Microeconomic Theory for the Social Sciences

8/18/2019 Microeconomic Theory for the Social Sciences

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Preface

This book covers microeconomic theory at the level of intermediate/advancedundergraduates, but I also intend it to be an introduction for those with otherintellectual backgrounds, who do not necessarily agree to what so-called ”main-stream economists” say, but at least feel it OK to know how they think and seethings.

Stance of this bookI tried to give thorough explanations of denitions and assumptions which thetheory is based upon. Also I tried to give thorough accounts of motivations andreservations behind the theory.

Professional work in economic theory is presented as a sequence of deni-tions, assumptions and their implications. Its result is presented as a theorem,which is a statement in the form ”If A is true, then B is true.” It is vital fortheorists to share the understanding of what assumptions the present theory isrelying on, because there is no conclusion without assumption. If you think youare free from any assumption, it is either that you don’t know what assumptionyour argument is relying on or that you know it and you are hiding it.

Introductory teaching of economics, on the other hand, tends to omit givingthorough explanations of underlying assumptions and reservations. There is agood educational reason to do so, because teachers don’t want to make theirstudents bored before getting into ”useful” stuff. This causes a danger, however,that learners do not care about the underlying assumptions and the logicalprocess of how the assumptions lead to the conclusion. As a result, learnersquite often abuse theory by applying it to situations in which its assumptiondoes not hold, or criticize theory on the ground that its conclusion is wrongagain by applying it to situations in which its assumption does not hold.

My aim is to help the readers to get able to draw a precise line betweenwhat economic theory says (or can say) and what it does not (or cannot). Ihope this is helpful for economics majors as well as those from other disciplines.I am more than happy if my attempt is successful.

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On the mixture and order of formal presentation, verbal

discussion and examplesTo achieve the above goal at the level of intermediate/advanced undergraduatetextbook, I tried to nd the best mixture and order of formal presentation of themodel, verbal discussions on its motivation and reservations, and explanationsthrough examples.

The mixture and order vary across the topics. When I nd it helpful to putemphasis on formal presentation as I believe it conveys the point more vividly(and it does not let you behind), I do so. When I nd that verbal discussion givesyou a richer understanding or helps you to avoid confusion which I expect mayhappen in the rst-look at the formal model, I put discussions before or alongwith it. When I nd it helpful to go over an example rst when I introduce a newnotion rather than presenting a formal denition or giving wordy discussions, I

do so.

On the degree of generality and simplicationTo achieve the above goal, I tried to nd the best level of generality and sim-plication at which the theory is presented.

I would say there are two kinds of simplication. One is made when we canmake the model more general and complex in order to get closer to the realityand it is feasible at a methodological level, but it does not essentially changethe argument or it is irrelevant to the issue you are looking at. This is calledsimplication without loss of generality .

For example, in many places in the book I assume that there are just two

goods in the economy. This of course does not mean that there are really onlytwo goods in the world, but simply means that in order to understand the pointit is enough to think of just two goods.

What we mean by ”without loss of generality” depends naturally on the typeof audiences and their interests. First possibility is that professional researchersare not content with the simplication since it abstracts away what they thinkis important in the professional works, while it should be allowed for certaineducational purposes.

There is an opposite possibility: theorists are quite often content with sim-plied illustration and want to concentrate on the main point, when they seethat its extension to more complicated settings is straightforward, while learnersor readers from outside of the discipline rather feel that relying on simpliedillustration is a cheat.

I take the latter case more seriously, and present certain topics at a higherlevel of generality when I believe it helps understanding more effectively.

The other kind of simplication is of course due to the limitation of ouranalytical ability. In such cases I put discussions on what are abstracted awayand what the present theory is missing, and I hope it helps you to go beyond.


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On terminologies

Microeconomics is related to the society, and because of this the words usedthere naturally overlap real-life wordings. This is actually dangerous.

For example, what do you imagine from the words such as ”utility,” ”perfectcompetition,” and ”efficiency?” If you imagine some kind of ”substance” fromthe word ”utility,” that’s wrong. If you imagine a situation like everybody killingeach other from the word ”perfect competition,” that’s wrong. If you imagine aone-dimensional criterion which ranks between all social alternatives from theword ”efficiency,” that’s wrong.

In economics these words are given precise boundaries in the form of de-nitions, and I relegate them to the corresponding chapters. What I like to sayto you here is that you should wipe away what you imagine from the usual lifeusages of those words.

Unfortunately, the terminologies like above are accepted ones, and it maybe embarrassing if I make up new ones, so I decided to accept most of themby putting adequate discussions when I introduce them. Nevertheless, I de-cided to use unconventional terminologies in some cases when I’m afraid usingthe accepted terminology causes a serious misunderstanding. I hope it doesn’tembarrass you too much.

On mathematical expositionsI use mathematical exposition as long as it is easier than the verbal one .Economic theorists use mathematics not because they like to mystify but be-cause it is the easiest way to share understandings precisely. It is the easiestway to precisely share denitions, assumptions and the process of deriving con-clusions from them, and to avoid confusions and errors which often happen inthe arguments by natural languages.

Of course what we mean by easier will depend on the audiences. I guess it ishard in the beginning, but I bet you will see it much easier as you proceed. AlsoI tried to give explanations to mathematical notions in a self-contained matteras much as possible when they are necessary for reading ahead.

He or SheBecause of the nature of the subject, I use third-person singular pronoun re-peatedly. It is always a problem for economic theorists if we should use he orshe.

There isn’t a ”gender-neutral” third-person singular pronoun in English (nei-ther in my mother language) which refers to an ”abstract individual,” and itwill be embarrassing if I make up such one. So I have to make a choice. Isometimes use ”she” in research papers, but given that I’m a male this mightbe somewhat articial. So I decided to use ”he.” I wish this doesn’t make mytexts look sexist.


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Acknowledgements

This book is largely based on my book published in Japanese (1st edition in2007, 2nd in 2013). I like to thank Minerva Shobo publisher and Mr. KentaroHorikawa for their cooperation which enabled the publication of that book. Mostof the materials are originally based on my lecture notes given at the Universityof Texas at Austin. Some topics are based on my lecture notes given at theUniversity of Glasgow.


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v

Mathematical Notation

I’m not actually using serious mathematics and the most of difficulties you mightface will be simply due to unfamiliar notations, which I use for the purpose of concision. Here I give brief description of them.

• x∈X is read as ”x belongs to X ” or ”x belonging to X .”• {x ∈ X : f (x)} denotes the set consisting of x belonging to X whichsatises proposition f (x).• ∀is universal quantier. ” ∀x; f (x)” is read as ”every x satises propositionf (x),” and ”∀x ∈ X ; f (x)” is read as ”every x belonging to X satisesproposition f (x).”

• ∃ is existential quantier. ”

∃

x; f (x)” is read as ”there exists at least onex which satises proposition f (x),” and ”∃x∈X ; f (x)” is read as ”thereexists at least one x belonging to X which satises proposition f (x).”

• =⇒ is the symbol of implication. For two propositions A, B , ”A =⇒ B ”is read as ”If A is true, then B is true.”• ⇐⇒ is the symbol of logical equivalence. For two propositions A, B ,”A⇐⇒B ” is read as ” A is true if and only if B is true.”• R denotes the set of real numbers.• R+ denotes the set of non-negative real numbers.

• R++ denotes the set of positive real numbers.

• Rn denotes the set of n -dimensional vectors of real numbers. Its elementis for example denoted by x = ( x1 , · · · , xn ), where its i-th coordinate isx i .• Rn+ = {x ∈ Rn : x i ≥ 0, i = 1 , · · · , n} denotes the set of n-dimensionalnon-negative vectors.• Rn++ = {x∈Rn : x i > 0, i = 1, · · · , n} denotes the set of n-dimensionalpositive vectors.• ∏ni =1 Ai denotes the product of n sets A1 , · · · , An , that is, A1 ×·· ·×An .


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Contents

I Individual Preference and Choice 1

1 On the concept of ”rationality” in economics 2

2 Choice objects and choice opportunities 112.1 Description of choice objects . . . . . . . . . . . . . . . . . . . . . 112.2 Opportunity sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Consumption set . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Preference 233.1 Preference relation . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Preference over consumptions . . . . . . . . . . . . . . . . . . . . 253.3 Marginal rate of substitution . . . . . . . . . . . . . . . . . . . . 313.4 Smooth preferences . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Convexity and diminishing marginal rate of substitution . . . . . 343.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 So-called utility function 364.1 ”Utility” representation of preference . . . . . . . . . . . . . . . . 364.2 Marginal utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Describing marginal rate of substitution by means of marginal

utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Ordinal utility and cardinal utility . . . . . . . . . . . . . . . . . 514.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Choice and demand 545.1 Maximal elements for preference . . . . . . . . . . . . . . . . . . 545.2 Smooth consumption choice . . . . . . . . . . . . . . . . . . . . . 575.3 The case of perfect substitution . . . . . . . . . . . . . . . . . . . 615.4 The case of perfect complementarity . . . . . . . . . . . . . . . . 635.5 Demand function . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.6 Consumption choice and demand in exchange economy . . . . . . 645.7 Describing choice as utility maximization . . . . . . . . . . . . . 65

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5.8 Expenditure minimization and compensated demand . . . . . . . 665.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Demand analysis 716.1 Normal and inferior goods . . . . . . . . . . . . . . . . . . . . . . 716.2 Ordinary and Giffen goods . . . . . . . . . . . . . . . . . . . . . . 726.3 Gross substitutes and gross complements . . . . . . . . . . . . . . 736.4 Elasticity of demand . . . . . . . . . . . . . . . . . . . . . . . . . 756.5 Substitution effect and income effect . . . . . . . . . . . . . . . . 776.6 Income evaluation of welfare change . . . . . . . . . . . . . . . . 816.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Willingness to pay and consumer surplus 897.1 Naive utility argument . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 The assumption of no income effect . . . . . . . . . . . . . . . . . 907.3 Marginal willingness to pay as marginal rate of substitution . . . 957.4 No income effect and inverse demand function . . . . . . . . . . . 977.5 Compensated variation, equivalent variation and consumer surplus 99

8 Intertemporal choice 1018.1 Intertemporal choice and intertemporal budget constraint . . . . 1018.2 How to deal with ination . . . . . . . . . . . . . . . . . . . . . . 1028.3 Discounted present value of streams . . . . . . . . . . . . . . . . 1038.4 Preference over consumption streams . . . . . . . . . . . . . . . . 1058.5 Intertemporal consumption choice . . . . . . . . . . . . . . . . . 1138.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9 Choice under risk 1169.1 Risk and uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 1169.2 Risk attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169.3 Expected utility representation: an experimental construction . . 1179.4 Expected utility representation: the formulation . . . . . . . . . 1199.5 Axiomatic characterization of expected utility representation . . 1209.6 ”Cardinal” properties of vNM indices . . . . . . . . . . . . . . . 1229.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.8 Violation of the expected utility theory . . . . . . . . . . . . . . . 1299.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10 Revealed preference 134

II Perfectly Competitive and Complete Market withComplete Information 139

11 Perfectly competitive and complete market with complete in-formation 14011.1 Perfect competition . . . . . . . . . . . . . . . . . . . . . . . . . . 140


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CONTENTS viii

11.2 Complete market . . . . . . . . . . . . . . . . . . . . . . . . . . . 14411.3 Complete information . . . . . . . . . . . . . . . . . . . . . . . . 146

12 Competitive equilibrium in exchange economies 14812.1 Exchange economy . . . . . . . . . . . . . . . . . . . . . . . . . . 14812.2 Competitive equilibrium . . . . . . . . . . . . . . . . . . . . . . . 14912.3 Interest rate in borrowing-lending economies . . . . . . . . . . . . 15212.4 Security exchange and security price . . . . . . . . . . . . . . . . 15712.5 Exerc ise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

13 Efficiency of competitive allocation 16513.1 Pareto improvement and Pareto efficiency . . . . . . . . . . . . . 16513.2 Efficiency of competitive equilibrium allocation . . . . . . . . . . 16913.3 Important remarks on Pareto efficiency . . . . . . . . . . . . . . . 171

13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17214 Production technology 173

14.1 1-input/1-output case . . . . . . . . . . . . . . . . . . . . . . . . 17314.2 2-input/1-output case . . . . . . . . . . . . . . . . . . . . . . . . 17514.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

15 Prot maximization and cost minimization 18115.1 Prot maximization when output price and input prices are given 18115.2 Cost minimization when input prices are given . . . . . . . . . . 18715.3 Long-run and short-run . . . . . . . . . . . . . . . . . . . . . . . 190

16 Cost curve and supply 193

16.1 Average cost and marginal cost . . . . . . . . . . . . . . . . . . . 19316.2 Prot maximization under perfect competition . . . . . . . . . . 19516.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

17 Competitive equilibrium in production economies 20017.1 Private ownership economy . . . . . . . . . . . . . . . . . . . . . 20017.2 The representative consumer/producer model . . . . . . . . . . . 20617.3 Interest rate determination in an intertemporal production economy21117.4 Efficiency of competitive equilibria . . . . . . . . . . . . . . . . . 21317.5 Socialist calculation debate . . . . . . . . . . . . . . . . . . . . . 220

18 Partial equilibrium analysis 22218.1 Competitive partial equilibrium . . . . . . . . . . . . . . . . . . . 22318.2 Pareto efficiency and maximal surplus . . . . . . . . . . . . . . . 22418.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228


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III Imperfect competition and game theory 229

19 Monopoly 23019.1 Monopoly equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 23119.2 Pareto inefficiency of monopoly equilibrium . . . . . . . . . . . . 23319.3 Price discrimination and monopolistic surplus extraction . . . . . 23419.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

20 Basic game theory I: normal-form games 24320.1 Description of strategic interdependence: normal-form games . . 24420.2 Dominant strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 24720.3 Iterated elimination of dominated strategies . . . . . . . . . . . . 24820.4 Rationalizable strategies . . . . . . . . . . . . . . . . . . . . . . . 25120.5 Nash equil ibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

20.6 Mixed strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 26020.7 Renement of Nash equilibria . . . . . . . . . . . . . . . . . . . . 26520.8 How should we think of multiple equilibria? . . . . . . . . . . . . 26820.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

21 Basic game theory II: extensive-form games 27221.1 Description of strategic interdependence: extensive-form games . 27221.2 Subgame-perfect Nash equilibrium . . . . . . . . . . . . . . . . . 27321.3 Extensive-form games with imperfect information . . . . . . . . . 27821.4 Bargaining game . . . . . . . . . . . . . . . . . . . . . . . . . . . 28021.5 Repeated games and sustainable cooperation . . . . . . . . . . . 28321.6 Exerc ise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

22 Oligopoly 28722.1 Simultaneous quantity setting (Cournot competition) . . . . . . . 28822.2 Sequential quantity setting: Stackelberg competition . . . . . . . 29122.3 Simultaneous price setting: Bertand competition . . . . . . . . . 29422.4 Sequential price setting . . . . . . . . . . . . . . . . . . . . . . . 29922.5 Convergence to perfect competition . . . . . . . . . . . . . . . . . 30222.6 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30422.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

IV Economic Analysis with Incomplete Information 308

23 Basic game theory III: games with incomplete information 30923.1 Bayesian game and Bayesian Nash equilibrium . . . . . . . . . . 30923.2 On the common prior assumption . . . . . . . . . . . . . . . . . . 31523.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316


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24 Auction 31724.1 Prominent auction formats . . . . . . . . . . . . . . . . . . . . . 31724.2 Information, timeline and the natures of values . . . . . . . . . . 31824.3 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31924.4 First-price auction . . . . . . . . . . . . . . . . . . . . . . . . . . 31924.5 Second-price auction . . . . . . . . . . . . . . . . . . . . . . . . . 32324.6 The revenue equivalence theorem . . . . . . . . . . . . . . . . . . 32524.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

25 Trade with incomplete information 32725.1 Adverse selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 32725.2 Moral hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33125.3 S ignaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33525.4 Speculative trade . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

25.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

V Market Failure and Normative Economic Analysis 344

26 Externality 34526.1 Market fai lure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34526.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

27 Public goods and the free-rider problem 35327.1 Public goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35327.2 Efficiency criterion: the Samuelson condition . . . . . . . . . . . 35327.3 The case of quasi-linear preferences . . . . . . . . . . . . . . . . . 357

27.4 The free-rider problem . . . . . . . . . . . . . . . . . . . . . . . . 35927.5 Strategy-proof mechanism . . . . . . . . . . . . . . . . . . . . . . 36027.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

28 Indivisibility and heterogeneity 36328.1 Allocation of indivisible objects . . . . . . . . . . . . . . . . . . . 36428.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36628.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

29 Efficiency, welfare comparison and fairness 37129.1 The Kaldor/Hicks criteria . . . . . . . . . . . . . . . . . . . . . . 37129.2 Fair allocation in exchange economies . . . . . . . . . . . . . . . 37729.3 Fairness in production economies . . . . . . . . . . . . . . . . . . 38029.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

30 Aggregation of preferences and social choice 38330.1 Motivations from welfare economics and political science . . . . . 38330.2 Axioms for aggregation of preferences . . . . . . . . . . . . . . . 38630.3 Arrow’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 38930.4 May’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389


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30.5 Borda rule again . . . . . . . . . . . . . . . . . . . . . . . . . . . 39030.6 Domain restriction and single-peaked preferences . . . . . . . . . 39130.7 Proof of Arrow’s theorem . . . . . . . . . . . . . . . . . . . . . . 393

31 Implementability of social choice ob jectives 39731.1 Social choice function and mechanism . . . . . . . . . . . . . . . 39731.2 Implementation in dominant strategy equilibrium . . . . . . . . . 39831.3 Implementation in Nash equilibrium and allowing multiple equi-

libria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40131.4 Appendix: Proof of the Gibbard-Satterthwaite theorem . . . . . 404

Postscripts 407

Solutions to the exercises 424


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Part I

Individual Preference andChoice

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Chapter 1

On the concept of ”rationality” in economics

Economics is very often criticized of making unrealistic assumptions. The mostcommon criticism will be about ”rationality,” saying that real human beings arenot rational as assumed in economics.

I agree to some of them eventually (probably not in the way the readersexpect), but let me give some clarications before I proceed, since the word”rationality” is broad — in daily life the word ”rational” or ”irrational” hasbeen used even as a convenient rhetoric to justify and praise or criticize and dissomebody’s choice or action while maintaining the appearance of being value-

neutral.It is obviously a hard problem to summarize the notion of rationality ineconomics so that everybody can agree. Let me try, however, to summarize whatI understand is consistently underling economic theory. I would say ”rationality”in economics refers to that

1. an individual has certain consistent subjective criterion of value (calledpreference);

2. he takes all the relevant contingencies into account and perceive themcorrectly;

3. he goes through ”logically correct” reasonings; and

4. he fullls the criterion up to the maximum.

That is, the notion of ”rationality” here is purely a formal one at an individuallevel. As far as the above conditions are met we have to say that one is ”rational”even if he is a vicious killer. At the same time, this notion of rationality does notpresume that one is ”selsh,” and does not exclude altruism to be a componentof individual’s subjective criterion at all, as far as it remains to be consistent.

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CHAPTER 1. ”RATIONALITY” 3

Put differently, some action being ”rational” for an individual and its beingsocially desirable are different issues.

Of course this claried notion of rationality again faces criticisms, sayingthat real human beings are not consistent or knowledgeable or precise or smartas described above. It is an ”idealized” principle which is impossible if you takeit literally.

To borrow an outdated analogy (it’s a shame but I can come up with onlythis one), this is analogous to how physics starts its rst-step argument byassuming vacuum and no friction. There is no vacuum or frictionless situationin reality, but such assumption helps us to build the rst several laws in classicalmechanics, which kicks us upstairs so that we can understand more realisticsituations.

I view that in social sciences it is not only helpful to start with such idealizedassumption but rather necessary, in that we cannot see or understand ”reality asit is” without standing on such ”baseline.” Reality is of course different from thebaseline, but it can be understood only by seeing how it is different or distantfrom the baseline.

A natural question arising here is what is the postulate for a good choice of such baseline. A most extreme form of ”positivist” view says that it doesn’t haveto have anything to do with reality and it should be the simplest assumptionunder the simplest setting which can derive predictions consistent with realphenomena as many as possible, and it is rather better as it is more unrealistic.It says for example that it is meaningless to test whether individuals are reallysolving their maximization problems rationally, and what is important is that

their behaviors are explained ”as if” they are solving maximization problemsrationally.I don’t take this view, however, because it does not say anything about

the necessity of particular assumptions, as there may be several equally simpleprinciples which can explain the reality in the ”as if” way. Why do we have tochoose the above assumptions over the others?

Also, economists have a task to do welfare analysis and provide normativearguments, which critically depend on how much individuals are responsiblefor their rationality. If an economist takes the above ”pure positivist” view heshould not be able to draw any normative implication from his positive analysis.If he does so it must be a deception. A typical deception is that in ”positive”analysis one describes individuals’ choices ”as if” they are acting rationally andin its normative implication he switches the interpretation implicitly so that theindividuals are indeed rational and responsible for their choices.

Another view about rationality often invoked is an evolutionary story, whichsays that if one is not rational he would die or perish either in the social orbiological sense, meaning that there is little to lose by assuming that those whoare living (that is, who have survived by now) are rational.


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I don’t take this view either, since it says at most that certain characteristicsmake one the ”ttest” and more likely to survive under certain environment.We cannot draw any normative implication from this either, while applicationof evolutionary arguments to social sciences often falls in the confusion thatsuch characteristics are desirable and those with such characteristics should be dominant in the society, even in a modern civilized society — it is in thebeginning strange to insist that, since if some people are really the ttest theywould have been already dominant before saying ”should.”

I would say, the primary role of economics is to provide a consistent andmeaningful understanding of commensurability between different indi-vidual values each of which is solid, stable and deliberate, and howto realize such commensuration . It is not to provide an explanation orprediction of ”behavior in general,” nor to grab an organic formation of valuesentiments in the society ”as a whole.”

The choice of baseline should serve this objective, and certain abstraction isnecessary in order that we can clearly see solid, stable and deliberate individualvalues.

Note that such abstraction is purely a formal one, in the sense that weidentify time horizons, spaces and contingencies over which the notion of solid,stable and deliberate individual values makes sense, by abstracting away certainranges of idiosyncratic details of choice situations. It is not selecting a particularcontent of social life over another, such as selecting ”economic rationality” and”abstracting away” the other ones such as political, social and cultural.

Of course, in this sense, we should note that economics has an ”imperialisticambition” which tries to apply its methodology to any aspect of social life that

possesses the same formal structure.It is natural that you wonder if such rationality approach works (or should

work). So let me briey explain the nature of the approach, what types of abstraction are carried out there, and list challenges to it.

To illustrate, let x denote input, which is an observable external elementgiven to the individual (such as constraint or situation), and let y denote output,which is his observed behavior. If we take the most extreme stance of so-called behavioralism, then we consider only a functional relationship which holdsbetween input and output. Denote such functional relationship by f , then therelation between input and output is denoted by

y = f (x).

In the empirical side, given data consisting of pairs of input and output

(x1 , y2), (x2 , y2), · · · , (xn , yn ),


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By the way, I put quotation marks on ”logically correct” in Postulate 3in the above formulation. This obviously suggests that I don’t want to beseen as naively believing that ”logic” is single and universal. Yes, the waysof reasonings used in this book are limited to the standard formal logic andprobability theory. However, legal reasonings such as argumentum e contrarioand analogy are not quite a logical operation in the sense of formal logic, butthey may be a convincing ”logic” indeed. There may be ”rational” choices basedon ”logically correct” reasonings which are different from the standard formallogic, and it is one of the jobs of theorists to investigate the consequences of being ”rational” in such sense.

The ”rationality” approach has always been facing criticisms.The rst thing one can think of in order to absorb the criticisms is to in-

troduce ”noise” or ”unobserved heterogeneity,” denoted ε here, and extend theabove rationality model like

y = g(θ, x) + ε

When ”noise” ε is zero ”on average” we can say that the individual is ”rationalon average.” This is exactly the sense in which I wrote ”individually solid andstable” above. 1

Now, how should we think when a deviation from ”rationality” looks sys-tematic and cannot be explained by ”errors” or ”noise?” Let me bring up someissues which I like to share before proceeding.

Issue 1: Preference is not xed, but it rather changes over time.

In order to deal with this issue we need to select the most relevant time horizon to the given problem. For, in my view, most of the cases which are regarded asshowing ”irrationality” or ”inconsistency” of human choices are simply due tothe observer’s misspecication of the relevant time horizon.

For example, even when somebody ate pasta for lunch yesterday and ham-burger for lunch today we won’t say his choice over foods is inconsistent, sincewe know that the problem of what to eat today is not identical to or separatefrom the problem of what to eat on another day. Much less when we spend ourlunch budget on weakly or monthly basis.

That is, what we observe here is not a contradiction like

pasta > hamburger , hamburger > pasta

but a revealed ranking

(pasta , hamburger) > (pasta , pasta) , (hamburger , hamburger) , (hamburger , pasta) ,

1 Of course it is no more than an analogy at this point since we don’t yet have the denitionof ”+.”


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where (pasta , hamburger) denotes the choice of eating pasta on the rst day andhamburger on the second day (although we cannot specify the ranking amongthe three on the right-hand-side from this observation alone).

It is natural to wonder, ”As you set the time horizon longer you can absorbvariation of choices across periods into the length of the time horizon. Doesn’tit mean that by taking time horizon arbitrarily long we can explain anythingas ”rational” choice generated by a ”xed” preference over arbitrarily long objects?” In other words, if we take it literally that life is just once anything is”rational” since there is only one sample.

This is ultimately a problem for the outside observer, who judges how longthe time horizon should be taken so that observed choice data are seen as a ”rep-etition of some complete problem” and it is meaningful to think of consistencyand inconsistency across samples. For example, you can think of proportions of kinds of lunch meals during one month, and see the data as a repetition of suchmonthly summary.

What if the life is not a repetition of an identical problem, and like a ”wholelife” only one dynamic choice problem is given to an individual and we canobserve just one sample of his life path? In statistical treatment of such dynamicchoice problems, we usually consider that people are ”ex-ante identical” andtake different life paths because of different inputs, which are observable, andunobservable noises.

Now, how should we think if we see inconsistencies of choices even after se-lecting the time horizon as adequately as possible? Let us think of the followingexample.

Example 1.1 You have a choice of starting cocaine or not. There are threepossible paths:

• A: Start it and quit it later.• B: Start it and continue.• C: Don’t take it at all.As you have not started cocaine yet and you are curious, your preference

over the paths is

A > C > B

However, because of the nature of addiction, once you start cocaine you becomea different person, literally, and the preference of your new personality is

B > A

That is, there are two different ”selves,” before and after taking cocaine, whohave contradicting preferences.


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In such cases one preference cannot simply determine the choice. Thereare at least two ways of choice. One is that the current self makes choice by(mistakenly) believes that he can control his future selves. It is called naivedecision. In the above example, the naive decision is to start taking cocaine,intending to stop later, and does not actually stop it later. The other oneis that the current self foresees how future selves behave, and makes choice bytaking how future selves behave as a given constraint. It is called sophisticateddecision. In the above example the sophisticated decision is not to take cocaineat all, given that the future self cannot stop taking cocaine once he starts it.

The cocaine example may be a bit too extreme, but this type of problemoften occurs in choice with habit formation.

Let us think of one more example.

Example 1.2 Consider the following two choice problems.Problem A

A1: Receiving 1000 dollars after one year.

A2: Receiving 1050 dollars after one year and one week.

Problem B

B1: Receiving 1000 dollars now.

B2: Receiving 1050 dollars after one week.

The example as presented like above is somewhat misleading since you can

save money, so assume that you have to spend the money immediately afterreceiving. Then, (in more carefully designed experiments) the pair of choiceslike A2 and B1 is frequently observed.

What’s the problem with this? Suppose you choose A2 in A and B1 in B.Then you would choose to sign a contract to receive 1050 dollars after one yearand one week, rather than a contract to 1000 dollars after one year. After oneyear, you will regret, since you like to receive 1000 dollars immediately ratherthan to wait for one more week. Thus, there is a conict between current self and self after one year.

How do economists think when such ”successive selves” are present? Mostlywe then take an individual as a ”society” consisting of different ”selves,” andgame theory or social choice theory to such micro-society. Although, we shouldbe careful about the applicability of these theories to the micro-society, since”successive selves” are not totally different persons from each other. I will cometo this issue in the last part of the postscripts.

Issue 2: It is untrue that an individual chooses the best available thing for him.There are cases in which he gives up selsh choice, due to certain socialreasons.


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Consider the following example. Suppose you receive 100 dollars from some-body, and you can either spend all of that for yourself or give half of that toyour brother. If you care only about your consumption in the current periodyou will spend all for your self, but many of you may give half of that to yourbrother.

We can explain this in two ways.

1. The apparent departure from ”rationality” is again due to misspecicationof time horizon and relevant contingencies. There is nothing wrong in thata rational action causes loss in the short-run whereas it realizes gains inthe long run.Also, there is nothing wrong in that a rational action causes loss underparticular state, while it is protable in expectation from the ex-ante view-point. Insurance is a typical example.In such a way, we can explain mutual help as a collection of individuals’selsh behaviors in the long run or under uncertainty.

2. Altruism and care for social status are nothing but a part of preference. Itappears that an individual is not choosing what he likes because the out-side observer is mis-specifying his preference or captures it only partially.In such cases, an individual compares between satisfaction of is his ”self-ish” motive and satisfaction of his altruistic motive and his taste for socialstatus, and after ”weighing” he makes the total decision.

By going through the above ways we can extend the standard choice theory,often borrowing helps of game theory to be covered in the later part of the

book. I would say that economics puts priority on the rst way, since allowingthe second way without discipline may lead to ”anything goes.” In any case, fromthe viewpoint of ”rationality” as summarized above it is not essential whetheran individual is ”selsh” or not.

Issue 3: Individual’s choice criterion does not exist independently of choicesituation.

In the above model, the structural parameter θ is supposed to exist prior toand independently of x being given. That is, preference is supposed to existindependently of choice situation. The ”rationality” approach then considersthat behavior is a function of preference and choice situation.

It is shown by many experimental studies, however, that individual’s choice

criterion depends on how choice opportunities are given. The following exampleis due to Benartzi and Thaler [3].

Example 1.3 Consider coin ipping and the following two choice problems.

Problem 1: Split 100 dollars between two securities below.


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Security A: It doubles the allocated money if it shows head, and it becomes junk if tail.

Security B: It doubles the allocated money if it shows tail, and it becomes junkif head.

Problem 2: Split 100 dollars between two securities as below.

Security A: It doubles the allocated money if it shows head, and it becomes junk if tail.

Security C: It returns the allocated money as it is regardless of the coin-ipoutcome.

Since head and tails are equally likely it is natural to split money equallybetween A and B in Problem 1. However, it is reported that they tend to splitmoney between A and C in Problem 2 as well. This is strange from ”rational”viewpoint, since splitting money equally between A and B is nothing but buyingSecurity C, and there is no point in allocating money to A in Problem 2.

Such deviation from rational choice, which is inconsistent but non-randomand has certain tendency, is called an anomaly .

How does a ”die-hard” rational choice theorist handle this problem? Hewould go one step back, and take an ex-ante viewpoint. In the above example,he would consider probability distribution of security choice problems to befaced by the individual, and takes ex-ante evaluation of such distribution, bymeans of expectation calculation.

The last criticism will be,

4: Humans are not ”choosing.”

I would say, if humans are not ”choosing,” all what we can say is ”it is what itis” and we cannot talk about ”value” or what is ”good” or ”bad” or ”better”or ”worse.”

I know I’m taking the order of causality upside-down, and it must be absurdto say that the nature has to work so that the rationality approach allows us totalk about ”values.” But the problem is obviously beyond my intellect.

Let me stop here for now. I know there are many undiscussed problems, butI think it is better to share them after digesting the main body of this book. Iwill come back to this at the end of the postscript.


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Chapter 2

Choice objects and choiceopportunities

2.1 Description of choice objectsThe method (not substantive contents conveyed by it) covered in this bookcan be applied to general kinds of spheres of society, not only to individualand material consumptions. It allows for example that one’s consumption mayaffect other ones’ consumptions (externality), and that there may be a goodwhich many people can use at the same time (public good). Also it is notlimited to material consumption but can be applied to non-material kinds of social actions such as political or social or cultural choice.

In the beginning, the set of choice objects X is just an abstract set. Forexample, the set of choice objects in US presidential election is let’s say

X = {Obama , Clinton , Romney , Palin , McCain , · · ·},and the set of choice objects in the problem of which school to attend is let’ssay

X = {Univ. A , Univ. B , Univ. C , Univ. D , Univ. E , · · ·},and the set of choice objects in the problem of which company to work for islet’s say

X =

{Co. A, Co. B, Co. C, Co. D, Co. E,

· · ·}.

I guess the readers wonder here. ”How can we choose from them even whennot all of them can be the presidential candidates?” ”How can we choose fromthem even when not all of those schools make offers to me?” ”How can we choosefrom them even when not all of those companies make offers to me?” Here Itake X to be the set of all the conceivable and potentially available objects,putting it aside which ones are actually available to choose.

11


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CHAPTER 2. OBJECTS AND OPPORTUNITIES 12

The rst step in microeconomics is to write down the right set of choiceobjects according to the interest. Consider again the example

X = {Co.A, Co.B, Co.C, Co.D, Co.E , · · ·}.Here it is implicitly assumed that salary is already a xed component of eachcompany’s feature. However, one can consider that salary is an explicit variableas well, then the set of choice objects is

X = {Co.A, Co.B, Co.C , Co.D, Co.E , · · ·}×R+ ,where R+ denotes the non-negative half line. Its element is for example (Co.D , m ),which means working for Company D for salary m.

Also, one can consider that which city to work in is also an explicit variableas well. Then the set of choice objects is

X = {Co.A, Co.B, Co.C , Co.D, Co.E , · · ·}×R+ ×{City α, City β , City γ , · · ·},and its element is for example (Co.D ,m,γ ), which means working for CompanyD for salary m and living in City γ .

Also, consider what to eat for lunch then the set of choice objects is

X = {pizza, humburger , pasta , sandwitch , sh and chips , · · ·}but if you are talking not just about lunch for today lunch but also about lunchfor tomorrow, the right description is

X = {pizza, hamburger , pasta , sandwich , sh and chips , · · ·}×{pizza, hamburger , pasta , sandwich , sh and chips , · · ·},

and its element is for example (sandwich , pizza), which says ”eating sandwichtoday and pizza tomorrow.”

2.2 Opportunity setsAs explained above, the set of choice objects X consists of all the potentiallyavailable ones. However, in actual choice opportunities we are given only asubset of it. Let us call it an opportunity set . Denote it let’s say by B , thenit must satisfy B

⊂X and B

̸

=

∅

.In the example of school choice, given the set of all schools

X = {Univ. A , Univ. B , Univ. C , Univ. D , Univ. E , · · ·},the set of schools one can be admitted to is let’s say

B = {Univ. C , Univ. E , Univ. J }.


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In the example choosing which company to work for, given the set of all com-panies

X = {Co. A, Co. B, Co. C, Co. D, Co. E, · · ·}.the set of companies from which one can get an offer is let’s say

B = {Co. A, Co. K, Co. M, Co. Q}.Here let B denote the family of opportunity sets which are institutionallypossible. The simplest form of B will consist of all the non-empty subsets of X , but it is not always the case institutionally. For example, in US presidential

election because no more than one candidate can run from one party we cannothave a choice opportunity like

B =

{Obama , Clinton

}.

2.3 Consumption setSo far, the set of choice object X or choice objects x and y can be anything.However, in the rst half of this book we consider individual consumptionsmostly, as far as we are concerned with market theory.

In this context the set of choice objects, which consists of all the potentially possible combination of consumptions, is called consumption set . Again, be-cause consumers are constrained by their incomes not all of them are alwaysavailable to choose. I need to explain consumption set rst, however.

2.3.1 Standard consumption setTo simplify the explanation, we mostly assume that there are just two goods.Of course this does not mean that there are really only two goods in the world,and it is simply that the two-good illustration is enough for understanding of the contents covered in this book. Here let me call them Good 1 and Good 2.

Also, we mostly assume that each good is homogeneous and divisible .Like gasoline, we consider that this 1 gallon of it and that 1 gallon of it areidentical and we can buy it in arbitrarily ne quantities such as 1.367... gallons.Of course actual accounting does now allow this but let us consider that wecan do something like this as closely as possible. On the other hand, thishouse and that house are typically different. They are heterogeneous . Alsotypically we cannot buy 0.47 units of house. It is indivisible . We will consider

heterogeneous goods and indivisible goods in the next section and in a laterchapter.As we assume each of the two goods is homogeneous and divisible, the con-

sumption set is given as the non-negative quadrant of the 2-dimensional planeR2+ . Its element x = ( x1 , x2) is called consumption vector . When the con-sumer is receiving x = ( x1 , x2) it means he is receiving x1 units of Good 1 andx2 units of Good 2 (see Figure 2.1). For example, when Good 1 is gasoline and


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x

x1

x2

Good 1

Good 2

Figure 2.1: 2-dimensional consumption set

Good 2 is water, consumption vector x = ( x1 , x2) refers to x1 units of gasolineand x2 units of water.

We assume that consumption of each good is non-negative, but dependingon interpretation it is possible to think of negative consumption, and I will cometo it when it is necessary to consider.

2.3.2 Indivisible goodsThere are variations in how to describe heterogeneity and indivisibility, but here

let me pick a simple illustration: Good 1 is indivisible but homogeneous andGood 2 is homogeneous and divisible. I will take heterogeneity and indivisibilitymore seriously in Chapter 28.

Good 1, which is homogeneous but indivisible, allows only integer amountsof consumption. Good 2 is the same as before. Then the consumption set isZ+ ×R+ . See Figure 2.2, where the rst-coordinate consists only of integervalues. For example, x = ( x1 , x2) is in the consumption set because x1 = 3is an integer. On the other hand, y = ( y1 , y2) is not in the consumption setbecause y1 is not an integer.

2.3.3 Labor and leisureIn the above formulation of consumption set we have assumed that unless theconsumer is constrained by his budget we may consider arbitrarily large amountsof consumptions. This will be inadequate for the case of labor and leisure, asone cannot work more than 24 hours a day in the beginning. Therefore, whenwe analyze the choice of labor and leisure we assume that available hours perperiod are limited in the outset, after excluding minimal necessary hours forsubsistence such as sleeping hours.


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x

y

Good 1

Good 2

Figure 2.2: Good 1 is indivisible

In economics we mostly take the description that the consumer choosesleisure hours out of his disposable hours, and out the rest to labor (thoughit doesn’t have to be). For simplicity let the disposable hours be 1. On theother hand, let us assume that there is just one consumption good or thatall the consumption goods have been aggregated into one. This is enough forunderstanding tradeoffs between consumption and leisure.

Then the set of possible combinations of leisure and consumption is [0 , 1]×R+as depicted in Figure 2.3. When its element ( l, c) is given it means that theconsumer puts l into leisure and 1

−l into labor, and consumes c units of the

consumption good.Of course one can think that labor and leisure are an indivisible. Right, in

reality one can be either employed or unemployed rather than he can choosebetween working 35 hours or 40 hours per week. Also, where to work willmatter as well. I will come to such problem of indivisibility and heterogeneityin Chapter 28.

2.3.4 Consumption over timeWhat is important in economics is that even if goods are materially the samethey are treated as different goods if they are to be consumed at different timeperiods and different contingencies. For example, gasoline to be consumed todayand gasoline to be consumed tomorrow are different goods. Saving is an actionto buy future consumptions by means of selling current consumptions.

The simplest model of such intertemporal consumption is 2-period model.Assume that there are just two periods, Period 1 and Period 2, and there is justone material good in each period. It might look too simple, but this is enoughfor the understanding and it can be extended to many periods.

Then the consumption set is the non-negative quadrant R2+ . That is, when


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(l, c)

Leisure

Consumption

Figure 2.3: Leisure/labor and consumption

a consumption vector x = ( x1 , x2) is given to the consumer it means the heconsumes x1 units in Period 1 and x2 units in Period 2. We call this a con-sumption stream . It may be too short to be called a ”stream” but we adoptthis word in order to emphasize the relevance of time.

2.3.5 Consumption under uncertainty: state-contingentconsumption

Same argument holds for uncertainty as well. For example, 1 gallon of gasolinewhen Republicans win the US presidential election is a different good than onegallon of gasoline when Democrats win. If you have to make some investmentdecision before the election your choice = bet is described in the form of state-contingent consumption .

To simplify, focus on the case that there are just two possible states of theworld, like ”Republicans or Democrats” and ”hot summer or cold summer.” Callthe rst one State 1 and the second one State 2, and there is just one materialgood at each state. It might look too simple again, but this is enough for theunderstanding and it can be extended to many states.

Then the set of state-contingent consumption vectors is described by the non-negative quadrant R2+ . That is, when a vector of state-contingent consumptionx = ( x1 , x2) is given it means that the consumer receives x1 units of consumption

at State 1 and x2 units st State 2.

2.4 Budget constraintConsumption set describes all possible combinations of consumptions which arepotentially available for the consumer. Not all of them are actually affordable,however, and he is constrained by his budget according to this income and


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y p1x1 + p2x2 = w

w/p 1

w/p 2

Good 1

Good 2

Figure 2.4: Budget constraint

prices of the goods. That is, the set of consumptions affordable under thebudget constraint is a subset of the consumption set.

Budget constraint in the standard form

Let us think of the simplest case, which I call standard form. When a pair of prices (called price vector ) p = ( p1 , p2) and income w are given, any affordablecombination of consumption x = ( x1 , x2) must satisfy

p1x1 + p2x2 ≦ w.

As the left-hand-side is expenditure and the right-hand-side is income, the aboveinequality says that expenditure should not exceed income.

Graphically speaking, any affordable consumption vector cannot go outsideof the triangle depicted in Figure 2.4. For example, consumption vector y =(y1 , y2) is not affordable.

Remark 2.1 One may naturally ask, ”where does the price p = ( p1 , p2) comefrom, and how is it determined?” Please delay this question until the chapterson market. Here I’m just talking about how consumers respond to given prices.

Given a price vector p = ( p1 , p2) and income w, denote the set of consump-tion vectors satisfying the budget constraint by

B ( p, w) = {x∈R2+ : p1x1 + p2x2 ≦ w}This is called budget set . Graphically, B( p, w) corresponds to the area sur-rounded by the triangle as in Figure 2.4. Its upper-left face is called budgetline . When the consumer spends all his income his consumption vector mustline on the budget line. Budget line is described by the equality

p1x1 + p2x2 = w,


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which is called budget equation . When a consumption vector is strictly belowthe budget line it means the consumer is not spending all his income.

Remark 2.2 In most cases we consider that our consumer spends all his incomeand chosen consumption vector lies on the budget line, and that the budget con-straint is met with equality. One may say, ”that’s wrong, it excludes saving.”This does not take future consumption into account as a different good. In eco-nomics goods to be consumed at different time periods are taken to be differentgoods. As you take all the relevant time periods into account it is without lossof generality to assume that the consumer spends all his income in the end.

One may ask, ”what about bequest?” Yes, this should be counted as differenta consumption good.

By the way, let us consider that prices of all goods and income are both

doubled, that is, consider a change from ( p, w) = ( p1 , p2 , w) to (2 p, 2w) =(2 p1 , 2 p2 , 2w). How does the budget constraint change?You will immediately see that it does not make any difference. The new

budget constraint is2 p1x1 + 2 p2x2 ≦ 2w,

but as you divide both sides by 2 it is the same as the original budget constraint

p1x1 + p2x2 ≦ w.

In general, for all positive number λ > 0 we have B (λp, λw ) = B( p, w).

Opportunity cost

What you have to give up when you choose something is called its opportunitycost . As the slope of budget line is − p1 p2 , when you like to increase the amountof Good 1 you have to give up p1 p2 units of Good 2. Thus opportunity cost of extra 1 unit of Good 1 is p1 p2 units of Good 2. Likewise, opportunity cost of extra 1 unit of Good 2 is p2 p1 units of Good 1.

2.4.1 Budget constraint in exchange economyIn the above budget constraint in the standard form I did not specify the sourceof income w. Income may have many sources in reality, such as sales of goods,wage, returns from assets, dividend from rm shares, and so on. Here let meconsider the simplest one, income in an exchange economy.

In an exchange economy each consumer brings her initial endowmente = ( e1 , e2) to the market. Then he either sells Good 1 and buys Good 2, orsells Good 2 and buys Good 1, or sells or buys nothing. Given a price vector p = ( p1 , p2), his income is the market valuation of his initial endowment

p1e1 + p2e2 .


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x e

p1x1 + p2x2 = p1e1 + p2e2

Good 1

Good 2

Figure 2.5: Budget constraint in an exchange economy

Hence the budget constraint is

p1x1 + p2x2 ≦ p1e1 + p2e2 .

Here we assume that initial e is xed, and only p = ( p1 , p2) is variable. Thendenote the set of consumption vectors meeting the budget constraint by

B ( p) = {x∈R2+ : p1x1 + p2x2 ≦ p1e1 + p2e2}.Note that here once price p is given income is determined as well.

Likewise, the budget line is described by

p1x1 + p2x2 = p1e1 + p2e2

Note that the initial endowment point is always in the budget line as in Figure2.5.

If the consumer chooses a point like x on the budget line which is left to theendowment point as in Figure 2.5, that is, if it holds

x1 < e 1 , x2 > e 2

then he is selling Good 1 and buying Good 2. Similarly for the opposite direc-tion.

Numeraire

By the way, it is immediate to see that the budget constraint p1x1 + p2x2 ≦ p1e1 + p2e2 is equivalent to

p1 p2

x1 + x2 ≦ p1 p2

e1 + e2 .


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That is, in an exchange economy only relative price does matter, not itsabsolute level. Therefore it is OK to normalize the price of some good equal to1. Such good is called numeraire . Any good can be a numeraire, but here let’ssay it is Good 2, and let p denote the relative price of Good 1 for Good 2, thenthe budget constraint it.

px1 + x2 ≦ pe1 + e2 .

2.4.2 Labor and consumptionThe labor-consumption model can be treated as a special case of the model of exchange economy. Here the consumer’s initial endowment is 1 unit of disposablehours and e unit of the consumption good. In the consumption space it is (1 , e).

Let q denote wage and p denote the price of the consumption good. If theconsumer puts l unit hours into leisure he works for 1

−l unit hours. Then he

earns income from labor q (1 −l). His income consists of this and the marketvalue of his initial holding of the consumption good e, which is pe. Thereforehis consumption c must follow the constraint

pc≦ q (1 −l) + pe.Note that this is equivalent to

ql + pc≦ q + pe,

in which the right-hand-side is the market value of initial endowment (1 , e).As only relative price does matter in exchange economies, here let’s take the

consumption good to be the numeraire and divide the both sides of the above

by p, then we get q p

l + c ≦ q p

+ e.

Here q p is the wage measure by the consumption good, which is the real wage .

Here if the consumer wants to increase 1 extra unit of leisure then he has togive up q p units of consumption. Thus, the opportunity cost of extra 1 unit of leisure is q p units of the consumption good.

2.4.3 Saving and borrowingLet me repeat that even if goods are materially the same they are treated asdifferent goods if they are to be consumed at different time periods. We describethis by the two-period model, and let me introduce budget constraint here.

In the two-period model initial endowment is interpreted as earning stream .That is, when the consumer has initial endowment e = ( e1 , e2) it means thathe earns e1 units of the consumption good in the current period and e2 units inthe future period. It might be too short to be called a ”stream,” but let me gowith this.


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Denote the pure interest rate by r. Assume there is no ination, as I willcome to it in Chapter 9.

Now suppose one consumes x1 units of the consumption good in the currentperiod. Then he saves e1 −x1 units, which can be negative and in that casehe is borrowing. Return from saving (or repayment for borrowing) to comein the future period is obtained by multiplying the gross interest rate 1 + rto e1 −x1 , which is (1 + r )(e1 −x1). Thus the upper limit of consumptionin the future period consists of this plus earning in the future e2 , hence it ise2 + (1 + r )(e1 −x1).Therefore consumption in the future period x2 has to obey

x2 ≦ e2 + (1 + r )(e1 −x1).By rewriting this we obtain

(1 + r )x1 + x

2 ≦ (1 + r )e

1 + e

2.

Here the right-hand-side is the amount of consumption in the future periodwhich is obtained when the consumer saves all the earning in the current period.As it is the lifetime earning measured by future consumption it is called futurevalue of lifetime earning.

Note that this is a special case of the budget constraint in standard form,where future consumption is taken to be the numeraire, that is, p1 = 1 + r and p2 = 1.

On the other hand, as we divide both sides of the above by 1 + r we obtain

x1 + 11 + r

x2 ≦ e1 + 11 + r

e2 .

Here the right-hand-side is the amount of current consumption one can obtainif he borrows up to limit against his lifetime income. As it is the lifetimeearning measured by current consumption it is called present value of lifetimeearning

Again note that this is a special case of the budget constraint in standardform, where current consumption is taken to be the numeraire, that is, p1 = 1and p2 = 11+ r .

Future value of lifetime earning corresponds to the x2-intercept of the budgetline, and present value of it corresponds to the x1-intercept of the budget line. Ineither formulation, the (absolute value of) slope of the budget line is p1 p2 = 1+ r ,which means that gross interest rate is the relative price of current consumptionfor future consumption. In other words, gross interest rate is the opportunitycost of 1 extra unit of current consumption measured by future consumption,

as the consumer has to give up 1 + r units of future consumption as he gets oneextra unit of current consumption.

2.5 ExercisesExercise 1 You have 120 units of income. Price of Good 1 is 4, that of Good2 is 3.


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(i) Write down the budget constraint.(ii) What is the relative price of Good 1 for Good 2?(iii) Suppose Good 1 is taxed 20% per price and Good 2 is taxed 0.5 per unit.Then write down the new budget constraint.


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Chapter 3

Preference

3.1 Preference relationPreference relation describes an individual’s subjective ranking over choice objects. It is denoted by ≿,≻,∼. To help understanding you may take an analogyto inequality and equality symbols ≧, > , =, while this analogy is not quite rightas seen below.

Let X be the set of choice objects, which is at this point abstract and it mayconsist of anything . Then, given choice objects x, y ∈X , the relation

x ≿ y

is read as ” x is at least as good as y for the individual.”Likewise,

x≻y

is read as ” x is better than y for the individual.”Also,

x∼y

is read as ” x is as good as y for the individual” or ”the individual is indifferentbetween x and y.”

We will need ≿ only, under the completeness condition introduced below,because x≻y may be dened by ” y ≿ x is not true,” and x∼y may be denedby ”both x ≿ y and y ≿ x are true.”

Above I wrote that the analogy to ≧, > , = is not quite right. This is becausetwo difference objects can be equally preferable. That is, the relation x∼y canbe true for two different objects x and y. On the other hand, the equalityrelation x = y can be true only when x and y are an identical object.

Throughout the book we assume that individual preference relation satisesthe following two properties. One is

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CHAPTER 3. PREFERENCE 24

Completeness For every x, y ∈X , it holds at least one of x ≿ y and y ≿ x.Note that this allows both x ≿ y and y ≿ x hold, and we have x∼y then.Completeness says that the individual can compare any two alternatives. Itmight sound obvious, but it is not. Consider the following example.

Example 3.1 Mr. S has double selves. Call his one self S1 and the other S2.S1 has preference ≿1 and S2 has ≿2 , each of which satises completeness. Giventhese, Mr. S forms unied preference ≿ by

x ≿ y holds if and only if both x ≿1 y and x ≿2 y hold.

Each of his selves has complete preference, but the preference of his unied self fails to satisfy completeness. When x ≿1 y but y ≻2 x, the unied self cannotdetermine which one is better or if they are equally preferable. Similarly for thecase that y ≻1 x and x ≿2 y.

The other assumption is transitivity

Transitivity : If x ≿ y and y ≿ z, then x ≿ z

This requires that preference does not cycle, which means that it is consistent.The following example shows violation of it.

Example 3.2 Mr. N has triple selves. Call his selves N1, N2 and N3, respec-tively. N1 has preference ≿1 , N2 had preference ≿2 , and N3 has preference ≿3 ,each of which satises transitivity.

Given these, Mr. N forms his unied preference ≿ by majority voting among

the three selves.Even if each of his selves satises transitivity, the preference of his unied self fails to satisfy it. For example, consider a prole

x≻1 y ≻1 zz ≻2 x≻2 yy ≻3 z ≻3 x

Then x beats y by 2 vs. 1, y beats z by 2 vs. 1, and z beats x by 2 vs. 1, andit violates transitivity.

As I acknowledged in Chapter 1 I will proceed with assuming that at leastindividual preferences satisfy completeness and transitivity. Of course, as sug-gested by the above examples if we attempt to aggregate individual preferencesinto a ”social preference” it is non-obvious if that can satisfy completeness andtransitivity (see Chapter 30 for details).


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x

A

B C

Good 1

Good 2

Figure 3.1: An indifference curve

3.2 Preference over consumptions3.2.1 Indifference curvesBelow, choice object x is not an abstract point but refers to a 2-dimensionalconsumption vector x = ( x1 , x2).

In the 2-dimensional consumption space, you can can think of preference justlike you think of level curves of a mountain. For example, the set of consumptionvectors which are better than x = ( x

1, x

2) is described by area A in Figure 3.1.

In other words, it is the area ”higher than” x.Likewise, the set of consumption vectors worse than x is described by area

B. It is the area ”lower than” x.And, the border between A and B consists of consumption vectors which are

equally preferable to x. We call this an indifference curve . It is just like alevel curve of a mountain. When you draw an indifference curve passing throughanother consumption vector y = ( y1 , y2) which is preferred to x, it looks likein Figure 3.2. This is a level curve which is passing above x. Likewise, we candraw a series of indifference curves like in Figure 3.2. Because we can actuallydraw indenitely many indifference curves what I drew in the gure is only apart of that.

Here I took an analogy of level curves of a mountain, but in our case thereare no numbers which describe the ”height” of the mountain. This is becauseindifference curves describe only a relative ordering about which one is better orworse. We can know if something is better than another for a given consumer,but cannot know ”how much she is happy,” and such quantitative statementdoes not have economic content. Therefore our level curves, i.e., indifferencecurves, do not accompany numbers signifying heights.

Notice that under Transitivity indifference curves do not cross. Suppose


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x y

Good 1

Good 2

Figure 3.2: Indifference curves

indifference curve C and C ′ cross as in Figure 3.3, and denote the intersectionby x. Then, since y in the gure is above x across C we have y ≻x. Likewise,since z is above y across C ′ we have z ≻y. However, since x is above z acrossC we have x≻z, which leads to a cycle and contradicts Transitivity.

Let me give you some examples of preference. The simplest one consists of parallel and straight indifference curves as in Figure 3.4. Here the two goodsare said to be perfect substitutes of each other. Here the slope of indifferencecurves being

−3 in the graph means ( x1 , x2)

∼ (x1 + t, x 2

−3t) for any t. In

other words, the consumer is willing to give up 3 units of Good 2 per one extraunit of Good 1. Thus the slope of indifference curves express the consumer’ssubjective rate of exchange between two goods. We call this marginal rate of substitution of Good 2 for Good 1, while its more general denition will begiven later.

Next example consists of L-shaped indifference curves located parallel alongan upward-sloping straight line passing through the origin. Here the two goodsare said to be perfect complements of each other. In this graph the L-shapedindifference curves are located parallel along the line x1 = 2x2 . This means theconsumer sticks to some xed proportion between Good 1 and Good 2, which is2:1 here, and any extras have no value for him. So for example when he originallyhas (8 , 4), receives extra 6 units of Good 1 and ends up with (14 , 4), becausethe extra 6 units of Good 1 have no value we have (8 , 4)

∼ (14, 4). Likewise,

when we add 5 units of Good 2 to (8 , 4) so as to obtain (8 , 9), again becausethe extra 5 units of Good 2 have no value and thus we have (8 , 4)∼(8, 9).In this book I refer to perfect substitution and perfect complementarity asextreme cases mostly. More exible preferences will be between the two.


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8 14

4

9

Good 1

Good 2

Figure 3.5: Perfect complementarity

3.2.2 MonotonicityNext I introduce two assumptions which are natural for preferences over con-sumptions. One is monotonicity, which says more is better.

Strong Monotonicity : For any x = ( x1 , x2) and y = ( y1 , y2), if x1 ≧ y1 ,x2 ≧ y2 and if at least one of these inequalities are strict, then x≻y.

This means the consumer is better off when the consumptions of both goodsincrease or the consumption of one good increases while that of the other staysthe same. Therefore our ”mountain” does not have a peak, and extends upwardto the north-east direction. Under Monotonicity, the set of consumption vectorsbetter than x contains the quadrant of north-east direction and the indifferencecurves are always downward-sloping (see Figure 3.6).

Of course you can think of preferences which violate monotonicity. Considerfor example that the consumer dislikes some commodity, which is a bad for him,then his preference violates monotonicity. Also it is violated when the consumergets ”full” and consumption more than that makes him sick.

Monotonicity is pretty innocuous, however. If there is a harmful commodityone can trade it for a negative price and it is equivalent to trading the ”right toput that away” for a positive price, in which monotonicity is taken to hold withregard to such right. What about the case of becoming ”full?” It is a matter of how long we take one period to be. If we take it to be short we may have a case

that the consumer becomes full, but it we take to be sufficiently long then theconsumer’s preference satises the property that more is better.

Now, as I put the word ”strong” in the above denition it suggests that thereis a weaker denition.

Weak Monotonicity : For all x = ( x1 , x2), y = ( y1 , y2), if x1 > y1 andx2 > y2 , then x≻y.


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x

Good 1

Good 2

Figure 3.6: Monotonicity

This says it is better if you increase the amounts of both goods, and it leaves thepossibility that you don’t get strictly better off when you increase the amountof just one good.

To illustrate the difference, consider the case of perfect complementarity.When you increase the amounts of both Good 1 and Good 2 at ( x1 , x2) andobtain ( y1 , y2), we have ( y1 , y2) ≻ (x1 , x2) and weak monotonicity is met. Onthe other hand, when we increase the amount of Good 1 only, let’s say by t , aswe just move along the same indifference curve we have ( x1 + t, x 2)∼(x1 , x2),which says strong monotonicity fails. Likewise, when we increase the amount of

Good 2 only, let’s say by t, as we just move along the same indifference curvewe have (x1 , x2 + s)∼(x1 , x2), which again says strong monotonicity fails.

3.2.3 ConvexityThe other assumption is convexity, which says taking middle is better.

Strict Convexity : For any x = ( x1 , x2) and y = ( y1 , y2), if x∼y then for all0 < λ < 1 it holdsλx + (1 −λ)y ≻x∼y,

where λx + (1 −λ)y = ( λx 1 + (1 −λ)y1 , λx 2 + (1 −λ)y2).Given two equally preferable consumption vectors x and y, consider any pointin the middle, λx + (1 −λ)y. Then convexity says such middle point is betterthan the two extreme points (seet Figure 3.7).

Convexity looks somewhat articial as compared to monotonicity. It is avery natural assumption, however, in the context of intertemporal consumptionand consumption under uncertainty. In the setting of intertemporal consump-tion, taking middle corresponds to reducing uctuation of consumption betweenperiods. For example, while consumption stream (10 , 0) refers to consuming 10


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x

y λx + (1 −λ)y

Good 1

Good 2

Figure 3.7: Convexity

now and 0 in the future, (0 , 10) refers to consuming 0 now and 10 in the future,the midpoint (5 , 5) refers to consuming 5 both now and in the future. It isthus natural to prefer the midpoint when the consumer dislikes uctuation overtime. 1

In the setting of consumption under uncertainty, taking middle correspondsto hedging uncertainty. For example, while state-contingent vector (10 , 0) refersto consuming 10 if Republicans win and 0 if Democrats win, (0 , 10) refers toconsuming 0 if Republicans win and 10 if Democrats win, the mid point (5 , 5)refers to consuming 5 regardless of the election outcome. It is thus natural to

prefer the midpoint when the consumer dislikes uncertainty.

Now, as I put the word ”strict” in the above denition it suggests that thereis a weaker denition.

Weak Convexity : For any x = ( x1 , x2) and y = ( y1 , y2), if x∼y then for all0 < λ < 1 it holdsλx + (1 −λ)y ≿ x∼y.

This means that taking middle of any two equally preferable points does notmake the consumer worse off. The difference here is that the consumer may notget strictly better off.

To illustrate, consider the case of perfect substitution. Because indifferencecurves are straight here, any point between any two equally preferable pointsis again equally preferable to those, which fails to satisfy strict convexity whilethe weak one is met.

1 We need to be a bit more careful, since typically a consumer is not indifferent between10 units to be received now and 10 units to be received in the future, since he is normallyimpatient. I will come to the issue of impatience in Chapter 8, and let us pretend here thatthe consumer is patient and 10 units now and 10 units in the future are equally valuable.


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3.3 Marginal rate of substitutionMarginal rate of substitution is the subjective rate of exchange between goods.It refers to relative importance of a good which is measured in terms of anothergood.

Consider the following question.

How important is Good 1 for you as compared to Good 2?

Note that I’m not asking ”How important is Good 1 for you?,” which is not aneconomically meaningful question, as importance of something can be measuredonly by something else. In other words, value of something for one can berevealed only by telling how much of something else he can sacrice.

In any case, let me restate the above question as follows:

In order to get extra one unit of Good 1 how many units of Good 2 can yougive up?

Answer to this question is called marginal rate of substitution of Good2 for Good 1 , which is the subjective relative value of Good 1 measured bymeans of Good 2.

Of course one can dene ”marginal rate of Good 1 for Good 2”=”the amountsof Good 1 which can be given up in order to get extra one unit of Good 2,”and it is the inverse of the above one. In order to avoid confusion, however,we adopt the rst one throughout the book, where Good 1 is what is measuredand Good 2 is what measures. Hence we omit the part ”of Good 2 for Good 1”hereafter.

To illustrate, look again at the case of perfect substitution as depicted inFigure 3.4. Here the slope of indifference curves is −3, which means that ( x1 , x2)and ( x1 + t, x 2 −3t) are equally preferable. In other words the consumer is willto sacrice up to 3 t units of Good 2 in order to get t extra units of Good 1. Thatis, the slope corresponds to how many units of Good 2 one can give up in orderto get 1 extra unit of Good 1. Thus, the marginal rate of substitution of Good2 for Good 1 is given by the (absolute value) of the slope of the indifferencecurves. In this example it is 3.

Now, why do we use the term ”marginal” rate of substitution, not just ”rateof substitution?” This is because indifference curves in general are not straightor parallel, and it rather looks like in Figure 3.8 typically. Here the slope of indifference curves varies across points. Therefore we need to look at localslope of the indifference curves, and marginal rate substitution is dened bythe absolute value of such local slope.

Suppose we are at point x = ( x1 , x2) on an indifference curve as in Figure3.8. Now suppose we add a ”slight amount” of Good 1. Denote this ”slightamount” by ∆ x1 . Let ∆ x2 the amount of Good 2 the consumer can give upin order of get extra ∆ x1 of Good 1. Then (x1 , x2) and ( x1 + ∆ x1 , x2 + ∆ x2)


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Good 1

Good 2

Figure 3.9: Diminishing MRS

Condition 2 says that when Good 1 is indenitely scarce it becomes indenitelyvaluable compared to Good 2 and the amount of Good 2 the consumer cangive up in order to get extra one unit of Good 1 becomes indenitely large,and also says that when Good 2 is indenitely scarce it becomes indenitelyvaluable compared to Good 1 and the amount of Good 2 the consumer can giveup in order to get extra one unit of Good 1 becomes indenitely small. Thatis, marginal rate of substitution varies exibly between 0 and innity, like inFigure 3.8 and 3.9

For example, preference exhibiting perfect complementarity between two

goods has kinks on their indifference curves and therefor is not smooth. Also,preference exhibiting perfect substitution between two goods satises condition1 because its indifference curves are parallel and straight, which have no kinks,while it fails to meet condition 2 because the indifference curves hit both axis.In contract, Cobb-Douglass preference is smooth.

3.5 Convexity and diminishing marginal rate of substitution

Let us rethink the meaning of convexity in terms of marginal rate of substitu-tion. Indifference curves generated by convex preference are steeper as Good 1

quantity is smaller, and atter as Good 1 quantity is larger, as in Figure 3.9.Equivalently, marginal rate of substitution MRS (x) is larger as x1 is smaller,and smaller as x1 is larger. That is, when Good 1 is scarce the amount of Good2 one can give for one extra unit of it is larger, and when Good 1 is abundantthe amount of Good 2 one can give for one extra unit of it is smaller. This iscalled the law of diminishing marginal rate of substitution . I’ll come thothis in relation to so called ”the law of diminishing marginal utility.”


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3.6 ExercisesExercise 2 Let Good 1 be consumption good at Pe

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