HAL R. VARIAN 1 NORTON To my parents Copyright @ 1992, 1984, 1978 by W. W. Norton & Company, Inc. All rights reserved Printed in the United States of America THIRD EDITION Library of Congress Cataloging-in-Publication Data Varian, Hal R. Mlcroeconon~lc analysis / Hal R. Varian. -- 3rd ed. p. an Includes blbllographlcal references and index. 1. Mlcroeconomlcs. 1. Title. HB172.V35 1992 338.5--dc20 ISBN 0-393-95735-7 W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110 W. W. Norton & Company, Ltd., 10 Coptic Street, London WClA 1PU CONTENTS PREFACE 1 Technology Measurement of inputs and outputs 1 Specification of technology 2 Example: Input requzrement set Example: Isoquant Example: Short- run productzon posszbzlztzes set Example: Pt-oductzon functzon Exam- ple: Transformatzon functzon Example: Cobb-Douglas technology Ex- ample: Leontzef technology Activity analysis 5 Monotonic technologies 6 Convex technologies 7 Regular technologies 9 Parametric rep- resentations of technology 10 The technical rate of substitution 11 Example: T R S for a Cobb-Douglas technology The elasticity of substitu- tion 13 Example: The elastzczty of substztutzon for the Cobb-Douglas productzon functzon Returns to scale 14 Example: Returns to scale and the Cobb-Douglas technology Homogeneous and homothetic technolo- gies 17 Example: The CES productzon functzon Exercises 21 2 Profit Maximization . Profit maximization 25 Difficulties 28 Example: The profit functzon for Cobb-Douglas technology Properties of demand and supply functions 31 Comparative statics using the first-order conditions 32 Compara- tive statics using algebra 35 Recoverability 36 Exercises 39 VI CONTENTS 3 Profit Function Properties of the profit function 40 Example: The eflects of price stabilization Supply and demand functions from the profit function 43 The envelope theorem 45 Comparative statics using the profit function 46 Example: The LeChatelier principle Exercises 48 4 Cost Minimization Calculus analysis of cost minimization 49 More on second-order condi- tions 52 Difficulties 53 Example: Cost function for the Cobb-Douglas technology Example: The cost function for the CES technology Exam- ple: The cost function for the Leontief technology Example: The cost function for the linear technology Conditional factor demand functions 58 Algebraic approach to cost minimization 61 Exercises 63 5 Cost Function . . I Average and marginal costs 64 Example: The short-run Cobb-Douglas cost functions Example: Constant returns to scale and the cost function The geometry of costs 67 Example: The Cobb-Douglas cost curves Long-run and short-run cost curves 70 Factor prices and cost functions 71 The envelope theorem for constrained optimization 75 Example: Marginal cost revisited Comparative statics using the cost function 76 Exercises 77 6 Duality Duality 82 Sufficient conditions for cost functions 84 Demand func- tions 86 Example: Applying the duality mapping Example: Constant returns to scale and the cost function Example: Elasticity of scale and the cost function Geometry of duality 89 Example: Production func- tions, cost functions, and conditional factor demands The uses of duality 91 Exercises 93 CONTENTS VII 7 Utility Maximization 3 .: n Consumer preferences ,941 Example: The existence of a utility function Example: The marginal rate of substitution Consumer behavior 98 In- direct utility 102 Some important identities 105 The money met- ric utility functions 108 Example: The Cobb-Douglas utzlity function L Example: The CES utility function Appendix 113 Exercises 114 8 Choice Comparative statics 116 Example: Excise and income taxes The Slut- sky equation 119 Example: The Cobb-Douglas Slutsky equation Prop erties of demand functions 122 Comparative statics using the first-order conditions 123 The integrability problem 125 Example: Integrabil- zty wzth two goods Example: Integrability with several goods Duality in consumption 129 Example: Solving for the direct utility function Revealed preference 131 Sufficient conditions for maximization 133 Comparative statics using revealed preference 135 The discrete version of the Slutsky equation 137 Recoverability 138 Exercises 140 9 Demand Endowments in the budget constraint 144 Labor supply Homothetic utility functions 146 Aggregating across goods 147 Hicksian sepa- rability The two-good model Functional separability Aggregating across consumers 152 Inverse demand functions 155 Continuity of demand functions 158 Exercises 157 10 Consumers' Surplus Compensating and equivalent variations 160 Consumer's surplus 163 , Quasilinear utility 164 Quasilinear utility and money metric utility 166 Consumer's surplus as an approximation 167 Aggregation 168 Nonparametric bounds 170 Exercises 171 Wf . CONTENTS 11 Uncertainty Lotteries 172 Expected utility 173 Uniqueness of the expected utility function 175 Other notations for expected utility 176 Risk aversion 177 Example: The demand for insurance Global risk aversion 181 Example: Comparative statics of a simple portfolio problem Example: Asset pricing Relative risk aversion 188 Example: Mean-variance utility State dependent utility 190 Subjective probability theory 190 Example: The Allais paradox and the Ellsberg paradox The Allais paradox The Ellsberg paradox Exercises 194 I . r t 12 Econometrics The optimization hypothesis 198 Nonparametric testing for maximizing behavior 199 Parametric tests of maximizing behavior 200 Impos- ing optimization restrictions 201 Goodness-of-fit for optimizing mod- els 201 Structural models and reduced form models 202 Estimating technological relationships 204 Estimating factor demands 207 More complex technologies 207 Choice of functional form 209 Example: The Diewert cost function Example: The translog cost function Esti- mating consumer demands 210 Demand functions for a single good Multiple equations Example: Linear expenditure system Example: Almost Ideal Demand System Summary 213 13 Cornpetltive Markets The competitive firm 215 The profit maximization problem 216 The industry supply function 218 Example: Different cost functions Ex- ample: Identical cost functions Market equilibrium 219 Example: Identical firms Entry 220 Example: Entry and long-run equilibrium Welfare economics 221 Welfare analysis 222 Several consumers 224 Pareto efficiency 225 Efficiency and welfare 226 The discrete good model 227 Taxes and subsidies 228 Exercises 230 CONTENTS IX 14 Monopoly -I L Special cases 236 Comparative statics 236 Welfare and output 238 Quality choice 239 Price discrimination 241 First-degree price dis- crimination 243 Second-degree price discrimination 244 Example: A graphical treatment Third-degree price discrimination 248 Welfare effects Exercises 253 15 Came Theory Description of a game 260 Example: Matching pennies Example: The Pmsoner's Dzlemma Example: Cournot duopoly Example: Bertrand duopoly Economic modeling of strategic choices 263 Solution concepts 264 Nash equilibrium 265 Example: Calculatzng a Nash equilibmum Interpretation of mixed strategies 268 Repeated games 269 Exam- ple: Maintaining a cartel Refinements of Nash equilibrium 271 Domi- nant strategies 272 Elimination of dominated strategies 272 Sequen- tial games 273 Example: A szmple bargaining model Repeated games and subgame perfection 278 Games with incomplete information 279 Example: A sealed-bid auction Discussion of Bayes-Nash equilibrium 281 Exercises 282 16 Oligopoly Cournot equilibrium 285 Stability of the system Comparative stat- ics 288 Several firms 289 Welfare . Bertrand equilibrium 291 _- Example: A model of sales Complements and substitutes 294 Quan- tity leadership 295 Price leadership 298 Classification and choice of models 301 Conjectural variations 302 Collusion 303 Repeated oligopoly games 305 Sequential games 307 Limit pricing 308 Ex- ercises 310 17 Exchange Agents and goods 314 Walrasian equilibrium 315 Graphical analysis 316 Existence of Walrasian equilibria 317 Existence of an equilib- rium 319 Example: The Cobb-Douglas Economy The first theorem of welfare economics 323 The second welfare theorem 326 A revealed preference argument a Pareto efficiency and calculus 329 Welfare max- imization 333 Exercises 336 18 Production Firm behavior 338 Difficulties 340 Consumer behavior 341 Labor supply Dzstnbutzon of profits Aggregate demand 342 Existence of an equilibrium 344 Welfare properties of equilibrium 345 A revealed preference argument a Welfare analysis in a productive economy 348 Graphical treatment 349 Example: The Cobb-Douglas constant returns economy Example: A decreasing-returns-to-scale economy The Nonsubstitution Theorem 354 Industry structure in general equilibrium 356 Exercises 357 19 Time Intertemporal preferences 358 Intertemporal optimization with two pe- riods 359 Intertemporal optimization with several periods 361 Ex- ample: Loganthrnic utzlity General equilibrium over time 363 Infinity General equilibrium over states of nature 365 Exercises 366 20 Asset Markets Equilibrium with certainty 368 Equilibrium with uncertainty 369 Notation 370 The Capital Asset Pricing Model 371 The Arbitrage Pricing Theory 376 Two factors a Asset-speczfic nsk a Expected utility 379 Example: Expected utzlity and the APT Complete markets 382 Pure arbitrage 383 Appendix 385 Exercises 386 CONTENTS XI 21 Equilibrium Analysis The core of an exchange economy 387 Convexity and size 393 Unique- ness of equilibrium 394 Gross substitutes . Index analysis . General equilibrium dynamics 398 Tatonnement processes 398 Nontaton- nement processes 401 Exercises 402 22 Welfare The compensation criterion 404 Welfare functions 409 Optimal tax- ation 410 Exercises 413 23 Public goods Efficient provision of a discrete public good 415 Private provision of a discrete public good 417 Voting for a discrete public good 417 Effi- cient provision of a continuous public good 418 Example: Solving for the efficient provision of a public good Private provision of a continuous public good 420 Example: Solving for Nash equilibrium provzsion Vot- ing 424 Example: Quasilinear utility and voting Lindahl allocations 425 Demand revealing mechanisms 426 Demand revealing mecha- nisms with a continuous good 429 Exercises 430 24 Externalities An example of a production externality 433 Solutions to the external- ities problem 433 Pigovian taxes * Missing markets . Property rights The compensation mechanism 436 Efficiency conditions in the presence of externalities 438 Exercises 439 XI1 CONTENTS 25 Information The principal-agent problem 441 Full information: monopoly solution 442 Full information: competitive solution 444 Hidden action: mon- opoly solution 445 Agent's action can be observed o Analysis of the optimal incentive scheme Example: Comparative statics Example: Principal-agent model with mean-variance utility Hidden actions: compet- itive market 455 Example: Moral hazard in insurance markets Hidden information: monopoly 457 Market equilibrium: hidden information 464 Example: An algebraic example Adverse selection 466 The lemons market and adverse selection 468 Signaling 469 Educational signaling 470 Exercises 471 26 Mathematics Linear algebra 473 Definite and semidefinite matrices 475 Tests for definite matrices Cramer's rule 477 Analysis 477 Calculus 478 Higher-order derivatives Gradients and tangent planes 480 Limits 481 Homogeneous functions 481 Affine functions 482 Convex sets 482 Separating hyperplanes 483 Partial differential equations 483 Dynamical systems 484 Random variables 485 27 Optimization Single variable optimization 487 First-order and second-order condi- tions Example: First- and second-order conditions. Concavity The envelope theorem Example: The value function Example: The envelope theorem Comparative statics Example: Comparative statics for a particular problem Multivariate maximization 493 First- and second-order conditions Comparative statics Example: Compara- tive statics Convexity and concavity Quasiconcave and quasiconvex functions Constrained maximization 497 An alternative second-order condition 498 How to remember the second-order conditions The envelope theorem Constrained maximization with inequality constraints 503 Setting up Kuhn-Tucker problems 504 Existence and continuity of a maximum 506 References l A1 Answers to Odd-Numbered Exercises l A9 Index 1 A37 PREFACE The first edition of Microeconomic Analysis was published in 1977. Af- ter 15 years, I thought it was time for a major revision. There are two types of changes I have made for this third edition, structural changes and substantive changes. The structural changes involve a significant rearrangement of the mate- rial into "modular" chapters. These chapters have, for the most part, the same titles as the corresponding chapters in my undergraduate text, In- termedzate Mzcroeconomzcs. This makes it easy for the student to go back to the undergraduate book to review material when appropriate. It also works the other way around: if an intermediate student wants to pursue more advanced work on a topic, it is easy to turn to the appropriate chap- ter in Mzcroeconomzc Analyszs. I have found that this modular structure also has two further advantages: it is easy to traverse the book in various orders, and it makes it more convenient to use the book for reference. In addition to this reorganization, there are several substantive changes. First, I have rewritten substantial sections of the book. The material is now less terse, and, I hope, more accessible. Second, I have brought a lot of material up to date. In particular, the material on monopoly and oligopoly has been completely updated, following the major advances in the theory of industrial organization during the eighties. Third, I have added lots of new material. There are now chapters on game theory, asset markets, and information. These chapters can serve as an appropriate introduction to this material for first-year economics students. I haven't tried to provide in-depth treatments of these topics since I've found that is better pursued in the second or third year of graduate XIV PREFACE studies, after facility with the standard tools of economic analysis have been mastered. Fourth, I've added a number of new exercises, along with complete an- swers to all odd-numbered problems. I must say that I am ambivalent about putting the answers in the book-but I hope that most graduate students will have sufficient willpower to avoid looking at the answer until they have put some effort into solving the problems for themselves. Organization of the book As I mentioned above, the book is organized into a number of short chap- ters. I suspect that nearly everyone will want to study the material in the first half of the book systematically since it describes the fundamental tools of microeconomics that will be useful to all economists. The material in the second half of the book consists of introductions to a number of t o p ics in microeconomics. Most people will want to pick and choose among these topics. Some professors will want to emphasize game theory; others will want to emphasize general equilibrium. Some courses will devote a lot of time to dynamic models; others will spend several weeks on welfare economics. It would be impossible to provide in-depth treatment of all of these top- ics, so I have decided Lo provide introductions to the subjects. I've tried to use the notation and methods described in the first part of the book so that these chapters can pave the way to a more thorough treatment in books or journal articles. Luckily, there are now several book-length treat- ments of asset markets, game theory, information economics, and general equilibrium theory. The serious student will have no shortage of materials in which he or she can pursue the study of these topics. Production of the book In the process of rewriting the book, I have moved everything over to Donald Knuth's Tj$ system. I think that the book now looks a lot better; furthermore, cross-referencing, equation numbering, indexing, and so on are now a lot easier for both the author and the readers. Since the cost to the author of revising the book is now much less, the reader can expect to see more frequent revisions. (Perhaps that last sentence can be turned into an exercise for the next edition.. . ) Part of the book was composed on MS-DOS equipment, but the majority of it was composed and typeset on a NeXT computer. I used Emacs as the primary editor, operating in Kresten Thorup's auc-tex mode. I use ispell for spell-checking, and the standard makeindex and bibtex tools for indexing and bibliographic management. Tom Rokicki's Tj$view was PREFACE XV the tool of choice for previewing and printing. Preliminary versions of the diagrams were produced using Designer and Top Draw. An artist rendered final versions using FreeHand and sent me the Encapsulated Postscript files which were then incorporated into the TJ$ code using Trevor Darrell's psfig macros. I owe a special debt of gratitude to the authors of these software tools, many of which have been provided to users free of charge. Acknowledgments Many people have written to me with typos, comments, and suggestions over the years. Here is a partial list of names: Tevfik Aksoy, Jim Andreoni, Gustavo Angeles, Ken Binmore, Soren Blomqvist, Kim Border, Gordon Brown, Steven Buccola, Mark Burkey, Lea Verdin Carty, Zhiqi Chen, John Chilton, Francisco Armando da Costa, Giacomo Costa, David W. Craw- ford, Peter Diamond, Karen Eggleston, Maxim Engers, Sjur Flam, Mario Forni, Marcos Gallacher, Jon Hamilton, Barbara Harrow, Kevin Jackson, Yi Jiang, John Kennan, David Kiefer, Rachel Kranton, Bo Li, George Mailath, David Malueg, Duhamel Marc, John Miller, V. A. Noronha, Mar- tin Osborne, Marco Ottaviani, Attila Ratfai, Archie Rosen, Jan Rutkowski, Michael Sandfort, Marco Sandri, Roy H. M. Sembel, Mariusz Shatba, Bert Schoonbeek, Carl Simon, Bill Sjostrom, Gerhard Sorger, Jim Swan- son, Knut Sydsater, A. J. Talman, Coenraad Vrolijk, Richard Woodward, fiances Wooley, Ed Zajac, and Yong Zhu. If my filing system were better, there would probably be several more names. I appreciate being notified of errata and will usually be able to correct such bugs in the next printing. You can send me e-mail about bugs at Hal. VarianQumich . edu. Several people have contributed suggestions on the new third edition, including Eduardo Ley, Pat Reagan, John Weymark, and Jay Wilson. Ed- uardo Ley also provided some of the exercises and several of the answers. Finally, I want to end with a comment to the student. As you read this work, it is important to keep in mind the immortal words of Sir Richard Steele (1672-1729): "It is to be noted that when any part of this paper appears dull there is a design in it." > Ann Arbor November 1991 Microeconomic Analysis Third ~dition CHAPTER 1 TECHNOLOGY The simplest and most common way to describe the technology of a firm is the production function, which is generally studied in intermediate courses. However, there are other ways to describe firm technologies that are both more general and more useful in certain settings. We will discuss several of these ways to represent firm production possibilities in this chap- ter, along with ways to describe economically relevant aspects of a firm's technology. -- 1.1 Measurement of inputs and outputs A firm produces outputs from various combinations of inputs. In order to study firm choices we need a convenient way to summarize the production possibilities of the firm, i.e., which combinations of inputs and outputs are technologically feasible. It is usually most satisfactory to think of the inputs and outputs as being measured in terms of BOWS: a certain amount of inputs per time period are used to produce a certain amount of outputs per unit time period. It is a good idea to explicitly include a time dimension in a specification of inputs