we would say that capital is the binding constraint in this production process. The employment of more labor would not raise output, and hence the marginal product of labor is zero; additional labor is superﬂuous in this case. Similarly, if ak > bl, then labor is the binding constraint on output and additional capital is superﬂuous. When ak ¼ bl, both inputs are fully utilized. When this happens, k=l ¼ b=a, and production takes place at a vertex on the isoquant map. If both inputs are costly, this is the only cost-minimizing place to operate. The locus of all such vertices is a straight line through the origin with a slope given by b=a. The ﬁxed-proportions production function has a wide range of applications.8 Many machines, for example, require a certain number of people to run them, but any excess labor is superﬂuous. Consider combining capital (a lawn mower) and labor to mow a lawn. It will always take one person to run the mower, and either input without the other is not able to produce any output at all. It may be that many machines are of this type and require a ﬁxed complement of workers per machine.9

Case 3: Cobb-Douglas (σ ¼ 1) The production function for which σ ¼ 1, called a Cobb-Douglas production function,10 provides a middle ground between the two polar cases previously discussed. Isoquants for

With the form reﬂected by Equation 9.35, the ﬁxed-proportions production function exhibits constant returns to scale because

8

f ðtk, tlÞ ¼ minðatk, btlÞ ¼ t minðak, blÞ ¼ tf ðk, lÞ for any t > 1. As before, increasing or decreasing returns can be easily incorporated into the functions by using a nonlinear transformation of this functional form—such as ½ f ðk, lÞγ , where γ may be greater than or less than 1. 9

The lawn mower example points up another possibility, however. Presumably there is some leeway in choosing what size of lawn mower to buy. Hence, prior to the actual purchase, the capital-labor ratio in lawn mowing can be considered variable: Any device, from a pair of clippers to a gang mower, might be chosen. Once the mower is purchased, however, the capital-labor ratio becomes ﬁxed.

10 Named after C. W. Cobb and P. H. Douglas. See P. H. Douglas, The Theory of Wages (New York: Macmillan Co., 1934), pp. 132–f35.

Chapter 9 Production Functions

the Cobb-Douglas case have the “normal” convex shape and are shown in panel (c) of Figure 9.4. The mathematical form of the Cobb-Douglas production function is given by q ¼ f ðk, lÞ ¼ Ak a l b ,

(9.37)

where A, a, and b are all positive constants. The Cobb-Douglas function can exhibit any degree of returns to scale, depending on the values of a and b. Suppose all inputs were increased by a factor of t . Then f ðtk, tlÞ ¼ AðtkÞa ðtlÞb ¼ At aþb ka l b ¼ t aþb f ðk, lÞ.

(9.38)

Hence, if a þ b ¼ 1, the Cobb-Douglas function exhibits constant returns to scale because output also increases by a factor of t . If a þ b > 1 then the function exhibits increasing returns to scale, whereas a þ b < 1 corresponds to the decreasing returns-to-scale case. It is a simple matter to show that the elasticity of substitution is 1 for the Cobb-Douglas function.11 This fact has led researchers to use the constant returns-to-scale version of the function for a general description of aggregate production relationships in many countries. The Cobb-Douglas function has also proved to be quite useful in many applications because it is linear in logarithms: ln q ¼ ln A þ a ln k þ b ln l.

(9.39)

The constant a is then the elasticity of output with respect to capital input, and b is the elasticity of output with respect to labor input.12 These constants can sometimes be estimated from actual data, and such estimates may be used to measure returns to scale (by examining the sum a þ b) and for other purposes.

Case 4: CES production function A functional form that incorporates all of the three previous cases and allows σ to take on other values as well is the constant elasticity of substitution (CES) production function ﬁrst introduced by Arrow et al. in 1961.13 This function is given by q ¼ f ðk, lÞ ¼ ½k ρ þ l ρ γ=ρ

(9.40)

for ρ 1, ρ 6¼ 0, and γ > 0. This function closely resembles the CES utility function discussed in Chapter 3, though now we have added the exponent γ=ρ to permit explicit introduction of returns-to-scale factors. For γ > 1 the function exhibits increasing returns to scale, whereas for γ < 1 it exhibits diminishing returns.

11

For the Cobb-Douglas function, RTS ¼

fl bAka l b1 b k ¼ ¼ fk aAka1 l b a l

or ln RTS ¼ lnðb=aÞ þ lnðk=lÞ. Hence σ¼ 12

∂ ln k=l ¼ 1. ∂ ln RTS

See Problem 9.5.

K. J. Arrow, H. B. Chenery, B. S. Minhas, and R. M. Solow, “Capital-Labor Substitution and Economic Efﬁciency,” Review of Economics and Statistics (August 1961): 225–50. 13

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Part 3 Production and Supply

Direct application of the deﬁnition of σ to this function14 gives the important result that σ¼

1 . 1ρ

(9.41)

Hence the linear, ﬁxed-proportions, and Cobb-Douglas cases correspond to ρ ¼ 1, ρ ¼ ∞, and ρ ¼ 0, respectively. Proof of this result for the ﬁxed proportions and CobbDouglas cases requires a limit argument. Often the CES function is used with a distributional weight, β ð0 β 1Þ, to indicate the relative signiﬁcance of the inputs: q ¼ f ðk, lÞ ¼ ½βkρ þ ð1 βÞl ρ γ=ρ :

(9.42)

With constant returns to scale and ρ ¼ 0, this function converges to the Cobb-Douglas form q ¼ f ðk, lÞ ¼ k β l 1β .

(9.43)

EXAMPLE 9.3 A Generalized Leontief Production Function Suppose that the production function for a good is given by pﬃﬃﬃﬃﬃﬃﬃﬃ q ¼ f ðk, lÞ ¼ k þ l þ 2 k ⋅ l .

(9.44)

This function is a special case of a class of functions named for the Russian-American economist Wassily Leontief.15 The function clearly exhibits constant returns to scale because pﬃﬃﬃﬃﬃ f ðtk, tlÞ ¼ tk þ tl þ 2t kl ¼ tf ðk, lÞ. (9.45) Marginal productivities for the Leontief function are fk ¼ 1 þ ðk=lÞ0:5 , fl ¼ 1 þ ðk=lÞ0:5 .

(9.46)

Hence, marginal productivities are positive and diminishing. As would be expected (because this function exhibits constant returns to scale), the RTS here depends only on the ratio of the two inputs RTS ¼

fl 1 þ ðk=lÞ0:5 ¼ . fk 1 þ ðk=lÞ0:5

(9.47)

This RTS diminishes as k=l falls, so the isoquants have the usual convex shape.

14

For the CES function we have RTS ¼

fl ðγ=ρÞ ⋅ q ðγρÞ=γ ⋅ ρl ρ1 ¼ ¼ fk ðγ=ρÞ ⋅ q ðγρÞ=γ ⋅ ρk ρ1

ρ1 1ρ l k ¼ . k l

Applying the deﬁnition of the elasticity of substitution then yields σ¼

∂ lnðk=lÞ 1 ¼ . ∂ ln RTS 1 ρ

Notice in this computation that the factor ρ cancels out of the marginal productivity functions, thereby ensuring that these marginal productivities are positive even when ρ is negative (as it is in many cases). This explains why ρ appears in two different places in the deﬁnition of the CES function. 15 Lenotief was a pioneer in the development of input-output analysis. In input-output analysis, production is assumed to take place with a ﬁxed-proportions technology. The Leontief production function generalizes the ﬁxed-proportions case. For more details see the discussion of Leontief production functions in the Extensions to this chapter.

Chapter 9 Production Functions

There are two ways you might calculate the elasticity of substitution for this production function. First, you might notice that in this special case the function can be factored as pﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃ q ¼ k þ l þ 2 kl ¼ ð k þ l Þ2 ¼ ðk 0:5 þ l 0:5 Þ2 , (9.48) which makes clear that this function has a CES form with ρ ¼ 0:5 and γ ¼ 1. Hence the elasticity of substitution here is σ ¼ 1=ð1 ρÞ ¼ 2. Of course, in most cases it is not possible to do such a simple factorization. A more exhaustive approach is to apply the deﬁnition of the elasticity of substitution given in footnote 6 of this chapter: σ¼ ¼

fk fl ½1 þ ðk=lÞ0:5 ½1 þ ðk=lÞ0:5 pﬃﬃﬃﬃﬃ ¼ f ⋅ fkl q ⋅ ð0:5= kl Þ 2 þ ðk=lÞ0:5 þ ðk=lÞ0:5 1 þ 0:5ðk=lÞ0:5 þ 0:5ðk=lÞ0:5

¼ 2:

(9.49)

Notice that in this calculation the input ratio ðk=lÞ drops out, leaving a very simple result. In other applications, one might doubt that such a fortuitous result would occur and hence doubt that the elasticity of substitution is constant along an isoquant (see Problem 9.7). But here the result that σ ¼ 2 is intuitively reasonable, because that value represents a compromise between the elasticity of substitution for this production function’s linear part ðq ¼ k þ l, σ ¼ ∞Þ and its Cobb-Douglas part ðq ¼ 2k0:5 l 0:5 , σ ¼ 1Þ. QUERY: What can you learn about this production function by graphing the q ¼ 4 isoquant? Why does this function generalize the ﬁxed proportions case?

TECHNICAL PROGRESS Methods of production improve over time, and it is important to be able to capture these improvements with the production function concept. A simpliﬁed view of such progress is provided by Figure 9.5. Initially, isoquant q0 records those combinations of capital and labor that can be used to produce an output level of q0 . Following the development of superior production techniques, this isoquant shifts to q 00 . Now the same level of output can be produced with fewer inputs. One way to measure this improvement is by noting that with a level of capital input of, say, k1 , it previously took l2 units of labor to produce q0 , whereas now it takes only l1 . Output per worker has risen from q0 =l2 to q0 =l1 . But one must be careful in this type of calculation. An increase in capital input to k2 would also have permitted a reduction in labor input to l1 along the original q0 isoquant. In this case, output per worker would also rise, although there would have been no true technical progress. Use of the production function concept can help to differentiate between these two concepts and therefore allow economists to obtain an accurate estimate of the rate of technical change.

Measuring technical progress The ﬁrst observation to be made about technical progress is that historically the rate of growth of output over time has exceeded the growth rate that can be attributed to the growth in conventionally deﬁned inputs. Suppose that we let q ¼ Aðt Þf ðk, lÞ

(9.50)

be the production function for some good (or perhaps for society’s output as a whole). The term AðtÞ in the function represents all the inﬂuences that go into determining q other than k (machine-hours) and l (labor-hours). Changes in A over time represent technical progress.

311

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Part 3 Production and Supply

FIGURE 9.5

Technical Progress Technical progress shifts the q0 isoquant toward the origin. The new q0 isoquant, q 00 , shows that a given level of output can now be produced with less input. For example, with k1 units of capital it now only takes l1 units of labor to produce q0 , whereas before the technical advance it took l2 units of labor. k per period

k2

k1 q0

q′0 l1

l2

l per period

For this reason, A is shown as a function of time. Presumably dA=dt > 0; particular levels of input of labor and capital become more productive over time. Differentiating Equation 9.50 with respect to time gives dq dA df ðk, lÞ ¼ ⋅ f ðk, l Þ þ A ⋅ dt dt dt

dA q q ∂f dk ∂f dl þ þ . ¼ ⋅ ⋅ ⋅ dt A f ðk, lÞ ∂k dt ∂l dt

(9.51)

Dividing by q gives dq=dt dA=dt ∂f =∂k dk ∂f =∂l dl ¼ þ þ ⋅ ⋅ q A f ðk, lÞ dt f ðk, lÞ dt

(9.52)

dq=dt dA=dt ∂f k dk=dt ∂f l dl=dt ¼ þ þ . ⋅ ⋅ ⋅ ⋅ q A ∂k f ðk, lÞ k ∂l f ðk, lÞ l

(9.53)

or

Now, for any variable x, (dx=dt )/x is the proportional rate of growth of x per unit of time. We shall denote this by Gx .16 Hence, Equation 9.53 can be written in terms of growth rates as

Two useful features of this deﬁnition are: (1) Gx ⋅ y ¼ Gx þ Gy —that is, the growth rate of a product of two variables is the sum of each one’s growth rate; and (2) Gx=y ¼ Gx Gy .

16

Chapter 9 Production Functions

Gq ¼ GA þ

∂f k ∂f l ⋅ ⋅ Gk þ ⋅ ⋅ Gl , ∂k f ðk, lÞ ∂l f ðk, lÞ

(9.54)

but ∂f k ∂q k ¼ ¼ elasticity of output with respect to capital input ⋅ ⋅ ∂k f ðk, lÞ ∂k q ¼ eq, k and ∂f l ∂q l ¼ ⋅ ⋅ ¼ elasticity of output with respect to labor input ∂l f ðk, lÞ ∂l q ¼ eq;l .

Growth accounting Therefore, our growth equation ﬁnally becomes Gq ¼ GA þ eq, k Gk þ eq, l Gl .

(9.55)

This shows that the rate of growth in output can be broken down into the sum of two components: growth attributed to changes in inputs (k and l) and other “residual” growth (that is, changes in A) that represents technical progress. Equation 9.55 provides a way of estimating the relative importance of technical progress (GA ) in determining the growth of output. For example, in a pioneering study of the entire U.S. economy between the years 1909 and 1949, R. M. Solow recorded the following values for the terms in the equation:17 Gq ¼ 2:75 percent per year, Gl ¼ 1:00 percent per year, Gk ¼ 1:75 percent per year, eq, l ¼ 0:65, eq, k ¼ 0:35. Consequently, GA ¼ Gq eq, l Gl eq, k Gk ¼ 2:75 0:65ð1:00Þ 0:35ð1:75Þ ¼ 2:75 0:65 0:60 ¼ 1:50.

(9.56)

The conclusion Solow reached, then, was that technology advanced at a rate of 1.5 percent per year from 1909 to 1949. More than half of the growth in real output could be attributed to technical change rather than to growth in the physical quantities of the factors of production. More recent evidence has tended to conﬁrm Solow’s conclusions about the relative importance of technical change. Considerable uncertainty remains, however, about the precise causes of such change.

R. M. Solow, “Technical Progress and the Aggregate Production Function,” Review of Economics and Statistics 39 (August 1957): 312–f20.

17

313

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Part 3 Production and Supply

EXAMPLE 9.4 Technical Progress in the Cobb-Douglas Production Function The Cobb-Douglas production function provides an especially easy avenue for illustrating technical progress. Assuming constant returns to scale, such a production function with technical progress might be represented by q ¼ Aðt Þf ðk, lÞ ¼ Aðt Þk α l 1α .

(9.57)

If we also assume that technical progress occurs at a constant exponential (θ), then we can write Aðt Þ ¼ Ae θt and the production function becomes q ¼ Ae θt kα l 1α .

(9.58)

A particularly easy way to study the properties of this type of function over time is to use “logarithmic differentiation”: ∂ ln q ∂ ln q ∂q ∂q=∂t ∂½ln A þ θt þ α ln k þ ð1 αÞ ln l ¼ ¼ ¼ Gq ¼ ⋅ ∂t ∂q ∂t q ∂t ∂ ln k ∂ ln l (9.59) þ ð1 αÞ ⋅ ¼ θ þ αGk þ ð1 − αÞGl . ¼θþα⋅ ∂t ∂t So this derivation just repeats Equation 9.55 for the Cobb-Douglas case. Here the technical change factor is explicitly modeled, and the output elasticities are given by the values of the exponents in the Cobb-Douglas. The importance of technical progress can be illustrated numerically with this function. Suppose A ¼ 10, θ ¼ 0:03, α ¼ 0:5 and that a ﬁrm uses an input mix of k ¼ l ¼ 4. Then, at t ¼ 0, output is 40ð¼ 10 ⋅ 40:5 ⋅ 40:5 Þ. After 20 years ðt ¼ 20Þ, the production function becomes q ¼ 10e 0:03⋅20 k0:5 l 0:5 ¼ 10 ⋅ ð1:82Þk 0:5 l 0:5 ¼ 18:2k0:5 l 0:5 .

(9.60)

In year 20 the original input mix now yields q ¼ 72:8. Of course, one could also have produced q ¼ 72:8 in year 0, but it would have taken a lot more inputs. For example, with k ¼ 13:25 and l ¼ 4, output is indeed 72.8 but much more capital is used. Output per unit of labor input would rise from 10 (q=l ¼ 40=4) to 18:2 ð¼ 72:8=4) in either circumstance, but only the ﬁrst case would have been true technical progress. Input-augmenting technical progress. It is tempting to attribute the increase in the average productivity of labor in this example to, say, improved worker skills, but that would be misleading in the Cobb-Douglas case. One might just as well have said that output per unit of capital rose from 10 to 18.2 over the 20 years and attribute this rise to improved machinery. A plausible approach to modeling improvements in labor and capital separately is to assume that the production function is q ¼ Aðe φt kÞα ðe εt lÞ1α ,

(9.61)

where φ represents the annual rate of improvement in capital input and ε represents the annual rate of improvement in labor input. But, because of the exponential nature of the Cobb-Douglas function, this would be indistinguishable from our original example: q ¼ Ae ½αφþð1αÞεt kα l 1α ¼ Ae θt kα l 1α ,

(9.62)

where θ ¼ αφ þ ð1 αÞε. Hence, to study technical progress in individual inputs, it is necessary either to adopt a more complex way of measuring inputs that allows for improving quality or (what amounts to the same thing) to use a multi-input production function.

Chapter 9 Production Functions

315

QUERY: Actual studies of production using the Cobb-Douglas tend to ﬁnd α 0.3. Use this ﬁnding together with Equation 9.62 to discuss the relative importance of improving capital and labor quality to the overall rate of technical progress.

SUMMARY In this chapter we illustrated the ways in which economists conceptualize the production process of turning inputs into outputs. The fundamental tool is the production function, which—in its simplest form—assumes that output per period (q) is a simple function of capital and labor inputs during that period, q ¼ f ðk, lÞ. Using this starting point, we developed several basic results for the theory of production. •

If all but one of the inputs are held constant, a relationship between the single-variable input and output can be derived. From this relationship, one can derive the marginal physical productivity (MP) of the input as the change in output resulting from a one-unit increase in the use of the input. The marginal physical productivity of an input is assumed to decline as use of the input increases.

•

The entire production function can be illustrated by its isoquant map. The (negative of the) slope of an isoquant is termed the marginal rate of technical substitution (RTS), because it shows how one input can be substituted for another while holding output constant. The RTS is the ratio of the marginal physical productivities of the two inputs.

•

Isoquants are usually assumed to be convex—they obey the assumption of a diminishing RTS. This assumption cannot be derived exclusively from the assumption of diminishing marginal physical productivities. One must also be concerned with the effect of changes in one input on the marginal productivity of other inputs.

•

The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs. If output increases proportionately with input use, there are constant returns to scale. If there are greater than proportionate increases in output, there are increasing returns to scale, whereas if there are less than proportionate increases in output, there are decreasing returns to scale.

•

The elasticity of substitution ðσÞ provides a measure of how easy it is to substitute one input for another in production. A high σ implies nearly linear isoquants, whereas a low σ implies that isoquants are nearly L-shaped.

•

Technical progress shifts the entire production function and its related isoquant map. Technical improvements may arise from the use of improved, more-productive inputs or from better methods of economic organization.

PROBLEMS 9.1 Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 24-inch blade and are used on lawns with many trees and obstacles. The larger mowers are exactly twice as big as the smaller mowers and are used on open lawns where maneuverability is not so difﬁcult. The two production functions available to Power Goat are:

Output per Hour (square feet)

Capital Input (# of 2400 mowers)

Labor Input

Large mowers

8000

2

1

Small mowers

5000

1

1

a. Graph the q ¼ 40,000 square feet isoquant for the ﬁrst production function. How much k and l would be used if these factors were combined without waste?

316

Part 3 Production and Supply b. Answer part (a) for the second function. c. How much k and l would be used without waste if half of the 40,000-square-foot lawn were cut by the method of the ﬁrst production function and half by the method of the second? How much k and l would be used if three fourths of the lawn were cut by the ﬁrst method and one fourth by the second? What does it mean to speak of fractions of k and l? d. On the basis of your observations in part (c), draw a q ¼ 40,000 isoquant for the combined production functions.

9.2 Suppose the production function for widgets is given by q ¼ kl 0:8k 2 0:2l 2 , where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input. a. Suppose k ¼ 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that k ¼ 10, graph the MPl curve. At what level of labor input does MPl ¼ 0? c. Suppose capital inputs were increased to k ¼ 20. How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

9.3 Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by q ¼ 0:1k 0:2 l 0:8 , where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. Upon hearing that Norm’s plan will save money, Cliff argues that Sam should put the savings into more bar stools in order to provide seating to more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff’s plan? d. Carla worries that Cliff’s suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10–bar stool plan?

Chapter 9 Production Functions

9.4 Suppose that the production of crayons ðqÞ is conducted at two locations and uses only labor as an input. 0.5 The production function in location 1 is given by q1 ¼ 10l 0.5 1 and in location 2 by q2 ¼ 50l 2 : a. If a single ﬁrm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations in order to do so? Explain precisely the relationship between l1 and l2 : b. Assuming that the ﬁrm operates in the efﬁcient manner described in part (a), how does total output ðqÞ depend on the total amount of labor hired ðlÞ?

9.5 As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by q ¼ f ðk, lÞ ¼ Akα l β , where 0 < α < 1 and 0 < β < 1: For this production function: a. Show that fk > 0, fl > 0, fkk < 0, fll < 0, and fkl ¼ flk > 0. b. Show that eq, k ¼ α and ee, l ¼ β: c. In footnote 5, we deﬁned the scale elasticity as eq, t ¼

∂f ðtk, tlÞ t , ⋅ ∂t f ðtk, tlÞ

where the expression is to be evaluated at t ¼ 1: Show that, for this Cobb-Douglas function, eq, t ¼ α þ β: Hence, in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9). d. Show that this function is quasi-concave. e. Show that the function is concave for α þ β 1 but not concave for α þ β > 1:

9.6 Suppose we are given the constant returns-to-scale CES production function q ¼ ½k ρ þ l ρ 1=ρ . a. Show that MPk ¼ ðq=kÞ1ρ and MPl ¼ ðq=lÞ1ρ : b. Show that RTS ¼ ðl=kÞ1ρ ; use this to show that σ ¼ 1=ð1 ρÞ: c. Determine the output elasticities for k and l, and show that their sum equals 1. d. Prove that q ¼ l and hence that ln

q l

∂q ∂l

σ

¼ σ ln

∂q . ∂l

Note: The latter equality is useful in empirical work, because we may approximate ∂q=∂l by the competitively determined wage rate. Hence, σ can be estimated from a regression of lnðq=lÞ on ln w:

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Part 3 Production and Supply

9.7 Consider a generalization of the production function in Example 9.3: pﬃﬃﬃﬃﬃ q ¼ β0 þ β1 kl þ β2 k þ β3 l, where 0 βi 1,

i ¼ 0,…,3.

a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters β0 , . . . , β3 ? b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0. c. Calculate σ in this case. Although σ is not in general constant, for what values of the β’s does σ ¼ 0, 1, or ∞?

9.8 Show that Euler’s theorem implies that, for a constant returns-to-scale production function ½q ¼ f ðk, lÞ, q ¼ fk ⋅ k þ fl ⋅ l: Use this result to show that, for such a production function, if MPl > APl then MPk must be negative. What does this imply about where production must take place? Can a ﬁrm ever produce at a point where APl is increasing?

Analytical Problems 9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eq, t ¼ ∂f ðtk, tlÞ=∂t ⋅ t =q evaluated at t ¼ l: a. Show that if the production function exhibits constant returns to scale then eq, t ¼ 1: b. We can deﬁne the output elasticities of the inputs k and l as eq, k eq, l

∂f ðk, lÞ ⋅ ∂k ∂f ðk, lÞ ¼ ⋅ ∂l ¼

k , q l . q

Show that eq, t ¼ eq, k þ eq, l : c. A function that exhibits variable scale elasticity is q ¼ ð1 þ k1 l 1 Þ1 : Show that, for this function, eq, t > 1 for q < 0.5 and that eq, t < 1 for q > 0.5: d. Explain your results from part (c) intuitively. Hint: Does q have an upper bound for this production function?

Chapter 9 Production Functions

319

9.10 Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we showed that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by σ¼

fk fl . f ⋅ fkl

Suppose now that we deﬁne the homothetic production function F as F ðk, lÞ ¼ ½ f ðk, lÞγ , where f ðk, lÞ is a constant returns-to-scale production function and γ is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f : b. Show how this result can be applied to both the Cobb-Douglas and CES production functions.

9.11 More on Euler’s theorem Suppose that a production function f ðx1 , x2 , …, xn Þ is homogeneous of degree k: Euler’s theorem X shows that i xi fi ¼ k f , and this fact can be used to show that the partial derivatives of f are homogeneous of degree k 1: a. Prove that

Xn

i¼1

Xn

j¼1 xi xj fij

¼ kðk 1Þf :

b. In the case of n ¼ 2 and k ¼ 1, what kind of restrictions does the result of part (a) impose on the second-order partial derivative f12 ? How do your conclusions change when k > 1 or k < 1? c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobbn α Douglas production function f ðx1 , x2 , …, xn Þ ¼ ∏i¼1 x i i for αi 0?

SUGGESTIONS FOR FURTHER READING Clark, J. M. “Diminishing Returns.” In Encyclopaedia of the Social Sciences, vol. 5. New York: Crowell-Collier and Macmillan, 1931, pp. 144–f46.

Mas-Collell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995.

Lucid discussion of the historical development of the diminishing returns concept.

Chapter 5 provides a sophisticated, if somewhat spare, review of production theory. The use of the proﬁt function (see Chapter 11) is quite sophisticated and illuminating.

Douglas, P. H. “Are There Laws of Production?” American Economic Review 38 (March 1948): 1–f41.

Shephard, R . W. Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press, 1978.

A nice methodological analysis of the uses and misuses of production functions.

Extended analysis of the dual relationship between production and cost functions.

Ferguson, C. E. The Neoclassical Theory of Production and Distribution. New York: Cambridge University Press, 1969.

Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGrawHill, 2001.

A thorough discussion of production function theory (as of 1970). Good use of three-dimensional graphs.

Fuss, M., and McFadden, D. Production Economics: A Dual Approach to Theory and Application. Amsterdam: NorthHolland, 1980. An approach with a heavy emphasis on the use of duality.

Thorough analysis of the duality between production functions and cost curves. Provides a proof that the elasticity of substitution can be derived as shown in footnote 6 of this chapter.

Stigler, G. J. “The Division of Labor Is Limited by the Extent of the Market.” Journal of Political Economy 59 (June 1951): 185–f93. Careful tracing of the evolution of Smith’s ideas about economies of scale.

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EXTENSIONS Many-Input Production Functions Most of the production functions illustrated in Chapter 9 can be easily generalized to many-input cases. Here we show this for the Cobb-Douglas and CES cases and then examine two quite ﬂexible forms that such production functions might take. In all of these examples, the β’s are nonnegative parameters and the n inputs are represented by x1 , …, xn :

E9.1 Cobb-Douglas The many-input Cobb-Douglas production function is given by n Y β xi i . (i) q¼

function is generally not used in econometric analyses of microeconomic data on ﬁrms. However, the function has a variety of general uses in macroeconomics, as the next example illustrates. The Solow growth model The many-input Cobb-Douglas production function is a primary feature of many models of economic growth. For example, Solow’s (1956) pioneering model of equilibrium growth can be most easily derived using a two-input constant-returns-to-scale Cobb-Douglas function of the form Y ¼ AK α L 1α ,

i¼1

a.

This function exhibits constant returns to scale if n X

βi ¼ 1.

(ii)

where A is a technical change factor that can be represented by exponential growth of the form A ¼ e at .

i¼1

b. In the constant-returns-to-scale Cobb-Douglas function, βi is the elasticity of q with respect to input xi : Because 0 β < 1; each input exhibits diminishing marginal productivity. c. Any degree of increasing returns to scale can be incorporated into this function, depending on ε¼

n X

βi .

(iii)

i¼1

d. The elasticity of substitution between any two inputs in this production function is 1. This can be shown by using the deﬁnition given in footnote 7 of this chapter: σij ¼

∂ lnðxi =xj Þ ∂ lnð fj =fi Þ

.

Here fj fi

β 1

¼

βj x j j

β

∏i6¼j x i i

β β 1 βi x i i ∏j 6¼i x j j

¼

βj βi

⋅

xi . xj

Hence, ! fj βj xi ¼ ln þ ln ln fi βi xj and σij ¼ 1: Because this parameter is so constrained in the Cobb-Douglas function, the

(iv)

(v)

Dividing both sides of Equation iv by L yields y ¼ e at kα ,

(vi)

where y ¼ Y =L and k ¼ K =L. Solow shows that economies will evolve toward an equilibrium value of k (the capital-labor ratio). Hence cross-country differences in growth rates can be accounted for only by differences in the technical change factor, a: Two features of Equation vi argue for including more inputs in the Solow model. First, the equation as it stands is incapable of explaining the large differences in per capita output ðyÞ that are observed around the world. Assuming α ¼ 0:3, say (a ﬁgure consistent with many empirical studies), it would take cross-country differences in K =L of as much as 4,000,000-to-1 to explain the 100-to-1 differences in per capita income observed—a clearly unreasonable magnitude. By introducing additional inputs, such as human capital, these differences become more explainable. A second shortcoming of the simple Cobb-Douglas formulation of the Solow model is that it offers no explanation of the technical change parameter, a—its value is determined “exogenously.” By adding additional factors, it becomes easier to understand how the parameter a may respond to economic incentives.

Chapter 9 Production Functions

This is the key insight of literature on “endogenous” growth theory (for a summary, see Romer, 1996).

E9.2 CES The many-input constant elasticity of substitution (CES) production function is given by hX iε=ρ , ρ 1. (vii) q¼ βi x ρi By substituting mxi for each output, it is easy to show that this function exhibits constant returns to scale for ε ¼ 1: For ε > 1, the function exhibits increasing returns to scale. b. The production function exhibits diminishing marginal productivities for each input because ρ 1: c. As in the two-input case, the elasticity of substitution here is given by

a.

σ¼

1 , 1ρ

(viii)

and this elasticity applies to substitution between any two of the inputs. Checking the Cobb-Douglas in the Soviet Union One way in which the multi-input CES function is used is to determine whether the estimated substitution parameter ðρÞ is consistent with the value implied by the Cobb-Douglas ðρ ¼ 0, σ ¼ 1Þ: For example, in a study of ﬁve major industries in the former Soviet Union, E. Bairam (1991) ﬁnds that the Cobb-Douglas provides a relatively good explanation of changes in output in most major manufacturing sectors. Only for food processing does a lower value for σ seem appropriate. The next three examples illustrate ﬂexible-form production functions that may approximate any general function of n inputs. In the Chapter 10 extensions, we examine the cost function analogues to some of these functions, which are more widely used than the production functions themselves.

E9.3 Nested production functions In some applications, Cobb-Douglas and CES production functions are combined into a “nested” single function. To accomplish this, the original n primary inputs are categorized into, say, m general classes of inputs. The speciﬁc inputs in each of these categories are then aggregated into a single composite input, and

321

the ﬁnal production function is a function of these m composites. For example, assume there are three primary inputs, x1 , x2 , x3 : Suppose, however, that x1 and x2 are relatively closely related in their use by ﬁrms (for example, capital and energy) whereas the third input (labor) is relatively distinct. Then one might want to use a CES aggregator function to construct a composite input for capital services of the form x4 ¼ ½γx ρ1 þ ð1 γÞx ρ2 1=ρ :

(ix)

Then the ﬁnal production function might take a Cobb-Douglas form: q ¼ x α3 x β4 :

(x)

This structure allows the elasticity of substitution between x1 and x2 to take on any value ½σ ¼ 1=ð1 ρÞ but constrains the elasticity of substitution between x3 and x4 to be one. A variety of other options are available depending on how precisely the embedded functions are speciﬁed. The dynamics of capital/energy substitutability Nested production functions have been widely used in studies that seek to measure the precise nature of the substitutability between capital and energy inputs. For example, Atkeson and Kehoe (1999) use a model rather close to the one speciﬁed in Equations ix and x to try to reconcile two facts about the way in which energy prices affect the economy: (1) Over time, use of energy in production seems rather unresponsive to price (at least in the short-run); and (2) across countries, energy prices seem to have a large inﬂuence over how much energy is used. By using a capital service equation of the form given in Equation ix with a low degree of substitutability ðρ ¼ 2:3Þ—along with a Cobb-Douglas production function that combines labor with capital services—they are able to replicate the facts about energy prices fairly well. They conclude, however, that this model implies a much more negative effect of higher energy prices on economic growth than seems actually to have been the case. Hence they ultimately opt for a more complex way of modeling production that stresses differences in energy use among capital investments made at different dates.

E9.4 Generalized Leontief q¼

n X n X i¼1 j ¼1

where βij ¼ βji .

βij

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ xi xj ,

322

Part 3 Production and Supply

a.

The function considered in Problem 9.7 is a simple case of this function for the case n ¼ 2: For n ¼ 3, the function would have linear terms in the three inputs along with three radical terms representing all possible cross-products of the inputs. b. The function exhibits constant returns to scale, as can be shown by using mxi . Increasing returns to scale can be incorporated into the function by using the transformation q0 ¼ qε ,

ε > 1.

c.

Because each input appears both linearly and under the radical, the function exhibits diminishing marginal productivities to all inputs. d. The restriction βij ¼ βji is used to ensure symmetry of the second-order partial derivatives.

E9.5 Translog ln q ¼ β0 þ

n X

βi ln xi þ 0:5

i¼1

n X n X i¼1 j ¼1

βij ln xi ln xj ; βij ¼ βji:

a.

Note that the Cobb-Douglas function is a special case of this function where β0 ¼ βij ¼ 0 for all i, j : b. As for the Cobb-Douglas, this function may assume any degree of returns to scale. If n X i¼1

βi ¼ 1

and

n X j ¼1

βij ¼ 0

for all i, then this function exhibits constant returns to scale. The proof requires some care in dealing with the double summation.

c.

Again, the condition βij ¼ βji is required to ensure equality of the cross-partial derivatives.

Immigration Because the translog production function incorporates a large number of substitution possibilities among various inputs, it has been widely used to study the ways in which newly arrived workers may substitute for existing workers. Of particular interest is the way in which the skill level of immigrants may lead to differing reactions in the demand for skilled and unskilled workers in the domestic economy. Studies of the United States and many other countries (Canada, Germany, France, and so forth) have suggested that the overall size of such effects is modest, especially given relatively small immigration ﬂows. But there is some evidence that unskilled immigrant workers may act as substitutes for unskilled domestic workers but as complements to skilled domestic workers. Hence increased immigration ﬂows may exacerbate trends toward rising wage differentials. For a summary, see Borjas (1994).

References Atkeson, Andrew, and Patrick J. Kehoe. “Models of Energy Use: Putty-Putty versus Putty-Clay.” American Economic Review (September 1999): 1028–43. Bairam, Erkin. “Elasticity of Substitution, Technical Progress and Returns to Scale in Branches of Soviet Industry: A New CES Production Function Approach.” Journal of Applied Economics (January–March 1991): 91–f96. Borjas, G. J. “The Economics of Immigration.” Journal of Economic Literature (December 1994): 1667–f1717. Romer, David. Advanced Macroeconomics. New York: McGraw-Hill, 1996. Solow, R. M. “A Contribution to the Theory of Economic Growth.” Quarterly Journal of Economics (February 1956): 65–f94.

CHAPTER

10 Cost Functions In this chapter we illustrate the costs that a ﬁrm incurs when it produces output. In Chapter 11, we will pursue this topic further by showing how ﬁrms make proﬁt-maximizing input and output decisions.

DEFINITIONS OF COSTS Before we can discuss the theory of costs, some difﬁculties about the proper deﬁnition of “costs” must be cleared up. Speciﬁcally, we must differentiate between (1) accounting cost and (2) economic cost. The accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries. The economist’s deﬁnition of cost (which in obvious ways draws on the fundamental opportunity-cost notion) is that the cost of any input is given by the size of the payment necessary to keep the resource in its present employment. Alternatively, the economic cost of using an input is what that input would be paid in its next best use. One way to distinguish between these two views is to consider how the costs of various inputs (labor, capital, and entrepreneurial services) are deﬁned under each system.

Labor costs Economists and accountants regard labor costs in much the same way. To accountants, expenditures on labor are current expenses and hence costs of production. For economists, labor is an explicit cost. Labor services (labor-hours) are contracted at some hourly wage rate ðwÞ, and it is usually assumed that this is also what the labor services would earn in their best alternative employment. The hourly wage, of course, includes costs of fringe beneﬁts provided to employees.

Capital costs In the case of capital services (machine-hours), the two concepts of cost differ. In calculating capital costs, accountants use the historical price of the particular machine under investigation and apply some more-or-less arbitrary depreciation rule to determine how much of that machine’s original price to charge to current costs. Economists regard the historical price of a machine as a “sunk cost,” which is irrelevant to output decisions. They instead regard the implicit cost of the machine to be what someone else would be willing to pay for its use. Thus the cost of one machine-hour is the rental rate for that machine in its best alternative use. By continuing to use the machine itself, the ﬁrm is implicitly forgoing what someone else would be willing to pay to use it. This rental rate for one machine-hour will be denoted by v.1 1 Sometimes the symbol r is chosen to represent the rental rate on capital. Because this variable is often confused with the related but distinct concept of the market interest rate, an alternative symbol was chosen here. The exact relationship between v and the interest rate is examined in Chapter 17.

323

324

Part 3 Production and Supply

Costs of entrepreneurial services The owner of a ﬁrm is a residual claimant who is entitled to whatever extra revenues or losses are left after paying other input costs. To an accountant, these would be called proﬁts (which might be either positive or negative). Economists, however, ask whether owners (or entrepreneurs) also encounter opportunity costs by working at a particular ﬁrm or devoting some of their funds to its operation. If so, these services should be considered an input, and some cost should be imputed to them. For example, suppose a highly skilled computer programmer starts a software ﬁrm with the idea of keeping any (accounting) proﬁts that might be generated. The programmer’s time is clearly an input to the ﬁrm, and a cost should be inputted for it. Perhaps the wage that the programmer might command if he or she worked for someone else could be used for that purpose. Hence some part of the accounting proﬁts generated by the ﬁrm would be categorized as entrepreneurial costs by economists. Economic proﬁts would be smaller than accounting proﬁts and might be negative if the programmer’s opportunity costs exceeded the accounting proﬁts being earned by the business. Similar arguments apply to the capital that an entrepreneur provides to the ﬁrm.

Economic costs In this book, not surprisingly, we use economists’ deﬁnition of cost. DEFINITION

Economic cost. The economic cost of any input is the payment required to keep that input in its present employment. Equivalently, the economic cost of an input is the remuneration the input would receive in its best alternative employment. Use of this deﬁnition is not meant to imply that accountants’ concepts are irrelevant to economic behavior. Indeed, accounting procedures are integrally important to any manager’s decision-making process because they can greatly affect the rate of taxation to be applied against proﬁts. Accounting data are also readily available, whereas data on economic costs must often be developed separately. Economists’ deﬁnitions, however, do have the desirable features of being broadly applicable to all ﬁrms and of forming a conceptually consistent system. They therefore are best suited for a general theoretical analysis.

Two simplifying assumptions As a start, we will make two simpliﬁcations about the inputs a ﬁrm uses. First, we assume that there are only two inputs: homogeneous labor (l, measured in labor-hours) and homogeneous capital (k, measured in machine-hours). Entrepreneurial costs are included in capital costs. That is, we assume that the primary opportunity costs faced by a ﬁrm’s owner are those associated with the capital that the owner provides. Second, we assume that inputs are hired in perfectly competitive markets. Firms can buy (or sell) all the labor or capital services they want at the prevailing rental rates (w and v). In graphic terms, the supply curve for these resources is horizontal at the prevailing factor prices. Both w and v are treated as “parameters” in the ﬁrm’s decisions; there is nothing the ﬁrm can do to affect them. These conditions will be relaxed in later chapters (notably Chapter 16), but for the moment the price-taker assumption is a convenient and useful one to make.

Economic proﬁts and cost minimization Total costs for the ﬁrm during a period are therefore given by total costs ¼ C ¼ wl þ vk, (10.1) where, as before, l and k represent input usage during the period. Assuming the ﬁrm produces only one output, its total revenues are given by the price of its product ðpÞ times its

Chapter 10

Cost Functions

325

total output [q ¼ f ðk, lÞ, where f ðk, lÞ is the ﬁrm’s production function]. Economic proﬁts ðπÞ are then the difference between total revenues and total economic costs. Economic profits. Economic proﬁts ðπÞ are the difference between a ﬁrm’s total revenues and DEFINITION its total costs: π ¼ total revenue total cost ¼ pq wl vk ¼ pf ðk, lÞ wl vk.

(10.2)

Equation 10.2 shows that the economic proﬁts obtained by a ﬁrm are a function of the amount of capital and labor employed. If, as we will assume in many places in this book, the ﬁrm seeks maximum proﬁts, then we might study its behavior by examining how k and l are chosen so as to maximize Equation 10.2. This would, in turn, lead to a theory of supply and to a theory of the “derived demand” for capital and labor inputs. In the next chapter we will take up those subjects in detail. Here, however, we wish to develop a theory of costs that is somewhat more general and might apply to ﬁrms that are not necessarily proﬁt maximizers. Hence, we begin the study of costs by ﬁnessing, for the moment, a discussion of output choice. That is, we assume that for some reason the ﬁrm has decided to produce a particular output level (say, q0 ). The ﬁrm’s revenues are therefore ﬁxed at pq0 . Now we wish to examine how the ﬁrm can produce q0 at minimal costs.

COST-MINIMIZING INPUT CHOICES Mathematically, this is a constrained minimization problem. But before proceeding with a rigorous solution, it is useful to state the result to be derived with an intuitive argument. To minimize the cost of producing a given level of output, a ﬁrm should choose that point on the q0 isoquant at which the rate of technical substitution of l for k is equal to the ratio w=v: It should equate the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace. Suppose that this were not true. In particular, suppose that the ﬁrm were producing output level q0 using k ¼ 10, l ¼ 10, and assume that the RTS were 2 at this point. Assume also that w ¼ $1, v ¼ $1, and hence that w=v ¼ 1 (which is unequal to 2). At this input combination, the cost of producing q0 is $20. It is easy to show this is not the minimal input cost. For example, q0 can also be produced using k ¼ 8 and l ¼ 11; we can give up two units of k and keep output constant at q0 by adding one unit of l. But at this input combination, the cost of producing q0 is $19 and hence the initial input combination was not optimal. A contradiction similar to this one can be demonstrated whenever the RTS and the ratio of the input costs differ.

Mathematical analysis Mathematically, we seek to minimize total costs given q ¼ f ðk, lÞ ¼ q0 . Setting up the Lagrangian expression (10.3) ℒ ¼ wl þ vk þ λ½q0 f ðk, lÞ, the ﬁrst-order conditions for a constrained minimum are ∂ℒ ∂f ¼wλ ¼ 0, ∂l ∂l ∂ℒ ∂f (10.4) ¼vλ ¼ 0, ∂k ∂k ∂ℒ ¼ q0 f ðk, lÞ ¼ 0, ∂λ

326

Part 3 Production and Supply

or, dividing the ﬁrst two equations, w ∂f =∂l ¼ ¼ RTS ðl for kÞ. (10.5) v ∂f =∂k This says that the cost-minimizing ﬁrm should equate the RTS for the two inputs to the ratio of their prices.

Further interpretations These ﬁrst-order conditions for minimal costs can be manipulated in several different ways to yield interesting results. For example, cross-multiplying Equation 10.5 gives fk f ¼ l. (10.6) v w That is: for costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs. If increasing one input promised to increase output by a greater amount per dollar spent than did another input, costs would not be minimal—the ﬁrm should hire more of the input that promises a bigger “bang per buck” and less of the more costly (in terms of productivity) input. Any input that cannot meet the common beneﬁt-cost ratio deﬁned in Equation 10.6 should not be hired at all. Equation 10.6 can, of course, also be derived from Equation 10.4, but it is more instructive to derive its inverse: w v ¼ ¼ λ. (10.7) fl fk This equation reports the extra cost of obtaining an extra unit of output by hiring either added labor or added capital input. Because of cost minimization, this marginal cost is the same no matter which input is hired. This common marginal cost is also measured by the Lagrangian multiplier from the cost-minimization problem. As is the case for all constrained optimization problems, here the Lagrangian multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly. Because marginal cost plays an important role in a ﬁrm’s supply decisions, we will return to this feature of cost minimization frequently.

Graphical analysis Cost minimization is shown graphically in Figure 10.1. Given the output isoquant q0 , we wish to ﬁnd the least costly point on the isoquant. Lines showing equal cost are parallel straight lines with slopes w=v. Three lines of equal total cost are shown in Figure 10.1; C1 < C2 < C3 . It is clear from the ﬁgure that the minimum total cost for producing q0 is given by C1 , where the total cost curve is just tangent to the isoquant. The cost-minimizing input combination is l , k . This combination will be a true minimum if the isoquant is convex (if the RTS diminishes for decreases in k=l). The mathematical and graphic analyses arrive at the same conclusion, as follows. OPTIMIZATION PRINCIPLE

Cost minimization. In order to minimize the cost of any given level of input (q0 ), the ﬁrm should produce at that point on the q0 isoquant for which the RTS (of l for k) is equal to the ratio of the inputs’ rental prices ðw=vÞ.

Contingent demand for inputs Figure 10.1 exhibits the formal similarity between the ﬁrm’s cost-minimization problem and the individual’s expenditure-minimization problem studied in Chapter 4 (see Figure 4.6). In both problems, the economic actor seeks to achieve his or her target (output or utility) at minimal cost. In Chapter 5 we showed how this process is used to construct a theory of compensated demand for a good. In the present case, cost minimization leads to a demand for capital and labor input that is contingent on the level of output being produced. This is

Chapter 10

FIGURE 10.1

Cost Functions

Minimization of Costs Given q ¼ q0

A ﬁrm is assumed to choose k and l to minimize total costs. The condition for this minimization is that the rate at which k and l can be traded technically (while keeping q ¼ q0 ) should be equal to the rate at which these inputs can be traded in the market. In other words, the RTS (of l for k) should be set equal to the price ratio w=v. This tangency is shown in the ﬁgure; costs are minimized at C1 by choosing inputs k and l .

k per period

C1

C2

C3 k* q0 l*

l per period

not, therefore, the complete story of a ﬁrm’s demand for the inputs it uses because it does not address the issue of output choice. But studying the contingent demand for inputs provides an important building block for analyzing the ﬁrm’s overall demand for inputs, and we will take up this topic in more detail later in this chapter.

The ﬁrm’s expansion path A ﬁrm can follow the cost-minimization process for each level of output: For each q, it ﬁnds the input choice that minimizes the cost of producing it. If input costs (w and v) remain constant for all amounts the ﬁrm may demand, we can easily trace this locus of cost-minimizing choices. This procedure is shown in Figure 10.2. The line 0E records the cost-minimizing tangencies for successively higher levels of output. For example, the minimum cost for producing output level q1 is given by C1 , and inputs k1 and l1 are used. Other tangencies in the ﬁgure can be interpreted in a similar way. The locus of these tangencies is called the ﬁrm’s expansion path, because it records how input expands as output expands while holding the prices of the inputs constant. As Figure 10.2 shows, the expansion path need not be a straight line. The use of some inputs may increase faster than others as output expands. Which inputs expand more rapidly will depend on the shape of the production isoquants. Because cost minimization requires that the RTS always be set equal to the ratio w=v, and because the w=v ratio is assumed to be constant, the shape of the expansion path will be determined by where a particular RTS occurs on successively higher isoquants. If the production function exhibits constant returns to scale (or, more generally, if it is homothetic), then the expansion path will be a straight line because in that case the RTS depends only on the ratio of k to l. That ratio would be constant along such a linear expansion path.

327

FIGURE 10.2

The Firm’s Expansion Path The ﬁrm’s expansion path is the locus of cost-minimizing tangencies. Assuming ﬁxed input prices, the curve shows how inputs increase as output increases. k per period

E

q3 k1

q2 C1

0

FIGURE 10.3

l1

C2

C3

q1 l per period

Input Inferiority With this particular set of isoquants, labor is an inferior input, because less l is chosen as output expands beyond q2 . k per period

E

q4

q3

q2 q1 0

l per period

Chapter 10

Cost Functions

It would seem reasonable to assume that the expansion path will be positively sloped; that is, successively higher output levels will require more of both inputs. This need not be the case, however, as Figure 10.3 illustrates. Increases of output beyond q2 actually cause the quantity of labor used to decrease. In this range, labor would be said to be an inferior input. The occurrence of inferior inputs is then a theoretical possibility that may happen, even when isoquants have their usual convex shape. Much theoretical discussion has centered on the analysis of factor inferiority. Whether inferiority is likely to occur in real-world production functions is a difﬁcult empirical question to answer. It seems unlikely that such comprehensive magnitudes as “capital” and “labor” could be inferior, but a ﬁner classiﬁcation of inputs may bring inferiority to light. For example, the use of shovels may decline as production of building foundations (and the use of backhoes) increases. In this book we shall not be particularly concerned with the analytical issues raised by this possibility, although complications raised by inferior inputs will be mentioned in a few places. EXAMPLE 10.1 Cost Minimization The cost-minimization process can be readily illustrated with two of the production functions we encountered in the last chapter. 1. Cobb-Douglas: q ¼ f ðk, lÞ ¼ kα l β . For this case the relevant Lagrangian expression for minimizing the cost of producing, say, q0 is ℒ ¼ vk þ wl þ λðq0 kα l β Þ, and the ﬁrst-order conditions for a minimum are ∂ℒ ¼ v λαkα1 l β ¼ 0, ∂k ∂ℒ ¼ w λβkα l β1 ¼ 0, ∂l ∂ℒ ¼ q0 kα l β ¼ 0. ∂λ Dividing the second of these by the ﬁrst yields

(10.8)

(10.9)

w βkα l β1 β k ¼ ⋅ , ¼ (10.10) α1 β αk l v α l which again shows that costs are minimized when the ratio of the inputs’ prices is equal to the RTS. Because the Cobb-Douglas function is homothetic, the RTS depends only on the ratio of the two inputs. If the ratio of input costs does not change, the ﬁrms will use the same input ratio no matter how much it produces—that is, the expansion path will be a straight line through the origin. As a numerical example, suppose α ¼ β ¼ 0.5, w ¼ 12, v ¼ 3, and that the ﬁrm wishes to produce q0 ¼ 40. The ﬁrst-order condition for a minimum requires that k ¼ 4l. Inserting that into the production function (the ﬁnal requirement in Equation 10.9), we have q0 ¼ 40 ¼ k0.5 l 0.5 ¼ 2l. So the cost-minimizing input combination is l ¼ 20 and k ¼ 80, and total costs are given by vk þ wl ¼ 3ð80Þ þ 12ð20Þ ¼ 480. That this is a true cost minimum is suggested by looking at a few other input combinations that also are capable of producing 40 units of output: k ¼ 40, l ¼ 40, C ¼ 600, k ¼ 10, l ¼ 160, C ¼ 2,220, (10.11) k ¼ 160, l ¼ 10, C ¼ 600. Any other input combination able to produce 40 units of output will also cost more than 480. Cost minimization is also suggested by considering marginal productivities. At the optimal point (continued)

329

330

Part 3 Production and Supply

EXAMPLE 10.1 CONTINUED MPk ¼ fk ¼ 0.5k 0.5 l 0.5 ¼ 0.5ð20=80Þ0.5 ¼ 0.25,

(10.12) MPl ¼ fl ¼ 0.5k0.5 l 0.5 ¼ 0.5ð80=20Þ0.5 ¼ 1.0; hence, at the margin, labor is four times as productive as capital, and this extra productivity precisely compensates for the higher unit price of labor input. 2. CES: q ¼ f ðk, lÞ ¼ ðk ρ þ l ρ Þγ=ρ . Again we set up the Lagrangian expression ℒ ¼ vk þ wl þ λ½q0 ðk ρ þ l ρ Þγ=ρ , (10.13) and the ﬁrst-order conditions for a minimum are ∂ℒ ¼ v λðγ=ρÞðk ρ þ l ρ ÞðγρÞ=ρ ðρÞk ρ1 ¼ 0, ∂k ∂ℒ (10.14) ¼ w λðγ=ρÞðk ρ þ l ρ ÞðγρÞ=ρ ðρÞl ρ1 ¼ 0, ∂l ∂ℒ ¼ q0 ðk ρ þ l ρ Þðγ=ρÞ ¼ 0. ∂λ Dividing the ﬁrst two of these equations causes a lot of this mass of symbols to drop out, leaving ρ1 1ρ 1=σ w l k k k w σ ¼ ¼ , or : (10.15) ¼ ¼ v k l l l v Because the CES function is also homothetic, the cost-minimizing input ratio is independent of the absolute level of production. The result in Equation 10.15 is a simple generalization of the Cobb-Douglas result (when σ ¼ 1). With the Cobb-Douglas, the cost-minimizing capitallabor ratio changes directly in proportion to changes in the ratio of wages to capital rental rates. In cases with greater substitutability ðσ > 1Þ, changes in the ratio of wages to rental rates cause a greater than proportional increase in the cost-minimizing capital-labor ratio. With less substitutability ðσ < 1Þ, the response is proportionally smaller. QUERY: In the Cobb-Douglas numerical example with w=v ¼ 4, we found that the costminimizing input ratio for producing 40 units of output was k=l ¼ 80=20 ¼ 4. How would this value change for σ ¼ 2 or σ ¼ 0.5? What actual input combinations would be used? What would total costs be?

COST FUNCTIONS We are now in a position to examine the ﬁrm’s overall cost structure. To do so, it will be convenient to use the expansion path solutions to derive the total cost function. DEFINITION

Total cost function. The total cost function shows that, for any set of input costs and for any output level, the minimum total cost incurred by the ﬁrm is C ¼ C ðv, w, qÞ.

(10.16)

Figure 10.2 makes clear that total costs increase as output, q, increases. We will begin by analyzing this relationship between total cost and output while holding input prices ﬁxed. Then we will consider how a change in an input price shifts the expansion path and its related cost functions.

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Cost Functions

331

Average and marginal cost functions Although the total cost function provides complete information about the output-cost relationship, it is often convenient to analyze costs on a per-unit-of-output basis because that approach corresponds more closely to the analysis of demand, which focused on the price per unit of a commodity. Two different unit cost measures are widely used in economics: (1) average cost, which is the cost per unit of output; and (2) marginal cost, which is the cost of one more unit of output. The relationship of these concepts to the total cost function is described in the following deﬁnitions. Average and marginal cost functions. The average cost function (AC) is found by comDEFINITION puting total costs per unit of output: C ðv, w, qÞ . (10.17) average cost ¼ AC ðv, w, q Þ ¼ q The marginal cost function (MC) is found by computing the change in total costs for a change in output produced: marginal cost ¼ MC ðv, w, qÞ ¼

∂C ðv, w, qÞ : ∂q

(10.18)

Notice that in these deﬁnitions, average and marginal costs depend both on the level of output being produced and on the prices of inputs. In many places throughout this book, we will graph simple two-dimensional relationships between costs and output. As the deﬁnitions make clear, all such graphs are drawn on the assumption that the prices of inputs remain constant and that technology does not change. If input prices change or if technology advances, cost curves generally will shift to new positions. Later in this chapter, we will explore the likely direction and size of such shifts when we study the entire cost function in detail.

Graphical analysis of total costs Figures 10.4a and 10.5a illustrate two possible shapes for the relationship between total cost and the level of the ﬁrm’s output. In Figure 10.4a, total cost is simply proportional to output. Such a situation would arise if the underlying production function exhibits constant returns to scale. In that case, suppose k1 units of capital input and l1 units of labor input are required to produce one unit of output. Then (10.19) C ðq ¼ 1Þ ¼ vk1 þ wl1 . To produce m units of output, then, requires mk1 units of capital and ml1 units of labor, because of the constant returns-to-scale assumption.2 Hence C ðq ¼ mÞ ¼ vmk1 þ wml1 ¼ mðvk1 þ wl1 Þ ¼ m ⋅ C ðq ¼ 1Þ, (10.20) and the proportionality between output and cost is established. The situation in Figure 10.5a is more complicated. There it is assumed that initially the total cost curve is concave; although initially costs rise rapidly for increases in output, that rate of increase slows as output expands into the midrange of output. Beyond this middle range, however, the total cost curve becomes convex, and costs begin to rise progressively more rapidly. One possible reason for such a shape for the total cost curve is that there is some third factor of production (say, the services of an entrepreneur) that is ﬁxed as capital and labor usage expands. In this case, the initial concave section of the curve might be explained by the

2

The input combination ml1 , mk1 minimizes the cost of producing m units of output because the ratio of the inputs is still k1 =l1 and the RTS for a constant returns-to-scale production function depends only on that ratio.

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FIGURE 10.4

Total, Average, and Marginal Cost Curves for the Constant Returns-to-Scale Case In (a) total costs are proportional to output level. Average and marginal costs, as shown in (b), are equal and constant for all output levels. Total costs

C

(a)

Output per period

Average and marginal costs

AC = MC

Output per period (b)

increasingly optimal usage of the entrepreneur’s services—he or she needs a moderate level of production to utilize his or her skills fully. Beyond the point of inﬂection, however, the entrepreneur becomes overworked in attempting to coordinate production, and diminishing returns set in. Hence, total costs rise rapidly. A variety of other explanations have been offered for the cubic-type total cost curve in Figure 10.5a, but we will not examine them here. Ultimately, the shape of the total cost curve is an empirical question that can be determined only by examining real-world data. In the Extensions to this chapter, we illustrate some of the literature on cost functions.

Graphical analysis of average and marginal costs Information from the total cost curves can be used to construct the average and marginal cost curves shown in Figures 10.4b and 10.5b. For the constant returns-to-scale case (Figure 10.4), this is quite simple. Because total costs are proportional to output, average and marginal costs

Chapter 10

FIGURE 10.5

Cost Functions

Total, Average, and Marginal Cost Curves for the Cubic Total Cost Curve Case

If the total cost curve has the cubic shape shown in (a), average and marginal cost curves will be U-shaped. In (b) the marginal cost curve passes through the low point of the average cost curve at output level q .

Total costs C

Output per period

(a) Average and marginal costs

MC

AC

q*

Output per period

(b)

are constant and equal for all levels of output.3 These costs are shown by the horizontal line AC ¼ MC in Figure 10.4b. For the cubic total cost curve case (Figure 10.5), computation of the average and marginal cost curves requires some geometric intuition. As the deﬁnition in Equation 10.18 makes clear, marginal cost is simply the slope of the total cost curve. Hence, because of the assumed shape of the curve, the MC curve is U-shaped, with MC falling over the concave portion of the total cost curve and rising beyond the point of inﬂection. Because the slope is always positive, however, MC is always greater than 0. Average costs (AC) start out being equal to

3

Mathematically, because C ¼ aq (where a is the cost of one unit of output), AC ¼

C ∂C ¼a¼ ¼ MC . q ∂q

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marginal cost for the “ﬁrst” unit of output.4 As output expands, however, AC exceeds MC, because AC reﬂects both the marginal cost of the last unit produced and the higher marginal costs of the previously produced units. So long as AC > MC, average costs must be falling. Because the lower costs of the newly produced units are below average cost, they continue to pull average costs downward. Marginal costs rise, however, and eventually (at q ) equal average cost. Beyond this point, MC > AC, and average costs will be rising because they are being pulled upward by increasingly higher marginal costs. Consequently, we have shown that the AC curve also has a U-shape and that it reaches a low point at q , where AC and MC intersect.5 In empirical studies of cost functions, there is considerable interest in this point of minimum average cost. It reﬂects the “minimum efﬁcient scale” (MES) for the particular production process being examined. The point is also theoretically important because of the role it plays in perfectly competitive price determination in the long run (see Chapter 12).

COST FUNCTIONS AND SHIFTS IN COST CURVES The cost curves illustrated in Figures 10.4 and 10.5 show the relationship between costs and quantity produced on the assumption that all other factors are held constant. Speciﬁcally, construction of the curves assumes that input prices and the level of technology do not change.6 If these factors do change, the cost curves will shift. In this section, we delve further into the mathematics of cost functions as a way of studying these shifts. We begin with some examples. EXAMPLE 10.2 Some Illustrative Cost Functions In this example we calculate the cost functions associated with three different production functions. Later we will use these examples to illustrate some of the general properties of cost functions. 1. Fixed Proportions: q ¼ f ðk, lÞ ¼ minðak, blÞ. The calculation of cost functions from their underlying production functions is one of the more frustrating tasks for economics students. 4

Technically, AC ¼ MC at q ¼ 0. This can be shown by L’Hôpital’s rule, which states that if f ðaÞ ¼ gðaÞ ¼ 0 then lim

x !a

f ðxÞ f 0 ðxÞ . ¼ lim gðxÞ x !a g 0 ðxÞ

In this case, C ¼ 0 at q ¼ 0, and so lim AC ¼ lim q !0

q !0

C ∂C =∂q ¼ lim ¼ lim MC q !0 q !0 q 1

or AC ¼ MC at q ¼ 0, which was to be shown. 5

Mathematically, we can ﬁnd the minimum AC by setting its derivative equal to 0: ∂AC ∂ðC =qÞ q ⋅ ð∂C =∂qÞ C ⋅ 1 q ⋅ MC C ¼ ¼ 0, ¼ ¼ ∂q ∂q q2 q2

or q ⋅ MC C ¼ 0

or

MC ¼ C =q ¼ AC .

For multiproduct ﬁrms, an additional complication must be considered. For such ﬁrms it is possible that the costs associated with producing one output (say, q1 ) are also affected by the amount of some other output being produced ðq2 Þ. In this case the ﬁrm is said to exhibit “economies of scope,” and the total cost function will be of the form Cðq1 , q2 , w, vÞ. Hence, q2 must also be held constant in constructing the q1 cost curves. Presumably increases in q2 shift the q1 cost curves downward. 6

Chapter 10

Cost Functions

So, let’s start with a simple example. What we wish to do is show how total costs depend on input costs and on quantity produced. In the ﬁxed-proportions case, we know that production will occur at a vertex of the L-shaped isoquants where q ¼ ak ¼ bl. Hence, total costs are q q v w þ total costs ¼ C ðv, w, q Þ ¼ vk þ wl ¼ v þw ¼q . (10.21) a b a b This is indeed the sort of function we want because it states total costs as a function of v, w, and q only together with some parameters of the underlying production function. Because of the constant returns-to-scale nature of this production function, it takes the special form C ðv, w, qÞ ¼ qC ðv, w, 1Þ. (10.22) That is, total costs are given by output times the cost of producing one unit. Increases in input prices clearly increase total costs with this function, and technical improvements that take the form of increasing the parameters a and b reduce costs. 2. Cobb-Douglas: q ¼ f ðk, lÞ ¼ kα l β . This is our ﬁrst example of burdensome computation, but we can clarify the process by recognizing that the ﬁnal goal is to use the results of cost minimization to replace the inputs in the production function with costs. From Example 10.1 we know that cost minimization requires that w β k α w ¼ ⋅ and so k ¼ ⋅ ⋅ l. v α l β v Substitution into the production function permits a solution for labor input in terms of q, v, and w as α=ðαþβÞ α w α αþβ β l or l ¼ q 1=ðαþβÞ w α=ðαþβÞ v α=ðαþβÞ . (10.23) q ¼ kα l β ¼ ⋅ β v α A similar set of manipulations gives β=ðαþβÞ 1=ðαþβÞ α k¼q w β=ðαþβÞ vβ=ðαþβÞ . (10.24) β Now we are ready to derive total costs as C ðv, w, qÞ ¼ vk þ wl ¼ q 1=ðαþβÞ Bv α=ðαþβÞ w β=ðαþβÞ ,

(10.25)

where B ¼ ðα þ βÞαα=ðαþβÞ ββ=ðαþβÞ —a constant that involves only the parameters α and β. Although this derivation was a bit messy, several interesting aspects of this Cobb-Douglas cost function are readily apparent. First, whether the function is a convex, linear, or concave function of output depends on whether there are decreasing returns to scale ðα þ β < 1Þ, constant returns to scale ðα þ β ¼ 1Þ, or increasing returns to scale ðα þ β > 1Þ. Second, an increase in any input price increases costs, with the extent of the increase being determined by the relative importance of the input as reﬂected by the size of its exponent in the production function. Finally, the cost function is homogeneous of degree 1 in the input prices—a general feature of all cost functions, as we shall show shortly. 3. CES: q ¼ f ðk, lÞ ¼ ðkρ þ l ρ Þγ=ρ . For this case, your author will mercifully spare you the algebra. To derive the total cost function, we use the cost-minimization condition speciﬁed in Equation 10.15, solve for each input individually, and eventually get C ðv, w, qÞ ¼ vk þ wl ¼ q 1=γ ðvρ=ðρ1Þ þ w ρ=ðρ1Þ Þðρ1Þ=ρ ¼ q 1=γ ðv 1σ þ w 1σ Þ1=ð1σÞ , (10.26) where the elasticity of substitution is given by σ ¼ 1=ð1 ρÞ. Once again the shape of the total cost is determined by the scale parameter ðγÞ for this production function, and the cost function is increasing in both of the input prices. The function is also homogeneous of degree 1 in those prices. One limiting feature of this form of the CES function is that the (continued)

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EXAMPLE 10.2 CONTINUED inputs are given equal weights—hence their prices are equally important in the cost function. This feature of the CES is easily generalized, however (see Problem 10.7). QUERY: How are the various substitution possibilities inherent in the CES function reﬂected in the CES cost function in Equation 10.26?

Properties of cost functions These examples illustrate some properties of total cost functions that are quite general. 1. Homogeneity. The total cost functions in Example 10.3 are all homogeneous of degree 1 in the input prices. That is, a doubling of input prices will precisely double the cost of producing any given output level (you might check this out for yourself). This is a property of all proper cost functions. When all input prices double (or are increased by any uniform proportion), the ratio of any two input prices will not change. Because cost minimization requires that the ratio of input prices be set equal to the RTS along a given isoquant, the cost-minimizing input combination also will not change. Hence, the ﬁrm will buy exactly the same set of inputs and pay precisely twice as much for them. One implication of this result is that a pure, uniform inﬂation in all input costs will not change a ﬁrm’s input decisions. Its cost curves will shift upward in precise correspondence to the rate of inﬂation. 2. Total cost functions are nondecreasing in q, v, and w. This property seems obvious, but it is worth dwelling on it a bit. Because cost functions are derived from a costminimization process, any decline in costs from an increase in one of the function’s arguments would lead to a contradiction. For example, if an increase in output from q1 to q2 caused total costs to decline, it must be the case that the ﬁrm was not minimizing costs in the ﬁrst place. It should have produced q2 and thrown away an output of q2 q1 , thereby producing q1 at a lower cost. Similarly, if an increase in the price of an input ever reduced total cost, the ﬁrm could not have been minimizing its costs in the ﬁrst place. To see this, suppose the ﬁrm was using the input combination k1 , l1 and that w increases. Clearly that will increase the cost of the initial input combination. But if changes in input choices actually caused total costs to decline, that must imply that there was a lower-cost input mix than k1 , l1 initially. Hence we have a contradiction, and this property of cost functions is established.7 3. Total cost functions are concave in input prices. It is probably easiest to illustrate this property with a graph. Figure 10.6 shows total costs for various values of an input price, say, w, holding q and v constant. Suppose that initially a wage rate of w1 prevails 7

A formal proof could also be based on the envelope theorem as applied to constrained minimization problems. Consider the Lagrangian expression in Equation 10.3. As was pointed out in Chapter 2, we can calculate the change in the objective in such an expression (here, total cost) with respect to a change in a variable by differentiating the Lagrangian expression. Performing this differentiation yields ∂C ∂ℒ ¼ ¼ λ ð¼ MC Þ 0, ∂q ∂q ∂C ∂ℒ ¼ k 0, ¼ ∂v ∂v ∂C ∂ℒ ¼ l 0. ¼ ∂w ∂w Not only do these envelope results prove this property of cost functions, they also are quite useful in their own right, as we will show later in this chapter.

Chapter 10

FIGURE 10.6

Cost Functions

Cost Functions Are Concave in Input Prices

With a wage rate of w1 , total costs of producing q1 are Cðv, w1 , q1 Þ. If the ﬁrm does not change its input mix, costs of producing q1 would follow the straight line CPSEUDO . With input substitution, actual costs Cðv, w, q1 Þ will fall below this line, and hence the cost function is concave in w. Costs

C PSEUDO C(v,w,q1) C(v,w1,q1)

w1

w

and that the total costs associated with producing q1 are given by Cðv, w1 , q1 Þ. If the ﬁrm did not change its input mix in response to changes_in wages, then _ cost _ _ its total curve would be linear as reﬂected by the line CPSEUDO ð v, w, q1 Þ ¼ v k 1 þ wl 1 in the ﬁgure. But a cost-minimizing ﬁrm probably would change the input mix it uses to produce q1 when wages change, and these actual costs ½Cðv, w, q1 Þ would fall below the “pseudo” costs. Hence, the total cost function must have the concave shape shown in Figure 10.6. One implication of this ﬁnding is that costs will be lower when a ﬁrm faces input prices that ﬂuctuate around a given level than when they remain constant at that level. With ﬂuctuating input prices, the ﬁrm can adapt its input mix to take advantage of such ﬂuctuations by using a lot of, say, labor when its price is low and economizing on that input when its price is high. 4. Average and marginal costs. Some, but not all, of these properties of total cost functions carry over to their related average and marginal cost functions. Homogeneity is one property that carries over directly. Because Cðtv, tw, qÞ ¼ tCðv, w, qÞ, we have C ðtv, tw, qÞ tC ðv, w, qÞ ¼ ¼ tAC ðv, w, qÞ (10.27) AC ðtv, tw, q Þ ¼ q q and8 MC ðtv, tw, q Þ ¼

8

∂C ðtv, tw, qÞ t ∂C ðv, w, qÞ ¼ ¼ tMC ðv, w, qÞ. ∂q ∂q

(10.28)

This result does not violate the theorem that the derivative of a function that is homogeneous of degree k is homogeneous of degree k − 1, because we are differentiating with respect to q and total costs are homogeneous with respect to input prices only.

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The effects of changes in q, v, and w on average and marginal costs are sometimes ambiguous, however. We have already shown that average and marginal cost curves may have negatively sloped segments, so neither AC nor MC is nondecreasing in q. Because total costs must not decrease when an input price rises, it is clear that average cost is increasing in w and v. But the case of marginal cost is more complex. The main complication arises because of the possibility of input inferiority. In that (admittedly rare) case, an increase in an inferior input’s price will actually cause marginal cost to decline. Although the proof of this is relatively straightforward,9 an intuitive explanation for it is elusive. Still, in most cases, it seems clear that the increase in the price of an input will increase marginal cost as well.

Input substitution A change in the price of an input will cause the ﬁrm to alter its input mix. Hence, a full study of how cost curves shift when input prices change must also include an examination of substitution among inputs. To study this process, economists have developed a somewhat different measure of the elasticity of substitution than the one we encountered in the theory of production. Speciﬁcally, we wish to examine how the ratio of input usage (k=l) changes in response to a change in w=v, while holding q constant. That is, we wish to examine the derivative ∂ðk=lÞ ∂ðw=vÞ

(10.29)

along an isoquant. Putting this in proportional terms as ∂ðk=lÞ w=v ∂ ln k=l ¼ (10.30) s¼ ⋅ ∂ðw=vÞ k=l ∂ ln w=v gives an alternative and more intuitive deﬁnition of the elasticity of substitution.10 In the twoinput case, s must be nonnegative; an increase in w=v will be met by an increase in k=l (or, in the limiting ﬁxed-proportions case, k=l will stay constant). Large values of s indicate that ﬁrms change their input proportions signiﬁcantly in response to changes in relative input prices, whereas low values indicate that changes in input prices have relatively little effect.

Substitution with many inputs When there are only two inputs, the elasticity of substitution deﬁned in Equation 10.30 is identical to the concept we deﬁned in Chapter 9 (see Equation 9.32). This can be shown by remembering that cost minimization11 requires that the ﬁrm equate its RTS (of l for k) to the input price ratio w=v. The major advantage of the deﬁnition of the elasticity of substitution in Equation 10.30 is that it is easier to generalize to many inputs than is the deﬁnition based on the production function. Speciﬁcally, suppose there are many inputs to the production process ðx1 , x2 , …, xn Þ that can be hired at competitive rental rates ðw1 , w2 , …, wn Þ. Then the elasticity of substitution between any two inputs ðsij Þ is deﬁned as follows. 9

The proof follows the envelope theorem results presented in footnote 7. Because the MC function can be derived by differentiation from the Lagrangian for cost minimization, we can use Young’s theorem to show ∂MC ∂ð∂ℒ=∂qÞ ∂2 ℒ ∂2 ℒ ∂k ¼ ¼ ¼ ¼ . ∂v ∂v ∂v∂q ∂q∂v ∂q Hence, if capital is a normal input, an increase in v will raise MC whereas, if capital is inferior, an increase in v will actually reduce MC.

10

This deﬁnition is usually attributed to R. G. D. Allen, who developed it in an alternative form in his Mathematical Analysis for Economists (New York: St. Martin’s Press, 1938), pp. 504–9.

11 In Example 10.1 we found that, for the CES production function, cost minimization requires that k=l ¼ ðw=vÞσ , so lnðk=lÞ ¼ σ lnðw=vÞ and therefore sk, l ¼ ∂ lnðk=lÞ=∂ lnðw=vÞ ¼ σ.

Chapter 10

Cost Functions

339

Elasticity of substitution. The elasticity of substitution12 between inputs xi and xj is DEFINITION given by ∂ðxi =xj Þ wj =wi ∂ lnðxi =xj Þ , (10.31) ¼ si, j ¼ ⋅ ∂ðwj =wi Þ xi =xj ∂ lnðwj =wi Þ where output and all other input prices are held constant. The major advantage of this deﬁnition in a multi-input context is that it provides the ﬁrm with the ﬂexibility to adjust inputs other than xi and xj (while holding output constant) when input prices change. For example, a major topic in the theory of ﬁrms’ input choices is to describe the relationship between capital and energy inputs. The deﬁnition in Equation 10.31 would permit a researcher to study how the ratio of energy to capital input changes when relative energy prices rise while permitting the ﬁrm to make any adjustments to labor input (whose price has not changed) that would be required for cost minimization. Hence this would give a realistic picture of how ﬁrms actually behave with regard to whether energy and capital are more like substitutes or complements. Later in this chapter we will look at this deﬁnition in a bit more detail, because it is widely used in empirical studies of production.

Quantitative size of shifts in cost curves We have already shown that increases in an input price will raise total, average, and (except in the inferior input case) marginal costs. We are now in a position to judge the extent of such increases. First, and most obviously, the increase in costs will be inﬂuenced importantly by the relative signiﬁcance of the input in the production process. If an input constitutes a large fraction of total costs, an increase in its price will raise costs signiﬁcantly. A rise in the wage rate would sharply increase home-builders’ costs, because labor is a major input in construction. On the other hand, a price rise for a relatively minor input will have a small cost impact. An increase in nail prices will not raise home costs very much. A less obvious determinant of the extent of cost increases is input substitutability. If ﬁrms can easily substitute another input for the one that has risen in price, there may be little increase in costs. Increases in copper prices in the late 1960s, for example, had little impact on electric utilities’ costs of distributing electricity, because they found they could easily substitute aluminum for copper cables. Alternatively, if the ﬁrm ﬁnds it difﬁcult or impossible to substitute for the input that has become more costly, then costs may rise rapidly. The cost of gold jewelry, along with the price of gold, rose rapidly during the early 1970s, because there was simply no substitute for the raw input. It is possible to give a precise mathematical statement of the quantitative sizes of all of these effects by using the elasticity of substitution. To do so, however, would risk further cluttering the book with symbols.13 For our purposes, it is sufﬁcient to rely on the previous intuitive discussion. This should serve as a reminder that changes in the price of an input will have the effect of shifting ﬁrms’ cost curves, with the size of the shift depending on the relative importance of the input and on the substitution possibilities that are available.

Technical change Technical improvements allow the ﬁrm to produce a given output with fewer inputs. Such improvements obviously shift total costs downward (if input prices stay constant). Although 12

This deﬁnition is attributed to the Japanese economist M. Morishima, and these elasticities are sometimes referred to as “Morishima elasticities.” In this version, the elasticity of substitution for substitute inputs is positive. Some authors reverse the order of subscripts in the denominator of Equation 10.31, and in this usage the elasticity of substitution for substitute inputs is negative.

13 For a complete statement see Ferguson, Neoclassical Theory of Production and Distribution (Cambridge: Cambridge University Press, 1969), pp. 154–60.

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the actual way in which technical change affects the mathematical form of the total cost curve can be complex, there are cases where one may draw simple conclusions. Suppose, for example, that the production function exhibits constant returns to scale and that technical change enters that function as described in Chapter 9 (that is, q ¼ Aðt Þf ðk, lÞ where Að0Þ ¼ 1Þ. In this case, total costs in the initial period are given by (10.32) C0 ¼ C0 ðv, w, qÞ ¼ qC0 ðv, w, 1Þ. Because the same inputs that produced one unit of output in period 0 will produce Aðt Þ units of output in period t , we know that Ct ðv, w, Aðt ÞÞ ¼ Aðt ÞCt ðv, w, 1Þ ¼ C0 ðv, w, 1Þ; therefore, we can compute the total cost function in period t as

(10.33)

qC0 ðv, w, 1Þ C0 ðv, w, qÞ ¼ . (10.34) Aðt Þ Aðt Þ Hence, total costs fall over time at the rate of technical change. Note that in this case technical change is “neutral” in that it does not affect the ﬁrm’s input choices (so long as input prices stay constant). This neutrality result might not hold in cases where technical progress takes a more complex form or where there are variable returns to scale. Even in these more complex cases, however, technical improvements will cause total costs to fall. Ct ðv, w, q Þ ¼ qCt ðv, w, 1Þ ¼

EXAMPLE 10.3 Shifting the Cobb-Douglas Cost Function In Example 10.2 we computed the Cobb-Douglas cost function as (10.35) C ðv, w, qÞ ¼ q 1=ðαþβÞ Bv α=ðαþβÞ w β=ðαþβÞ , α=ðαþβÞ β=ðαþβÞ where B ¼ ðα þ βÞα β . As in the numerical illustration in Example 10.1, let’s assume that α ¼ β ¼ 0.5, in which case the total cost function is greatly simpliﬁed: (10.36) C ðv, w, qÞ ¼ 2qv 0.5 w 0.5 . This function will yield a total cost curve relating total costs and output if we specify particular values for the input prices. If, as before, we assume v ¼ 3 and w ¼ 12, then the relationship is pﬃﬃﬃﬃﬃﬃ C ð3, 12, qÞ ¼ 2q 36 ¼ 12q, (10.37) and, as in Example 10.1, it costs 480 to produce 40 units of output. Here average and marginal costs are easily computed as C ¼ 12, AC ¼ q (10.38) ∂C MC ¼ ¼ 12. ∂q As expected, average and marginal costs are constant and equal to each other for this constant returns-to-scale production function. Changes in input prices. If either input price were to change, all of these costs would change also. For example, if wages were to increase to 27 (an easy number with which to work), costs would become pﬃﬃﬃﬃﬃﬃ C ð3, 27, qÞ ¼ 2q 81 ¼ 18q, AC ¼ 18,

(10.39)

MC ¼ 18. Notice that an increase in wages of 125 percent raised costs by only 50 percent here, both because labor represents only 50 percent of all costs and because the change in input prices encouraged the ﬁrm to substitute capital for labor. The total cost function, because it is

Chapter 10

Cost Functions

derived from the cost-minimization assumption, accomplishes this substitution “behind the scenes”—reporting only the ﬁnal impact on total costs. Technical progress. Let’s look now at the impact that technical progress can have on costs. Speciﬁcally, assume that the Cobb-Douglas production function is (10.40) q ¼ Aðt Þk 0.5 l 0.5 ¼ e .03t k0.5 l 0.5 . That is, we assume that technical change takes an exponential form and that the rate of technical change is 3 percent per year. Using the results of the previous section (Equation 10.34) yields C0 ðv, w, qÞ (10.41) ¼ 2qv 0.5 w 0.5 e . 03t : Aðt Þ So, if input prices remain the same then total costs fall at the rate of technical improvement— that is, at 3 percent per year. After, say, 20 years, costs will be (with v ¼ 3, w ¼ 12) pﬃﬃﬃﬃﬃﬃ C20 ð3, 12, qÞ ¼ 2q 36 ⋅ e . 60 ¼ 12q ⋅ ð0.55Þ ¼ 6.6q, (10.42) AC20 ¼ 6.6, MC20 ¼ 6.6. Consequently, costs will have fallen by nearly 50 percent as a result of the technical change. This would, for example, more than have offset the wage rise illustrated previously. Ct ðv, w, q Þ ¼

QUERY: In this example, what are the elasticities of total costs with respect to changes in input costs? Is the size of these elasticities affected by technical change?

Contingent demand for inputs and Shephard’s lemma As we described earlier, the process of cost minimization creates an implicit demand for inputs. Because that process holds quantity produced constant, this demand for inputs will also be “contingent” on the quantity being produced. This relationship is fully reﬂected in the ﬁrm’s total cost function and, perhaps surprisingly, contingent demand functions for all of the ﬁrm’s inputs can be easily derived from that function. The process involves what has come to be called Shephard’s lemma,14 which states that the contingent demand function for any input is given by the partial derivative of the total cost function with respect to that input’s price. Because Shephard’s lemma is widely used in many areas of economic research, we will provide a relatively detailed examination of it. The intuition behind Shephard’s lemma is straightforward. Suppose that the price of labor (w) were to increase slightly. How would this affect total costs? If nothing else changed, it seems that costs would rise by approximately the amount of labor ðlÞ that the ﬁrm was currently hiring. Roughly speaking, then, ∂C=∂w ¼ l, and that is what Shephard’s lemma claims. Figure 10.6 makes roughly the same point graphically. Along the “pseudo” cost function all inputs are held constant, so an increase in the wage increases costs in direct proportion to the amount of labor used. Because the true cost function is tangent to the pseudo-function at the current wage, its slope (that is, its partial derivative) also will show the current amount of labor input demanded. Technically, Shephard’s lemma is one result of the envelope theorem that was ﬁrst discussed in Chapter 2. There we showed that the change in the optimal value in a constrained optimization problem with respect to one of the parameters of the problem can be found by

14

Named for R. W. Shephard, who highlighted the important relationship between cost functions and input demand functions in his Cost and Production Functions (Princeton, NJ: Princeton University Press, 1970).

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differentiating the Lagrangian expression for that optimization problem with respect to this changing parameter. In the cost-minimization case, the Lagrangian expression is _ ℒ ¼ vk þ wl þ λ½ q f ðk, lÞ (10.43) and the envelope theorem applied to either input is ∂C ðv, w, qÞ ∂ℒðv, w, q, λÞ ¼ ¼ kc ðv, w, qÞ, ∂v ∂v ∂C ðv, w, qÞ ∂ℒðv, w, q, λÞ ¼ ¼ l c ðv, w, qÞ, ∂w ∂w

(10.44)

where the notation is intended to make clear that the resulting demand functions for capital and labor input depend on v, w, and q. Because quantity produced enters these functions, input demand is indeed contingent on that variable. This feature of the demand functions is also reﬂected by the “c” in the notation.15 Hence, the demand relations in Equation 10.44 do not represent a complete picture of input demand because they still depend on a variable that is under the ﬁrm’s control. In the next chapter, we will complete the study of input demand by showing how the assumption of proﬁt maximization allows us to effectively replace q in the input demand relationships with the market price of the ﬁrm’s output, p. EXAMPLE 10.4 Contingent Input Demand Functions In this example, we will show how the total cost functions derived in Example 10.2 can be used to derive contingent demand functions for the inputs capital and labor. 1. Fixed Proportions: Cðv, w, qÞ ¼ qðv=a þ w=bÞ. For this cost function, contingent demand functions are quite simple: ∂C ðv, w, qÞ q ¼ , k c ðv, w, qÞ ¼ ∂v a (10.45) ∂C ðv, w, qÞ q c l ðv, w, qÞ ¼ ¼ . ∂w b In order to produce any particular output with a ﬁxed proportions production function at minimal cost, the ﬁrm must produce at the vertex of its isoquants no matter what the inputs’ prices are. Hence, the demand for inputs depends only on the level of output, and v and w do not enter the contingent input demand functions. Input prices may, however, affect total input demands in the ﬁxed proportions case because they may affect how much the ﬁrm can sell. 2. Cobb-Douglas: Cðv, w, qÞ ¼ q 1=ðαþβÞ Bv α=ðαþβÞ wβ=ðαþβÞ . In this case, the derivation is messier but also more instructive: ∂C α ¼ kc ðv, w, qÞ ¼ ⋅ q 1=ðαþβÞ Bv β=ðαþβÞ w β=ðαþβÞ ∂v αþβ w β=ðαþβÞ α ¼ , ⋅ q 1=ðαþβÞ B αþβ v (10.46) ∂C β c 1=ðαþβÞ α=ðαþβÞ α=ðαþβÞ ¼ Bv w l ðv, w, qÞ ¼ ⋅q ∂w αþβ w α=ðαþβÞ β ¼ : ⋅ q 1=ðαþβÞ B αþβ v

15 The notation mirrors that used for compensated demand curves in Chapter 5 (which were derived from the expenditure function). In that case, such demand functions were contingent on the utility target assumed.

Chapter 10

Cost Functions

Consequently, the contingent demands for inputs depend on both inputs’ prices. If we assume α ¼ β ¼ 0.5 (so B ¼ 2), these reduce to w 0.5 w 0.5 ¼q , k c ðv, w, q Þ ¼ 0.5 ⋅ q ⋅ 2 ⋅ v (10.47) w vw 0.5 0.5 l c ðv, w, q Þ ¼ 0.5 ⋅ q ⋅ 2 ⋅ ¼q . v v With v ¼ 3, w ¼ 12, and q ¼ 40, Equations 10.47 yield the result we obtained previously: that the ﬁrm should choose the input combination k ¼ 80, l ¼ 20 to minimize the cost of producing 40 units of output. If the wage were to rise to, say, 27, the ﬁrm would choose the input combination k ¼ 120, l ¼ 40=3 to produce 40 units of output. Total costs would rise from 480 to 520, but the ability of the ﬁrm to substitute capital for the now more expensive labor does save considerably. For example, the initial input combination would now cost 780. 3. CES: Cðv, w, qÞ ¼ q 1=γ ðv 1σ þ w1σ Þ1=ð1σÞ . The importance of input substitution is shown even more clearly with the contingent demand functions derived from the CES function. For that function, ∂C 1 σ=ð1σÞ ¼ ð1 σÞv σ kc ðv, w, qÞ ¼ ⋅ q 1=γ ðv1σ þ w 1σ Þ ∂v 1σ ¼ q 1=γ ðv1σ þ w 1σ Þσ=ð1σÞ vσ , ∂C 1 σ=ð1σÞ ð1 σÞw σ l c ðv, w, qÞ ¼ ¼ ⋅ q 1=γ ðv1σ þ w 1σ Þ ∂w 1σ

(10.48)

¼ q 1=γ ðv1σ þ w 1σ Þσ=ð1σÞ w σ . These functions collapse when σ ¼ 1 (the Cobb-Douglas case), but we can study examples with either more ðσ ¼ 2Þ or less ðσ ¼ 0.5Þ substitutability and use Cobb-Douglas as the middle ground. If we assume constant returns to scale ðγ ¼ 1Þ and v ¼ 3, w ¼ 12, and q ¼ 40, then contingent demands for the inputs when σ ¼ 2 are k c ð3, 12, 40Þ ¼ 40ð31 þ 121 Þ2 ⋅ 32 ¼ 25:6,

(10.49) l c ð3, 12, 40Þ ¼ 40ð31 þ 121 Þ2 ⋅ 122 ¼ 1:6: That is, the level of capital input is 16 times the amount of labor input. With less substitutability ðσ ¼ 0.5Þ, contingent input demands are kc ð3, 12, 40Þ ¼ 40ð30:5 þ 120:5 Þ1 ⋅ 30:5 ¼ 120,

(10.50) l c ð3, 12, 40Þ ¼ 40ð30:5 þ 120:5 Þ1 ⋅ 120:5 ¼ 60. So, in this case, capital input is only twice as large as labor input. Although these various cases cannot be compared directly because different values for σ scale output differently, we can, as an example, look at the consequence of a rise in w to 27 in the low-substitutability case. With w ¼ 27, the ﬁrm will choose k ¼ 160, l ¼ 53.3. In this case, the cost savings from substitution can be calculated by comparing total costs when using the initial input combination (¼ ð3Þ120 þ 27ð60Þ ¼ 1980) to total costs with the optimal combination (¼ ð3Þ160 þ 27ð53:3Þ ¼ 1919). Hence, moving to the optimal input combination reduces total costs by only about 3 percent. In the Cobb-Douglas case, cost savings are over 20 percent. QUERY: How would total costs change if w increased from 12 to 27 and the production function took the simple linear form q ¼ k þ 4l? What light does this result shed on the other cases in this example?

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SHEPHARD’S LEMMA AND THE ELASTICITY OF SUBSTITUTION One especially nice feature of Shephard’s lemma is that it can be used to show how to derive information about input substitution directly from the total cost function through differentiation. Using the deﬁnition in Equation 10.31 yields si, j ¼

∂ lnðxi =xj Þ ∂ lnðwj =wi Þ

¼

∂ lnðCi =Cj Þ ∂ lnðwj =wi Þ

,

(10.51)

where Ci and Cj are the partial derivatives of the total cost function with respect to the input prices. Once the total cost function is known (perhaps through econometric estimation), information about substitutability among inputs can thus be readily obtained from it. In the Extensions to this chapter, we describe some of the results that have been obtained in this way. Problems 10.11 and 10.12 provide some additional details about ways in which substitutability among inputs can be measured.

SHORT-RUN, LONG-RUN DISTINCTION It is traditional in economics to make a distinction between the “short run” and the “long run.” Although no very precise temporal deﬁnition can be provided for these terms, the general purpose of the distinction is to differentiate between a short period during which economic actors have only limited ﬂexibility in their actions and a longer period that provides greater freedom. One area of study in which this distinction is quite important is in the theory of the ﬁrm and its costs, because economists are interested in examining supply reactions over differing time intervals. In the remainder of this chapter, we will examine the implications of such differential response. To illustrate why short-run and long-run reactions might differ, assume that capital input is held ﬁxed at a level of k1 and that (in the short run) the ﬁrm is free to vary only its labor input.16 Implicitly, we are assuming that alterations in the level of capital input are inﬁnitely costly in the short run. As a result of this assumption, the short-run production function is q ¼ f ðk1 , lÞ, (10.52) where this notation explicitly shows that capital inputs may not vary. Of course, the level of output still may be changed if the ﬁrm alters its use of labor.

Short-run total costs Total cost for the ﬁrm continues to be deﬁned as C ¼ vk þ wl (10.53) for our short-run analysis, but now capital input is ﬁxed at k1 . To denote this fact, we will write (10.54) SC ¼ vk1 þ wl, where the S indicates that we are analyzing short-run costs with the level of capital input ﬁxed. Throughout our analysis, we will use this method to indicate short-run costs, whereas long-run costs will be denoted by C, AC, and MC. Usually we will not denote the level of capital input explicitly, but it is understood that this input is ﬁxed.

16 Of course, this approach is for illustrative purposes only. In many actual situations, labor input may be less ﬂexible in the short run than is capital input.

Chapter 10

Cost Functions

345

Fixed and variable costs The two types of input costs in Equation 8.53 are given special names. The term vk1 is referred to as (short-run) ﬁxed costs; because k1 is constant, these costs will not change in the short run. The term wl is referred to as (short-run) variable costs—labor input can indeed be varied in the short run. Hence we have the following deﬁnitions. Short-run fixed and variable costs. Short-run ﬁxed costs are costs associated with inputs DEFINITION that cannot be varied in the short run. Short-run variable costs are costs of those inputs that can be varied so as to change the ﬁrm’s output level. The importance of this distinction is to differentiate between variable costs that the ﬁrm can avoid by producing nothing in the short run and costs that are ﬁxed and must be paid regardless of the output level chosen (even zero).

Nonoptimality of short-run costs It is important to understand that total short-run costs are not the minimal costs for producing the various output levels. Because we are holding capital ﬁxed in the short run, the ﬁrm does not have the ﬂexibility of input choice that we assumed when we discussed cost minimization earlier in this chapter. Rather, to vary its output level in the short run, the ﬁrm will be forced to use “nonoptimal” input combinations: The RTS will not be equal to the ratio of the input prices. This is shown in Figure 10.7. In the short run, the ﬁrm is constrained to use k1 units of capital. To produce output level q0 , it therefore will use l0 units of labor. Similarly, it will use l1 units of labor to produce q1 and l2 units to produce q2 . The total costs of these input combinations are given by SC0 , SC1 , and SC2 , respectively. Only for the input combination k1 , l1 is output being produced at minimal cost. Only at that point is the RTS equal to the ratio of the input prices. From Figure 10.7, it is clear that q0 is being produced with “too much” capital in this short-run situation. Cost minimization should suggest a southeasterly movement along the q0 isoquant, indicating a substitution of labor for capital in production. Similarly, q2 is being produced with “too little” capital, and costs could be reduced by substituting capital for labor. Neither of these substitutions is possible in the short run. Over a longer period, however, the ﬁrm will be able to change its level of capital input and will adjust its input usage to the cost-minimizing combinations. We have already discussed this ﬂexible case earlier in this chapter and shall return to it to illustrate the connection between long-run and short-run cost curves.

Short-run marginal and average costs Frequently, it is more useful to analyze short-run costs on a per-unit-of-output basis rather than on a total basis. The two most important per-unit concepts that can be derived from the short-run total cost function are the short-run average total cost function (SAC) and the shortrun marginal cost function (SMC). These concepts are deﬁned as total costs SC ¼ , SAC ¼ total output q (10.55) change in total costs ∂SC SMC ¼ ¼ , change in output ∂q where again these are deﬁned for a speciﬁed level of capital input. These deﬁnitions for average and marginal costs are identical to those developed previously for the long-run, fully ﬂexible case, and the derivation of cost curves from the total cost function proceeds in exactly the same way. Because the short-run total cost curve has the same general type of cubic shape as did the total cost curve in Figure 10.5, these short-run average and marginal cost curves will also be U-shaped.

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FIGURE 10.7

“Nonoptimal” Input Choices Must Be Made in the Short Run Because capital input is ﬁxed at k, in the short run the ﬁrm cannot bring its RTS into equality with the ratio of input prices. Given the input prices, q0 should be produced with more labor and less capital than it will be in the short run, whereas q2 should be produced with more capital and less labor than it will be. k per period

SC2 SC 0 SC1 = C

k1 q2 q1 q0 l0

l1

l2

l per period

Relationship between short-run and long-run cost curves It is easy to demonstrate the relationship between the short-run costs and the fully ﬂexible long-run costs that were derived previously in this chapter. Figure 10.8 shows this relationship for both the constant returns-to-scale and cubic total cost curve cases. Short-run total costs for three levels of capital input are shown, although of course it would be possible to show many more such short-run curves. The ﬁgures show that long-run total costs ðCÞ are always less than short-run total costs, except at that output level for which the assumed ﬁxed capital input is appropriate to long-run cost minimization. For example, as in Figure 10.7, with capital input of k1 the ﬁrm can obtain full cost minimization when q1 is produced. Hence, short-run and long-run total costs are equal at this point. For output levels other than q1 , however, SC > C, as was the case in Figure 10.7. Technically, the long-run total cost curves in Figure 10.8 are said to be an “envelope” of their respective short-run curves. These short-run total cost curves can be represented parametrically by short-run total cost ¼ SC ðv, w, q, kÞ, (10.56) and the family of short-run total cost curves is generated by allowing k to vary while holding v and w constant. The long-run total cost curve C must obey the short-run relationship in Equation 10.56 and the further condition that k be cost minimizing for any level of output. A ﬁrst-order condition for this minimization is that

FIGURE 10.8

Two Possible Shapes for Long-Run Total Cost Curves

By considering all possible levels of capital input, the long-run total cost curve (C) can be traced. In (a), the underlying production function exhibits constant returns to scale: in the long run, though not in the short run, total costs are proportional to output. In (b), the long-run total cost curve has a cubic shape, as do the short-run curves. Diminishing returns set in more sharply for the short-run curves, however, because of the assumed ﬁxed level of capital input. Total costs

SC (k2) SC (k1) SC (k0)

q0

C

q2 Output per period

q1

(a) Constant returns to scale Total costs SC (k2)

C

SC (k1) SC (k0)

q0

q1

q2

Output per period

(b) Cubic total cost curve case

∂SC ðv, w, q, kÞ ¼ 0. (10.57) ∂k Solving Equations 10.56 and 10.57 simultaneously then generates the long-run total cost function. Although this is a different approach to deriving the total cost function, it should give precisely the same results derived earlier in this chapter—as the next example illustrates.

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EXAMPLE 10.5 Envelope Relations and Cobb-Douglas Cost Functions Again we start with the Cobb-Douglas production function q ¼ kα l β , but now we hold capital input constant at k1 . So, in the short run, q ¼ k α1 l β

or

α=β

l ¼ q 1=β k 1

,

(10.58)

and total costs are given by α=β

SC ðv, w, q, k1 Þ ¼ vk1 þ wl ¼ vk1 þ wq 1=β k1 . (10.59) Notice that the ﬁxed level of capital enters into this short-run total cost function in two ways: (1) k1 determines ﬁxed costs; and (2) k1 also in part determines variable costs because it determines how much of the variable input (labor) is required to produce various levels of output. To derive long-run costs, we require that k be chosen to minimize total costs: ∂SC ðv, w, q, kÞ α 1=β ðαþβÞ=β ¼ 0. (10.60) ¼vþ ⋅ wq k ∂k β Although the algebra is messy, this equation can be solved for k and substituted into Equation 10.59 to return us to the Cobb-Douglas cost function: C ðv, w, qÞ ¼ Bq 1=ðαþβÞ vα=ðαþβÞ w β=ðαþβÞ .

(10.61)

Numerical example. If we again let α ¼ β ¼ 0.5, v ¼ 3, and w ¼ 12, then the short-run cost function is (10.62) SC ð3, 12, q, kÞ ¼ 3k1 þ 12q 2 k1 1 . In Example 10.1 we found that the cost-minimizing level of capital input for q ¼ 40 was k ¼ 80. Equation 10.62 shows that short-run total costs for producing 40 units of output with k ¼ 80 is 1 3q 2 ¼ 240 þ 20 80 (10.63) ¼ 240 þ 240 ¼ 480, which is just what we found before. We can also use Equation 10.62 to show how costs differ in the short and long run. Table 10.1 s