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Microeconomic Theory 3
Course Notes by Lutz-Alexander Busch Dept.of Economics University of Waterloo
Revised 2004
c °Lutz-Alexander Busch, 1994,1995,1999,2001,2002,2004 Do not quote or redistribute without permission
Contents
1 Preliminaries
1
1.1
About Economists . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
What is (Micro—)Economics? . . . . . . . . . . . . . . .
2
1.3
Economics and Sex . . . . . . . . . . . . . . . . . . . . . . . .
5
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Review 2.1
2.2
9
Modelling Choice Behaviour
1
. . . . . . . . . . . . . . . . . .
9
2.1.1
Choice Rules . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2
Preferences . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3
What gives? . . . . . . . . . . . . . . . . . . . . . . . . 13
Consumer Theory . . . . . . . . . . . . . . . . . . . . . . . . . 15 Special Utility Functions . . . . . . . . . . . . . . . . . . . . . 18 2.2.1
Utility Maximization . . . . . . . . . . . . . . . . . . . 20
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 2.3 1
Expenditure Minimization and the Slutsky equation . . 25
General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 27
This material is based on Mas-Colell, Whinston, Green, chapter 1
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2.4
2.3.1
Pure Exchange . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2
A simple production economy . . . . . . . . . . . . . . 33
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Inter-temporal Economics 3.1
May 2004
39
The consumer’s problem . . . . . . . . . . . . . . . . . . . . . 39 3.1.1
Deriving the budget set
. . . . . . . . . . . . . . . . . 40
No Storage, No Investment, No Markets . . . . . . . . . . . . 40 Storage, No Investment, No Markets . . . . . . . . . . . . . . 40 Storage, Investment, No Markets . . . . . . . . . . . . . . . . 41 Storage, No Investment, Full Markets . . . . . . . . . . . . . . 42 Storage, Investment, Full Markets . . . . . . . . . . . . . . . . 43 3.1.2
Utility maximization . . . . . . . . . . . . . . . . . . . 44
3.2
Real Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3
Risk-free Assets . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4
3.3.1
Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2
More on Rates of Return . . . . . . . . . . . . . . . . . 49
3.3.3
Resource Depletion . . . . . . . . . . . . . . . . . . . . 51
3.3.4
A Short Digression into Financial Economics . . . . . . 52
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Uncertainty 4.1
57
Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.1
Comparing degrees of risk aversion . . . . . . . . . . . 65
Contents iii 4.2
Comparing gambles with respect to risk
4.3
A first look at Insurance . . . . . . . . . . . . . . . . . . . . . 69
4.4
The State-Preference Approach . . . . . . . . . . . . . . . . . 72
4.5
4.6
. . . . . . . . . . . . 67
4.4.1
Insurance in a State Model . . . . . . . . . . . . . . . . 74
4.4.2
Risk Aversion Again . . . . . . . . . . . . . . . . . . . 76
Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.5.1
Diversification . . . . . . . . . . . . . . . . . . . . . . . 77
4.5.2
Risk spreading . . . . . . . . . . . . . . . . . . . . . . 78
4.5.3
Back to Asset Pricing . . . . . . . . . . . . . . . . . . . 79
4.5.4
Mean-Variance Utility . . . . . . . . . . . . . . . . . . 81
4.5.5
CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Information
91
5.1
Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2
Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3
Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4
The Principal Agent Problem . . . . . . . . . . . . . . . . . . 105 5.4.1
The Abstract P-A Relationship . . . . . . . . . . . . . 105
6 Game Theory 6.1
113
Descriptions of Strategic Decision Problems . . . . . . . . . . 117 6.1.1
The Extensive Form . . . . . . . . . . . . . . . . . . . 117
6.1.2
Strategies and the Strategic Form . . . . . . . . . . . . 121
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Solution Concepts for Strategic Decision Problems . . . . . . . 126 6.2.1
Equilibrium Concepts for the Strategic Form . . . . . . 127
6.2.2
Equilibrium Refinements for the Strategic Form . . . . 131
6.2.3
Equilibrium Concepts and Refinements for the Extensive Form . . . . . . . . . . . . . . . . . . . . . . . . . 134
Signalling Games . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Review Question Answers
145
7.1
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
List of Figures 2.1
Consumer Optimum: The tangency condition . . . . . . . . . 22
2.2
Corner Solutions: Lack of tangency . . . . . . . . . . . . . . . 23
2.3
An Edgeworth Box . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4
Pareto Optimal Allocations . . . . . . . . . . . . . . . . . . . 30
3.1
Storage without and with markets, no (physical) investment . 44
4.1
Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2
Risk Neutral and Risk Loving . . . . . . . . . . . . . . . . . . 63
4.3
The certainty equivalent to a gamble . . . . . . . . . . . . . . 64
4.4
Comparing two gambles with equal expected value . . . . . . . 67
4.5
An Insurance Problem in State-Consumption space . . . . . . 75
4.6
Risk aversion in the State Model . . . . . . . . . . . . . . . . 76
4.7
Efficient Portfolios and the Market Portfolio . . . . . . . . . . 83
5.1
Insurance for Two types . . . . . . . . . . . . . . . . . . . . . 99
5.2
Impossibility of Pooling Equilibria . . . . . . . . . . . . . . . . 101
5.3
Separating Contracts could be possible . . . . . . . . . . . . . 102
5.4
Separating Contracts definitely impossible . . . . . . . . . . . 103
v
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6.1
Examples of valid and invalid trees . . . . . . . . . . . . . . . 118
6.2
A Game of Perfect Recall and a Counter-example . . . . . . . 120
6.3
From Incomplete to Imperfect Information . . . . . . . . . . . 121
6.4
A Matrix game — game in strategic form . . . . . . . . . . . . 124
6.5
A simple Bargaining Game . . . . . . . . . . . . . . . . . . . . 125
6.6
The 4 standard games . . . . . . . . . . . . . . . . . . . . . . 126
6.7
Valid and Invalid Subgames . . . . . . . . . . . . . . . . . . . 135
6.8
The “Horse” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.9
A Signalling Game . . . . . . . . . . . . . . . . . . . . . . . . 139
6.10 A Minor Perturbation? . . . . . . . . . . . . . . . . . . . . . . 140
Chapter 1 Preliminaries These notes have been written for Econ 401 as taught by me at the University of Waterloo. As such the list of topics reflects the course material for that particular course. It is assumed that the student has mastered the prerequisites and little or no time is spent on them, aside from a review of standard consumer economics and general equilibrium in chapter 1. I assume that the student has access to a standard undergraduate micro theory text book. Any of the books commonly used will do and will give introductions to the topics covered here, as well as allowing for a review, if necessary, of the material from the pre-requisites. These notes will not give references. The material covered is by now fairly standard and can be found in one form or another in most micro texts. I wish to acknowledge two books, however, which have served as references: the most excellent book by Mas-Colell, Whinston, and Green, Microeconomic Theory, as well as the much more concise Jehle and Reny, Advanced Microeconomic Theory. I also would like to acknowledge my teachers Don Ferguson and Glen MacDonald, who have done much to bring microeconomics alive for me. This preliminary chapter contains extensive quotes which I have found informative, amusing, interesting, and thought provoking. Their sources have been indicated.
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1.1
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About Economists
By now you may have heard many jokes about economists and noticed that modern economics has a bad reputation in some circles. If you mention your field of study in the bar, you are quite possibly forced to defend yourself against various stereotypical charges (focusing on assumptions, mostly.) The most eloquent quotes about economists that I know of are the following two quotes reproduced from the Economist, Sept.4, 1993, p.25: No real Englishman, in his secret soul, was ever sorry for the death of a political economist, he is much more likely to be sorry for his life. You might as well cry at the death of a cormorant. Indeed how he can die is very odd. You would think a man who could digest all that arid matter; who really preferred ‘sawdust without butter’; who liked the tough subsistence of rigid formulae, might defy by intensity of internal constitution all stomachic or lesser diseases. However they do die, and people say that the dryness of the Sahara is caused by a deposit of similar bones. (Walter Bagehot (1855)) Are economists human? By overwhelming majority vote, the answer would undoubtedly be No. This is a matter of sorrow for them, for there is no body of men whose professional labours are more conscientiously, or consciously, directed to promoting the wealth and welfare of mankind. That they tend to be regarded as blue-nosed kill-joys must be the result of a great misunderstanding. (Geoffrey Crowther (1952))
1.2
What is (Micro—)Economics?
In Introductory Economics the question of what economics is has received some attention. Since then, however, this question may have received no further coverage, and so I thought to collect here some material which I to use to start a course. It is meant to provide a background for the field as well as a defense, of sorts, of the way in which micro economics is practiced. Malinvaud sees economics as follows:
Preliminaries 3 Here we propose the alternative, more explicit definition: economics is the science which studies how scarce resources are employed for the satisfaction of the needs of men living in society: on the one hand, it is interested in the essential operations of production, distribution and consumption of goods, and on the other hand, in the institutions and activities whose object it is to facilitate these operations. [..] The main object of the theory in which we are interested is the analysis of the simultaneous determination of prices and the quantities produced, exchanged and consumed. It is called microeconomics because, in its abstract formulations, it respects the individuality of each good and each agent. This seems a necessary condition a priori for logical investigation of the phenomena in question. By contrast, the rest of economic theory is in most cases macroeconomic, reasoning directly on the basis of aggregates of goods and agents. [E. Malinvaud, Lectures on Microeconomic Theory, revised, N-H, 1985, p.1-2.] This gives us a nice description of what economics is, and in particular what micro theory entails. In following the agenda laid out by Malinvaud a certain amount of theoretical abstraction and rigor have been found necessary, and one key critique heard often is the “attempt at overblown rigor” and the “unrealistic assumptions” which micro theory employs. Takayama and Hildenbrand both address these criticisms in the opening pages of their respective books. First Takayama: The essential feature of modern economic theory is that it is analytical and mathematical. Mathematics is a language that facilitates the honest presentation of a theory by making the assumptions explicit and by making each step of the logical deduction clear. Thus it provides a basis for further developments and extensions. Moreover, it provides the possibility for more accurate empirical testing. Not only are some assumptions hidden and obscured in the theories of the verbal and “curve-bending” economic schools, but their approaches provide no scope for accurate empirical testing, simply because such testing requires explicit and mathematical representations of the propositions of the theories to be tested. [..] But yet, economics is a complex subject and involves many things that cannot be expressed readily in terms of mathematics.
4 L-A. Busch, Microeconomics Commenting on Max Planck’s decision not to study economics, J.M. Keynes remarked that economics involves the “amalgam of logic and intuition and wide knowledge of facts, most of which are not precise.” In other words, economics is a combination of poetry and hard-boiled analysis accompanied by institutional facts. This does not imply, contrary to what many poets and institutionalists feel, that hard-boiled analysis is useless. Rather, it is the best way to express oneself honestly without being buried in the millions of institutional facts. [..] Mathematical economics is a field that is concerned with complete and hard-boiled analysis. The essence here is the method of analysis and not the resulting collection of theorems, for actual economies are far too complex to allow the ready application of these theorems. J.M. Keynes once remarked that “the theory of economics does not furnish a body of settled conclusions immediately applicable to policy. It is a method rather than a doctrine, an apparatus of the mind, a technique of thinking, which helps its possessor to draw conclusions.” An immediate corollary of this is that the theorems are useless without explicit recognition of the assumptions and complete understanding of the logic involved. It is important to get an intuitive understanding of the theorems (by means of diagrams and so on, if necessary), but this understanding is useless without a thorough knowledge of the assumptions and proofs. [Akira Takayama, Mathematical Economics, 2nd ed., Cambridge, 1985, p. xv.] Hildenbrand offers the following: I cannot refrain from repeating here the quotation from Bertrand Russell cited by F. Hahn in his inaugural lecture in Cambridge: “Many people have a passionate hatred of abstraction, chiefly, I think, because of its intellectual difficulty; but as they do not wish to give this reason they invent all sorts of other that sound grand. They say that all abstraction is falsification, and that as soon as you have left out any aspect of something actual you have exposed yourself to the risk of fallacy in arguing from its remaining aspects alone. Those who argue that way are in fact concerned with matters quite other than those that concern science.” (footnote 2, p.2, with reference)
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Preliminaries 5 Let me briefly recall the main characteristics of an axiomatic theory of a certain economic phenomenon as formulated by Debreu: First, the primitive concepts of the economic analysis are selected, and then, each one of these primitive concepts is represented by a mathematical object. Second, assumptions on the mathematical representations of the primitive concepts are made explicit and are fully specified. Mathematical analysis then establishes the consequences of these assumptions in the form of theorems. [Werner Hildenbrand, Twenty Papers of Gerard Debreu, Econometric Society Monograph 4, Cambridge, 1983, page 4, quoted with omissions.]
1.3
Economics and Sex
I close this chapter with the following thought provoking excerpt from Mark Perlman and Charles R. McCann, Jr., “Varieties of uncertainty,” in Uncertainty in Economic Thought, ed. Christian Schmidt, Edward Elgar 1996, p 9-10. The problem as perceived As this is an opening paper, let us begin with what was once an established cultural necessity, namely a reference to our religious heritage. What we have in mind is the Biblical story of the Fall of Man, the details of which we shall not bore you with. Rather, we open consideration of this difficult question by asking what was the point of that Book of Genesis story about the inadequacy of Man. We are told that apparently whatever were God’s expectations, He became disappointed with Man. Mankind and particularly Womankind1 did not live up to His expectations.2 In any case, Adam and Eve were informed 1 Much has been made of the failure of women, perhaps that is because men wrote up the history. We should add, in order to avoid deleterious political correctness (and thereby cut off provocation and discussion), that since Eve was the proximate cause of the Fall, and Eve represents sexual attraction or desire, some (particularly St Paul, whose opinion of womankind was problematic) have considered that sexual attraction was in some way even more responsible for the Fall than anything else. Put crudely, even if economics is not a sexy subject, its origins were sexual. 2 What that says about His omniscience and/or omnipotence is, at the very least, para-
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that they had ‘fallen’ from Grace, and all of us have been made to suffer ever since. From our analytical standpoint there are two crucial questions: 1. What was the sin; and 2. What was the punishment? The sin seems to have been something combining (1) an inability to follow precise directions; (2) a willingness to be tempted, particularly when one could assert that ‘one was only doing what everyone else (sic) was doing;3 (3) a greed involving things (something forbidden) and time (instant gratification); (4) an inability to leave well enough alone; and (5) an excessive Faustian curiosity. Naturally, as academic intellectuals, we fancy the fifth reason as best. But what interests us directly is the second question. It is ‘What was God’s punishment for Adam and Eve’s vicarious sin, for which all mankind suffers?’ Purportedly a distinction has been made between what happened to Man and Woman, but, the one clear answer, particularly as seen by Aquinas and by most economists ever since, was that man is condemned to live with the paradigm of scarcity of goods and services and with a schedule of appetites and incentives which are, at best, confusing. In the more modern terms of William Stanly Jevons, ours is a world of considerable pain and a few costly pleasures. We are driven to produce so that we can consume, and production is done mostly by the ‘sweat of the brow’ and the strength of the back. The study of economics — of the production, distribution and even the consumption of goods and services — it follows, is the result of the Original Sin. When Carlyle called Economics the ‘Dismal Science’, he was, if anything, writing in euphemisms; Economics per se, is the Punishment for Sin. doxical. 3 Cf. Genesis, 3:9-12,16,17. [9] But the Lord God called to the man and said to him, ‘ Where are you?’ [10] He replied, ‘I heard the sound as you were walking the garden , and I was afraid because I was naked, and I hid myself.’ [11] God answered, ‘ Who told you that you were naked? Have you eaten from the tree which I forbade you?’ [12] The man said, ‘The woman you gave me for a companion, she gave me fruit from the tree and I ate.’ [Note: The story, as recalled, suggests that Adam was dependent upon Eve (for what?), and the price of that dependency was to be agreeable to Eve (‘It was really all her fault — I only did what You [God] had laid out for me.’)] ([16] and [17] omitted) [Again, for those civil libertarians amongst us, kindly note that God forced Adam to testify against himself. Who says that the Bill of Rights is an inherent aspect of divine justice? Far from it, in the Last Judgment, pleading the Fifth won’t do at all.]
Preliminaries 7 But, it is another line of analysis, perhaps novel, which we put to you. Scarcity, as the paradigm, may not have been the greatest punishment, because scarcity, as such, can usually be overcome. Scarcity simply means that one has to allocate between one’s preferences, and the thinking man ought to be able to handle the situation. We use our reasoning power, surely tied up with Free Will, to allocate priorities and thereby overcome the greater disasters of scarcity. What was the greater punishment, indeed the greatest punishment, is more basic. Insofar as we are aware, it was identified early on by another Aristotelian, one writing shortly before Aquinas, Moses Maimonides. Maimonides suggested that God’s real punishment was to push man admittedly beyond the limits of his reasoning power. Maimonides held that prior to the Fall, Adam and Eve (and presumably mankind, generally) knew everything concerning them; after the Fall they only had opinions.4 Requisite to the wise use of power is understanding and full specification; what was lost was any such claim previously held by man to complete knowledge and the full comprehension of his surroundings. In other words, what truly underlies the misery of scarcity is neither hunger nor thirst, but the lack of knowledge of what one’s preference schedule will do to one’s happiness. For if one had complete knowledge (including foreknowledge) one could compensate accordingly. If one pursues Maimonides’ line of inquiry, it seems that uncertainty (which is based not on ignorance of what can be known with study of data collection, but also on ignorance tied to the unknowable) is the real punishment.
4
[omitted]
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Notation: Blackboard versions of these symbols may differ slightly.
I will not distinguish between vectors and scalars by notation. Generally all small variables (x, y, p) are column vectors (even if written in the notes as row vectors to save space.) The context should clarify the usage. Capital letters most often denote sets, as in the consumption set X, or budget set B. Sets of sets are denoted by p capital P 2script letters, such as X = {X1 , X2 , . . . , Xk }, where Xi = {x ∈ 1 + r. Then what I should do is to borrow money at the interest rate r, say I dollars, and use those funds to buy the good in question, i.e., purchase I/p0 units. Then wait and sell the goods in the next period. That will yield Ip1 /p0 dollars. I also have to pay back my loan, at
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(1 + r)I dollars, and thus I have a profit of p1 /p0 − (1 + r) per dollar of loan. Note that the optimal loan size would be infinite. However, the resulting large demand would certainly drive up current prices (while also lowering future prices, since everybody expects a flood of the stuff tomorrow), and this serves to reduce the profitability of the exercise. In a zero-arbitrage equilibrium we p1 therefore must have (1 + r) = p1 /p0 , or, more tellingly, p0 = . The 1+r correct market price for an asset is its discounted future value! This discussion has an application to the debate about pricing during supply or demand shocks. For example, gasoline prices during the Gulf war, or the alleged price-gouging in the ice-storm areas of Quebec and Ontario: What should the price of an item be which is already in stock? Many people argue that it is unfair to charge a higher price for in-stock items. Only the replacement items, procured at higher cost, should be sold at the higher cost. While this may be “ethical” according to some, it is easily demonstrated to violate the above rule: The price of the good tomorrow will be determined by demand and supply tomorrow, and apparently all are agreed that that price might well be higher due to a large shift out in the demand and/or reduction in supply. Currently I own that good, and have therefore the choice of selling it tomorrow or selling it today. I would want to obtain the appropriate rate of return on the asset, which has to be equal between the two options. Thus I am only willing to part with it now if I am offered a higher price which foreshadows tomorrows higher price. Should I be forced not to do so I am forced to give money away against my will and better judgment. This would normally be considered unethical by most (just try and force them to give you money.) Of course, assets are not usually all the same, and we will see this later when we introduce uncertainty. For example, a house worth $100,000 and $100,000 cash are not equivalent, since the cash is immediately usable, while the house may take a while to sell — it is less “liquid.” The same is true for thinly traded stocks. Such assets may carry a liquidity premium — an illiquidity punishment, really — and will have a higher rate of return in order to compensate for the potential costs and problems in unloading them. This can, of course, be treated in terms of risk, since the realization of the house’s value is a random variable, at least in time, if not in the amount. Of course, there are other kinds of risk as well, and in general the future price of the asset is not known. (Note that bonds are an exception to some degree. If you choose to hold the bond all the way to the maturity date you do know the precise stream of payments. If you sell early, you face the uncertain sale price which depends on the interest rate at that point in time.)
Inter-temporal 51 Assets may also yield consumption returns while you hold them: a car or house are examples, as are dividend payments of stocks or interest payments of bonds. For one period this is still simple to deal with: The asset will generate benefit (say rent saved, or train tickets saved) of b and we thus p1 − p 0 + b compute the rate of return as . If the consumer holds multiple p0 assets in equilibrium, then we again require that this be equal to the rate of return on other assets. Complicating things in the real world is the fact that assets often differ in their tax treatment. For example, if the house is a principal residence any capital gains (the tax man’s term for p1 − p0 , and to add insult to injury they ignore inflation) are tax free. For another asset, say a painting, this is not true. Equilibrium requires, of course, that the rates of return as perceived by the consumer are equalized, and thus we may have to use an after tax rate for one asset and set it equal to an untaxed rate for another.
3.3.3
Resource Depletion
The simple discounting rules above can also be applied to gain some first insights into resource economics. We can analyse the question of simple resource depletion: at what rate should we use up a non-renewable resource. We can also analyse when a tree (or forest) should to be cut down. Assume a non-renewable resource currently available at quantity S. For simplicity, first assume a fixed annual demand D. It follows that there are S/D years left, after which we assume that an alternative has to be used which costs C. Thus the price in the last year should be pS/D = C. Arbitrage implies that pt+1 = (1 + r)pt , so that p0 = C/(1 + r)S/D . Note that additional discoveries of supplies lower the price since they increase the time to depletion, as do reductions in demand. Lowering the price of the alternative also lowers the current price. Finally, increases in the discount rate lower the price. This approach has a major flaw, however. It assumes demand and supply to be independent of price. So instead, let us assume some current price p0 as a starting value and let us focus on supply. When will the owner of the resource be willing to sell? If the market rate of return on other assets is r then the resource, which is just another asset, will also have to generate that rate of return. Therefore p1 = (1 + r)p0 , and in general we’d have to expect pt = (1 + r)t p0 . Note that the price of the resource is therefore increasing with time, which, in general equilibrium, means two things: demand will
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fall as customers switch to alternatives, and substitutes will become more competitive. Furthermore we might expect more substitutes to be developed. We will ultimately run out of the resource, but it is nearly always wrong to simply use a linear projection of current use patterns. This fact has been established over and over with various natural resources such as oil, tin, copper, titanium, etc. What about renewable resources? Consider first the ‘European’ model of privately owned land for timber production as an example. Here we have a company who owns an asset — a forest — which it intends to manage in order to maximize the present value of current and future profits. When should it harvest the trees? Each year there is the decision to harvest the tree or not. If it is cut it generates revenue right away. If it continues to grow it will not generate this revenue but instead generate more revenue tomorrow (since it is growing and there will be more timber tomorrow.) It follows that the two rates of return should be equalized, that is, the tree should be cut once its growth rate divided by its current size has slowed to the market interest rate. This fact has a few implications for forestry: Faster growing trees are a better investment, and thus we see mostly fast growing species replanted, instead of, say, oaks, which grow only slowly. (This discussion is ceteris paribus — ignoring general equilibrium effects.) Furthermore, what if you don’t own the trees? What if you are the James Bond of forestry, with a (time-limited) license to kill? In that case you will simply cut the trees down either immediately or before the end of your license, depending on the growth rate. Of course, in Canada most licenses are for mature forests, which nearly by definition have slow or no growth — thus the thing to do is to clear cut and get out of there. The Europeans, critical of clear-cutting, forget that they have long ago cut nearly all of their mature forests and are now in a harvesting model with mostly high growth forests. As a final note, notice that lack of ownership will also impact the replanting decision. As we will see later in the course, if we treat the logger as an agent of the state, the state has serious incentive problems to overcome within this principal agent framework.
3.3.4
A Short Digression into Financial Economics
I thought it might be useful to provide you with a short refresher or introduction to multi-period present value and compound interest computations. For starters, assume you put $1 in the bank at 5% interest, computed yearly, and that all interest income is also reinvested at this 5% rate. How much
Inter-temporal 53 money will you have in each of the following years? The answer is 1.05, 1.052 , 1.053 , . . . 1.05t . The important fact about this is that a simple interest rate and a compounded interest rate are not the same, since with compounding there is interest on interest. For example, if you get a loan at 12%, it matters how often this is compounded. Let us assume it is just simple interest; You then owe $1.12 for every dollar you borrowed at the end of one year. What you will quickly find out is that banks don’t normally do that. They at least compound semiannually, and normally monthly. Monthly compounding would mean that )12 = 1.1268. On a million dollar loan this would be you will owe (1 + .12 12 a difference of $6825.03. In other words, you are really paying not a 12% interest rate but a 12.6825% simple interest rate. It is therefore very important to be sure to know what interest rate applies and how compounding is applied (semi-annual, monthly, etc.?) Here is a handy little device used in many circles: the rule of 72, sometimes also referred to as the rule of 69. It is used to find out how long it will take to double your money at any given interest rate. The idea is that it will approximately take 72/r periods to double your money at an interest rate of r percent. The proof is simple: we want to solve for the t for which ¶t ¶ µ µ r% r% = ln2. = 2 ⇒ tln 1 + 1+ 100 100 However, for small x we know that ln(1 + x) ∼ x, thus t
100ln2 69.3147 r% ∼ ln2 ⇒ t ∼ = 100 r% r%
but of course 72 has more divisors and is much easier to work with. The power of compounding also comes into play with mortgages or other installment loans. A mortgage is a promise to pay every period for a specified length (typically 25 years, i.e., 300 months) a certain payment p. This is also known as a simple annuity. What is the value of such a promise, i.e., its present value? We need to compute the value of the following sum: δp + δ 2 p + δ 3 p + . . . + δ n p. Here δ = 1/(1 + r), where r is the interest rate we use per period. Thus ¡ ¢ PV = δ p + δp + δ 2 p + . . . + δ n−1 p ¡ ¢ = δp 1 + δ + δ 2 + . . . + δ n−1
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1 − δn 1−δ 1 − (1 + r)−n PV = p r = δp
(Recall in the above derivation that for δ < 1 we have
P∞
i=1
δ i = 1/(1 − δ).)
The above equation relates four variables: the principal amount, the payment amount, the payment periods, and the interest rate. If you fix any three this allows you to derive the fourth after only a little bit of manipulation. A final note: In Canada a mortgage can be at most compounded semi-annually. Thus the effective interest rate per month is derived by solving (1 + r/2)2 = (1 + rm )12 . If you are quoted a 12% interest rate per year the monthly rate is therefore (1.06)1/6 − 1 = 0.975879418%. The effective yearly interest rate in turn is (1.06)2 − 1 = 12.36%, and by law the bank is supposed to tell you about that too. Given the above, and the fact that nearly all mortgages are computed for a 25 year term (but seldom run longer than 5 years, these days), the monthly payment at a 10% yearly interest rate for an additional $1000 on the mortgage is $8.95. Before you engage in mortgages it would be a good idea to program your spreadsheet with these formulas and convince yourself how bi-weekly payments reduce the total interest you pay, how important a couple of percentage points off the interest rate are to your monthly budget, etc.
3.4
Review Problems
Question 1: There are three time periods and one consumption good. The consumer’s endowments are 4 units in the first period, 20 units in the second, and 1 unit in the third. The money price for the consumption good is known to be p = 1 in all periods (no inflation.) Let rij denote the (simple, nominal) interest rate from period i to j. a) State the restrictions on r12 , r23 and r13 implied by zero arbitrage. b) Write down the consumer’s budget constraint assuming the restriction in (a) holds. Explain why it is useful to have this condition hold (i.e., point out what would cause a potential problem in how you’ve written the budget if the condition in (a) fails. c) Draw a diagrammatic representation of the budget constraint in periods 2 and 3, being careful to note how period 1 consumption influences this diagram. Question 2: There are two goods, consumption today and tomorrow. Joe
Inter-temporal 55 has an initial endowment of (100, 100). There exists a credit market which allows him to borrow or lend against his initial endowment at market interest rates of 0%. A borrowing constraint exists which prevents him from borrowing against more than 60% of his period 2 endowment. Joe also possesses an investment technology which is characterized by a production function √ x2 = 10 x1 . That is, an investment of x1 units in period 1 will lead to x2 units in period 2. a) What is Joe’s budget constraint? A very clearly drawn and well labelled diagram suffices, or you can give it mathematically. Also give a short explanatory paragraph how the set is derived. b) Suppose that Joe’s preferences can be represented by the function U (c1 , c2 ) = exp(c41 c62 ). (Here exp() denotes the exponential function.) What is Joe’s final consumption bundle, how much does he invest, and what are his transactions in the credit market. Question 3: Anna has preferences over her consumption levels in two periods which can be represented by the utility function ¶ ¾ ½ µ 13 23 12 c1 + c 2 , c1 + c 2 . u(c1 , c2 ) = min 22 10 10 a) Draw a carefully labelled representation of her indifference curve map. b) What is her utility maximizing consumption bundle if her initial endowment is (9, 8) and the interest rate is 25%. c) What is her utility maximizing consumption bundle if her initial endowment is (5, 12) and the interest rate is 25%. d) Assume she can lend money at 22% and borrow at 28%. What would her endowment have to be for her to be a lender, a borrower? e) Assume she can lend money at 18% and borrow at 32%. Would Anna ever trade at all? (Explain.) Question 4: Alice has preferences over consumption in two periods represented by the utility function uA (c1 , c2 ) = lnc1 + αlnc2 , and an endowment of (12, 6). Bob has preferences over consumption in two periods represented by the utility function uB (c1 , c2 ) = c1 + βc2 , and an endowment of (8, 4). a) Draw an appropriately labelled representation of this exchange economy in order to “prime” your intuition. (Indicate the indifference maps and the Contract Curve.) b) Assuming, of course, that both α and β lie strictly between zero and one, what is the equilibrium interest rate and allocation?
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Chapter 4 Uncertainty So far, it has been assumed that consumers would know precisely what they were buying and getting. In real life, however, it is often the case that an action does not lead to a definite outcome, but instead to one of many possible outcomes. Which of these occurs is outside the control of the decision maker. It is determined by what is referred to as “nature.” These situations are ones of uncertainty — it is uncertain what happens. Often, however, the probabilities of the different possibilities are known from past experience, or can be estimated in some other way, or indeed are assumed based on some personal (subjective) judgment. Economists then speak of risk. Note that our “normal” model is already handling such cases if we take it at its most general level: commodities in the model were supposed to be fully specified, and could, in principle, be state contingent. We will develop that interpretation further later on in this chapter. First, however, we will develop a more simple model which is designed to bring the role of probabilities to the fore. One of the key facts about situations involving risk/uncertainty is that the consumer’s wellbeing does not only depend on the various possible outcomes, and which occurs in the end, but also on how likely each outcome is. The standard model of chapter 2 does not allow an explicit role for such probabilities. They are somehow embedded in the utility function and prices. In order to compare situations which differ only in the probabilities, for example, it would be nice to have probabilities explicitly in the model formulation. A particularly simple model that does this holds the outcomes fixed, they will all be assumed to lie in some set of alternatives X, and focuses on the different probabilities with which they occur. We call such a list 57
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of the probabilities for each outcome a lottery. Definition 1 A simple lottery is a list L = (p1P , p2 , . . . , pN ) of probabilities for the N different outcomes in X, with pi ≥ 0, N i=1 pi = 1. If we have a suitably defined continuous space of outcomes, for example 0. We are allowed to scale the utility index and to change its slope, but we are not allowed to change its curvature. The ˆ reason for this should be clear. Suppose we compare two lotteries, L and L,
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which differ only in that probability is shifted between outcomes k and j and ˆ so: outcomes m and n. Suppose U (L) > U (L), U (L) = n X i=1
pi u(xi ) −
n X
i=1 n X
ˆ = pi u(xi ) > U (L)
n X i=1
pˆi u(xi ) > 0
i=1
(pk − pˆk )u(xk ) + (pj − pˆj )u(xj )(pm − pˆm )u(xm ) + (pn − pˆn )u(xn ) > 0 (pk − pˆk )(u(xk ) − u(xj )) + (pm − pˆm )(u(xm ) − u(xn )) > 0 This comparison clearly depends on both, the differences in probabilities as well as the differences in the utility indices of the outcomes. If we multiply u(·) by a constant, it will factor out of the last line above. If, however, we were to transform the function u(·), even by a monotonic transformation, we would change the difference between outcome utilities, and this could change the above comparison. In fact, as we shall see later, the curvature of the Bernoulli utility index u(·) is crucial in determining the consumer’s behaviour with respect to risk, and will be used to measure the consumer’s risk aversion. Before we proceed to that, some famous paradoxes relating to uncertainty and our assumptions. Allais Paradox: The Allais paradox shows that consumers may not satisfy the axioms we had assumed. It considers the following case: Consider a space of outcomes for a lottery given by C = (25, 5, 0) in hundred thousands of dollars. Subjects are then asked which of two lotteries they would prefer, LA = (0, 1, 0) or LB = (.1, .89, .01). Often consumers will indicate a preference for LA , probably because they foresee that they would regret to have been greedy if they end up with nothing under lottery B. On the other hand, if they are asked to choose between LC = (0, .11, .89) or LD = (.1, 0, .9) the same consumers often indicate a preference for lottery D. Note that there is little regret possible here, you simply get a lot larger winning in exchange for a slightly lower probability of winning under D. These choices, however, violate our assumptions. This is easily checked by assuming the existence of some u(·): The preference for A over B then indicates that u(5) > .1u(25) + .89u(5) + .01u(0) .11u(5) > .1u(25) + .01u(0) .11u(5) + .89u(0) > .1u(25) + .9u(0)
pˆi u(xi )
Uncertainty 61 and the last line indicates that lottery C is preferred to D! Ellsberg Paradox: This paradox shows that consumers may not be consistent in their assessment of uncertainty. Consider an urn with 300 balls in it, of which precisely 100 are known to be red. The other 200 are blue or green in an unknown proportion (note that this is uncertainty: there is no information as to the proportion available.) The consumer is again offered the choice between two pairs of gambles: ½ LA : $1000 if a drawn ball is red. Choice 1 = LB : $1000 if a drawn ball is blue. ½ LC : $1000 if a drawn ball is NOT red. Choice 2 : = LD : $1000 if a drawn ball is NOT blue. Often consumers faced with these two choices will choose A over B and will choose C over D. However, letting u(0) be zero for simplicity, this means that p(R)u(1000) > p(B)u(1000) ⇒ p(R) > p(B) ⇒ (1 − p(R)) < (1 − p(B)) ⇒ p(¬R) < p(¬B) ⇒ p(¬R)u(1000) < p(¬B)u(1000). Thus the consumer should prefer D to C if choice were consistent. Other problems with expected utility also exist. One is the intimate relation of risk aversion and time preference which is imposed by these preferences. There consequently is a fairly active literature which attempts to find a superior model for choice under uncertainty. These attempts mostly come at the expense of much higher mathematical requirements, and many still only address one or the other specific problem, so that they too are easily ‘refuted’ by a properly chosen experiment.
4.1
Risk Aversion
We will now restrict the outcome space X to be one-dimensional. In particular, assume that X is simply the wealth/total consumption of the consumer in each outcome. With this simplification, the basic attitudes of a consumer concerning risk can be obtained by comparing two different lotteries: one that gives an outcome for certain (a degenerate lottery), and another that has the same expected value, but is non-degenerate. So, let L be a lottery ¯ given by the probability density f (x). It generates an expected on [0, X] R X¯ value of wealth of 0 xf (x)dx = C. We can now compare the consumer’s utility from obtaining C for certain, and that from the lottery L (which has
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expected wealth C.) Compare U (L) =
Z
¯ X
u(x)f (x)dx to u 0
ÃZ
¯ X
!
xf (x)dx . 0
Definition 2 A risk-averse consumer is one for whom the expected utility of any lottery is lower than the utility of the expected value of that lottery: ! ÃZ ¯ Z ¯ X
X
xf (x)dx .
u(x)f (x)dx < u
0
0
Utility u(w2)
3
♦
2.5
u(E(w)) 2 ♦ E(u()) ♦ 1.5 u(w1)
u(w)
♦ ♦ ♦
1 0.5
♦
0 0
w1
♦
♦
5 E(w)
10
w2
15
20 wealth
Figure 4.1: Risk Aversion The astute reader may notice that this is Jensen’s inequality, which is one way to define a concave function, in this case u(·) (see Fig. 4.1.) This is also the reason why only affine transformations were allowed for expected utility functions. Any other transformation would affect the curvature of the Bernoulli utility function u(·), and thus would change the risk-aversion of the consumer. Clearly, consumers with different risk aversion do not have the same preferences, however.2 Note that a concave u(·) has a diminishing marginal utility of wealth, an assumption which is quite familiar from introductory courses. Risk aversion therefore implies (and is implied by) the fact 2
To belabour the point, consider preferences over wealth represented by u(w) = w. In the standard framework√of chapter 1 positive monotonic transformations are ok, so that the functions w 2 and w both represent identical preferences. It is easy to verify that these two functions lead to a quite different relationship between the expected utility and the utility of the expected wealth than the initial one, however. Thus they cannot represent the same preferences in a setting of uncertainty/risk.
Uncertainty 63 U 14 12 10 E(u())= 8 u(E(w)) 6 ♦ 4 u(w1) 2 ♦ 0 ♦ 0 2 4 w1 u(w2)
♦ ♦ ♦
♦
6 8 10 12 14 E(w) w2
U 120 100 80 u(w2) 60 E(u()) 40 ♦ u(E(w)) 20 ♦ u(w1) 0 w 0
u(w)
♦ ♦ ♦ ♦ 1 w1
♦ 2
♦
3 4 E(w) w2
5w
Figure 4.2: Risk Neutral and Risk Loving that additional units of wealth provide additional utility, but at a decreasing rate. Of course, consumers do not have to be risk-averse. Risk neutral and risk loving are defined in the obvious way: The first requires that ¶ µZ Z u(x)f (x)dx = u
xf (x)dx .
while the second requires ¶ µZ Z xf (x)dx . u(x)f (x)dx > u
There is a nice diagrammatic representation of these available if we consider only two possible outcomes (Fig. 4.2). There are two other ways in which we might define risk aversion, and both reveal interesting facts about the consumer’s economic behaviour. The first is by using the concept of a certainty equivalent. It is the answer to the question “how much wealth, received for certain, is equivalent (in the consumer’s eyes, according to preferences) to a given gamble/lottery?” In other words: Definition 3 The certainty equivalent C(f, u) for a lottery with probability distribution f (·) under the (Bernoulli) utility function u(·) is defined by the equation Z u (C(f, u)) =
u(x)f (x)dx.
Again a diagram for the two-outcome case might help (Fig. 4.3).
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3
♦
2.5
u(w)
2 E(u()) u(w1)
1.5
♦
♦
♦
♦
♦
♦
1 0.5
♦
0 0
w1 C()5 E(w)
♦ 10
w2
15
20 wealth
Figure 4.3: The certainty equivalent to a gamble A risk averse consumer is one for whom the certainty equivalent of any gamble is less than the expected value of that gamble. One useful economic interpretation of this fact is that the consumer is willing to pay (give up expected wealth) in order to avoid having to face the gamble. InRdeed, the maximum amount which the consumer would pay is the difference wf (w)dw − C(f, u). This observation basically underlies the whole insurance industry: risk-averse consumers are willing to pay in order to avoid risk. A well diversified insurance company will be risk neutral, however, and therefore is willing to provide insurance (assume the risk) as long as it guarantees the consumer not more than the expected value of the gamble: Thus there is room to trade, and insurance will be offered. (More on that later.) Another way to look at risk aversion is to ask the following question: If I were to offer a gamble to the consumer which would lead either to a win of ² or a loss of ², how much more than fair odds do I have to offer so that the consumer will take the bet? Note that a fair gamble would have an expected value of zero (i.e., 50/50 odds), and thus would be rejected by the (risk averse) consumer for sure. This idea leads to the concept of a probability premium. Definition 4 The probability premium π(u, ², w) is defined by u(w) = (0.5 + π(·)) u(w + ²) + (0.5 − π(·)) u(w − ²). A risk-averse consumer has a positive probability premium, indicating that the consumer requires more than fair odds in order to accept a gamble.
Uncertainty 65 It can be shown that all three concepts are equivalent, that is, a consumer with preferences that have a positive probability premium will be one for whom the certainty equivalent is less than the expected value of wealth and for whom the expected utility is less than the utility of the expectation. This is reassuring, since the certainty equivalent basically considers a consumer with a property right to a gamble, and asks what it would take for him to trade to a certain wealth level, while the probability premium considers a consumer with a property right to a fixed wealth, and asks what it would take for a gamble to be accepted.
4.1.1
Comparing degrees of risk aversion
One question we can now try to address is to see which consumer is more risk averse. Since risk aversion apparently had to do with the concavity of the (Bernoulli) utility function it would appear logical to attempt to measure its concavity. This is indeed what Arrow and Pratt have done. However, simply using the second derivative of u(·), which after all measures curvature, will not be such a good idea. The reason is that the second derivative will depend on the units in which wealth and utility are measured.3 Arrow and Pratt have proposed two measures which largely avoid this problem: Definition 5 The Arrow-Pratt measure of (absolute) risk aversion is u00 (w) rA = − 0 . u (w) The Arrow-Pratt measure of relative risk aversion is u00 (w)w . rR = − 0 u (w) Note that the first of these in effect measures risk aversion with respect to a fixed amount of gamble (say, $1). The latter, in contrast, measures risk aversion for a gamble over a fixed percentage of wealth. These points can be demonstrated as follows: Consider a consumer with initial wealth w who is faced with a small fair bet, i.e., a gain or loss of some small amount ² with equal probability. 3
You can easily verify this by thinking of the units attached to the second derivative. If the first derivative measures change in utility for change in wealth, then its units must be u/w, while the second derivative is like a rate of acceleration. Its units are u/w 2 .
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How much would the consumer be willing to pay in order to avoid this bet? Denoting this payment by I we need to consider (note that w − I is the certainty equivalent) 0.5u(w + ²) + 0.5u(w − ²) = u(w − I). Use a Taylor series expansion in order to approximate both sides: 0.5(u(w) + ²u0 (w) + 0.5²2 u00 (w)) + 0.5(u(w) − ²u0 (w) + 0.5²2 u00 (w)) ∼ u(w) − Iu0 (w) . Collecting terms and simplifying gives us 0.5²2 u00 (w)) ∼ −Iu0 (w)
⇒
I∼
²2 −u00 (w) × 0 . 2 u (w)
Thus the required payment is proportional to the absolute coefficient of risk aversion (and the dollar amount of the gamble.) On the other hand, u00 w du0 w du0 /u0 %∆u0 = = ∼ . u0 dw u0 dw/w %∆w Thus the relative coefficient of risk-aversion is nothing but the elasticity of marginal utility with respect to wealth. That is, it measures the responsiveness of the marginal utility to wealth changes. Comparing across consumers, a consumer is said to be more risk averse than another if (either) Arrow-Pratt coefficient of risk aversion is larger. This is equivalent to saying that he has a lower certainty equivalent for any given gamble, or requires a higher probability premium. We can also compare the risk aversion of a given consumer for different wealth levels. That is, we can compute these measures for the same u(·) but different initial wealth. After all, rA is a function of w. It is commonly assumed that consumers have (absolute) risk aversion which is decreasing with wealth. Sometimes the stronger assumption of decreasing relative risk aversion is made, however. Note that a constant absolute risk aversion implies increasing relative risk aversion. Finally, note also that the only functional form for u(·) which has constant absolute risk aversion is u(w) = −e(−aw) . You may wish to verify that a consumer exhibiting decreasing absolute risk aversion will have a decreasing difference between initial wealth and the certainty equivalent (a declining maximum price paid for insurance) on the one hand, and a decreasing probability premium on the other.
Uncertainty 67 Utility
3 2.5
♦
2 E(u(3,4)) ♦ E(u(1,2)) ♦ 1.5
♦
♦
♦ ♦
♦
♦
♦
♦
♦
w1
5 w3 E(w)
10 w4
w2
u(w)
♦
1 0.5 0 0
15
20 wealth
Figure 4.4: Comparing two gambles with equal expected value
4.2
Comparing gambles with respect to risk
Another type of comparison of interest is not across consumers or wealth levels, as above, but across different gambles. Faced with two gambles, when do we want to say that one is riskier than the other? We could try to approach this question with purely statistical measures, such as comparisons of the various moments of the two lotteries’ distributions. This has the major problem, however, that the consumer may in general be expected to be willing to trade off a higher expected return for higher variance, say. Because of this, a definition based directly on consumer preferences is preferable. Two such measures are commonly employed in economics, first and second order stochastic dominance. Let us first focus on lotteries with the same expected value. For example, consider the two gambles depicted in Fig. 4.4. The first is a gamble over w1 and w2 . The second is a gamble over w3 and w4 . Both have an identical expected value of E(w). Nevertheless a risk averse consumer clearly will prefer the second to the first, as inspection of the diagram verifies. Note that in Fig. 4.4 E(w) − w1 > E(w) − w3 and w2 − E(w) > w4 − E(w). This clearly indicates that the second lottery has a lower variance, and thus that a risk averse consumer prefers to have less variability for a given mean. With multiple possible outcomes the question is not so simple anymore, however. One could construct an example with two lotteries that have the same mean and variance, but which differ in higher moments. What are the “obvious” preferences of a risk averse consumer about skurtosis, say?
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This has lead to a more general definition for comparing distributions which have the same mean: Definition 6 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). Let F (·) have the same mean as G(·). F (·) is said to dominate G(·) according to second order stochastic dominance if for every non-decreasing concave u(x): Z Z u(x)dF (x) ≥ u(x)dG(x) In words, a distribution second order stochastically dominates another if they have the same mean and if the first is preferred by all risk-averse consumers. This definition has economic appeal in its simplicity, but is one of those definitions that are problematic to work with due to the condition that for all possible concave functions something is true. In order to apply this definition easily we need to find other tests. Lemma 1 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). F (·) dominates G(·) according to second order stochastic dominance if Z Z Z x Z x tg(t)dt = tf (t)dt, and G(t)dt ≥ F (t)dt, ∀x. 0
0
I.e., if they have the same mean and there is more area under the cdf G(·) than under the cdf F (·) at any point of the distribution.4 A concept related to second order stochastic dominance is that of a mean preserving spread. Indeed it can be shown that the two are equivalent. Definition 7 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). G(·) is a mean preserving spread compared to F (·) if x is distributed according to F (·) and G(·) is the distribution of the R random variable x + z, where z is distributed according to some H(·) with zdH(z) = 0. 4
Note that the condition of identical means also implies a restriction on the total Rx Rx areas below the cumulative distributions. After all, x tdF (t) = [tF (t)]xx − x F (t)dt = Rx x − x F (t)dt.
Uncertainty 69 The above gives us an easy way to construct a second order stochastically dominated distribution: Simply add a zero mean random variable to the given one. While it is nice to be able to rank distributions in this manner, the condition of equal means is restrictive. Furthermore, it does not allow us to address the economically interesting question of what the trade off between mean and risk may be. The following concept is frequently employed in economics to deal with such situations. Definition 8 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). F (·) is said to dominate G(·) according to first order stochastic dominance if for every nondecreasing u(x): Z Z u(x)dF (x) ≥
u(x)dG(x)
This is equivalent to the requirement that F (x) ≤ G(x), ∀x.
Note that this requires that any consumer, risk averse or not, would prefer F to G. It is often useful to realize two facts: One, a first order stochastically dominating distribution F can be obtained form a distribution G by shifting up outcomes randomly. Two, first order stochastic dominance implies a higher mean, but is stronger than just a requirement on the mean. The other moments of the distribution get involved too. In other words, just because the mean is higher for one distribution than another does not mean that the first dominates the second according to first order stochastic dominance!
4.3
A first look at Insurance
Let us use the above model to investigate a simple model of insurance. To be concrete, assume an individual with current wealth of $100,000 who faces a 25% probability to loose his $20,000 car through theft. Assume the individual has vN-M expected utility. The individual’s expected utility then is U (·) = 0.75u(100, 000) + .25u(80, 000). Now assume that the individual has access to an insurance plan. Insurance works as follows: The individual decides on an amount of coverage, C. This
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coverage carries a premium of π per dollar. The contract specifies that the amount C will be paid out if the car has been stolen. (Assume that this is all verifiable.) How would our individual choose the amount of coverage? Simple: maximize expected utility. Thus maxC {0.75u(100, 000 − πC) + 0.25u(80, 000 − πC + C)}. The first order condition for this problem is (−π)0.75u0 (100, 000 − πC) + (1 − π)0.25u0 (80, 000 − πC + C) = 0. Before we further investigate this equation let us verify the second order condition. It requires (−π)2 0.75u00 (100, 000 − πC) + (1 − π)2 0.25u00 (80, 000 − πC + C) < 0. Clearly this is only satisfied if u(·) is concave, in other words, if the consumer is risk averse. So, what does the first order condition tell us? Manipulation yields the condition that u0 (100, 000 − πC) (1 − π) = 0 u (80, 000 − πC + C) 3π which gives us a familiar looking equation in that the LHS is a ratio of marginal utilities. It follows that total consumption under each circumstance is set so as to set the ratio of marginal utility of wealth equal to some fraction which depends on price and the probabilities. Even without knowing the precise function we can say something about the insurance behaviour, however. To do so, let us compute the actuarially fair premium. The expected loss is $5,000, so that an insurance premium which collects that amount for the $20,000 insured value would lead to zero expected profits for the insurance firm: 0.75πC + 0.25(πC − C) = 0 ⇒ π = 0.25. An actuarially fair premium simply charges the odds (there is a 1 in 4 chance of a loss, after all.) If we use this fair premium in the above first order condition we obtain u0 (100, 000 − πC) = 1. u0 (80, 000 − πC + C)
Since the utility function is strictly concave it can have the same slope only at the same point, and we conclude that5 (100, 000 − πC) = (80, 000 − πC + C) ⇒ C = 20, 000. 5
Ok, read that sentence again. Do you understand the usage of the word ‘Since’ ? I am not “cancelling” the u0 terms, because those indicate a function. Instead the equation tells us that numerator and denominator must be the same. But for what values of the independent variable wealth does the function u(·) have the same derivative? For none, if u(·) is strictly concave. Therefore the function must be evaluated at the same level of the independent variable.
Uncertainty 71 This is one of the key results in the analysis of insurance: at actuarially fair premiums a risk averse consumer will fully insure. Note that the consumer will not bear any risk in this case: wealth will be $95,000 independent of if the car is stolen, since a $5,000 premium is due in either case, and if the car is actually stolen it will be replaced. As we have seen before, this will make the consumer much better off than if he is actually bearing the gamble with this same expected wealth level. If you draw the appropriate diagram you can verify that the consumer does not have to pay any of the amount he would be willing to pay (the difference between the expected value and the certainty equivalent.) If we had a particular utility function we could now also compute the maximal amount the consumer would be willing to pay. We have to be careful, however, how we set up this problem, since simply increasing π will reduce the amount of coverage purchased! So instead, let us approach the question as follows: What fee would the consumer be willing to pay in order to have access to actuarially fair insurance? Let F denote the fee. Then we have the consumer choose between u(95, 000 − F ) and 0.25u(80, 000) + 0.75u(100, 000). (Note that I have skipped a step by assuming full insurance. The left term is the expected utility of a fully insured consumer who pays the fee, the right term is the expected utility of an uninsured consumer. You should verify that the lump sum fee does not stop the consumer from fully insuring at a fair premium.) For example, if u(·) = ln(·) then simple manipulation yields F ∼ 426. It is important to note why we have set up the problem this way. Consider the alternative (based on these numbers and the logarithmic function) and assume that the total payment of $5,426 which is made in the above case of a fair premium plus fee, were expressed as a premium. Then we get that π = 5426/20000 = 0.2713. The first order condition for the choice of C then requires that (recall that ∂ln(x)/∂x = 1/x) (80, 000 + 0.7287C) 0.7287 = = 0.895318835 (100, 000 − 0.2713C) 0.8139
⇒ C = 9, 810.50.
As you can see, if the additional price is understood as a per dollar charge for insured value, the consumer will not insure fully. Of course this is an implication of the previous result — the consumer now faces a premium which is not actuarially fair. Indeed, we could also compute the premium for which the consumer will cease to purchase any insurance. For logarithmic utility like this we would want to compute (remember, we are trying to find
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when C = 0 is optimal) 1−π 80, 000 = 100, 000 3π
⇒ π = 0.2941.
As indicated before, there is room to trade between insurance providers and risk averse consumers. Indeed, as you can verify in one of the questions at the end of the chapter, there is room for trade between two risk averse consumers if they face different risk or if they differ in their attitudes towards risk (degree of risk aversion.)
4.4
The State-Preference Approach
While the above approach lets us focus quite well on the role of probabilities in consumer choice, it is different in character to the ‘maximize utility subject to a budget constraint’ approach we have so much intuition about. In the first order condition for the insurance problem, for example, we had a ratio of marginal utilities on the one side — but was that the slope of an indifference curve? As mentioned previously, we can actually treat consumption as involving contingent commodities, and will do so now. Let us start by assuming that the outcomes of any random event can be categorized as something we will refer to as the states of the world. That is, there exists a set of mutually exclusive states which are adequate to describe all randomness in the world. In our insurance example above, for example, there were only two states of the world which mattered: Either the car was stolen or it was not. Of course, in more general settings we could think of many more states (such as the car is stolen and not recovered, the car is stolen but recovered as a write off, the car is stolen and recovered with minor damage, etc.) In accordance with this view of the world we now will have to develop the idea of contingent commodities. In the case of our concrete example with just two states, a contingent commodity would be delivered only if a particular state (on which the commodity’s delivery is contingent) occurs. So, if there are two states, good and bad, then there could be two commodities, one which promises consumption in the good state, and one which promises consumption in the bad state. Notice that you would have to buy both of these commodities if you wanted to consume in both states. Notice also that nothing requires that the consumer purchase them in equal amounts. They are, after all, different commodities now, even if the underlying good which gets delivered in each state is the same. Finally, note that if one of these commodities were missing
Uncertainty 73 you could not assure consumption in both states (which is why economists make such a fuss about “complete markets” — which essentially means that everything which is relevant can be traded. It does not have to be traded, of course, that is up to people’s choices, but it should be available should someone want to trade.) Of course, after the fact (ex post in the lingo) only one of these states does occur, and thus only the set of commodities contingent on that state are actually consumed. Before the fact (before the uncertainty is resolved, called ex ante) there are two different commodities available, however. Once we have this setting we can proceed pretty much as before in our analysis. To be concrete let there be just two states, good and bad. We will now index goods by a subscript b or g to indicate the state in which they are delivered. We will further simplify things by having just one good, consumption (or wealth). Given that there are two states, that means that there are two distinct (contingent) commodities, cg and cb . We may now assume that the consumer has our usual vN-M expected utility.6 If the individual assessed a probability of π to the good state occurring, then we would obtain an expected utility of consumption of U (cg , cb ) = πu(cg ) + (1 − π)u(cb ). This expression gives us the expected utility of the consumer. The consumers’ objective is to maximize expected utility, as before. It might be useful at this point to assess the properties of this function. As long as the utility index applied to consumption in each state, u(·), is concave, this is a concave function. It will be increasing in each commodity, but at a decreasing rate. We can also ask what the marginal rate of substitution between the commodities will be. This is easily derived by taking the total derivative along an indifference curve and rearranging: πu0 (cg )dcg + (1 − π)u0 (cb )dcb = 0,
πu0 (cg ) dcb =− . dcg (1 − π)u0 (cb )
Note the fact that the MRS now depends not only on the marginal utility of wealth but also on the (subjective) probabilities the consumer assesses for each state! Even more importantly, we can consider what is known as the certainty line, that is, the locus of points where cg = cb . Since the marginal utility of consumption then is equal in both states (we have state independent utility here, after all, which means that the same u(·) applies in each state), it follows that the slope of an indifference curve on the certainty 6
Note that this is somewhat more onerous than before now: imagine the states are indexed by good health and poor health. It is easy to imagine that an individual would evaluate material wealth differently in these two cases.
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line only depends on the probability the consumer assesses for each state. In this case, it is π/(1 − π). The other ingredient is the budget line, of course. Since we have two commodities, each might be expected to have a price, and we denote these by pg , pb respectively. The consumer who has a total initial wealth of W may therefore consume any combination which lies on the budget line pg cg + pb cb = W , while a consumer who has an endowment of consumption given by (wg , wb ) may consume anything on the budget line pg cg + pb cg = pg wg + pb wb . Where do these prices come from? As before, they will be determined by general equilibrium conditions. But if contingent markets are well developed and competitive, and there is general agreement on the likelihood of the states, then we might expect that a dollar’s worth of consumption in a state will cost its expected value, which is just the dollar times the probability that it needs to be delivered. (I.e., a kind of zero profit condition for state pricing.) Thus we might expect that pg = π and pb = (1 − π). The budget line also has a slope, of course, which is the rate at which consumption in one state can be transferred into consumption in the other state. Taking total derivatives of the budget we obtain that the budget slope is dcb /dcg = pg /pb . Combining this with our condition on “fair” pricing in the previous paragraph, we obtain that the budget allows transformation of consumption in one state to the other according to the odds.
4.4.1
Insurance in a State Model
So let us reconsider our consumer who was in need of insurance in this framework. In order to make this problem somewhat neater, we will reformulate the insurance premium into what is known as a net premium, which is a payment which only accrues in the case there is no loss. Since the normal insurance contract specifies that a premium be paid in either case, we usually have a payment of premium × Amount in order to obtain a net benefit of Amount − premium × Amount. One dollar of consumption added in the state in which an accident occurs will therefore cost premium/(1−premium) dollars in the no accident state. Thus, let pb = 1 and let pg = P , the net premium. The consumer will then solve maxcb ,cg {πu(cg ) + (1 − π)u(cb )} s.t. P cg + cb = P (100, 000) + 80, 000. The two first order conditions for the consumption levels in this problem are πu0 (cg ) − λP = 0 and (1 − π)u0 (cb ) − λ = 0.
Uncertainty 75 loss
20 cert. 15
10
5 @ endowment 0 0
5
10
15
20 no loss
Figure 4.5: An Insurance Problem in State-Consumption space Combining them in the usual way we obtain πu0 (cg ) = P. (1 − π)u0 (cb ) Now, as we have just seen the LHS of this is the slope of an Indifference curve. The RHS is the slope of the budget, and so this says nothing but the familiar “there must be a tangency”. We have also derived P = π/(1 − π) for a fair net premium before. Thus we get that πu0 (cg ) π = , 0 (1 − π)u (cb ) 1−π
which requires that
u0 (cg ) =1 u0 (cb )
⇒
cg = 1. cb
Thus this model shows us, just as the previous one, that a risk averse consumer faced with a fair premium will choose to fully insure, that is, choose to equalize consumption levels across the states. A diagrammatic representation of this can be found in diagram 3.5, which is standard for insurance problems. The consumer has an endowment which is off the certainty (45-degree) line. The fair premium defines a budget line along which the consumer can reallocate consumption from the good (no loss) state to the bad (loss) state. Optimum occurs where there is a tangency, which must occur on the certainty line since then the slopes are equalized. The picture looks perfectly “normal”, that is, just as we are used from introductory economics.
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4.4.2
May2004
Risk Aversion Again
Given the amount of time spent previously on risk-aversion, it is interesting to see how risk-aversion manifests itself in this setting. Intuitively it might be apparent that a more risk averse consumer will have indifference curves which are more curved, that is, exhibit less substitutability (recall that a straight line indifference curve means that the goods are perfect substitutes, while a kinked Leontief indifference curve means perfect complements.) It therefore stands to reason that we might be interested in the rate at which the MRS is falling. It is, however, much easier to think along the lines of certainty equivalents: Consider two consumers with different risk aversion, that is, curvature of indifference curves. For simplicity, let us consider a point on the certainty line and the two indifference curves for our consumers through that common point (see Fig. 4.6). loss
20 cert. B A
15
10
5
0 0
5
10
15
20 no loss
Figure 4.6: Risk aversion in the State Model Assume further that consumer B’s indifference curve lies everywhere else above consumer A’s. We can now ask how much consumption we have to add for each consumer in order to keep the consumer indifferent between the certain point and a consumption bundle with some given amount less in the bad state. Clearly, consumer B will need more compensation in order to accept the bad state reduction. Looked at it the other way around, this means that consumer B is willing to give up more consumption in the good state in order to increase bad state consumption. Note that both assess the same probabilities on the certainty line, since the slopes of their ICs are the same. How does this relate to certainty equivalents? Well, a budget line at π fair odds will have the slope − 1−π . Consider three such budget lines which are all parallel and go through the certain consumption point and the two
Uncertainty 77 gambles which are equivalent for the consumer to the certain point. Clearly (from the picture) consumer B’s budget is furthest out, followed by consumer A’s, and furthest in is the budget through the certain point. But we know that parallel budgets differ only in the income/wealth they embody. Thus there is a larger reduction in wealth possible for B without reducing his welfare, compared to A. The wealth reduction embodied in the lower budget is the equivalent of the certainty equivalent idea before. (The expected value of a given gamble on such a budget line is given by the point on the certainty line and that budget, after all.)
4.5
Asset Pricing
Any discussion of models of uncertainty would be incomplete without some coverage of the main area in which all of this is used, which is the pricing of assets. As we have seen before, if there is only time to contend with but returns or future prices are known, then asset pricing reduces to a condition which says that the current price of an asset must relate to the future price through discounting. In the “real world” most assets do not have a future price which is known, or may otherwise have returns which are uncertain — stocks are a good example, where dividends are announced each year and their price certainly seems to fluctuate. Our discussion so far has focused on the avoidance of risk. Of course, even a risk averse consumer will accept some risk in exchange for a higher return, as we will see shortly. First, however, let us define two terms which often occur in the context of investments.
4.5.1
Diversification
Diversification refers to the idea that risk can be reduced by spreading one’s investments across multiple assets. Contrary to popular misconceptions it is not necessary that their price movements be negatively correlated (although that certainly helps.) Let us consider these issues via a simple example. Assume that there exists a project A which requires an investment of $9,000 and which will either pay back $12,000 or $8,000, each with equal probability. The expected value of this project is therefore $10,000. Now assume that a second project exists which is just like this one, but (and this is important) which is completely independent of the first. How much each pays back in no way depends on the other. Two investors now could each invest $4,500 in each project. Each investor then has again a total
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investment of $9,000. How much do the projects pay back? Well, each will pay an investor either $6,000 or $4,000, each with equal probability. Thus an investor can receive either $12,000, $10,000, or $8,000. $12,000 or $8,000 are received one quarter of the time, and half the time it is $10,000. The total expected return thus is the same. BUT, there is less risk, since we know that for a risk-averse consumer 0.5u(12)+0.5u(8) < 0.25u(12)+0.25u(8)+0.5u(10) since 2(0.25u(12) + 0.25u(8)) < 2(0.5u(10)). Should the investor have access to investments which have negatively correlated returns (if one is up the other is down) risk may be able to be eliminated completely. All that is needed is to assume that the second project above will pay $8,000 when the first pays $12,000, and that it will pay $12,000 if the first pays $8,000. In that case an investor who invests half in each will obtain either $6,000 and $4,000 or $4,000 and $6,000: $10,000 in either case. The expected return has not increased, but there is no risk at all now, a situation which a risk-averse consumer would clearly prefer.
4.5.2
Risk spreading
Risk spreading refers to the activity which lies at the root of insurance. Assume that there are 1000 individuals with wealth of $35,000 and a 1% probability of suffering a $10,000 loss. If the losses are independent of one another then there will be an average of 10 losses per period, for a total $100,000 loss for all of them. The expected loss of each individual is $100, so that all individuals have an expected wealth of $34,900. A mutual insurance would now collect $100 from each, and everybody would be reimbursed in full for their loss. Thus we can guarantee the consumers their expected wealth for certain. Note that there is a new risk introduced now: in any given year more (or less) than 10 losses may occur. We can get rid of some of this by charging the $100 in all years and retaining any money which was not collected in order to cover higher expenses in years in which more than 10 losses occur. However, there may be a string of bad luck which might threaten the solvency of the plan: but help is on the way! We could buy insurance for the insurance company, in effect insuring against the unlikely event that significantly more than the average number of losses occurs. This is called re-insurance. Since an insurance company has a well diversified portfolio of (independent) risks, the aggregate risk it faces itself is low and it will thus be able to get fairly cheap insurance.
Uncertainty 79 These kind of considerations are also able to show why there may not be any insurance offered for certain losses. You may recall the lament on the radio about the fact that homeowners in the Red River basin were not able to purchase flood insurance. Similarly, you can’t get earth-quake insurance in Vancouver, and certain other natural disasters (and man-made ones, such as wars) are excluded from coverage. Why? The answer lies in the fact that all insured individuals would have either a loss or no loss at the same time. That would mean that our mutual insurance above would either require no money (no losses) or $10,000,000. But the latter requires each participant to pay $10,000, in which case you might as well not insure! (A note aside: often the statement that no insurance is available is not literally correct: there may well be insurance available, but only at such high rates that nobody would buy it anyways. Even at low rates many people do not carry insurance, often hoping that the government will bail them out after the fact, a ploy which often works.)
4.5.3
Back to Asset Pricing
Before we look at a more general model of asset pricing, it may be useful to verify that a risk-averse consumer will indeed hold non-negative amounts of risky assets if they offer positive returns. To do so, let us assume the simple most case, that of a consumer with a given wealth w who has access to a risky asset which has a return of rg or rb < 0 < rg . Let x denote the amount invested in the risky asset. Wealth then is a random variable and will be either wg = (w − x) + x(1 + rg ) or wb = (w − x) + x(1 + rb ). Suppose the good outcome occurs with probability π. What will be the choice of x? max0≤x≤w {πu(w + rg x) + (1 − π)u(w + rb x)} . The first and second order conditions are rg πu0 (w + rg x) + rb (1 − π)u0 (w + rb x) = 0 rg2 πu00 (w + rg x) + rb2 (1 − π)u00 (w + rb x) < 0 The second order condition is satisfied trivially if the consumer is risk averse. To show under what circumstances it is not optimal to have a zero investment consider the FOC at x = 0: rg πu0 (w) + rb (1 − π)u0 (w) ? 0. The LHS is only positive if πrg + (1 − π)rb > 0, that is, if expected returns are positive. Notice also that in that case there will be some investment! Of
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course this is driven by the fact that not investing guarantees a zero rate of return. Investing is a gamble, which the consumer dislikes, but also increases returns. Even a risk-averse consumer will take some risk for that higher return! Now let us consider a more general model with many assets. Assume that there is a risk-free asset (one which yields a certain return) and many risky ones. Let the return for the risk-free asset be denoted by R0 and the ˜ i , each of which is a random returns for the risky assets be denoted by R variable with some distribution. Initial wealth of the consumer is w. Finally, we can let xi denote the fraction of wealth allocated to asset i = 0, . . . , n. In the second period (we will ignore time discounting for simplicity and clarity) wealth will be a random variable the distribution of which on how P depends ˜ i , with the much is invested in eachPasset. In particular, w˜ = w0 ni=0 xi R budget constraint that ni=0 xi = 1. We can transform this expression as follows: ' # ' # n n n X X X ˜ i = w R0 + ˜ i − R0 ) . w˜ = w (1 − xi )R0 + xi R xi ( R i=1
i=1
i=1
The consumer’s goal, of course, is to maximize expected utility from this wealth by choice of the investment fractions. That is, ( Ã ' #!) n X ˜ i − R0 ) max{x} {Eu (w)} ˜ = max{x} Eu w R0 + xi ( R . i
i
i=1
Differentiation yields the first order conditions ˜ i − R0 ) = 0, Eu0 (w)( ˜ R
∀i.
Now we will do some manipulation of this to make it look more presentable and informative. You may recall that the covariance of two random variables, ˜i = X, Y , is defined as COV(X, Y ) = EXY −EXEY. It follows that Eu0 (w) ˜ R 0 ˜ 0 ˜ i . Using this fact and distributing the subtraction in COV(u , Ri )+Eu (w)E ˜ R the FOC across the equal sign, we obtain for each risky asset i the following equation: ˜ i + COV(u0 (w), ˜ i ). Eu0 (w)R ˜ 0 = Eu0 (w)E ˜ R ˜ R From this it follows that in equilibrium the expected return of asset i must satisfy 0 ˜i) ˜ R ˜ i = R0 − COV(u (w), . ER Eu0 (w) ˜
Uncertainty 81 This equation has a nice interpretation. The first term is clearly the riskfree rate of return. The second part therefore must be the risk-premium which the asset must garner in order to be held by the consumer in a utility maximizing portfolio. Note that if a return is positively correlated with wealth — that is, if an asset will return much if the consumer is already rich — then it is negatively correlated with the marginal utility of wealth, since that is decreasing in wealth. Thus the expected return of such an asset must exceed the risk free return if it is to be held. Of course, assets which pay off when wealth otherwise would be low can have a lower return than the risk-free rate since they, in a sense, provide insurance.
4.5.4
Mean-Variance Utility
The above pricing model required us to know the covariance and expectation of marginal utility, since, as we have seen before, it is the fact that marginal utility differs across outcomes which in some sense causes risk-aversion. A nice simplification of the model is possible if we specify at the outset that our consumer likes the mean but dislikes the variance of random returns, i.e., the mean is a good, the variance is a bad. We can then specify a utility function directly on those two characteristics of the distribution. (The normal distribution, for example is completely described by these two moments. If distributions differ in higher moments, this formulation would not be able to pick that up, however.) Recall that for a set of outcomes (w1 , w2 , . . . , wn ) with probabilities (π1 , π2 , . . . , πn ) n X The mean is µw = π i wi , i=1
and the variance is
σw2
=
n X i=1
πi (wi − µw )2 .
We now define utility directly on these: u(µw , σw ), although it is standard practice to actually use the standard deviation as I just have done. Risk aversion is now expressed through the fact that we assume that ∂u(·) = u1 (·) > 0 while ∂µw
∂u(·) = u2 (·) < 0. ∂σw
We will now focus on two portfolios only (the validity of this approach will be shown in a while.) The risk free asset has a return of rf , the risky
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asset (the pPhas a return of ms with probability πs . Let P “market portfolio”) rm = πs ms and σm = πs (ms − rm )2 . Assume that a fraction x of wealth is to be invested in the risky asset (the market). The expected return for a fraction x invested will be X X rx = πs (xms + (1 − x)rf ) = (1 − x)rf + x πs ms .
The variance of this portfolio is X X 2 σx2 = πs (xms + (1 − x)rf − rx )2 = πs (xms − xrm )2 = x2 σm . The investor/consumer will maximize utility by choice of x:
maxx {u(xrm + (1 − x)rf , xσm )} FOC u1 (·)[rm − rf ] + u2 (·)σm = 0 2 + 2σm [rm − rf ]u12 (·) ≤ 0 SOC u11 (·)[rm − rf ]2 + u22 (·)σm Assuming that the second order condition holds, we note that we will require [rm − rf ] > 0 since u2 (·) is negative by assumption. We may also note that we can rewrite the FOC as rm − r f −u2 (·) = . u1 (·) σm The LHS of this expression is the MRS between the mean and the standard deviation, that is, the slope of an indifference curve. The RHS can be seen to be the slope of the budget line since the budget is a mix of two points, (rf , 0) and (rm , σm ), which implies that the tradeoff of mean for standard deviation is rise over run: (rm − rf )/σm . In a general equilibrium everybody has access to the same market and the same risk free asset. Thus, everybody who does hold any of the market will have the same MRS — a result analogous to the fact that in our usual general equilibria everybody will have the same MRS. Of course, this is just a requirement of Pareto Optimality. In this discussion we had but one risky asset. In reality there are many. As promised, we derive here the justification for considering only the so-called market portfolio. The basic idea is simple. Assume a set of risky assets. Since we are operating in a two dimensional space of mean versus standard deviation, one can certainly ask what combination of assets (also known as a portfolio) will yield the highest mean for a given standard deviation, or, which is often easier to compute, the lowest standard deviation for a given mean.
Uncertainty 83 mean
20 budget efficient risk
15 market
10
5 risk free 0 0
5
10
15
20 standard dev.
Figure 4.7: Efficient Portfolios and the Market Portfolio P Let x denote the vector of shares in each of the assets so that I x = 1. Define the mean and standard deviation of the portfolio x as µ(x) and σ(x). Then it is possible to solve maxx {µ(x) + λ(s − σ(x))} or minx {σ(x) + λ(m − µ(x)} . For each value of the constraint there will be a portfolio (an allocation of wealth across the different risky assets) which achieves the optimum. It turns out (for reasons which I do not want to get into here: take a finance course or do the math) that this will lead to a frontier which is concave to the standard variation axis. Now, by derivation, any proportion of wealth which is held in risky assets will be held according to one of these portfolios. But which one? Well, the consumer can combine any one of these assets with the risk free asset in order to arrive at the final portfolio. Since this is just a linear combination, the resulting “budget line” will be a straight line and have a positive slope of (µx − µf )/σx , where µx , σx are drawn from the efficient frontier. A simple diagram suffices to convince us that the highest budget will be that which is just tangent to the efficient frontier. The point of tangency defines the market portfolio we used above! Note a couple more things from this diagram. First of all it is clear from the indifference curves drawn that the consumer gains from the availability of risky assets on the one hand, and of the risk-free asset on the other. Second, the market portfolio with which the risk free asset will be mixed will depend on the return from the risk free asset. Imagine sliding the risk free return
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up in the diagram. The budget line would have to become flatter and this means a tangency to the efficiency locus further to the right. Finally, the precise mix between market and risk free will depend on preferences. Indeed, there might be people who would want to have more than their entire wealth in the market portfolio, that is, the tangency to the budget would occur to the right of the market portfolio. This requires a “leveraged” portfolio in which the consumer is allowed to hold negative amounts of certain assets (short-selling.)
4.5.5
CAPM
The above shows that all investors face the same price of risk (in terms of the variance increase for a given increase in the mean.) It does not tell us anything about the risk-return trade-off for any given asset, however. Any risk unrelated to the market risk can be diversified away in this setting, however, so that any unsystematic risk will not attract any excess returns. An asset must, however, earn higher returns to the extent that it contributes to the market risk, since if it did not it would not be held in the market portfolio. Consideration of this problem in more detail (assume a portfolio with a small amount of this asset held and the rest in the market, compute the influence of the asset on the portfolio return and variance, rearrange) will yield the famous CAPM equation involving the asset’s ‘beta’ a number which is published in the financial press: σX,M µx = µf + (µm − µf ) 2 . σM Here x denotes asset x, m the market, f the risk free asset, and σX,M is the σX,M covariance between asset x and the market. The ratio 2 is referred to as σM the asset’s beta.
4.6
Review Problems
Question 1: Demonstrate that all risk-averse consumers would prefer an investment yielding wealth levels 24, 20, 16 with equal probability to one with wealth levels 24 or 16 with equal probability. Question 2: Compute the certainty equivalent for an expected utility maxi√ mizing consumer with (Bernoulli) utility function u(w) = w facing a gamble over $3600 with probability α and $6400 with probability 1 − α.
Uncertainty 85 Question 3: Determine at what wealth levels a consumer with (Bernoulli) utility function u(w) = lnw has the same absolute risk aversion as a consumer √ with (Bernoulli) utility function u(w) = 2 w. How do their relative risk aversions compare at that level. What does that mean? Question 4: Consider an environment with two states — call them rain, R, and shine, S, — with the probability of state R occurring known to be π. Assume that there exist two consumers who are both risk-averse, vN-M expected utility maximizers. Assume further that the endowment of consumer A is (10, 5) — denoting 10 units of the consumption good in the case of state R and 5 units in the case of S — and that the endowment of consumer B is (5, 10). What are the equilibrium allocation and price? (Provide a well labelled diagram and supporting arguments for any assertions you make.) Question 5: Anna has $10,000 to invest and wants to invest it all in one or both of the following assets: Asset G is gene-technology stock, while asset B is stock in a bible-printing business. There are only two states of nature to worry about, both of which occur with equal probability. One is that gene-technology is approved and flourishes, in which case the return of asset B is 0% while the return of asset G is 80%. The other is that religious fundamentalism takes hold, and gene-technology is severely restricted. In that case the return of asset B will be 40% but asset G has a return of (40%). Anna is a risk-averse expected-utility maximizer with preferences over wealth represented by u(w). a) State Anna’s choice problem mathematically. b) What proportion of the $10,000 will she invest in the gene-technology stock? (I.e. solve the above maximization problem.) Question 6∗ : Assume that consumers can be described by the following preferences: they are risk averse over wealth levels, but they enjoy gambling for its consumption attributes (i.e., while they dislike the risk on wealth which gambling implies, they get some utility out of partaking in the excitement (say, they get utility out of daydreaming about what they could do if they won.)) Let us further assume that consumers only differ with respect to their initial wealth level and their consumption utility from gambling, but that all consumers have the same preferences over wealth. In order to simplify these preferences further, assume that wealth and gambling are separable. We can represent these preferences over wealth, w, and gambling, g ∈ {0, 1} by some u(w, g) = v(w) + µi g, where v(w) is strictly concave, and µi is an individual parameter for each consumer. Finally, they are assumed to be expected utility maximizers. a) Assume for now that the gambling is engaged in via a lottery, in
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which consumers pay a fixed price p for one lottery ticket, and the ticket is either a “Try Again” (no prize) or “Winner” (Prize won.) Also assume for simplicity that they can either gamble once or not at all (i.e., each consumer can only buy one ticket.) i) First verify that in such an environment the government can make positive profits. (I.e., verify that some consumers will buy a ticket even if the ticket price exceeds the expected value of the ticket. ii) If consumers have identical µi , how does the participation of consumers then depend on their initial wealth level if their preferences exhibit {constant| decreasing} {absolute |relative} risk aversion? (The above notation means: consider all (sensible) permutations.) iii) Assume now that preferences are characterized by decreasing absolute and constant relative √ risk aversion. Verify that the utility functions v(w) = ln w and v(w) = w satisfy this assumption. Also assume that the consumption enjoyment of gambling is decreasing in wealth, that is, consumers with high initial wealth have low µi . (They know what pain it is to be rich and don’t daydream as much about it.) Who would gamble then? Question 7∗ : Assume that a worker only cares about the income he generates from working and the effort level he expends at work. Also assume that the worker is risk averse over income generated and dislikes effort. His √ preferences can be represented by u(w, e) = w − e2 . The worker generates income by accepting a contract which specifies an effort level e and the associated wage rate w(e). (Since leisure time does not enter in his preferences he will work full time (supply work inelastically) and we can normalize the wage rate to be per period income.) The effort level of the worker is not observed directly by the firm, and thus the worker has an incentive to expend as little effort as possible. The firm, however, cares about the effort expended, since it affects the marginal product it gets from employing the worker. It can conduct random tests of the worker’s effort level with some probability α. These tests reveal the true effort level employed by the worker. If the worker is not tested, then it is assumed that he did indeed work at the specified effort level and will receive the contracted wage. If he is tested and found to have shirked (not supplied the correct effort level) then a penalty p is assessed and deducted from the wage of the worker. a) What relationship has to hold between w(e), e, α and p in order for a worker to provide the correct effort level if p is a constant (i.e., not dependent on either the contracted nor the observed effort level)? b) If we can make p depend on the actual deviation in effort which we observe, what relationship has to be satisfied then? c) Is there any economic insight hidden in this? Think about how these problems would change if the probability of detection somehow depended on
Uncertainty 87 the effort level (i.e., the more I deviate from the correct effort level, the more likely I might be caught.) Question 8: A consumer is risk averse. She is faced with an uncertain consumption in the future, since she faces the possibility of an accident. Accidents occur with probability π. If an accident occurs, it is either really bad, in which case she looses B, or minor, in which case her loss is M < B. Bad accidents represent only 1/5 of all accidents, all other accidents are minor. Without the accident her consumption would be W > B. a) Derive her optimal insurance purchases if the magnitude of the loss is publicly observable and verifiable and insurance companies make zero profits. More explicitly, if the type of accident is verifiable then a contract can be written contingent on the type of loss. The problem thus is equivalent to a problem where there are two types of accident, each of which occurs with a different probability and can be insured for separately at fair premiums. b) Now consider the case if the amount of loss is private information. In this case only the fact that there was an accident can be verified (and hence contracted on), but the insurer cannot verify if the loss was minor or major, and hence pays only one fixed amount for any kind of accident. Assume zero profits for the insurer, as before, and show that the consumer now overinsures for the minor loss but under-insures for the bad loss, and that her utility thus is lowered. (Note that the informational distortion therefore leads to a welfare loss.) Question 9: Assume a mean-variance utility model, and let µ denote the expected level of wealth, and σ its variance. √ Take the boundary of the efficient risky asset portfolios to be given by µ = σ − 16. Assume further that there exists a risk-free asset which has mean zero and standard deviation zero (if this bothers you, you can imagine that this is actually measuring the increase in wealth above current levels.) Let there be two consumers who have meanσ2 σ2 variance utility given by u(µ, σ) = µ− and u(µ, σ) = 3µ− respectively. 64 64 Derive their optimal portfolio choice and contrast their decisions. Question 10: Fact 1: The asset pricing formula derived in class states that ˜ i = R0 − ER
˜i) Cov(U 0 (w), ˜ R . EU 0 (w) ˜
Fact 2: A disability insurance contract can be viewed as an asset which pays some amount of money in the case the insured is unable to generate income from work. Use Facts 1 and 2 above to explain why disability insurance can have
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a negative return, (that is, why the price of the contract may exceed the expected value of the payment) if viewed as an asset in this way. Question 11: TRUE/FALSE/UNCERTAIN: Provide justification for your answers via proof, counter-example or argument. 1) The optimal amount of insurance a risk-loving consumer purchases is characterized by the First Order Condition of his utility maximization problem, which indicates a tangency between an indifference curve and a budget line. 2) For a consumer who is initially a borrower, the utility level will definitely fall if interest rates increase substantially. 3) The utility functions (3000 ∗ lnx1 + 6000 ∗ lnx2 ) 1/3 2/3 + 2462 and Exp(x1 x2 ) 12 represent the same consumer preferences. 4) A risk-averse consumer will never pay more than the expected value of a gamble for the right to participate in the gamble. A risk-lover would, on the other hand. 5) The market rate of return is 15%. The stock of Gargleblaster Inc. is known to increase to $117 next period, and is currently trading for $90. This market (and the current stock price of Gargleblaster Inc.) is in equilibrium. 6) Under Risk-Variance utility functions, all consumers who actually hold both the risk-free asset and the risky asset will have the same Marginal Rate of Substitution between the mean and the variance, but may not have the same investment allocation. Question 12∗ : Suppose workers have identical preferences√ over wealth only, which can be represented by the utility function u(w) = 2 w. Workers are also known to be expected utility maximizers. There are three kinds of jobs in the economy. One is a government desk job paying $40,000.00 a year. This job has no risk of accidents associated with it. The second is a bus-driver. In this job there is a risk of accidents. The wage is $44,100.00 and if there is an accident the monetary loss is $11,700.00. Finally a worker could work on an oil rig. These jobs pay $122,500.00 and have a 50% accident probability. These are all market wages, that is, all these jobs are actually performed in equilibrium. a) What is the probability of accidents in the bus driver occupation? b) What is the loss suffered by an oil rig worker if an accident occurs there? c) Suppose now that the government institutes a workers’ compensation scheme. This is essentially an insurance scheme where each job pays a fair premium for its accident risk. Suppose that workers can buy this insurance
Uncertainty 89 in arbitrary amounts at these premiums. What will the new equilibrium wages for bus-drivers and oil rig workers be? Who gains from the workers’ compensation? d) Now suppose instead that the government decides to charge only one premium for everybody. Suppose that of the workers in risky jobs 40% are oil rig workers and 60% are bus drivers. Suppose that they can buy as much or little insurance as they wish. How much insurance do the two groups buy? Who is better off, who is worse off in this case (at the old wages)? Can we say what the new equilibrium wages would have to be? Question 13∗ : Prove that state-independent expected utility is homothetic if the consumer exhibits constant relative risk aversion. (This question arises since indifference curves do have the same slope along the certainty line. So could they have the same slope along any ray from the origin? In that case they would be homothetic.)
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Chapter 5 Information In the standard model of consumer choice discussed in chapter 1, as well as the model of uncertainty developed in chapter 3, it was assumed that the decision maker knows all relevant information. In chapter 1 this meant that the consumer knows the price of all goods, as well as the precise features of each good (all characteristics relevant to the consumer.) In chapter 3 this in particular implied that the consumer has information about the probabilities of states or outcomes. Not only that, this information is symmetric, so that all parties to a transaction have the same information. Hence in chapter 3 the explicit assumption that the insurance provider has the same knowledge of the probabilities as the consumer. What if these assumptions fail? What if there is no complete and symmetric information? Fundamentally, one of the key problems is asymmetric information — when one party to a transaction knows something relevant to the transaction which the other party does not know. This quite clearly will lead to problems, since Pareto efficiency necessitates that all available information is properly incorporated. Consider, for example, the famous “Lemon’s Problem”:1 Suppose a seller knows the quality of her used car, which is either high or low. The seller attaches values of $5000 or $1000 to the two types of car, respectively. Buyers do not know the quality of a used car and have no way to determine it before purchase. Buyers value good used cars at $6000 and bad used cars at $2000. Note that it is Pareto efficient in either case for the car to be sold. Will the market mechanism work in this case? Suppose that it is known by everybody that half of all cars are good, and half are bad. To keep it simple, suppose buyers and sellers are risk 1
This kind of example is due to Akerlof (1970) Quarterly Journal of Economics, a paper which has changed economics.
91
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neutral. Buyers would then be willing to pay at most $4000 for a gamble with even odds in which they either receive a good or a bad used car. Now suppose that this were the market price for used cars. At this price only bad cars are actually offered for sale, since good car buyers rather keep theirs, since their valuation of their own car exceeds the price. It follows that this price cannot be an equilibrium price. It is fairly easy to verify that the only equilibrium would have bad cars trade at a price somewhere between $1000 and $2000 while good cars are not traded at all. This is not Pareto efficient. Of course, it is not necessary that there be asymmetric information. Suppose that there exist many restaurants, each offering slightly different combinations of food and ambiance. Consumers have tastes over the characteristics of the meals (how spicy, what kind of meat, if any meat at all; Italian, French, eastern European, Japanese, Egyptian, etc.) as well as the kind of restaurant (formal, romantic, authentic, etc.) as well as the general quality of the cooking within each category. In a Pareto efficient general equilibrium each consumer must frequent (subject to capacity constraints) the most preferred restaurant, or if that is full the next preferred one. Can we expect this to be the equilibrium?2 To see why the general answer may be “No” consider a risk averse consumer in an environment where a dining experience is necessary in order to find out all relevant characteristics of a restaurant. Every visit to a new place carries with it the risk of a really unpleasant experience. If the expected benefit of finding a better place than the one currently known does not exceed the expected cost of having a bad experience, the consumer will not try a new restaurant, and hence will not find the best match! What seems to be important then, are two aspects of information: One, can all relevant information be acquired before the transactions is completed?; Two, is the information symmetric or asymmetric? Aside from a classification of problems into asymmetric or symmetric information, it is common to distinguish between three classes of goods, based on the kind of informational problems they present: search goods, experience goods, and (less frequently) faith goods. A search good is one for which the consumer is lacking some information, be it price or some attribute of the good, which can be fully determined before purchase of the good. Anytime 2 This is a question similar to one very important in labour economics: are workers matched to the correct jobs, that is, are the characteristics of workers properly matched to the required characteristics of the job? These kind of problems are analysed in the large matching literature. In this literature you will find interesting papers on stable marriages — is everybody married to their most preferred partner, or could a small coalition swap partners and increase their welfare?
Information 93 the consumer can discover all relevant aspects before purchasing we speak of a search good. Supposing that search is costly, which seems reasonable, we can then model the optimal search behaviour of consumers by considering the (marginal) benefits and (marginal) costs of search. Applying such thinking to labour markets we can study the efficiency effects of unemployment insurance; or we can apply it to advertising, which for such goods focuses on supplying the missing information, and is therefore possibly efficiency enhancing. For some goods such a determination of all relevant characteristics may not be possible. Nobody can explain to you how something tastes, for example; you will have to consume the good to find out. Similarly for issues of quality. Inasmuch as this refers to how long a durable good lasts, this can only be determined by consuming the good and seeing when (and if) it breaks. Such goods are called experience goods. The consumer needs to purchase the good, but the purchase and consumption of the good (or service!) will fully inform the consumer. This situation naturally leads to consumers trying a good, but maybe not necessarily finding the best match for them. Advertising will be designed to make the consumer try the product — free samples could be used.3 Why would the consumer not necessarily find the best product? If there is a cost to unsatisfactory consumption experiences this will naturally arise. As in the restaurant example above. Similar examples can be constructed for hair cuts, and many other services. What if the consumer never finds out if the product performs its function? This is the natural situation for many medical and religious services. The consumer will not discover the full implications until it is (presumably) too late. Such goods are termed faith goods and present tremendous problems to markets. In our current society the spiritual health of consumers is not judged to be important, and so the market failure for religious services is not addressed.4 Health, in contrast, is judged important — since consumers value it, for one, and since there are large externalities in a society with a social safety net — and thus there is extensive regulation for health care services in most societies.5 Since education also has certain attributes of a 3
It used to be legal for cigarette manufacturers to distribute “sample packs” for free, allowing consumers to experience the good without cost. The fact that nicotine is addictive to some is only helpful in this regard, as any local drug pusher knows: they also hand out free sample packs in schools. 4 A convincing argument can be made that societies which prescribe one state religion do so not in an attempt to maximize consumer welfare but tax revenue and control. The Inquisition, for example, probably had little to do with a concern for the welfare of heretics. Note also that I am speaking especially with respect to religions in which the “afterlife” plays a large role. We will encounter them again in the chapter on game theory. 5 Interesting issues arise when the provision of health care services is combined with the
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faith good — in the sense that it is very costly to unlearn, or to repeat the education — we also see strong regulation of education in most societies.6 Note that religion and health care probably differ in another dimension: in health care there is information asymmetry between provider and consumer; it may be argued that in religion both parties are equally uninformed.7 Presumably the fact that information is asymmetric makes it easier to exploit the uninformed party on the one hand, and to regulate on the other.8 The government attempts to combat this informational asymmetry by certifying the supply. Aside from all the above, there are additional problems with information. These days everybody speaks of the “information economy”. Clearly information is supposed to have some kind of value and therefore should have a price. However, while that is clear conceptually, it is far from easy to incorporate information into our models. Information is not a good like most others. For one, it is hard to measure. There are, of course, measurements which have been derived in communications theory — but they often measure the amount of meaningful signals versus some noise (as in the transmission of messages.) These measurements measure if messages have been sent, and how many. Economists, in contrast, are concerned with what kind of message actually contains relevant information and what kind may be vacuous. Much care is therefore taken to define the informational context of any given decision problem (we will encounter this again in the game theory part, where we will use the term information set to denote all situations in which the same information has been gathered, loosely speaking.) Aside from the problem of defining and measuring information, it is also a special good since it is not rivalrous (a concept you may have encountered in ECON 301): the fact that I possess some information in no way impedes your ability to have the same information. Furthermore, the fact that I “consume” the information somehow (let’s say by acting on it) does not stop you from consuming it or me from using it again later. There are therefore provision of spiritual services. 6 This is regulation for economically justifiable reasons. Because education is so costly to undo or repeat it is also frequently meddled with for “societal engineering” reasons. 7 I am not trying to belittle beliefs here, just pointing to the fact that while a medical doctor may actually know if a prescribed treatment works (while the patient does not), neither the religious official or the follower of a faith know if the actions prescribed by the faith will “work”. 8 Note the weight loss and aphrodisiac markets, or cosmetics, for example. Little difference between these and the “snake oil cures” of the past seems to exist. Regulation can take the simple form that only “verifiable” claims may be made.
Information 95 large public good aspects to information which require special consideration. 9 The long and short of this is that standard demand-supply models often don’t work, and that markets will in general misallocate if information is involved, which makes it even more important to have a good working model. A complete study of this topic is outside the scope of these notes, however. In what follows we will only outline some specific issues in further detail.
5.1
Search
One of the key breakthroughs in the economics of information was a simple model of information acquisition. The basic idea is a simple one (as all good ideas are.) A consumer lacks information — say about the price at which a good may be bought. Clearly it is in the consumer’s interest not to be “fleeced,” which requires him to have some idea about what the market price is. In general the consumer will not know what the lowest price for the product is, but can go to different stores and find out what their price is. The more stores the consumer visits the better his idea about what the correct price might be — the better his information — but of course the higher his cost, since he has to visit all these stores. An optimizing consumer may be expected to continue to find new (hopefully) lower prices as long as the marginal benefit of doing so exceeds the marginal cost of doing so. Therefore we “just” need to define benefits and costs and can then apply our standard answer that the optimum is achieved if the marginal cost equals the marginal benefit! The problem with this is the fact that we will have to get into sampling distributions and other such details (most of which will not concern us here) to do this right. The reason for this is that the best way to model this kind of problem is as the consumer purchasing messages (normally: prices) which are drawn from some distribution. For example: let us say that the consumer knows that prices Rare distributed according to some cumulative z distribution function F (z) = 0 f (p)dp, where f (p) is the (known) density. If the consumer where to obtain n price samples (draws) from this distribution 9
This is the problem in the Patent protection fight: Once somebody has invented (created the information for) a new drug the knowledge should be freely available — but this ignores the general equilibrium question of where the information came from. The incentives for its creation will depend on what property rights are enforceable later on. For while two firms both may use the information, a firm may only really profit from it if it is the sole user.
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then the probability that any given sample (say p0 ) is the lowest will be [1 − F (p0 )]n−1 f (p0 ). (This formula should make sense to you from Econometrics or Statistics: we are dealing with n independent random variables.) From this it follows that the expected value of the lowest price after having taken a sample of n prices is Z ∞ n plow = p[1 − F (p)]n−1 f (p)dp. 0
Note that this expected lowest sample decreases as n increases, but at a decreasing rate: the difference between sampling n times and n − 1 times is Z ∞ n−1 n plow − plow = − pF (p)[1 − F (p)]n−2 f (p)dp < 0. 0
So additional sampling leads to a benefit (lower expected price) but with a diminishing margin. What about cost? Even with constant (marginal) cost of sampling we would have a well defined problem and it is easy to see that individuals with higher search costs will have lower sample sizes and thus pay higher expected prices. Also note that the lowest price paid is still a random variable, and hence consumers do not buy at the same price (which is inefficient, in general!) Computing the variance you would observe that dispersion of the lowest price is decreasing in n — that means that the lower the search costs the ‘better’ the market can be expected to work. Indeed, competition policy is concerned with this fact in some places and attempts to generate rules which require that prices be posted (lowering the cost of search). Any discussion of search would be lacking if we did not point out that search is normally sequential. In the above approach n messages where bought, with n predetermined. This is the equivalent of visiting n stores for a quote irrespective of what the received quotes are. The dynamic problem is the much more interesting one and has been quite well studied. We will attempt to distill the most important point and demonstrate it by way of a fairly simple example. The key insight into this kind of problem is that it is often optimal to employ a stopping rule.10 That is, to continue sampling until a price has been obtained which is below some preset limit, at which point search is abandoned and the transaction occurs (a sort of “good enough” attitude.) The price at which one decides to stop is the reservation 10
It does depend on the distributional assumptions we make on the samples — independence makes what follows true.
Information 97 price — the highest price one is willing to pay! In order to derive this price we will have to specify if one can ‘go back’ to previous prices, or if the trade will have fallen through if one walks away. The latter is the typical setup in the matching literature in labour economics, the former is a bit easier and we will consider it first. Suppose the cost of another sample (search effort) is c. The outcome of the sample is a new price p, which will only be of benefit if it is lower than the currently known minimum price. Evaluated at the optimal reservation price pR , the expected gain from an additional sample is therefore the savings pR − p, “expected over” all prices p < pR . If these expected savings are equal to the cost of the additional sample, then the consumer is just indifferent between buying another sample or not, and thus the reservation price is found: Z pR pR satisfies (pr − p)f (p)dp = c. 0
Next, consider the labour market, where unemployed workers are searching for jobs. This is the slightly more complex case where the consumer cannot return to a past offer. Also, the objective is to find the highest price. First determine the value of search, V , which is composed of the cost, the expected gain above the wage currently on the table, and the fact that a lower wage might arise which would indicate that another search is needed. Thus, assuming a linear utility function for simplicity (no risk aversion), and letting p stand for wages R∞ Z pR Z ∞ −c + pR pf (p)dp f (p)dp =⇒ V = . pf (p)dp + V V = −c + 1 − F (pR ) 0 pR Note that I assume stationarity here and the fact that one pR will do (i.e., the fact that the reservation wage is independent of how long the search has been going on.) All of these things ought to be shown for a formal model. We now will ask what choice of pR will maximize the value above (which is the expected utility from searching.) Taking a derivative we get R∞ −pR f (pR )(1 − F (pR )) + f (pR )(−c + pR pf (p)dp) = 0. (1 − F (pR ))2 Simplifying and rearranging we obtain Z ∞ pf (p)dp − pR = c − pR F (pR ). pR
The LHS of this expression is the expected increase in the wage, the RHS is the cost of search, which consists of the actual cost of search and the fact
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that the current wage is foregone if a lower value is obtained. What now will be the effect of a decrease in the search cost c, for example if the government decides to subsidize non-working (searching) workers? This would lower c, and the LHS would have to fall to compensate, which will occur only if a higher pR is chosen. Of course, a higher pR lowers the RHS further. In mathematical terms it is easy to compute that −1 dpR = . dc 1 − F (pR ) Unemployment insurance will increase the reservation wage (and thus unemployment — note the pun: it ensures unemployment!) The reason is that our workers can be more choosy and search for the “right” job. They become more discriminating in their job search. Note that this is not necessarily bad. If the quality of the match (suitability of worker and firm with each other) is reflected in a higher wage, then this leads to better matches (fewer Ph.D. cab drivers). This may well be desirable for the general equilibrium efficiency properties of the model. Now, this model is quite simplistic. More advanced models might take into account eligibility rules. In those models unemployment insurance can be shown to cause some workers to take bad jobs (because that way they can qualify for more insurance later.) Similar models can also be used to analyse other matching markets. The market for medical interns comes to mind, or the marriage market. In closing let us note that certain forms of advertising will lower search costs (since consumers now can determine cheaply who has what for sale at which price) and thus are efficiency enhancing (less search, less resources spent on search, and lower price dispersion in the market.) Other forms of advertising (image advertising) do not have this function, however, and will have to be looked at in a different framework. This is where the distinction between search goods and experience goods comes in.
5.2
Adverse Selection
Most problems which will concern us in this course are actually of a different nature than the search for information above. What we are interested in most are the problems which arise because information is asymmetric. This means that two parties to a transaction do not have the same information, as is the case if the seller knows more about the quality (or lack thereof) of
Information 99 20 cert.
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Figure 5.1: Insurance for Two types his good than the buyer, or if the buyer knows more about the value of the good to himself than the seller. In these types of environments we run into two well known problems, that of adverse selection (which you may think of as “hidden property”) and moral hazard (“hidden action.”) We will deal with the former in this section. Adverse selection lies at the heart of Akerlof’s Lemons Problem. These kind of markets lead naturally to the question if the informed party can send some sort of signal to reveal information. But how could they do so? Simply stating the fact that, say, the car is of good quality will not do, since such a statement is free and would also be made by sellers of bad cars (it is not incentive compatible.) Sometimes this problem can be fixed via the provision of a warranty, since a warranty makes the statement that the car is good more costly to sellers of bad cars than of cars which are, in fact, good. Let us examine these issues in our insurance model. Assume two states, good and bad, and assume two individuals who both have wealth endowment (wg , wb ); wg > wb . Suppose that these individuals are indistinguishable to the insurance company, but that one individual is a good risk type who has a probability of the good state occurring of πH , while the other is a bad risk type with probability of the good state of only πL < πH . To be stereotypical and simplify the presentation below, assume that the good risk is female, the bad risk male, so that grammatical pronouns can distinguish types in what follows. As we have seen before, the individuals’ indifference curves will have a slope of −πi /(1 − πi ) on the certainty line.
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If there were full information both types could buy insurance at fair premiums and would choose to be on the certainty line on their respective budget lines (with the high risk type on a lower budget line and with a lower consumption in each state.) However, with asymmetric information this is not a possible outcome. The bad risk cannot be distinguished from the good risk a priori and therefore can buy at the good risk premium. Now, if he were to maximize at this premium we know that he would over insure — and this action would then distinguish him from the good risk. The best he can do without giving himself away is to buy the same insurance coverage that our good risk would buy, in other words to mimic her. Thus both would attempt to buy full insurance for a good type. We now have to ask if this is a possible equilibrium outcome. The answer is NO, since the insurance company now would make zero (expected) profits on her insurance contract, but would loose money on his insurance contract. Consider the profit function (and recall that pi = πi ): (1 − πH )(wg − wb ) − (1 − πL )(wg − wb ) < 0. Foreseeing this fact, the insurance company would refuse to sell insurance at these prices. Well then, what is the equilibrium in such a market? There seem to be two options: either both types buy the same insurance (this is called pooling behaviour) or they buy different contracts (this is called separating behaviour.) Does there exist a pooling equilibrium in this market? Consider a larger market with many individuals of each of the two types. Let fH denote the proportion of the good types (πH ) in the market. An insurer would be making zero expected profits from selling a common policy for coverage of I at a premium of p to all types if and only if fH (πH pI − (1 − πH )(1 − p)I) + (1 − fH )(πL pI − (1 − πL )(1 − p)I) = 0. This requires a premium p = (1 − πL ) − fH (πH − πL ). Note that at this premium the good types subsidize the bad types, since the former pay too high a premium, the latter too low a premium. In Figure 5.2, this premium is indicated by a zero profit locus (identical to the consumers’ budget) at an intermediate slope (labelled ‘market’.) Any proposed equilibrium with pooling would have to lie on this line and be better than no insurance for both types. Such a point might be point M in the figure. However, this cannot be an equilibrium. In order for it to be an equilibrium
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15
20 no loss
Figure 5.2: Impossibility of Pooling Equilibria nobody must have any actions available which they could carry out and prefer to what is proposed. Consider, then, any point in the area to the right of the zero profit line, below the bad type’s indifference curve, and above the good type’s indifference curve: a point such as D. This contract, if proposed by an insurance company, offers less insurance but at a better price. Only the good type would be willing to take it (she would end up on a higher indifference curve). Since it is not all the way over on the good type zero profit line, the insurance company would make strictly positive profits. Of course, all bad types would remain at the old contract M , and since this is above the zero profit line for bad types whoever sold them this insurance would make losses. Notice that the same arguments hold whatever the initial point. It follows that a pooling equilibrium cannot exist. Well then, does a separating equilibrium exist? We now would need two contracts, one of which is taken by all bad types and the other by all good types. Insurance offerers would have to make zero expected profits. Of course, since each type takes a different contract the insurer will know who is who from their behaviour. This suggests that we look at contracts which insure the bad risks fully at fair prices. For the bad risk types to accept this type of contract the contract offered to the good risk type must be on a lower indifference curve. It must also be on the fair odds line for good types for there to be zero profits. Finally it must be acceptable to the good types. This suggests a point such as A in the Figure 5.3. There now are two potential problems with this. One is that the insurer and insured of good type would like to move to a different point, such as B, after the type
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20 cert.
loss
good market 15
10 bad * B
5 * A endowment * 0 0
5
10
15
20 no loss
Figure 5.3: Separating Contracts could be possible is revealed. But if that would indeed happen then it can’t, in equilibrium, because the bad types would foresee it and pretend to be good in order then to be mistaken for good. The other problem is that the stability of such separating equilibria depends on the mix of types in the market. If, as in the Figure 5.3, the market has a lot of bad types this kind of equilibrium works. But what if there are only a few bad types? In that case an insurer could deviate from our proposed contract and offer a pooling contract which makes strictly positive profits and is accepted by all in favour over the separating contract. This is point C in Figure 5.4. Of course, while this deviation destroys our proposed equilibrium it is itself not an equilibrium (we already know that no pooling contract can be an equilibrium.) This shows that a small proportion of bad risks who can’t be identified can destroy the market completely! Notice the implications of these findings on life or health insurance markets when there are people with terminal diseases such as AIDS or Hepatitis C, or various forms of cancer. The patient may well know that he has this problem, but the insurance company sure does not. Of course, it could require testing in order to determine the risk it will be exposed to. Our model shows that doing so would make sure that everybody has access to insurance at fair rates and that there is no cross-subsidization. This is clearly efficient. However, it will reduce the welfare of consumers with these diseases. Indeed, given that AIDS, for example, means a very high probability of seriously expensive treatment, the insurance premiums would be very high (justifiably so, by the way, since the risk is high.) What ought society do
Information 103 20 cert.
loss
good market 15
10 bad C * 5 * A endowment * 0 0
5
10
15
20 no loss
Figure 5.4: Separating Contracts definitely impossible about this? Economics is not able to answer that question, but it is able to show the implications of various policies. For example, banning testing and forcing insurance companies to bear the risk of insuring such customers might make the market collapse, in the worst scenario, or will at least lead to inefficiency for the majority of consumers. It would also most likely tend to lead to “alternative” screening devices — the insurance company might start estimating consumers’ “lifestyle choices” and then discriminate based on that information. Is that an improvement? An alternative would be to let insurance operate at fair rates but to directly subsidize the income of the affected minority. (This may cause “moral” objections by certain segments of the population.)
5.3
Moral Hazard
Another problem which can arise in insurance markets, and indeed in any situation with asymmetric information, is that of hidden actions being taken. In our insurance example it is often possible for the insured to influence the probability of a loss: is the car locked? Are there anti-theft devices installed? Are there theft and fire alarms, sprinklers? Do the tires have enough tread depth, and are they the correct tire for the season (in the summer a dedicated summer tire provides superior breaking and road holding to a M&S tire, while in the winter a proper winter tire is superior.) Some of these actions are observable and will therefore be included in the conditions of the insurance
104 L-A. Busch, Microeconomics
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contract. For example, in Germany your insurance will refuse to pay if you do not have “approved” tires, or if the car was not locked and this can be established. Indeed, insurance for cars without anti-theft devices is now basically unaffordable since theft rates are so high since the ‘iron curtain’ opened.11 If you insure your car as parked in a closed parking garage and then leave it out over night your insurance may similarly refuse to pay! Other actions are not so easily verified, however. How aggressively do you drive? How many risks do you decide to take on a given day? This is often not observable but nevertheless in your control. If the avoidance of accidents is costly to the insured in any way, then he can be expected to pick the optimal level of the (unobservable) action — optimal for himself, not the insurance company, that is. As a benchmark, let us consider an individual who faces the risk of loss L. The probability of the loss occurring depends on the amount of preventive action, A, taken and is denoted π(A). It would appear logical to assume that π 0 (A) < 0, π 00 (A) > 0. The activity costs money, and cost is c(A) with c0 (A) > 0, c00 (A) > 0. In the absence of insurance the individual would choose A so as to maxA {π(A)u(w − L − c(A)) + (1 − π(A))u(w − c(A))}. The first order condition for this problem is π 0 (A)u(w − L − c(A)) − c0 (A)π(A)u0 (w − L − c(A)) − π 0 (A)u(w − c(A)) − c0 (A)(1 − π(A))u0 (w − c(A)) = 0. Thus the optimal A∗ satisfies c0 (A∗ ) =
π 0 (A∗ )(u(w − L − c(A)) − u(w − c(A∗ ))) . π(A∗ )u0 (w − L − c(A∗ )) − (1 − π(A∗ ))u0 (w − c(A∗ ))
Now consider the consumer’s choice when insurance is available. To keep it simple we will assume that the only available contract has a premium of π(A∗ ) and is for the total loss L. Note that this would be a fair premium if the consumer continues to choose the level A∗ of abatement activity. However, the maximization problem the consumer now faces becomes (the consumer will assume the premium fixed) maxA {π(A)u(w − pL − c(A)) + (1 − π(A))u(w − pL − c(A))}. The first order condition for this problem is π 0 (A)u(·) − c0 (A)π(A)u0 (·) − π 0 (A)u(·) − c0 (A)(1 − π(A))u0 (·) = 0. 11
Yes, I am suggesting that most stolen cars end up in former eastern block countries — it is a fact.
Information 105 Thus the optimal A∗ satisfies ˆ = 0. c0 (A) Clearly (and expectedly) the consumer will now not engage in the activity at all. It follows that the probability of a loss is higher and that the insurance company would loose money. If we were to recompute the level of A at a fair premium for π(0) we would find that the same is true: no effort is taken. The only way to elicit effort is to expose the consumer to the right incentives: there must be a reward for the hidden action. A deductible will accomplish this to some extent and will lead to at least some effort being taken. This is due to the fact that the consumer continues to face risk (which is costly.) The amount of effort will in general not be the optimal, however.
5.4
The Principal Agent Problem
To conclude the chapter on information here is an outline of the leading paradigm for analysing asymmetric information problems. Most times one (uninformed) party wishes to influence the behavior of another (informed) party, the problem can be considered a principal-agent problem. This kind of problem is so frequent that it deserved its own name and there are books which nearly exclusively deal with it. We will obviously have only a much shorter introduction to the key issues in what follows. First note that this is a frequent problem in economics, to say the least. Second, note that at the root of the problem lies asymmetric information. We are, generically, concerned with situations in which one party — the principal — has to rely on some other party — the agent — to do something which affects the payoff of the first party. This in itself is not the problem, of course. What makes it interesting is that the principal cannot tell if the agent did what he was asked to do, that is, there is a lack of information, and that the agent has incentives not to do as asked. In short, there is a moral-hazard problem. The principal-agent literature explores this problem in detail.
5.4.1
The Abstract P-A Relationship
We will frame this analysis in its usual setting, which is that of a risk-neutral principal who tries to maximize (expected) profits, and one agent, who will
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have control over (unobservable) actions which influence profits. Call the principal the owner and the agent the manager. The agent/manager will be assumed to have choice of a single item, what we shall call his effort level e ∈ [e, e]. This effort level will be assumed to influence the level of profits before manager compensation, called gross-profits, which you may think of as revenue if there are no other inputs. More general models could have multiple dimensions at this stage. The manager’s preferences will be over monetary reward (wages/income) and effort: U (w, e). We assume that the manager is an expected utility maximizer. Also assume that the manager is risk-averse (that is u1 (·, ·) > 0, u11 (·, ·) < 0) and that effort is a bad (u2 (·, ·) < 0.) The owner can not observe the manager’s effort choice. Instead, only the realization of gross profits is observed. Note at this stage that this in itself does not yet cause a moral-hazard problem, since if the relationship between effort and profit is known we can simply invert the profit function to deduce the effort level. Therefore we need to introduce some randomness into the model. Let ρ be a random variable which also influences profits and which is not directly observable by the owner either. It could be known to the manager, but we will have to assume that it will become known only after the fact, so that the manager cannot condition his effort choice on the realization of ρ. Let the relationship between effort, profits and the random variable be denoted by Π(e, ρ). Note that Π(e, ρ) will be a random variable itself. All expectations in what follows will be taken with respect to the distribution of ρ. EΠ(e, ρ), for example, will be the expected profits for effort level e. Since the owner can only observe profits, the most general wage contract he can offer the manager will be a function w(Π). This formulation includes a fixed wage as well as all wage plus bonus schemes, or share contracts (where the manager gets a fixed share of profits.) Let us first, as a benchmark, determine the efficient level of effort, which would be provided under full information. In that case we would have to solve the Pareto problem, that is, solve maxe,w {EΠ(e, ρ) − w
s.t. U (w, e) = u} .
Here u is a level of utility sufficient to make the manager accept the contract. Note also that in formulating this problem like this I have already used the knowledge that a risk-neutral owner should fully insure a risk-averse manager by offering a constant wage. Assuming no corner solutions, it is easy to see that we would want to set the effort level such that the marginal benefit of effort (in terms of higher expected profits) is related to the marginal cost (in terms of higher disutility
Information 107 of effort, which will have to be compensated via the wages). In particular, we need that w EΠe (·, ·) = Ue (·, ·). Uw (·, ·) What if effort is not observable? In that case we will solve the model “backwards”: given any particular wage contract faced by the manager, we will determine the manager’s choice. Then we will solve for the optimal contract, “foreseeing” those choices. Assume in what follows that the owner wants to support some effort level eˆ (which is derived from this process.) So, our manager is faced with two decisions. One is to determine how much effort to provide given he accepts the contract: maxe {EU (w(Π(e, ρ)), e)}. This leads to FOC E [Uw (·, ·)w 0 (·)Πe (·, ·) + Ue (·, ·)] = 0, which determines the optimal e∗ . The other is the question if to accept the contract at all, which requires that maxe {EU (w(Π(e, ρ)), e)} = EU (w(Π(e∗ , ρ)), e∗ ) ≥ U0 . Here U0 is the level of utility the manager can obtain if he does not accept the contract but instead engages in the next best alternative activity (in other words, it is the manager’s opportunity cost.) Both of these have special names and roles in the principal-agent literature. The latter one is called the participation constraint, or individual rationality constraint. That is, any contract is constrained by the fact that the manager must willingly participate in it. Thus, the contract w ∗ (Π) must satisfy IR : maxe {EU (w(Π(e, ρ)), e)} ≥ U0 . The other constraint is that the manager’s optimal choice should, in equilibrium, correspond to what the owner wanted the manager to do. That is, if the owner wants to elicit effort level eˆ, then it should be true that the manager, in equilibrium, actually supplies that level. This is called the incentive compatibility constraint. Mathematically it says: IC : eˆ = argmaxe {EU (w(Π(e, ρ)), e)}. Here ‘argmax’ indicates the argument which maximizes. In other words, we want eˆ = e∗ .
108 L-A. Busch, Microeconomics
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Now we can write down the principal’s problem. The principal ‘simply’ wishes to maximize his own payoff subject to both, the participation and incentive compatibility constraints. This problem is easily stated maxe,w(Π) {E [Π(e, ρ) − w(Π(e, ρ)) +λP (U0 − U (w(Π(e, ρ)), e)) + λI (Uw (·, ·)w 0 (·)Πe (·, ·) + Ue (·, ·))]} , but hard to solve in general cases. We will, therefore, only look at three special cases in which some answers are possible. First, let us simplify by assuming a risk neutral manager. In that case we can have U (w, e) = w − v(e), where v(e) is the disutility of effort. Now note that we would not have to insure the manager in this case since he evaluates expected wealth the same as some fixed payment. Also note that there is no reason to over pay the manager compared to the outside option, that is, he needs to be given only U0 . Before we charge ahead and solve this brute force, let us think for a moment longer about the situation. We will not be insuring the manager. He will furthermore have the correct incentive to expend effort if he cares as much about profits as the owner. Thus it would stand to reason that if we were to sell him the profits he would be the owner and thus take the optimal action! But this is equivalent to paying him a wage which is equal to the profits less some fixed return for the owners. So, let us propose a wage contract of w = Π − p, and assume for the moment that the IR constraint can be satisfied (by choice of the correct p.) With such a contract the manager will choose an effort level such that Πe (e, ρ) − v 0 (e) = 0. Note that this is the efficient effort level under full information. The owner’s problem now simply is to choose the largest p such that the manager still accepts the contract! The other special case one can consider is that of an infinitely risk-averse manager. We can model this as a manager who has lexicographic preferences over his minimum wealth and effort. Independent of the effort levels the manager will prefer a higher minimum wealth, and for equal minimum wealth the manager will prefer the lower effort. For simplicity also assume that the lowest possible profit level is independent of effort (that is, only the probabilities of profits depend on the effort, not the level). In that case a manager will always face the same lowest wage for any wage function and thus will not be able to be enticed to provide more than the minimum level of effort.
Information 109 As you can see, we might expect anything between efficient outcomes and completely inefficient outcomes, largely depending on the precise situation. Let us return for a moment to the general setting above. We will restrict it somewhat by assuming that the manager’s preferences are separable: U (w, e) = u(w) − v(e) with the obvious assumptions of u0 (·) > 0, u00 (·) < 0, v 0 (·) ≥ 0, v 00 (·) > 0, v 0 (e) = 0, v 0 (e) = ∞. The last two are typical “Inada conditions” designed to ensure an interior solution. We will also assume that the cumulative distribution of profits exists and depends on effort, F (Π, e), and that this distribution is over Π ∈ [Π, Π], has a density f (Π, e) > 0, and satisfies first-order stochastic dominance. In this formulation the manager will solve (Z Π
maxe
Π
)
u(w(Π))f (Π, e)dΠ − v(e) .
This has first order condition Z Π u(w(Π))fe (Π, e)dΠ − v 0 (e) = 0. Π
We will ignore the SOC for a while. The participation constraint will be Z
Π Π
u(w(Π))f (Π, e)dΠ − v(e) ≥ U0 .
The owner will want to find a wage structure and effort level to (Z Π
maxw(·),e
Π
[(Π − w(Π))f (Π, e)
+λP (u(w(Π)) − v(e) − U0 )f (Π, e)
o + λC (u(w(Π))fe (Π, e) − v 0 (e)f (Π, e))] dΠ .
This will have to be differentiated, which is quite messy and, as pointed out before, hard to solve. However, consider the differentiation with respect to the wage function: −f (Π, e) + λP f (Π, e)u0 (w(Π)) + λC fe (Π, e)u0 (w(Π)) = 0. Rewriting we get something which is informative: 1 u0 (w(Π))
= λP + λC
fe (Π, e) . f (Π, e)
110 L-A. Busch, Microeconomics
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The left hand side of this is the inverse of the slope of the wealth utility function. It is therefore increasing in the wage. The right hand side consists of the constraint multipliers (which will both be positive since the constraints are binding) multiplied by some factors. Of particular interest is the last term, the ratio of the derivative of the density to the density. Let us further specialize the problem and suppose that there are only two effort levels, and that the owner wants to get the high effort level. In that case it is easy to see that the above equation will become ¶ µ fH (Π) − fL (Π) fL (Π) 1 . = λP + λC = λP + λC 1 − u0 (w(Π)) fH (Π) fH (Π) The term ffHL (Π) is called a likelihood ratio. It follows that the higher the (Π) relative probability that the effort was high for a given realized profit level, the higher the manager’s wage. Indeed, if the likelihood ratio is decreasing then then the wage function must be increasing with realized profits. What is going on here is that higher profits are a correct signal of higher effort, and thus we will make wages increase with the signal. Finally, assume that there are only two profit levels, with the high level of profits more likely under high effort. In that case it can be shown that 0
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formalism=' extension=' left=' advanced=' courses=' texts.br=' 118=' microeconomics=' acbr=' £b=' £=' bbr=' bcbr=' s£=' s=' bsbr=' ccbr=' bs=' ¡@=' @s=' s¡=' ¢a=' ¢=' aa=' ££=' s¢=' ¢sbr=' d=' cbr=' sbr=' s¡br=' ecbr=' ¤=' s¤=' cs¤=' £a=' c£=' s¢¢=' 6.1:=' valid=' invalid=' trees=' now=' ready=' defining=' bunch=' objects=' apply.=' capture=' need.=' information?=' play,=' i.e.,=' plays=' together=' player(s),=' knows=' making=' move,=' stage,=' exogenous=' exist,=' probability=' distribution=' formally,=' following:br=' 3=' n-player=' comprises:=' a;=' 2.=' associating=' vector=' length=' n=' γ;=' 3.=' partition=' {s0=' ,=' s1=' sn=' nonterminal=' nodes=' (the=' sets;)=' 4.=' s0=' immediate=' followers;=' 5.=' ∈=' {1,=' 2,=' n}=' si=' subsets=' sij=' (information=' sets),=' ∀b,=' 6.=' index=' iij=' 1-1=' map=' followers=' 119=' exhaustive=' list.=' notice=' (1)=' 1)=' sets.=' “nature”=' gets=' too.=' nature=' very=' concept.=' non-strategic=' choice=' environment,=' randomization=' (as=' accident=' choosing=' this.)=' (2)=' nature,=' non-strategic,=' (3)=' sets=' idea=' distinguish=' within=' every=' labels=' same.)=' restrictions=' implied,=' still=' draw=' important=' issues=' assumptions=' sets,=' words.=' restriction=' forget=' learn=' n-person=' recall=' never=' known,=' own=' previous=' moves:=' x,=' x0=' neither=' x=' nor=' predecessor=' one,=' xˆ=' xˆ,=' ),=' x˜=' leading=' x.=' bears=' close=' reading,=' quite=' intuitive=' practice:=' player’s=' set,=' follow=' forgets=' he=' himself=' moved=' previously=' furthermore,=' predecessors:=' true=' player,=' “the=' action”=' chosen=' nodes.=' otherwise,=' remember=' chose=' indistinguishable.=' always=' assume=' recall.=' words,=' moves,=' learned=' something=' opponents=' (such=' opponent=' took=' “up”=' moved)=' practice,=' sense.=' allow=' possibility=' know-br=' 120=' 1b=' ©=' hh=' hr=' ©©=' h=' rp=' ph=' r=' @=' ¡=' r¡=' @r=' 1,=' `=' @rbr=' 0,=' 0br=' 2br=' ³³=' ³=' ³br=' r³=' r¡br=' bppbr=' pp=' pbr=' pr2=' @pr=' 1p=' 6.2:=' counter-example=' ing=' simultaneously,=' his=' time.=' imperfect=' opposite=' singletons.=' crucial=' linguistic=' difference=' here:=' known=' more)=' players.=' order.=' all!=' non-trivial=' theorem=' harsanyi=' shows=' transform=' bayesian.6=' uncertainty=' 6br=' bayes’=' formula=' update=' beliefs.=' formulabr=' 121=' eb=' r©=' hrrm=' ight=' accom=' 10=' −2,=' r=' 4,=' ©br=' wolf=' lamb=' ep=' outbr=' m=' ac=' 10br=' 5br=' @rm=' 4br=' 6.3:=' outcomes,=' etc.,=' transformed=' only.=' that,=' construct=' information,=' equilibria=' coincide.=' entrant=' monopolist=' enjoys=' fighting=' entrant,=' types=' fight,=' playing,=' coincide=' entrant’s=' priors.7=' transformation=' specified=' 6.3.=' turns=' powerful=' results,=' makes=' tools=' lots=' nearly=' know.br=' 6.1.2br=' strategies=' analyze=' modelled=' employ=' concept=' strategies.=' complete,=' contingent=' plans=' just=' “move,”=' set.=' strategy=' plan,=' referee=' computer,=' according=' instructions,=' says=' event=' occurring=' times=' happening=' does,=' divided=' occurring:=' (a|b)=' 7br'="" (b|a)p=' (a)=' ∩=' b)=' .'="" (b)=' (b)br=' prior=' ex-ante=' belief=' ex-post=' posterior.br=' 122=' watch=' happens=' change=' mind=' point.=' submit=' plan=' before=' cover=' eventualities=' translates=' saying=' specify=' those=' rule=' out!=' σ=' associates=' element=' σi=' :=' 7→=' alternatively,=' randomize.8=' occur=' levels,=' strategies:=' 7=' behavioural=' βi=' elements=' 8=' mixed=' µi=' σi=' concepts,=' general.=' find=' corresponding=' strategy,=' (which=' properly=' associated=' introduce=' shortly.)=' decidedly=' easier=' with.=' are.=' only,=' now.=' giving=' σ2=' σn=' σ−i=' ).=' general,=' fully,=' nature.=' 8br=' somewhat=' controversial=' issue.=' people=' flip=' coins=' decisions?=' nevertheless=' generally=' accepted.=' circumstances=' seen=' equivalent=' bluffing:=' poker=' example.=' sometimes=' fold=' pair,=' stand,=' raise=' people.=' coin=' flip.=' instances=' randomizing=' explained=' person=' definite=' population=' probabilities=' correspond=' proportion=' strategy.=' worry=' necessary=' see!)br=' 123=' von=' neumann-morgenstern=' expected=' evaluate=' things.=' combination=' (vector)=' end,=' will,=' arrive=' precisely=' whatever=' vertex.=' played,=' implies=' evaluated=' using=' 9=' πi=' (σ).=' π(σ)='(π1' (σ),=' πn=' (σ)).=' γ.=' (it=' “normal=' form,”=' coming=' use.)=' fashion=' abbreviated=' representation=' cumbersome=' prevalent=' view=' nowadays,=' interpretation=' stressed=' “strategic=' form.”=' slightly=' viewpoint,=' socalled=' matrix-games=' analyzed=' first.=' thus,=' definition:=' 11=' g=' (normal)=' 3-tuple=' (n,=' s,=' u=' n},=' ×s2=' .×sn=' i’s=' rn=' ;=' implicit=' above.=' abstract=' suppressed,=' represented=' matrix=' (hence=' games.)=' row,=' column,=' third=' matrices.=' three=' clearly=' looses=' appeal.)=' 6.4=' related=' concept,=' abstract,=' here,=' outcomes=' specified,=' functions=' players.br=' 124=' 12br=' ←−=' 12=' rbr=' ubr=' (1,=' (2,=' 2)br=' 3,=' 3)br=' 1)br=' 0)br=' 4)br=' dbr=' (3,=' 0)=' 5)br=' 6.4:=' 12=' o)=' defined=' o=' couple=' things=' closely=' occur.=' principle,=' realizing=' indices=' arbitrary,=' relabel=' everything=' loss=' generality=' (does=' “option=' 1”?),=' with,=' say,=' eight=' entries=' each.=' four=' two.=' forms=' widely=' different,=' indeed.=' nevertheless,=' matrix.=' matter?=' say=' main=' don’t.=' point=' concerns=' convenient=' representation.=' 6.5=' gives=' announces=' 50=' 100=' dollars,=' same.=' add=' $100=' asked=' for,=' pay=' simple,=' ×=' 27=' matrix!9=' 9br=' why?=' announcement=' 3-tuple.=' 33=' vectors=' constructed.br=' 125=' (0,=' ..br=' 100)br=' 50br=' (50,=' 100br=' (100,=' (−1,=' −1)br=' 0»=' 2s»=' »»»=' 0=' ¡50=' @br=' 50)=' 100,=' »»=' »»»br=' »=' cxxbr=' 2sbr=' xxxbr=' ¡@br=' @100br=' ¡br=' −1=' −1br=' xx=' xxx=' xs=' @100=' 6.5:=' go=' on,=' 6.6=' below=' page=' listed=' beginning=' forms.=' am=' usual=' convention=' belongs=' row=' index.=' pennies=' t=' tbr=' sexes=' −1)=' 1bbr=' h¡¡=' @@t=' pr¡=' 2p=' p@=' −1,=' 1−1,=' mbr=' 30)br=' (5,=' (30,=' 50)br=' m¡=' @@s=' 50,=' 30br=' dilemma=' 5,=' 30,=' 3)=' 4)=' (4,=' c¡¡=' @@d=' 126=' “education=' game”=' b©©br=' r©©=' −3,=' cbbr=' ©hbr=' cbr=' hhn=' hrp=' (p,=' i)=' (i,=' )br=' nbbr=' −3)br=' i)br=' 6.6:=' gamesbr=' 6.2br=' problemsbr=' developed=' (“games”.)=' prediction=' “likely”=' this?10=' “solve”=' impose=' competition=' as,=' “markets=' clear”)=' planned=' buy=' sell=' desire=' price),=' (their=' game).=' equilibrium.=' impose,=' qualified=' name,=' “these=' constitute=' (bayes=' equilibrium,=' subgame=' cho-kreps=' criterion,..)=' nature.)=' general=' try=' “refine=' away”=' (lingo=' “discard”)=' appear=' sensible.=' 280=' select=' few=' gained=' wide=' acceptance=' easy=' (some=' others=' difficult=' apply=' game.)=' clear=' trying=' do:=' tell=' play.=' philosophical=' stake=' inclination!=' noted=' here=' manual=' interested=' iota=' modeled.br=' 127br=' 6.2.1br=' persuasive=' idea,=' old:=' why=' eliminate=' strictly=' worse=' do?=' elimination=' (strictly)=' dominated=' variants=' (iterated=' weakly=' strategies),=' principle=' do,=' 13=' dominates=' larger=' independent=' (a,=' s−i='> πi (b, s−i ) ∀s−i ∈ S−i . A similar definition can be made for weakly dominates if the strict inequality is replaced by a weak inequality. Other authors use the notion of a dominated strategy instead: Definition 14 Strategy a is is weakly (strictly) dominated if there exists a mixed strategy α such that πi (α, s−i ) ≥ (>)πi (a, s−i ), ∀s−i ∈ Σi and πi (α, s−i ) > πi (a, s−i ) for some s−i ∈ Σi . If we have a 2 × 2 game, then elimination of dominated strategies may narrow down our outcomes to one point. Consider the “Prisoners’ Dilemma” game, for instance. ‘Defect’ strictly dominates ‘Cooperate’ for both players, so we would expect both to defect. On the other hand, in “Battle of the Sexes” there is no dominated (dominating) strategy, and we would still not know what to predict. If a player has more than two strategies, we also do not narrow down the field much, even if there are dominated strategies. In that case, we can use Successive Elimination of Dominated Strategies, where we start with one player, then go to the other player, back to the first, and so on, until we can’t eliminate anything. For example, in the following game 12
(l, l)
(r, r)
(l, r)
(r, l)
L
(2, 0)
(2, 0)
R
(1, 0)
(2, −1)
(2, −1)
(3, 1)
(3, 1)
(1, 0)
128 L-A. Busch, Microeconomics
May2004
player 1 does not have a dominated strategy. Player 2 does, however, since (r, l) is strictly dominated by (l, r). If we also eliminate weakly dominated strategies, we can throw out (l, l) and (r, r) too, and then player 1 has a dominated strategy in L. So we would predict, after successive elimination of weakly dominated strategies, that the outcome of this game is (R, (l, r)). There are some criticisms about this equilibrium concept, apart from the fact that it may not allow any predictions. These are particularly strong if one eliminates weakly dominated strategies, for which the argument that a player should never choose those appears weak. For example you might know that the opponent will play that strategy for which you are indifferent between two strategies. Why then would you eliminate one of these strategies just because somewhere else in the game (where you will not be) one is worse than the other? Next, we will discuss the probably most widely used equilibrium concept ever, Nash equilibrium.11 This is the most universally accepted concept, but it is also quite weak. All other concepts we will see are refinements of Nash, imposing additional constraints to those imposed by Nash equilibrium. Definition 15 A Nash equilibrium in pure strategies is a set of strategies, one for each player, such that each player’s strategy maximizes that player’s payoff, taking the other players’ strategies as given: σ∗
is Nash iff
∀i, ∀σi ∈ Σi ,
∗ ∗ πi (σi∗ , σ−i ) ≥ πi (σi , σ−i ).
Note that the crucial feature of this equilibrium concept: each player takes the others’ actions as given and plays a best response to them. This is the mutual best response property we first saw in the Cournot equilibrium, which we can now recognize as a Nash equilibrium.12 Put differently, we only check against deviations by one player at a time. We do not consider mutual deviations! So in the Prisoners’ Dilemma game we see that one player alone cannot gain from a deviation from the Nash equilibrium strategies 11
Nash received the Nobel price for economics in 1994 for this contribution. He extended the idea of mutual best responses proposed by von Neumann and Morgenstern to n players. He did this in his Ph.D. thesis. von Neumann and Morgenstern had thought this problem too hard when they proposed it in their book Games and Economic Behaviour. 12 Formally we now have 2 players. Their strategies are qi ∈ [0, P −1 (0)]. Restricting attention to pure strategies, their payoff functions are πi (q1 , q2 ), so the strategic form is (π1 (q), π2 (q)). Denote by bi (q−i ) the best response function we derived in footnote 1 of this chapter. The Nash equilibrium for this game is the strategy vector (q∗ 1 , q∗2 ) = (b1 (q∗2 ), b2 (q∗1 )). This, of course, is just the computation performed in footnote 2.
Game Theory 129 (Defect,Defect). We do not allow or consider agreements by both players to defect to (Cooperate,Cooperate), which would be better! A Nash equilibrium in pure strategies may not exist, however. Consider, for example, the “Matching Pennies” game: If player 2 plays ‘H’ player 1 wants to play ‘H’, but given that, player 2 would like ‘T’, but given that 1 would like ‘T’, .. We may need mixed strategies to be able to have a Nash equilibrium. The definition for a mixed strategy Nash equilibrium is analogous to the one above and will not be repeated. All that changes is the definition of the strategy space. Since an equilibrium concept which may fail to give an answer is not that useful (hence the general disregard for elimination of dominated strategies) we will consider the question of existence next. Theorem 1 A Nash equilibrium in pure strategies exists for perfect information games. Theorem 2 For finite games a Nash equilibrium exists (possibly in mixed strategies.) Theorem 3 For (N, S, U ) with S ∈ Rn compact and convex and Ui : S 7→ R continuous and strictly quasi concave in si , a Nash equilibrium exists. Remarks: 1. Nash equilibrium is a form of rational expectations equilibrium (actually, a rational expectations equilibrium is a Nash equilibrium, formally.) As in a rational expectations equilibrium, the players can be seen to “expect” their opponent(s) to play certain strategies, and in equilibrium the opponents actually do, so that the expectation was justified. 2. There is an apparent contradiction between the first existence theorem and the fact that Nash equilibrium is defined on the strategic form. However, you may want to think about the way in which assuming perfect information restricts the strategic form so that matrices like the one for matching pennies can not occur. 3. If a player is to mix over some set of pure strategies {σi1 , σi2 , . . . , σik } in Nash equilibrium, then all the pure strategies in the set must lead
130 L-A. Busch, Microeconomics
May2004
to the same expected payoff (else the player could increase his payoff from the mixed strategy by changing the distribution.) This in turn implies that the fact that a player is to mix in equilibrium will impose a restriction on the other players’ strategies! For example, consider the matching pennies game: 12
H
T
H
(1, −1)
(−1, 1)
T
(−1, 1)
(1, −1)
For player 1 to mix we will need that π1 (H, µ2 ) = π1 (T, µ2 ). If β denotes the probability of player 2 playing H, then we need that β − (1 − β) = −β + (1 − β), or 2β − 1 = 1 − 2β, in other words, β = 1/2. For player 1 to mix, player 2 must mix at a ratio of 1/2 : 1/2. Otherwise, player 1 will play a pure strategy. But now player 2 must mix. For him to mix (the game is symmetric) we need that player 1 mixes also at a ratio of 1/2 : 1/2. We have, by the way, just found the unique Nash equilibrium of this game. There is no pure strategy Nash, and if there is to be a mixed strategy Nash, then it must be this. (Notice that we know there is a mixed strategy Nash, since this is a finite game!) The next equilibrium concept we mention is Bayesian Nash Equilibrium (BNE). This will be for completeness sake only, since we will in practice be able to use Nash Equilibrium. BNE concerns games of incomplete information, which, as we have seen already, can be modelled as games of imperfect information. The way this is done is by introducing “types” of one (or more) player(s). The type of a player summarizes all information which is not public (common) knowledge. It is assumed that each type actually knows which type he is. It is common knowledge what distribution the types are drawn from. In other words, the player in question knows who he is and what his payoffs are, but opponents only know the distribution over the various types which are possible, and do not observe the actual type of their opponents (that is, do not know the actual payoff vectors, but only their own payoffs.) Nature is assumed to choose types. In such a game, players’ expected payoffs will be contingent on the actual types who play the game, i.e., we need to consider π(σi , σ−i |ti , t−i ), where t is the vector of type realizations (potentially one for each player.) This implies that each player type will have a strategy, so that player i of type ti will have strategy σi (ti ). We then get the following:
Game Theory 131 Definition 16 A Bayesian Nash Equilibrium is a set of type contingent strategies σ ∗ (t) = (σi∗ (t1 ), . . . , σn∗ (tn )) such that each player maximizes his expected utility contingent on his type, taking other players’ strategies as given, and using the priors in computing the expectation: ∗ ∗ πi (σi∗ (ti ), σ−i |ti ) ≥ πi (σi (ti ), σ−i |ti ),
∀σi (ti ) 6= σi∗ (ti ), ∀i, ∀ti ∈ Ti .
What is the difference to Nash Equilibrium? The strategies in a Nash equilibrium are not conditional on type: each player formulates a plan of action before he knows his own type. In the Bayesian equilibrium, in contrast, each player knows his type when choosing a strategy. Luckily the following is true: Theorem 4 Let G be an incomplete information game and let G∗ be the complete information game of imperfect information that is Bayes equivalent: Then σ ∗ is a Bayes Nash equilibrium of the normal form of G if and only if it is a Nash equilibrium of the normal form of G∗ . The reason for this result is straight forward: If I am to optimize the expected value of something given the probability distribution over my types and I can condition on my types, then I must be choosing the same as if I wait for my type to be realized and maximize then. After all, the expected value is just a weighted sum (hence linear) of the conditional on type payoffs, which I maximize in the second case.
6.2.2
Equilibrium Refinements for the Strategic Form
So how does Nash equilibrium do in giving predictions? The good news is that, as we have seen, the existence of a Nash equilibrium is assured for a wide variety of games.13 The bad news is that we may get too many equilibria, and that some of the strategies or outcomes make little sense from a “common sense” perspective. We will deal with the first issue first. Consider the following game, which is a variant of the Battle of the Sexes game: 13
One important game for which there is no Nash equilibrium is Bertrand competition between 2 firms with different marginal costs. The payoff function for firm 1, say, is ( (p1 − c1 )Q(p1 ) if p1 < p2 π1 (p1 , p2 ) = α(p1 − c1 )Q(p1 ) if p1 = p2 0 otherwise which is not continuous in p2 and hence Theorem 3 does not apply.
132 L-A. Busch, Microeconomics
May2004
12
M
S
M
(6, 2)
(0, 0)
S
(0, 0)
(2, 6)
This game has three Nash equilibria. Two are in pure strategies — (M, M ) and (S, S) — and one is a mixed strategy equilibrium where µ1 (S) = 1/4 and µ2 (S) = 3/4. So what will happen? (Notice another interesting point about mixed strategies here: The expected payoff vector in the mixed strategy equilibrium is (3/2, 3/2), but any of the four possible outcomes can occur in the end, and the actual payoff vector can be any of the three vectors in the game.) The problem of too many equilibria gave rise to refinements, which basically refers to additional conditions which will be imposed on top of standard Nash. Most of these refinements are actually applied to the extensive form (since one can then impose restrictions on how information must be consistent, and so on.) However, there is one common refinement on the strategic form which is sometimes useful. Definition 17 A dominant strategy equilibrium is a Nash equilibrium in which each player’s strategy choice (weakly) dominates any other strategy of that player. You may notice a small problem with this: It may not exist! For example, in the game above there are no dominating strategies, so that the set of dominant strategy equilibria is empty. If such an equilibrium does exist, it may be quite compelling, however. There is another commonly used concept, that of normal form perfect equilibrium. We will not use this much, since a similar perfection criterion on the extensive form is more useful for what we want to do later. However, it is included here for completeness. Basically, normal form perfect will refine away some equilibria which are “knife edge cases.” The problem with Nash is that one takes strategies of the opponents as given, and can then be indifferent between one’s own strategies. Normal form perfect eliminates this by forcing one to consider completely mixed strategies, and only allowing pure strategies that survive after the limit of these completely mixed strategies is taken. This eliminates many of the equilibria which are only brought about by indifference. We first define an “approximate” equilibrium for completely mixed games, then take the limit:
Game Theory 133 Definition 18 A completely mixed strategy for player i is one that attaches positive probability to every pure strategy of player i: µi (si ) > 0 ∀si ∈ Si . Definition 19 A n-tuple µ(²) = (µ1 , . . . , µn ) is an ²-perfect equilibrium of the normal form game G if µi is completely mixed for all i ∈ {1, . . . , n}, and if µi (sj ) ≤ ² if πi (sj , µ−i ) ≤ πi (sk , µ−i ), sk 6= sj , ² > 0. Notice that this restriction implies that any strategies which are a poor choice, in the sense of having lower payoffs than other strategies, must be used very seldom. We can then take the limit as “seldom” becomes “never:” Definition 20 A Perfect Equilibrium is the limit point of an ²-perfect equilibrium as ² → 0. To see how this works, consider the following game: 12
T
B
t
(100, 0)
b
(100, 0)
(−50, −50) (100, 0)
The pure strategy Nash equilibria of this game are (t, T ), (b, B), and (b, T ). The unique normal form perfect equilibrium is (b, T ). This can easily be seen from the following considerations. Let α denote the probability with which player 1 plays t, and let β denote the probability with which player 2 plays T . 2’s payoff from T is zero independent of α. 2’s payoff from B is −50α, which is less than zero as long as α > 0. So, in the ²-perfect equilibrium we have to set (1 − β) < ², that is β > 1 − ² in any ²-perfect equilibrium. Now consider player 1. His payoff from t will be 150β − 50, while his payoff from b is 100. His payoff from t is therefore less than from b for all β, and we require that α < ². As ² → 0, both α and (1 − β) thus approach zero, and we have (b, T ) as the unique perfect equilibrium. While the payoffs are the same in the perfect equilibrium and all the Nash equilibria, the perfect equilibrium is in some sense more stable. Notice in particular that a very small probability of making mistakes in announcing or carrying out strategies will not affect the nPE, but it would lead to a potentially very bad payoff in the other two Nash equilibria.14 14
Note that an nPE is Nash, but not vice versa.
134 L-A. Busch, Microeconomics
6.2.3
May2004
Equilibrium Concepts and Refinements for the Extensive Form
Next, we will discus equilibrium concepts and refinements for the extensive form of a game. First of all, it should be clear that a Nash equilibrium of the strategic form corresponds one-to-one with a Nash equilibrium of the extensive form. The definition we gave applies to both, indeed. Since our extensive form game, as we have defined it so far, is a finite game, we are also assured existence of a Nash equilibrium as before. Consider the following game, for example, here given in both its extensive and strategic forms: 1b 21 U D ©H © HHD U© © HH (u, u) (6, 4) (1, 2) 2 r©© Hr2 u ¡@ d u ¡@ d (u, d) (6, 4) (8, 1) ¡ r¡
4, 6
@ @r
0, 4
¡ r¡
2, 1
@ @r
1, 8
(d, u)
(4, 0)
(1, 2)
(d, d)
(4, 0)
(8, 1)
This game has 3 pure strategy Nash equilibria: (D, (d, d)), (U, (u, u)), and (U, (u, d)).15 What is wrong with this? Consider the equilibrium (D, (d, d)). Player 1 moves first, and his move is observed by player 2. Would player 1 really believe that player 2 will play d if player 1 were to choose U , given that player 2’s payoff from going to u instead is higher? Probably not. This is called an incredible threat. By threatening to play ‘down’ following an ‘Up’, player 2 makes his preferred outcome, D followed by d, possible, and obtains his highest possible payoff. Player 1, even though he moves first, ends up with one of his worst payoffs.16 However, player 2, if asked to follow his strategy, would rather not, and play u instead of d if he finds himself after a move of U . The move d in this information set is only part of a best reply because under the proposed strategy for 1, which is taken as given in a Nash equilibrium, this information set is never reached, and thus it does not matter (to player 2’s payoff) which action is specified. This is a type of behaviour which we may want to rule out. This is done most easily by requiring all moves to be best replies for their part of the game, a concept we will now make more formal. (See also Figure 6.7) 15
There are also some mixed strategy equilibria, namely (U, (P21 (u) = 1, P22 (u) = α)) for any α ∈ [0, 1], and (D, (P21 (u) ≤ 1/4, P22 (u) = 0)). 16 It is sometimes not clear if the term ‘incredible threat’ should be used only if there is some actual threat, as for example in the education game when the parents threaten to punish. The more general idea is that of an action that is not a best reply at an information set. In this sense the action is not credible at that point in the game.
Game Theory 135
Bs©©©
¡ s E ¡ £B £ B £ B
¡A
©©
A As F £B £ B £ B
Ac ©H
HH
HH D Hs ¢A £B ¢ A £ B As ¢ £ Bs G¢ A ¡A H A ¡ ¢ A As ¡ A ¢ s I¢ A ¢ AK ¢ A ¢ A ¢ A ¢ A
C s
Subgames start at: A,E,F,G,D,H
No Subgames at: B,C,I,K
Figure 6.7: Valid and Invalid Subgames Definition 21 Let V be a non-terminal node in Γ, and let ΓV be the game tree comprising V as root and all its followers. If all information sets in Γ are either completely contained in ΓV or disjoint from ΓV , then ΓV is called a subgame. We can now define a subgame perfect equilibrium, which tries to exclude incredible threats by assuring that all strategies are best replies in all proper subgames, not only along the equilibrium path.17 Definition 22 A strategy combination is a subgame perfect equilibrium (SPE) if its restriction to every proper subgame is a subgame perfect equilibrium. In the example above, only (U, (u, d)) is a SPE. There are three proper subgames, one starting at player 2’s first information set, one starting at his second information set, and one which is the whole game tree. Only u is a best reply in the first, only d in the second, and thus only U in the last. Remarks: 1. Subgame Perfect Equilibria exist and are a strict subset of Nash Equilibria. 2. Subgame Perfect equilibrium goes hand in hand with the famous “backward induction” procedure for finding equilibria. Start at the end of the game, with the last information sets before the terminal nodes, 17
The equilibrium path is, basically, the sequence of actions implied by the equilibrium strategies, in other words the implied path through the game tree (along some set of arcs.)
136 L-A. Busch, Microeconomics
May2004
and determine the optimal action there. Then back up one level in the tree, and consider the information sets leading up to these last decisions. Since the optimal action in the last moves is now known, they can be replaced by the resulting payoffs, and the second last level can be determined in a similar fashion. This procedure is repeated until the root node is reached. The resulting strategies are Subgame Perfect. 3. Incredible Threats are only eliminated if all information sets are singletons, in other words, in games of perfect information. As a counterexample consider the following game: a³³³ (0,0) ³ 2³ (0,1) s ³ c A©©© a ÃÃ (1/2,0) 1c©©©B ÃÃÃ Ã s ` ` HH `c`` ` (-1,-1) H 2 H a C HHsP (1,0) PPc PP P (-1,-1) In this game there is no subgame starting with player 2’s information set after 1 chose B or C, and therefore the equilibrium concept reverts to Nash, and we get that (A, (c, c)) is a SPE, even though c is strictly dominated by a in the non-trivial information set. 4. Notwithstanding the above, Subgame Perfection is a useful concept in repeated games, where a simultaneous move game is repeated over and over. In that setting a proper subgame starts in every period, and thus at least incredible threats with regard to future retaliations are eliminated. 5. Subgame Perfection and normal Form Perfect lead to different equilibria. Consider the game we used before when we analyzed nPE: 1b ©H HHb © t© H © r© p © p p p p p p p p p 2p p p p p p p p H pH p r T ¡@ B T ¡@ B ¡ @ ¡ @ r¡ @r r¡ @r
100, 0 −50, −50
100, 0
100, 0
12
T
B
t
(100, 0)
b
(100, 0)
(−50, −50) (100, 0)
As we had seen before, the nPE is (b, T ), but since there are no subgames, the SPE are all the Nash equilibria, i.e., (b, T ), (t, T ) and (b, B).
Game Theory 137 1c
2s
A
D
l
s J
(3, 2, 2)
(1, 1, 1)
d 3
J
a
l
Jr JJ
(0, 0, 0)
(4, 4, 0)
s J
J
Jr JJ
(0, 0, 1)
Figure 6.8: The “Horse” As we can see, SPE does nothing to prevent incredible threats if there are no proper subgames. In order to deal with this aspect, the following equilibrium concept has been developed. For games of imperfect information we cannot use the idea of best replies in all subgames, since a player may not know at which node in an information set he is. We would like to ensure that the actions taken at an information set are best responses nevertheless. In order to do so, we have to introduce what the player believes about his situation at that information set. By introducing a belief system — which specifies a probability distribution over all the nodes in each of the player’s information sets — we can then require all actions to be best responses given the belief system. Consider the example in Figure 6.8, which is commonly called “The Horse.” This game has two pure strategy Nash equilibria, (A, a, r) and (D, a, l). Both are subgame perfect since there are no proper subgames at all. The second one, (D, a, l), is “stupid” however, since player 2 could, if he is actually asked to move, play d, which would improve his payoff from 1 to 4. In other words, a is not a best reply for player 2 if he actually gets to move. Definition 23 A system of beliefs φ is a vector of beliefs for each player, φi , where φi is a vector of probability distributions, φji , over the nodes in each of player i’s information sets Sij : φji
:
xjk
7→ [0, 1],
K X k=1
φji (xjk ) = 1; ∀xjk ∈ Sij .
Definition 24 An assessment is a system of beliefs and a set of strategies, (σ ∗ , φ∗ ).
138 L-A. Busch, Microeconomics
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Definition 25 An assessment (σ, φ) is sequentially rational if ¤ ¤ ∗ £ ∗ £ ∗ ∗ |Sij ) , ∀i, ∀σi ∈ Σi , ∀Sij . |Sij ) ≥ E φ πi (σi , σ−i E φ πi (σi∗ , σ−i Definition 26 An assessment (σ ∗ , φ∗ ) is consistent if (σ ∗ , φ∗ ) = limn→∞ (σn , φn ), where σn is a sequence of completely mixed behavioural strategies and φn are beliefs consistent with σn being played (i.e., obtained by Bayesian updating.) Definition 27 An assessment (σ ∗ , φ∗ ) is a sequential equilibrium if it is consistent and sequentially rational. As you can see, some work will be required in using this concept! Reconsider the horse in Figure 6.8. The strategy combination (A, a, r) is a sequential equilibrium with beliefs α = 0, where α denotes player 3’s probability assessment of being at the left node. You can see this by considering the following sequence of strategies: 1 plays D with (1/n)2 , which converges to zero, as required. 2 plays d with (1/n), also converging to zero. The consistent belief for three thus is given by (from Bayes’ Rule) α(n) =
(1/n)2 n 1 = = , (1/n)2 + (1 − (1/n)2 )(1/n) n2 + n − 1 n + 1 − 1/n
which converges to zero as n goes to infinity. As usual, the tougher part is to show that there is no sequence which can be constructed that will lead to (D, a, l). Here is a short outline of what is necessary: In order for 3 to play l we need beliefs which put at least a probability of 1/3 on being at the left node. We thus need that 1 plays down almost surely, since player 2 will play d any time 3 plays l with more than a 1/4 probability. But as weight shifts to l for 3, and 2 plays d, sequential rationality for 1 requires him to play A (4 > 3). This destroys our proposed setup.
Signalling Games This is a type of game used in the analysis of quality choice, advertising, warranties, or education and hiring. The general setup is that an informed party tries to convey information to an uninformed party. For example, the fact that I spend money advertising should convey the information that my product is of high quality to consumers who are not informed about the quality of my product. There are other sellers of genuinely low quality,
Game Theory 139 (3, 0)
l ´
(−4, −1) ´´
´
´
s ´α
L
1s
R
β QQ
l
s
Q
t¯ (1/2)
´r
(1, 2)
Q
r QQ
Q (−2, 1)
Root: eNature (−3, −2) Q Q (−2, −2)
Qr Q
l
Q
t (1/2) Q Qs
L
s
1
R
s´
´
´
r´
(−1, −1) ´´
´
l
(2, 3)
Figure 6.9: A Signalling Game however, and they will try to mimic my actions. In order to be credible my action should therefore be hard (costly) to mimic. The equilibrium concept used for this type of game will be sequential equilibrium, since we have to model the beliefs of the uninformed party. Consider the game in Figure 6.9. Nature determines if player 1 is a high type or a low type. Player 1 moves left or right, knowing his type. Player 2 then, without knowing 1’s true type, moves left or right also. The payoffs are as indicated. This type of game is called a game of asymmetric information, since one of the parties is completely informed while the other is not. The usual question is if the informed party can convey its private information or not. In the above game, the Nash equilibria are the following: Let player 1’s strategy vector (S1 , S2 ) indicate 1’s action if he is the low and high type, respectively, while player 2’s strategy vector (s1 , s2 ) indicates 2’s response if he observes L and R, respectively. Then we get that the pure strategy Nash equilibria are ((R, R), (r, r)), ((R, L), (r, r)), and ((L, R), (l, l)). Now introduce player 2’s beliefs. Let 2’s belief of facing a low type player 1 be denoted by α if 2 observes L, and by β if 2 observes R. We then can get two sequential equilibria: ((L, R), (l, r), (α = 1, β = 0)) and ((R, R), (r, r), (α = 0, β = 0.5)) (It goes without saying that you should try to verify this claim!). The first of these is a separating equilibrium. The action of player 1 completely conveys the private information of player 1. Only low types move Left, only high types move Right, and the move thus reveals the type of player. The second equilibrium, in contrast, is a pooling equilibrium. Both types take the same move in equilibrium, and no information is transmitted. Notice, however, that the belief that α = 0 is somewhat stupid. If you find yourself, as player 2, inadvertently in your first information set, what should you believe? Your beliefs here say that
140 L-A. Busch, Microeconomics
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you think it must have been the high type who made the mistake. This is “stupid”, since the high type’s move L is strictly dominated by R, while the low type’s move L is not dominated by R. It would be more reasonable to assume, therefore, that if anything it was the low type who was trying to “tell you something” by deviating (you are not on the equilibrium path if you observe L, remember!) There are refinements that impose restrictions like this last argument on the beliefs out of the equilibrium path, but we will not go into them here. Look up the Cho-Kreps criterion in any good game theory book if you want to know the details. The basic idea is simple: of the equilibrium path you should only put weight on types for whom the continuation equilibria off the equilibrium path are actually better than if they had followed the proposed equilibrium. The details are, of course, messy. Finally, notice a last problem with sequential equilibrium. Minor perturbations of the extensive form change the equilibrium set. In particular, the two games in Figure 6.10 have different sequential equilibria, even though the games would appear to be quite closely related. 1e ©© ¡ C ©© ¡ C L ©© ¡ M C R © ¡ C ©© © ¡ C 2 © © s¡ Cs ¡@ ¡@ (2, 2) ¡u d@ ¡u d@ ¡ @ ¡ @ (3, 3)
(1.5, 0) (0, 0)
(1, 1)
1e
L ©©
©© ©
(2, 2)
©
©©
©©
© s© © ¡@ @d ¡u @ ¡
(3, 3)
©©
(1.5, 0)
©
©©
Not L 1s ©
© ©© © M ©
2 ©©
s ©©
u ©©
(0, 0)
R
d
(1, 1)
Figure 6.10: A Minor Perturbation?
6.3
Review Problems
Question 1: Provide the definition of a 3-player game in extensive form. Then draw a well labelled example of such a game in which you indicate all the elements of the definition. Question 2: Define “perfect recall” and provide two examples of games
Game Theory 141 which violate perfect recall for different reasons. Question 3: Assume that you are faced with some finite game. Will this game have a Nash equilibrium? Will it have a Subgame Perfect Equilibrium? Why can you come to these conclusions? Question 4: Consider the following 3 player game in strategic form: Lef t 12
L
R
Player 3 C
Right 12 L
R
C
U
(1, 1, 1) (2, 1, 2)
(1, 3, 2)
U
(2, 2, 2) (4, 2, 4)
(2, 6, 4)
C
(1, 2, 1) (1, 1, 1)
(2, 3, 3)
C
(5, 0, 1) (1, 1, 1)
(0, 1, 1)
D
(2, 1, 2) (1, 1, 3)
(3, 1, 1)
D
(3, 2, 3) (2, 2, 4)
(4, 2, 2)
Would elimination of weakly dominated strategies lead to a good prediction for this game? What are the pure strategy Nash equilibria of this game? Describe in words how you might find the mixed strategy Nash equilibria. Be clear and concise and do not actually attempt to solve for the mixed strategies. Question 5: Find the mixed strategy Nash equilibrium of this game: 12
L
R
C
U
(1, 4) (2, 1)
(4, 2)
C
(3, 2) (1, 1)
(2, 3)
Question 6: Consider the following situation and construct an extensive form game to capture it. A railway line passes through a town. Occasionally, accidents will happen on this railway line and cause damage and impose costs on the town. The frequency of these accidents depends on the effort and care taken by the railway — but these are unobservable by the town. The town may, if an accident has occurred, sue the railway for damages, but will only be successful in obtaining damages if it is found that the railway did not use a high level of care. For simplicity, assume that there are only two levels of effort/care (high and low) and that the courts can determine with certainty which level was in fact used. Also assume that going to court costs the railway and the town money, that effort is costly for the railway (high effort reduces profits), that accidents cost the railway and the town money and that this cost is independent of the effort level (i.e., there is a “standard accident”). Finally, assume that if the railway is “guilty” it has to pay the town’s damages and court costs.
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Question 7: Consider the following variation of the standard Battle of the Sexes game: with probability α Juliet gets informed which action Romeo has taken before she needs to choose (with probability (1 − α) the game is as usual.) a) What is the subgame perfect equilibrium of this game? b) In order for Romeo to be able to insist on his preferred outcome, what would α have to be? Question 8: (Cournot Duopoly) Assume that inverse market demand is given by P (Q) = (Q − 10)2 , where Q refers to market output by all firms. Assume further that there are n firms in the market and that they all have zero marginal cost of production. Finally, assume that all firms are Cournot competitors. This means that they take the other firms’ P outputs as given and consider their own inverse demand to be pi (qi ) = ( j6=i qj + qi − 10)2 . Derive the Nash equilibrium output and price. (That is, derive the multilateral best response strategies for the output choices: Given every other firm’s output, a given firm’s output is profit maximizing for that firm. This holds for all firms.) Show that market output converges to the competitive output level as n gets large. (HINT: Firms are symmetric. It is then enough for now to focus on symmetric equilibria. One can solve for the so-called reaction function of one firm (it’s best reply function) which gives the profit maximizing level of output for a given level of joint output by all others. Symmetry then suggests that each firm faces the same joint output by its (n − 1) competitors and produces the same output in equilibrium. So we can substitute out and solve.) Question 9∗ : Assume that a seller of an object knows its quality, which we will take to be the probability with which the object breaks during use. For simplicity assume that there are only two quality levels, high and low, with breakdown probabilities of 0.1 and 0.4, respectively. The buyer does not know the type of seller, and can only determine if a good breaks, but not its quality. The buyer knows that 1/2 of the sellers are of high quality, and 1/2 of low quality. Assume that the seller receives a utility of 10 from a working product and 0 from a non-working product, and that his utility is linear in money (so that the price of the good is deducted from the utility received from the good.) If the seller does not buy the object he is assumed to get a utility level of 0. The cost of the object to the sellers is assumed to be 2 for the low quality seller and 3 for the high quality seller. We want to investigate if signalling equilibria exist. We also want to train our understanding of sequential equilibrium, so use that as the equilibrium concept in what follows. a) Assume that sellers can only differ in the price they charge. Show that no separating equilibrium exist.
Game Theory 143 b) Now assume that the sellers can offer a warranty which will replace the good once if it is found to be defective. Does a separating equilibrium exist? Does a pooling equilibrium exist? Question 10∗ : Assume a uniform distribution of buyers over the range of possible valuations for a good, [0, 2]. a) Derive the market demand curve. b) There are 2 firms with cost functions C1 (q1 ) = q1 /10 and C2 (q2 ) = q22 . Find the Cournot Equilibrium and calculate equilibrium profits. c) Assume that firm 1 is a Stackelberg leader and compute the Stackelberg equilibrium. (This means that firm 1 moves first and firm 2 gets to observe firm 1’s output choice. The Stackelberg equilibrium is the SPE for this game.) d) What is the joint profit maximizing price and output level for each firm? Why could this not be attained in a Nash equilibrium?
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Chapter 7 Review Question Answers 7.1
Chapter 2
Question 1: a) There are multiple C(·) which satisfy the Weak Axiom. Note, however, that you have to check back and forth to make sure that the WA is indeed satisfied. (I.e., C({x, y, z}) = {x}, C({x, y}) = {x, y} does not satisfy the axiom since while the check for x seems to be ok, you also have to check for y, and there it fails.) One choice structure that does work is C({x, y, z}) = {x}, C({x, z, w}) = {x}, C({y, w, z}) = {w}, C({y, w}) = {w}, C({x, z}) = {x}, C({x, w}) = {x}, C({x}) = {x}. b) Yes (I thought of that first, actually, in deriving the above) it is x º w º y º z. c) Yes, it is transitive. d) I was aiming for an application of out Theorem: our set of budget sets B does not contain all 2 and 3 element subsets of X. Missing are {x, y, w}, {x, y}, {y, z}, {w, z}. e) The best way to go about this one is to determine where we can possibly get this to work. Examination of the sets B shows that the two choices y, x only appear in one of the sets and thus must be our key if we want to satisfy the WA without having rational preferences. Some fiddling reveals that the following works: C({x, y, z}) = {x}, C({x, z, w}) = {w}, C({y, w, z}) = {y}, C({y, w}) = {y}, C({x, z}) = {x}, C({x, w}) = {w}, C({x}) = {x}. The problem is intransitivity, since the above implies that y º w º x º z but we also have x º y! Question 2: Here you have to make sure to maximize income for any given 145
146 L-A. Busch, Microeconomics work hrs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Part a 1 then 2 112 124 136 148 160 172 188 204 212 220 234 248 262 276 290 304
2 then 1 108 116 130 144 158 172 186 200 212 224 236 248 260 272 288 304
Max 112 124 136 148 160 172 188 204 212 224 236 248 262 276 290 304
May2004
Partb 1 then 2 2 then 1 112 108 124 116 136 130 32 148 145 31 160 160 172 174 32 188 189 31 204 204 212 216 220 228 234 32 240 249 31 252 264 264 2 278 3 276 293 31 292 308 308
Max 112 124 136 148 160 174 23 189 13 204 216 228 240 252 264 278 23 293 13 308
Table 7.1: Table 1: Computing maximal income amount of work. In parts (a) and (b) you have to choose to work either job 1 then job 2 (after 8 hours in job 1) or job 2 then job 1. Simply plotting the two and then taking the outer hull (i.e., the highest frontier) for each leisure level gives you the frontier. In (a) they only cross twice (at 9 and 12 hours of work) while in part (b) they cross 4 times. You can best see this effect by considering a table in which you tabulate total hours worked against total income, computed by doing job 1 first, and by doing job 2 first. This is shown in Table 1. In neither part a) nor in part b) is the budget set convex. c) This is a possibly quite involved problem. The intuitive answer is that it will not matter since marginal and average pay is (weakly) increasing in both jobs. Here is a more general treatment of these questions: We really are faced with an maximization problem, to max income given the constraints, for any given total amount worked. Let h1 and h2 denote hours worked in jobs 1 and 2, respectively. Then the objective function is I(h1 , h2 ) = h1 w1 (h1 ) + h2 w2 (h2 ), where wi (hi ) are½ the wage schedules. w1 if h1 ≤ C1 and The wage schedules have the general form w1 (h1 ) = w1 if h1 ≥ C1 ½ w2 if h2 ≤ C2 , where w i < wi . I ignore here that no hours w2 (h2 ) = w2 if h2 ≥ C2 above 8 are possible for either job, choosing to put that information into the constraints later.
Answers 147 Consider now the iso-income curves in (h1 , h2 ) space which result. We will have four regions to consider, namely A = {(h1 , h2 )|h1 ≤ C1 , h2 ≤ C2 }, B = {(h1 , h2 )|h1 ≤ C1 , h2 ≥ C2 }, C = {(h1 , h2 )|h1 ≥ C1 , h2 ≤ C2 }, D = {(h1 , h2 )|h1 ≥ C1 , h2 ≥ C2 }. The slope of the iso-income curves for the regions is easily seen to be the negative of the ratio of wages, so we have S(A) = −w 1 /w2 , S(B) = −w 1 /w2 , S(C) = −w 1 /w2 , S(D) = −w 1 /w2 . It is obvious that S(C) < S(D) and S(A) < S(B), as well as that S(C) < S(A) and S(D) < S(B). This implies, of course, that S(C) < S(A) < S(B) as well as that S(C) < S(D) < S(B), with the comparison of S(A) to S(D) indeterminate. (But luckily not needed in any case.) The important fact which follows from all of this is that the iso-income curves are all concave to the origin and piece-wise linear. Now superimpose the choice sets onto this. Note that without any restrictions H = h1 + h2 , that is, for any given number of hours H the hours in each job are “perfect substitutes”. These iso-hour curves are all straight lines with a slope of −1. (For our parameters all of S(A), S(C), S(D) are less than −1, while S(B) > −1, but his is not important.) For parts (a) and (b) the feasible set consists of the boundaries of the 8 × 8 square of feasible hours, where either hi = 0, hj < 8, or where hi = 8, 0 ≤ hj ≤ 8. The choice set is thus given by the intersection of the iso-hour lines with the feasible set (the box boundary). In part (c) this restriction is removed and the whole interior of the box is feasible. Due to the concavity to the origin of the iso-income lines this is of no relevance, however. Note how I have used our usual techniques of iso-objective curves and constraint sets to approach this problem. Works pretty well, doesn’t it! d) Now we “clearly” take up jobs in decreasing order of pay, starting with the highest paid and progressing to the lower paid ones in order. The resulting budget set will be convex. Question 3: The consumer will © ª 0.6 maxx x0.3 1 x2 + λ (m − x1 p1 − x2 p2 ) which leads to the first order conditions 0.3x−0.7 x0.6 1 2 = λp1 ,
−0.4 0.6x0.3 = λp2 , 1 x2
x1 p1 + x2 p2 = m.
The utility function is quasi-concave (actually, strictly concave in this case) and the budget set convex, so the second order conditions will be satisfied. Combining the first two first order conditions we get p1 0.3x2 = 0.6x1 p2
=⇒
x2 =
2p1 x1 . p2
148 L-A. Busch, Microeconomics
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Substitute into the budget constraint and simplify: x1 p 1 +
2p1 x1 p 2 = m p2
=⇒
x1 =
m . 3p1
³ ´ 2m m 2m Now use this to solve for x2 : x2 = 3p . So (x (p, m), x (p, m)) = , . 1 2 3p1 3p2 2 To find the particular quantity demanded, simply plug in the numbers and simplify: x1 = x2 =
412 + 24 436 3 × 412 + 1 × 72 = = ; 3×3 3 3
2(3 × 412 + 1 × 72) 2(412 + 24) = = 872. 3×1 1
Question 4: The key is to realize that this utility function is piece-wise linear with line segments at slopes −5, −1, −1/5, from left to right. The segments join at rays from the origin with slopes 3 and 1/3. Properly speaking, neither the Hicksian nor the Marshallian demands are functions. The function has either a perfect substitute or Leontief character. In the former the substitution effects approach infinity, in the latter they are zero. Demands are easiest derived from the price offer curve, which is a nice zigzag line. It starts at the intercept of the budget with the vertical axis (point A). It follows the indifference curve segment with −5 slope to the ray with slope 3. Call this point B. From there it follows the ray with slope 3 until that ray intersects a budget drawn from A with a slope of 1 (point C). It then continues on this budget and the coinciding indifference curve segment to the ray with slope 1/3 (point D). Up along that ray to an intersection with a budget from A with slope 1/5 (point E), along that budget to the intercept with the horizontal axis (point F), and then along the horizontal axis off to infinity. x2 AX £ X ³ ³ X £ XXX @ C ³³ ³ X ³ C @£ C XX XXX E ³³³ C £@ XX ³ ³³XXX C£ @ XXX ³³ BC£ ³ @ XX ³ D ³ XXX £C @ ³³ XXX ³ £ C ³³ @ XX ³ XX F £³ C @
x1 Now we can solve for the demands along the different pieces of the offer curve and get the Marshallian demand. Note that demand is either a whole range, or a “proper demand”. The ranges can be computed from the endpoints (i.e., A to B, C to D, E to F.) Along the rays demand is solved as for
Answers 149 a Leontief consumer: we know the ratio of consumption, we know the budget. So for example on the first ray segment (B to C) we know that x2 = 3x1 . Also, p1 x1 + p2 x2 = w. Hence x1 (p1 + 3p2 ) = w, and x1 = w/(p1 + 3p2 ). (For the Hicksian demand we simply need to fix one indifference curve and compute the points along it. We then get either a segment (like A to B above), or we stay at a kink for a range of prices.) The demands for good 1 therefore are 0, if p1 /p2 > 5; [0, 5w/(8p )] , if 5 = p1 /p2 ; 1 if 5 > p1 /p2 > 1; w/(3p2 + p1 ), x1 (p, w) = [w/(4p1 ), 3w/(4p1 )] , if p1 /p2 = 1; 3w/(3p1 + p2 ), if 1 > p1 /p2 > 1/5; [3w/(8p1 ), w/p1 ] , if p1 /p2 = 1/5; w/p1 , if 1/5 > p1 /p2 . 0, if p1 /p2 > 5; [0, u/8] , if p1 /p2 = 5; if 5 > p1 /p2 > 1; u/8, h1 (p, u) = [u/8, 3u/16] , if p1 /p2 = 1; 3u/16, if 1 > p1 /p2 > 1/5; [3u/16, u] , if p1 /p2 = 1/5; u, if 1/5 > p1 /p2 .
The demands for good 2 are similar and left as exercise. The income expansion paths and Engel curves can be whole regions at price ratios 1,5,1/5, otherwise the income expansion paths are the axes or rays, and the Engel curves are straight increasing lines.
Question 5: The elasticity of substitution measures by how much the consumption ratio changes as the price ratio changes (both measured in percentages.) In other words, as the price ratio changes the slope of the budget changes and we know this will cause a change in the ratio of the quantity demanded of the goods. But by how much? The higher the value of the elasticity, the larger the response in demands. Question 6: First we need to realize that the utility index which each function assigns to a given consumption point does not have to be the same. Instead, as long as the MRS is identical at every point, two utility functions represent the same preferences. So instead of taking the limit of the utility function directly, we will take the limits of the MRS and compare those to the MRSs of the other functions. M RS =
x21−ρ (1/ρ)(xρ1 + xρ2 )(1−ρ)/ρ (ρxρ−1 xρ−1 u1 1 ) 1 = . = = u2 (1/ρ)(xρ1 + xρ2 )(1−ρ)/ρ (ρxρ−1 xρ−1 x11−ρ 2 2 )
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The MRSs for the other functions are CD:
x2 ; x1
Perfect Sub: 1;
Leon: 0, or ∞.
So, consider the Leontief function min{x1 , x2 }. Its MRS is 0 or ∞. But as ρ → −∞ we see that (x2 /x1 )1−ρ → (x2 /x1 )∞ . But if x2 > x1 the fraction is greater than 1 and an infinite power goes to infinity. If x2 < x1 the fraction is less than one and the power goes to zero. The Cobb-Douglas function x1 x2 has MRS x2 /x1 . But as ρ → 0 the MRS of our function is just that. The perfect substitute function x1 + x2 has a constant MRS of 1. But as ρ → 1 the MRS of our function is (x2 /x1 )0 = 1. Therefore the CES function “looks like” those three functions for those choices of ρ. The parameter ρ essentially controls the curvature of the IC’s. Question 7: Set up the consumer’s optimization problem: maxx1 ,x2 ,x3 {x1 + lnx2 + 2lnx3 + λ(m − p1 x1 − p2 x2 − p3 x3 )} . The FOCs are 1 − λp1 = 0;
1 − λp2 = 0; x2
2 − λp3 = 0 x3
and the budget. The first of these allows us to solve for λ = 1/p1 . Therefore the second and third give us x2 = p1 /p2 and x3 = 2p1 /p3 . Combining this with the budget we get x1 = m/p1 − 3. Of course, this is only sensible if m > 3p1 . If it is not we must be at a corner solution. In that case x1 = 0 and all money is spent on x2 and x3 . The second and third FOC above tell us that x3 /x2 = 2p2 /p3 . Hence (remember x1 = 0 now) m = p2 x2 + 2p2 x2 and x2 = m/(3p2 ) while x3 = 2m/(3p3 ). So we get ´ ³ m − 3, p1 , 2 p1 if m > 3p1 p1 p2 ´ p3 x(p, m) = ³ 0, m , 2m if m ≤ 3p1 3p2 3p3 Question 8: Here we have a pure exchange economy with 2 goods and 2 consumers. We can best represent this in an Edgeworth box.
Answers 151 20
5
11.25
@ SSS S · S SSS S@ · sˆ S SSS S@ω · S SSS · S@ S SS· S@ S S S @ ·SSS S · SSS S x∗ -S· SSS S ω S S · S S 11 SSS · S S SSS · S SSS · S SSS · S SSS · S SSS · S SSS · S SSS · SSS · S SSS · S ·
OA
OB 20 8 Suppose x1 is on the horizontal axis and x2 is on the vertical, and let consumer B have the lower left hand corner as origin, consumer A the upper right hand corner. (I made this choice because I like to have the “harder” consumer oriented the usual way.) The dimensions of the box are 20 by 20 units. The first thing to do is to find the Pareto Set (the contract curve), since we know that any equilibrium has to be Pareto efficient. The MRS for person A is 4/3, the MRS for person B is 3x2 /(4x1 ). Therefore the Pareto Set is defined by x2 /x1 = 16/9 (in person B’s coordinates.) This is a straight ray from B’s origin with a slope greater than 1, and therefore above the main diagonal. The Pareto set is this ray and the portion of the upper boundary of the box from the ray’s intersection point to the origin of A. There now are 2 possibilities for the equilibrium. Either it is on the ray, and therefore must have a price ratio of 4/3. Or it is on the upper boundary of the box, in which case the price ratio must be below 4/3, but we know that B’s consumption level for good 2 is 20. In the first case we have 2 equations defining equilibrium. The ray, x2 = 16x1 /9, and the budget line (x2 − 11) = 4(8 − x1 )/3. From this we get 16x1 /9 − 11 = 32/3 − 12x1 /9 and from that 28x1 /9 = 65/3 and thus x1 = 195/28 < 20. It follows that x2 = (16 × 195)/(9 × 28) = 780/63 = 260/21 < 20. Since both of B’s consumption points are strictly within the interior of the box, we are done. All that remains is to compute A’s allocation. The equilibrium is therefore µ µ ¶ µ ¶¶ 4 195 780 195 780 A B (p∗, (x ), (x )) = , 20 − , 20 − , , . 3 28 63 28 63 Question 9: Again we have a square Edgeworth box, 20×20. Again I choose to put consumer B on the bottom left origin. B’s preferences are quasi-linear with respect to x1 , A’s are piece-wise linear with slopes 4/3 and 3/4 which
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meet at the kink line x2 = x1 (in A’s coordinates!) which coincides with the main diagonal. The MRS for B’s preferences is 3x2 /4. We have Pareto optimality when 3x2 /4 = 4/3 → x2 = 16/9 and when 3x2 /4 = 3/4 → x2 = 1. So, the Pareto Set is the vertical axis from B’s origin to xB 2 = 1, the horizontal line xB = 1 to the main diagonal (the point (1, 1) in other words), up the 2 B B main diagonal to the point (x1 , x2 ) = (16/9, 16/9), from there along the horizontal line xB 2 = 16/9 to the right hand edge of the Box, and then up that border to A’s origin. OA 20 Z ZS Z SZ ZZ Z ZSZ Z ZS Z ¡ Z Z SZZ Z S Z Z Z Z ¡ ZS Z Z ZS ZZZ Z Z¡ Z Z S 11 ωSS Z S S SS S¡ S S SS S S SS S S SS S S SS S S SSS S S S S S S S 16/9 ¡ x∗ S S SSS 1 SS S SSS
OB 8 20 By inspection, the most likely candidate for an equilibrium is a price ratio of 4/3 with an allocation on the second horizontal line segment. Let us attempt to solve for it. First, the budget equation (in B’s coordinates) is 3(x2 − 11) = 4(8 − x1 ). Second, we are presuming that x2 = 16/9. So we get 11 . Since this is less than 20 we have found 16/3 − 33 = 32 − 4x1 , or x1 = 14 12 an interior point and are done. The equilibrium is ¶ µ ¶¶ µ µ 4 1 164 11 16 A B , 5 , (p∗, (x ), (x )) = , 14 , . 3 12 9 12 9 Question 10: To prove this we need to show the implication in both directions: (⇐) : Suppose x  y. Then ∃B, x, y ∈ B with the property that x ∈ C(B), y ∈ / C(B). Consider all other B 0 ∈ B with the property that x, y ∈ B 0 . By the Weak Axiom ∃ 6 B 0 with y ∈ C(B 0 ) since otherwise the set B would violate the weak axiom (applied to the choice y with the initial set B 0 .) Therefore x Â∗ y. (⇒) : Let x Â∗ y. The first part of the definition requires ∃B, x, y, ∈ B, x ∈ C(B). By the weak axiom there are two possibilities: either all B 0 ∈ B with x, y ∈ B 0 have {x, y} ∈ C(B 0 ) or none have y ∈ C(B 0 ). The second part of
Answers 153 the definition requires us to be in the second case, but then y ∈ / C(B), and so x  y. If the WA fails a counter example suffices: Let X = {x, y, z}, B = {{x, y}, {x, y, z}}, C({x, y}) = {x}, C({x, y, z}) = {x, y}. This violates the WA. C({x, y}) = {x} demonstrates that x  y by definition. On the other hand it is not true that x Â∗ y (let B = {x, y} and B 0 = {x, y, z} in the definition of Â∗ . Question 11: a) This is another 20 by 20 box, with the endowment in the centre. Suppose B’s origin on the bottom left, A’s the top right. As in question 8, A’s indifference curves have a constant MRS of α and are perfect substitute type. B’s ICs have a MRS of βx2 /x1 and are Cobb-Douglas. The contract curve in the interior must have the MRSs equated, so it occurs where (in B’s coordinates) x2 /x1 = α/β. This is a straight ray from B’s origin and depending on the values of α and β it lies above or below the main diagonal. Since these cases are (sort of) symmetric we pick one, and assume that α/β > 1. The contract curve is this ray and then the part of the upper edge of the box to A’s origin. As in question 8 there are two cases for the competitive equilibrium. Either it occurs on the part of the Contract curve interior to the box, or it occurs on the boundary of the box. In the first case the slope of the budget and hence the equilibrium price must be α, since both MRSs have that slope along the ray and in equilibrium the price must equal the MRS. Note that the budget now coincides with A’s indifference curve through the endowment point. The equilibrium allocation is determined by the intersection of the contract curve and this budget/IC. So we have two equations in two unknowns: α=
x2 − 10 10 − x1
and x2 =
α x1 . β
Hence α(10 − x1 ) = αx1 /β or αβ10 = x1 (α + αβ) and thus x1 = β10/(1 + β) and x2 = α10/(1 + β). These are the consumption levels for B. A gets the B B B rest. The equilibrium thus would be (p∗ , (xA 1 , x2 ), (x1 , x2 )) = ¶ µ ¶¶ µ µ 2(1 + β) − α β10 α10 2+β , 10 , , α, 10 1+β 1+β 1+β 1+β which only makes sense if the allocation indeed is interior, that is, as long as 10α/(1 + β) < 20, or (α − β) < (2 + β). If that is not true we find ourselves in the other case. In that case we know that we are looking for an equilibrium on the upper boundary of the box and A thus know that xB 2 = 20 while x2 = 0. It remains to determine p and the allocations for good 1. At the equilibrium point the budget must be flatter
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than A’s IC (so that A chooses to only consume good 1). The allocation must also be the optimal choice for B and hence the budget must be tangent to B’s IC, since for B this is an interior consumption bundle (interior to B’s consumption set, that is.) So we again have to solve two equations in 2 unknowns: c2 − 10 βc2 = p and p = c1 10 − c1
while c2 = 20.
It follows that β20(10 − c1 ) = 10c1 and therefore c1 = 20β/(1 + 2β). A gets the rest. The equilibrium is therefore µ µ ¶ µ ¶¶ 1+β β20 ∗ A A B B (p , (x1 , x2 ), (x1 , x2 )) = 1 + 2β, 20 ,0 , , 20 . 1 + 2β 1 + 2β b) All endowments above and to the right of the line x2 = 40 − 2x1 in B’s coordinates will lead to a boundary equilibrium. All those on this line and below will lead to an interior equilibrium with p = 2. Question 12: a) The social planner’s problem is ¾ ½ √ 1 maxl ln(4 16 − l) + ln(l) 2 which has first order condition 1 2 1 √ − √ + = 0. 4 16 − l 16 − l 2l √ Hence 16 − l = l and so l∗ = 8, x∗ = 8, c∗ = 8 2. b) Since the profits, we solve for the firm √ consumer’s problem requires √ first. maxx {p4 x − wx} has FOC 2p/ x = w and leads to firm labour demand of x(p, w) = 4p2 /w2 , consumption good supply of c(p, w) = 8p/w, and profits of π(p, w) = 4p2 /w. The consumer will ½ ¾ 1 maxc,l lnc + lnl + λ (16w + π(p, w) − pc − wl) 2 which has first order conditions 1/c − λp = 0; 1/(2l) − λw = 0; 16w + π(p, w) = pc + wl. The first 2 imply that pc = 2lw. Substituting into the third and using the profits computed above yields demand of c(p, w) = 32w/(3p) + 8p/(3w) and leisure demand of l(p, w) = 16/3 + 4p2 /(3w2 ). We can now solve for the equilibrium price ratio. Take any one market and set demand equal to supply. For the goods market this implies 32w/(3p) +
Answers 155 8p/(3w) = 8p/w, and hence p2 /w2 = 2. Substituting into the demands and supplies this gives l∗ = 8, and hence all values are the same as in the social planner’s problem in part a). You may want to verify that you could have solved for the price ratio from the labour market. The complete statement of the general equilibrium is: The equilibrium price √ √ ratio is p/w = √ 2, the consumer’s allocation is (c, l) = (8 2, 8), and the firm produces 8 2 units consumption good from 8 units input. Note that we cannot state profits without fixing one √ of the prices. So let w = 1 (so that we use labour as numeraire), then p = 2 and profits are 8.
7.2
Chapter 3
Question 1: a) Zero arbitrage means that whichever way I move between periods, I get the same final answer. In particular I could lend in period 1 to collect in period three, or I could lend in period 1 to period 2, and then lend the proceeds to period 3. Hence the condition is (1 + r12 )(1 + r23 ) = (1 + r13 ). Note that if we where to treat r13 not as a simple interest rate but as a compounding one, we’d get (1 + r12 )(1 + r23 ) = (1 + r13 )2 instead. b) You have to adopt one period as your viewpoint and then put all other values in terms of that period (by discounting or applying interest). With period 3 as the viewpoint I use period 3 future values for everything: B = {(c1 , c2 , c3 )|(1 + r13 )c1 + (1 + r23 )c2 + c3 = (1 + r13 )m1 + (1 + r23 )m2 + m3 } Note that any other viewpoint is equally valid. The restriction in (a) means that it does not matter which interest rate I use to compute the forward value of c1 , say. Indeed, without that restriction I would get an infinite budget if it is possible to borrow infinite amounts. With some borrowing constraints in place I would have to compute the highest possible arbitrage profits for the various periods and compute the resulting budget. c) This is a standard downward sloping budget line in (c2 , c3 ) space with a slope of −(1 + r23 ). It does not necessarily have to go through the endowment point (m2 , m3 ), however. It will be below that point if c1 > m1 and above that point if c1 < m1 .
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c3
S{m1 > c1 } S S S ¡ {m1 < c1 } S S¡ S S¡ S S S m m3 S S SS SS
m2
c2
Question 2: a) The easy way to get this is to first ignore the technology. The market budget is a straight line with a slope of −1 through the point (100, 100), which is truncated at the point (160, 40), where the budget becomes vertical. Note that the gross rate of return is 1 since the interest rate is 0. Now consider the technology and the implications of zero arbitrage: Joe can move consumption from period 1 to period 2 in two ways, via the financial market, or via “planting”. Both must yield the same gross rate of return at the optimum (why? we know that at the optimum of a maximization problem the last dollar allocated to each option must yield the same marginal benefit.) The gross rate of return at the margin is nothing but the marginal product of the √ technology, however. So, compute the MP (5/ x1 ) and find the investment level at which the MP is 1. 5 √ =1 x1
−→
5=
√
x1
−→
x1 = 25.
At optimal use at an interior optimum Joe invests 25 units (and collects 50 in the next period.) This means that from any point on the financial market budget Joe can move left 25 and up 50. So that gives a straight line with slope −1 which starts at (0, 225) and goes to (135, 90). After this point there is a corner solution in technology choice: Joe cannot use the market any more. The technology therefore may give a higher return than the market. So the budget follows the (flipped over to the left) technology frontier down to the point (160, 40), and down to (160, 0) from there. b) First simplify the preferences (this step is not necessary!). Applying a natural logarithm gives the function Uˆ (c1 , c2 ) = c41 c62 which represents the .6 same preferences. Applying the 10th root gives U˜ (c1 , c2 ) = c.4 1 c2 which also represents the same preferences and is recognized as a Cobb-Douglas. Now you can either compute the MRS (2c2 /3c1 ) and set that equal to 1 (since most of the budget has a slope of −1 and we know that M RS = Slope at the optimum.) That gives you two equations in two unknowns, and we can solve: c2 = 225 − c1 , c2 = 3c1 /2
→
450 = 5c1
→
c1 = 90, c2 = 135.
Answers 157 We then double check that the assumption that we are on the −1 sloped portion of the budget was correct, which it is (by inspection.) Or you could use the demand function for C-D, so you know µ ¶ ¶ µ .4M .6M .4 × 225 .6 × 225 , , (c1 , c2 ) = = (90, 135). = p1 p2 1+0 1 Now this is his final consumption bundle. In order to get there he invested 25 units, so on the “market budget” line he must have started at (115, 85), and that required him to borrow 15 units. In summary, he borrows 15, giving him 115, of which he invests 25, so he has 90 left to consume. In the next period he gets 100 from his endowment, 50 from the investment, for a total of 150, of which he has to use 15 to pay back the loan, so he can consume 135! Question 3: I will not draw the diagram but describe it. You should refer to a rough diagram while reading these solutions to make sense of them. a) The indifference curves have two segments with a slope of −1.3 and −1.2 respectively. The switch (kink) occurs where µ ¶ µ ¶ 12 13 23 → c2 = (22×13−23×12)c1 /10 = c1 . c1 + c2 = 22 c1 + c 2 10 10 b) Note that the budget has a slope of 1.25 which is less than 1.3 and more than 1.2, so she consumes at the kink. Thus she is on the kink line and the budget: c1 = c2 and − 1.25 =
c2 − 8 c1 − 9
→
c1 = c2 = 77/9.
→
c1 = c2 = 73/9.
c) Again she consumes at the kink, so c1 = c2 and − 1.25 =
c2 − 12 c1 − 5
d) Here we need to work back. Note that at the slopes implied by the interest rates she continues to consume at her kink line. The reason is that both 1.25 and 1.28 are bigger than 1.2, the slope of her lower segment, but less than 1.3, the slope of the steep segment. Hence optimal consumption is at the kink and she borrows if she has less period 1 endowment than period 2 endowment. She lends money if she has larger period 1 endowment than period 2 endowment. So for all endowments above the main diagonal she is
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a borrower, for all endowments below a lender. e) Now she never trades. To lend money the budget slope is 1.18 which is less than either of her IC segment slopes. She would not want to lend ever at this rate no matter what her endowment. On the other hand, suppose she were to borrow. The budget slope is 1.32 which is steeper than even her steepest IC segment. She would not borrow. Thus she remains at the kink in her budget (the endowment point) no matter where it is. Question 4: Again I will not draw the diagram but describe it. You should refer to a rough diagram while reading these solutions to make sense of them. a) This is a 20 by 10 box. Suppose A’s origin on the bottom left, B’s the top right. A’s indifference curves have a MRS of c2 /(αc1 ) and are nice C-D type curves. B’s ICs have a MRS of 1/β and are straight lines. The contract curve in the interior must have the MRSs equated (from Econ 301: for differentiable utility functions an interior Pareto optimum has a tangency), so it occurs where c2 /c1 = α/β. This is a straight ray from A’s origin and depending on the values of α and β it lies above or below the main diagonal. Since these cases are (sort of) symmetric we pick one, and assume that α/β > 1/2. The contract curve is this ray and then the part of the upper edge of the box to B’s origin. b) There are two cases, either the Contract curve ray is shallow enough that the equilibrium occurs on it, or it is so steep that the equilibrium occurs on the top boundary of the box. In the first case the slope of the budget and hence the equilibrium price must be 1/β, since both MRSs have that slope along the ray and in equilibrium the price must equal the MRS. So the equilibrium interest rate is (1 − β)/β. Note that the budget now coincides with player B’s indifference curve through the endowment point. Hence the ray of the contract curve must intersect that, and it does so only if it intersects the top boundary to the right of the intersection of B’s IC with the boundary. The latter occurs at (4(3 − β), 10). The former occurs at (10β/α, 10). So the interior solution obtains if 10β/α > 4(3 − β), or if 10β > 12α − 4αβ. In that case the equilibrium allocations are derived by solving the intersection of the budget and the ray: c2 = αc2 /β and −
c2 − 6 1 = β c1 − 12
→
cA 1 =
6(2 + β) A α6(2 + β) , c2 = . 1+α β(1 + α)
B gets the remainder.
In the other case, when the ray fails to intersect B’s IC, we know that we are looking for an equilibrium on the upper boundary of the box (so B cA 2 = 10 and c2 = 0.) At this point we must have a budget flatter than B’s IC (so that B chooses to only consume good 1). It must also be tangent to
Answers 159 A’s IC, since for player A this is an interior consumption bundle (interior to his consumption set, that is.) So we require 1 + r = 10/(αc1 ) to have the tangency, and we require 1 + r = (10 − 6)/(12 − c1 ) in order to be on the budget line. These are two equations in two unknowns again, so we solve: cA 1 = 60/(5 + 2α) and r = (5 − 4α)/(6α). B gets the rest of good 1, of course.
7.3
Chapter 4
Question 1: We wish to show that for any concave u(x) 1 1 1 1 1 u(24) + u(20) + u(16) ≥ u(24) + u(16). 3 3 3 2 2 We can do the following: first bring the u(24) and u(16) to the RHS: 1 1 2 u(20) ≥ u(24) + u(16). 6 6 6 Then multiply both sides by 3: 1 1 u(20) ≥ u(24) + u(16). 2 2 The LHS of this represents a certain outcome of 20, the RHS a lottery with 2 equally likely outcomes. Now note that 1 1 24 + 16 = 20. 2 2 That is, the expected value of the lottery on the RHS of the last inequality above is equal to the expected value of the degenerate lottery on the LHS. Therefore this penultimate inequality must be true, since it coincides with the definition of a risk averse consumer. (utility of expectation greater than expectation of utility.) Question 2: The certainty equivalent is defined by Z X U (CE) = pi u(xi ) = u(x)dF (x). Using the particular function we are given: √ √ √ CE = α 3600 + (1 − α) 6400
CE = (α60 + (1 − α)80)2 = (80 − 20α)2 .
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Note that the expected value of the gamble is E(w) = α3600 + (1 − α)6400 = 6400 − α2800 and thus the maximal fee this consumer would pay for access to fair insurance would be the difference E(w) − CE = 400α(1 − α). Question 3: The coefficient of absolute risk aversion is defined as rA = −u00 (w)/u0 (w). Computing this for both functions we get u(w) = lnw −→ rA =
1 ; w
√ 1 u(w) = 2 w −→ rA = . 2w
Therefore the two consumers exhibit equal risk aversion if the second consumer has half the wealth of the first. Their relative risk aversion coefficients (defined as −u00 (w)w/u0 (w) are 1 and 1/2, respectively. That means that while, if the logarithm consumer has twice the wealth as the root consumer, he will have the same attitude towards a fixed dollar amount gamble, he will be more risk averse with respect to a gamble over a given proportion of wealth. (Note that the two statements don’t contradict one another: a $1 gamble represents half the percentage of wealth for a consumer with twice the wealth!) Question 4: Here we need an Edgeworth Box diagram, which is a square, 15 units a side. Suppose we have consumer A on the bottom left origin (B then goes top right). Suppose also that we put state R on the horizontal. Note that the certainty line is the main diagonal of the box! This observation is crucial, since it means that there is no aggregate risk! General equilibrium requires that demand is equal to supply for each good, but we can’t find those here (not knowing the consumers’ tastes), so it is not useful information. But we also know that in general equilibrium the price ratio must equal each consumer’s MRS (since GE is Pareto optimal and that requires MRSs to be equalized, at least for interior allocations.) Note that the two MRSs here are M RSA =
πu0A (cA R) 0 (1 − π)uA (cA S)
M RSB =
πu0B (cB R) 0 (1 − π)uB (cB S)
B B A On the certainty line (the main diagonal) cA S = cR and cS = cR , so M RSA = M RSB = π/(1 − π). In other words, the certainty line for each consumer coincides and together they are the set of Pareto optimal points.
Hence the equilibrium price ratio must be p∗ = π/(1 − π). The allocation is now easily computed: we know the price ratio and the endowment, hence the budget line for the consumers. We also know that
Answers 161 consumption is equal in both states. So π cA − 5 cA − 5 A → cA = S = S = cR = 5(1 + π) A 1−π 10 − c 10 − cA R B B A and since cB i = 15 − ci we get cS = cR = 5(2 − π).
Question 5:
a)
maxx {0.5u(10000(1 + 0.8x)) + 0.5u(10000(1.4 − 0.8x))}
b) The FOC for this is 0.5 × 0.8u0 (100000(1 + 0.8x)) − 0.5 × 0.8u0 (10000(1.4 − 0.8x)) = 0 implies : implies :
u0 (10000(1 + 0.8x)) = u0 (10000(1.4 − 0.8x)) 10000(1 + 0.8x) = 10000(1.4 − 0.8x)
since she is risk averse. It follows that 1 + .8x = 1.4 − .8x, and therefore that 1.6x = 0.4, so that x = 0.25. One quarter, or 25% are invested in gene technology. Question 6: i) Denote the probability with which a ticket wins by π and the prize by P . A fair price for this lottery ticket would have to be a fraction p per dollar of prize such that π(P − pP ) − (1 − π)pP = 0, or p = π. Let us start with this as a benchmark case (we know that normally such a lottery would not be accepted.) Utility maximization requires that for a gambling consumer v(w0 ) ≤ πv(w0 + (1 − p)P ) + (1 − π)v(w0 − pP ) + µi . Thus all consumers for whom µi ≥ v(w0 ) − πv(w0 + (1 − p)P ) − (1 − π)v(w0 − pP ) purchase a ticket. At a fair gamble this is µi ≥ v(w0 ) − πv(w0 + (1 − π)P ) − (1 − π)v(w0 − πP ) > v(w0 ) − v(π(w0 + (1 − π)P ) + (1 − π)(w0 − πP )) = v(w0 ) − v(w0 ) (the second strict inequality follows from the definition of risk aversion). Clearly a strictly positive µ is required. Can the government make money on this? Well, assume that the price p above is fair (p = π) and let there be an additional charge of q. Now all consumers gamble for whom µi ≥ v(w0 ) − πv(w0 + (1 − π)P − q) − (1 − π)v(w0 − πP − q). While such a µi is larger than before, it exists (for small q in any case) as long as things are sufficiently smooth and the µi go that high. Note that those who gamble have a high utility for it (a high taste parameter µi ) in this setting. Note that this implies that even though they lose money on average they have a higher
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welfare. (The anti-gambling arguments in public policy debates therefore come in two flavours: (i) your gambling is against my (religious) beliefs, and thus it ought to be banned, (ii) there are externalities: your lost money is really not yours but should have bought a lunch for your child/spouse/dog. Since your child/spouse/dog can’t make you stop, we will on their behalf.) ii) Now µ is fixed. Of course, the decision to gamble will still depend on the same inequality, namely µ > v(w0 ) − πv(w0 + (1 − π)P − q) − (1 − π)v(w0 − πP − q). We thus can translate this question into the question of how the right hand side depends on w0 and how this dependency relates to the different behaviours of risk aversion with wealth. So, is the right hand side increasing or decreasing with wealth, and is this a monotonic relationship? The right hand side is related, of course, to the utility loss from going to the expected utility from the expected value (ignoring q for a minute.) Intuitively, we would expect the difference to be declining in wealth for constant absolute risk aversion: Constant absolute risk aversion implies a constant difference between the expected value and the certainty equivalent.1 Let this difference be the base of a right triangle. Orthogonal to that we have the side which is the required distance between the two utilities. The third side must have a declining slope as wealth increases since it is related to the marginal utility of wealth at the certainty equivalent, which is declining in wealth by assumption. There you go, I’d expect the utility difference must fall with wealth. More formally, consider the original inequality again and approximate the RHS by its second order Taylor series expansion (that way we get first and second derivatives, which we want in order to form rA : v(w0 ) − πv(w0⊕ ) − (1 − π)v(w0 − πP − q) ≈ v(w0 ) − π(v(w0 ) + ⊕v 0 (w0 ) + ⊕2 v 00 (w0 )/2) − (1 − π)(v(w0 ) − ªv 0 (w0 ) + ª2 v 00 (w0 )/2) = −π ⊕ v 0 (w0 ) − π ⊕2 v 00 (w0 )/2(1 − π)(ªv 0 (w0 ) − ª2 v 00 (w0 )/2) = v 0 (w0 ) [(1 − π) ª (1 − ªrA /2) − π ⊕ (1 − ⊕rA /2)] . This looks more like it! Now note that we use ª and ⊕ as positive quantities (which are not equal: ª is larger!) Furthermore we know that (a) this quantity must be positive and (b) that π is probably a very small number. Now, if rA is constant then the term in brackets is constant, but of course 1
Is there a general proof for that? Note that constant rA has for example the functional form u(w) = −e−aw , for which the above is certainly true.
Answers 163 v 0 (w) falls with w and thus the right hand side of our initial inequality (way above) falls. Any given µ is therefore more likely to be larger than it. Thus rich consumers participate, poor consumers don’t if we have constant absolute risk aversion. If we have decreasing absolute risk aversion this effect is strengthened. Now, since relative risk aversion is just rA w, it follows that constant relative risk aversion requires a decreasing absolute risk aversion, and that decreasing relative risk aversion requires an even more decreasing absolute risk aversion. Thus in all cases the rich gamble and the poor don’t. (Note here that they are initially rich. Since they loose money on average they will become poor and stop gambling.) 2 iii) If v(w) = ln w then v 0 (w) = 1/w and v 00 (w) . Therefore √ √ = −1/w 0 rA = 1/w, with ∂rA /∂w < 0, and rR = 1. If v(w) = w then v (w) = 1/2 w and v 00 (w) = −w 3/2 /4. Therefore rA = 1/(2w) and rR = 1/2. We now know two pieces of information: the consumers’ risk aversion to a given size gamble is declining with wealth. This would, ceteris paribus make them more likely to purchase the gamble for a constant µ (see above). But µ now is also declining with wealth. The final outcome therefore depends on what declines faster, and we can’t make a definite statement. (As an aside note the following. Suppose we are talking stock market participation here. Then it might be reasonable to assume that the utility of participating in it is increasing in wealth, on average, and so we get higher participation by wealthier people. Now, if the stock market on average is a bad bet we get mean reversion in wealth, while if the stock market is on average more profitable than savings accounts etc we get the rich getting richer. If you now run a voting model where the mean voter wins, you get the desire to redistribute (i.e., tax the investing and profiting rich and give the cash to those who have a too high marginal utility of wealth to invest themselves.) Note also that progressive taxes reduce the returns of a given investment proportional to wealth, counteracting the above effect of more participation by wealthy individuals. . . . See how much fun you can have with these simple models and a willingness to extrapolate wildly?) Question 7: This question forms part of a typical incomplete information contracting environment. Here we focus only on the consumer’s behaviour. a) Assume that the worker has a contractual obligation to provide an effort level of E. Once he has signed the contract, however, he knows that his actual effort is not observable and thus would try to shirk. Expected utility is maximized for p p e∗ = argmax{α w(E) − p + (1 − α) w(E) − e2 }.
The first order condition for this problem is −2e = 0 if e 6= E. I.e., given the worker shirks he will go all the way (after all, the punishment does not
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depend on the severity of the crime in any way.) Thus wepneed to ensure that is the casep if w(E) − E 2 ≥ p the worker will notpshirk at all, 2which p α w(E) − p + (1 − α) w(E), or E ≤ α( w(E) − w(E) − p). If the wage function satisfies this inequality for all E, it will elicit the correct effort levels in all cases. b) Now we have a potentially variable punishment. Given some job with contractual obligation E, the worker now will maximize expected utility and set p p e∗ = argmax{α w(E) − p(E − e) + (1 − α) w(E) − e2 }. p The FOC for this problem is αp0 (·)(2 w(E) − p(E − e))−1 − 2e = 0. (There are also second order conditions which need to hold!) This implies that the worker will play off the cost of shirking against the gains from doing so. We need to make sure that this equation is only satisfied for e∗ = E, in which case he “voluntarily” chooses the contracted level. This clearly requires a p −1 0 0 positive p (). In particular, αp (0)(2 w(E)) − 2E = 0. Note: We could also vary the detection/supervision p probability and makepα depend on E. Then we get e∗ = argmax{α(E) w(E) − p + (1 − α(E)) w(E) − e2 }. As in (a), if the worker deviates he will go all the way here. So the problem is similar to (a), only the wage schedule is now different since α(E) can also vary now. What this shows us is that we tend to want a punishment and a detection probability which both depend on the deviation from the correct level. (This is going to be a question about the technology available: some technologies may be able to detect flagrant shirking more readily than slight shirking.) c) What this seems to indicate is that we would like to make punishments fit the crime. (So for example, if the punishment for a hold-up with a weapon is as severe as if somebody actually gets shot during it, then I might as well shoot people when I’m at it and I think that helps (and if it does not increase the effort the police put into finding me.)) Furthermore, if detection is a function of the actual effort level (the more you fudge the books the more likely will you be detected) then we need lower punishments, ceteris paribus, since the increasing risk will provide some disincentive to cheat anyways. Question 8: a) Let CB denote the coverage purchased for bad losses, and CM the coverage for minor losses. Zero profits imply that the premiums pB and pM for bad and minor losses, respectively, are pB = π/5 and pM = 4π/5. Hence the consumer’s expected utility maximization problem becomes ½ maxCB ,CM (1 − π)u(W − pM CM − pB CB )+ 1 π( u(W − pM CM − pB CB + CB − B)+ 5
Answers 165 4 u(W − pM CM − pB CB + CM − M )) 5
¾
The first order conditions for this problem are 4π π −pM (1 − π)u0 (n) − pM u0 (b) + (1 − pM ) u0 (m) = 0 5 5 π 4π −pB (1 − π)u0 (n) + (1 − pB ) u0 (b) − pB u0 (m) = 0 5 5 Using the fair premiums this simplifies to µ ¶ π 0 4π 0 −(1 − π)u (n) − u (b) + 1 − u0 (m) = 0 5 5 ³ 4π 0 π´ 0 u (b) − u (m) = 0 −(1 − π)u0 (n) + 1 − 5 5
Hence
¶ µ ³ π´ 0 π 4π 0 4π u (m) u (b) − u0 (m) − u0 (b) = 1 − 1− 5 5 5 5
and thus u0 (b) = u0 (m), which finally implies that u0 (n) = u0 (b) = u0 (m) and therefore that 0 = CB − B = CM − M. As expected, the consumer buys full insurance for each accident type separately. b) Now only one coverage can be purchased, denote it by C, and will be paid in case of either accident. Zero profits imply that the premium p is p = π. Hence the consumer’s expected utility maximization problem becomes ½ maxC (1 − π)u(W − pC)+ 1 π( u(W − pC + C − B)+ 5 ¾ 4 u(W − pC + C − M )) 5
The first order condition for this problem is π 4π −p(1 − π)u0 (n) + (1 − p) u0 (b) + (1 − p) u0 (m) = 0 5 5 Using the fair premium this simplifies to 1 4 u0 (n) = u0 (b) + u0 (m) 5 5
166 L-A. Busch, Microeconomics
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and hence either u0 (b) > u0 (n) > u0 (m) or u0 (b) < u0 (n) < u0 (m). Thus either W − B + (1 − π)C < W − πC < W − M + (1 − π)C or W − B + (1 − π)C > W − πC > W − M + (1 − π)C, but this implies either −B + 1C < 0 < −M + 1C or −B + 1C > 0 > −M + 1C. Since B > M by definition we obtain that B > C > M , the consumer over insures against minor losses, and is under insured against big losses. Question 9: From Figure 3.7 in the text, we can take the consumers’ budget line to be the line from the risk free asset point (the origin in this case) to a tangency with the efficient portfolio frontier. where √ Now this tangency √ occurs −1 the margin is equal to the average, so that σ − 16/σ = (2 σ − 16) . That means that the market portfolio has 2(σ − 16) = σ or σ = 32. Therefore µ = 4. The slope of the portfolio line thus is 4/32. For an optimal solution the consumer’s MRS must equal the slope of the portfolio line. For the two consumers given the MRS is σ/32 and σ/96. Thus the optima are σ = 4, µ = 1/2 and σ = 12, µ = 3/2. As expected, the consumer with the higher marginal utility for the mean will have a higher mean at the same prices (and given that both have the same disutility from variance.) Question 10: The asset pricing formula implies that the expected return of the insurance equals the expected risk-free return less a covariance term. If insurance has a lower expected return than the risk-free asset, this covariance term must be positive. In the denominator we have the expected marginal utility, guaranteed to be positive. Thus the numerator must be positive. This means that Cov(u0 (w), Ri ) > 0. But since u00 (w) < 0 this implies that the covariance between w and Ri is negative, that is, if wealth is low the return to the policy is high, if wealth is high, the return to the policy is low. That of course is precisely the feature of disability insurance which replaces income from work if and only if the consumer is unable to work. Question 11: 1) False. The second order condition would indicate a minimum as demonstrated here: maxC {πu(w − L − pC + C) + (1 − π)u(w − pC)} has FOC π(1−p)u0 (w−L+(1−p)C)−p(1−π)u0 (w−pC) = 0. The second order condition for a maximum is π(1−p)2 u00 (w−L+(1−p)C)+p2 (1−π)u00 (w−pC) ≤ 0. Note that 1 ≥ π, p ≥ 0, so that the SOC requires u00 (·) to be negative for at least one of the terms. A risk-lover has, by definition, u00 (·) > 0. 2) Uncertain. We can draw 2 diagrams to demonstrate. In both we have two intersecting budget lines, one steeper, one flatter. The flatter one corresponds to the initial situation. They intersect at the consumer’s endowment. Since the consumer is a borrower, the initial consumption point is below and
Answers 167 to the right of the endowment on the initial budget. The indifference curve through this point is tangent to this budget. It may, however, cut the new budget (so that the IC tangent to the new budget represents a higher level of utility) or lie everywhere above it (in which case utility falls.) 3) True. Apply the following positive monotonic transformations to the first function: −2462, ×12, collect terms in one logarithm, take exponential, take the 9000th root. What you get is the second function. R 4) True. A risk averse consumer is defined as having u( xg(x)dx) > R u(x)g(x)dx. Let the consumer have initial wealth w and suppose he could participate in a lottery which leads to a change in his initial wealth by x, distributed as f (x). Suppose the payment for this lottery is p. If this payment is equal to the expected value of the lottery then the consumer will not have a change in expected wealth, but will face risk. Thus by definition he would not buy this lottery. If the payment is less, then the expected value of wealth from participating in the lottery exceeds the initial wealth. Depending on by how much, the consumer may purchase. A risk loving consumer, of course, would already buy at when the expected net gain is zero. (This argument could be made more precise, and you should try to put it into equations!) 5) False. The market rate of return is 15%. Gargleblaster stock has a rate of return of (117 − 90)/90 = 30%. This violates zero arbitrage. 6) True. All consumers face the same budget line in mean-variance space. At an interior optimum (and assuming their MRS is defined) they all consume on this line where the tangency to their indifference curve occurs. This may be anywhere along the line, depending on tastes, but the slope is dictated by the market price for risk. Question 12: a) Since workers work as bus driver and at a desk job we require √ √ √ 2 40000 = α2 44100 − 11700 + (1 − α)2 44100 Therefore
√ √ 1 44100 − 40000 210 − 200 √ = . α= √ = 210 − 180 3 44100 − 32400
b) Since workers work on oil rigs and at a desk job we require √ √ √ 2 40000 = 0.5 × 2 122500 − Loss + 0.5 × 2 122500. √ √ Thus 400 = 122500 − Loss + 350 and hence 50 = 122500 − Loss or Loss = 120000. c) At fair premiums the workers will fully insure. That is, they suffer their expected loss for certain. For a bus driver the √ expected loss √ is 11700/3 = 3900. Thus the bus driver wage must satisfy 2 40000 = 2 w − 3900 and
168 L-A. Busch, Microeconomics
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hence it is $43900. For the oil rig worker the expected loss is $60000, and their wages will fall to $100000 under workers compensation. Note the the condition that workers take all jobs together with a fixed desk job wage fixes the utility level in equilibrium for workers. However, the wage premium for risky jobs will not have to be paid: the wages of the risky occupations fall which benefits the firms in those industries by lowering their wage costs. (This is why industries are in favour of workers’ compensation.) d) The average probability of an accident now is 0.4 × 0.5 + 0.6 × 1/3 = 0.4. If we were to use this as a fair premium (but see below!) this premium is too high for bus drivers, who will under insure, and too low for oil rig workers, who will over √ bus drivers will choose to buy insurance Cb √ insure. Indeed, the such that 6 44100 √ − 0.4Cb = 8 32400 +√.6Cb (Take the first order condition √ from for maxCb {(1/3) 32400 + 0.6Cb + (2/3) 44100 − 0.4Cb }, bring the the denominator into the numerators and loose the 1/30 on both sides.) Thus we require 9(44100−0.4Cb ) = 16(32400+0.6Cb ), or Cb = 10(9×44100−16× 32400)/(6×16+4×9) = −9204. What does this mean? It means that the bus drivers would like to bet on themselves having an accident buying negative amounts of insurance! (The ultimate in under insurance!) Note that the governments expected profit from bus drivers is −0.4×9204/3+1.2×9204/3 = 2454.40 > 0. The√oil rig workers would need to solve √ + 122500 − 0.4Co }, which leads to max √ Co { 2500 + 0.6Co √ 3 122500 − 0.4Co = 2 2500 + 0.6C and thus Co = 182083.33. Note that the govt looses money on them, since (0.5 × 0.4 − 0.5 × 0.6) × 182083.33 = −18208.30. Overall then the govt makes losses of 5810.68N , where N is the number of workers in risky occupations. At the old wages both groups are better off (and thus there would be an influx of desk workers and a reallocation towards oil rigs.) In order to break even the insurance rates would have to be changed, in particular raised. It also seems that the govt would ban the purchase of negative insurance amounts. In which case the bus drivers would find it optimal to buy no insurance, and then premiums would have to be 0.5 for the govt to break even. This would be deemed unjust by all involved, and so in practice the govt forces all workers to buy a fixed amount of insurance! In principle we could compute equilibrium wages if we treat the insurance purchase as a function of the wage. So, for example we know from the above√that Cb (w) = 10(9×w−16×(w−11700))/(6×16+4×9). We then solve p p for s 40000 = (2/3) w + (1 − 0.4)Cb (w) + (4/3) w − 11700 − 0.4Cb (w). The details are left to the reader.
Answers 169 The important point here is that it is important to charge the correct premiums. If that is not done things will work out funny. That in turn leads to real life plans which do not allow a choice — workers have to insure, the amount is dictated (often capped, that is, the insured amount is a function of the wage up to a maximum.) You can see that such plans can be quite complicated and that it can be quite complicated to figure out who would want to do what, what the distributional implications are, etc. Question 13: Let us translate the question into notation: We are to show u0 (c1 ) c2 u00 (w)w that 0 = k if = λ if the function u(·) satisfies 0 = a, ∀w. u (c2 ) c1 u (w) c2 u0 (c1 ) = k and =λ 0 u (c2 ) c1
=⇒
u0 (c1 ) = k(λ)u0 (λc1 ).
If the left and right hand side of that last expression are identical functions, then their derivatives must equal: u00 (c1 ) = k(λ)λu00 (λc1 ), but we know that k(λ) = u0 (c1 )/u0 (λc1 ), so that u00 (c1 ) =
u0 (c1 ) λu00 (λc1 ) u0 (λc1 )
=⇒
u00 (c1 ) u00 (λc1 ) = λ u0 (c1 ) u0 (λc1 )
Thus the MRS is constant for any consumption ratio λ if u00 (c1 )c1 u00 (λc1 )λc1 = u0 (c1 ) u0 (λc1 )
∀λ,
which is constant relative risk aversion.
7.4
Chapter 6
Question 1: A 3-player game in extensive form comprises a game tree, Γ, a payoff vector of length three for each terminal node, a partition of the set of non-terminal nodes into player sets S0 , S1 , S2 , S3 , a partition of the player sets S1 , S2 , S3 into information sets. Further, a probability distribution for each node in S0 over the set of immediate followers and for each SiJ an index set IiJ and a 1-1 mapping from Iij to the set of immediate followers of the nodes in Sij . Any carefully labelled game tree diagram will do. It does not even have to have nature (i.e., S0 could be empty.) Question 2: Perfect recall is when each player never forgets any of his own previous moves (so that for any two nodes within an information set one
170 L-A. Busch, Microeconomics
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may not be a predecessor of the other and any two nodes may not have a common predecessor in another information set of that player such that the arc leading to the nodes differs) and never forgets information once known (so that any two nodes in a player’s information set may not have predecessors in distinct previous information sets of this player.) Counter examples as in the text, or any game which violates these requirements. Question 3: Yes, any finite game has a Nash equilibrium, possibly in mixed strategies. This follows from the Theorem we have in the text. The game therefore will also have a SPE (they are a subset of the Nash equilibria, but keep in mind that the condition of subgame perfection may have no ‘bite’, in which case we revert to Nash.) Question 4: Player 1 has no weakly dominated strategies since u1 (D, L, Lef t) > u1 (U, L, Lef t) but u1 (D, R, Lef t) < u1 (U, R, Lef t), while u1 (C, L, Right) > u1 (U, L, Right). Player 3 does also not have a weakly dominated strategy. Depending on the opponents’ moves he gets a higher payoff sometimes in the left and sometimes in the right matrix. Player 2 does have weakly dominated strategies: Both L and R are weakly dominated by C. This does not leave us with a good prediction yet, aside from the fact that 2 can be argued to play C. However, if we now consider repeated elimination we can narrow down the answer to what is also the unique Nash equilibrium in pure strategies in this case, (D, C, Right). To find mixed strategy Nash we assign probabilities to the strategies for players, so let µ1 = P r(U ), µ2 = P r(C), γ1 = P r(L), γ2 = P r(R), and α = P r(Lef t). We can then compute the payoffs for players for each of their pure strategies. So for example u1 (U, γ, α) = α(γ1 + 2γ2 + (1 − γ1 − γ2 ) + (1 − α)(2γ1 + 4γ2 + 2(1 − γ1 − γ2 )). We then can ask, when is player 1, say, actually willing to mix? Only if the payoff from the pure strategies in the support of the mixed strategy are equal, so that the player does not care. Question 5: This one is made easier by the fact that strategy R is (strictly) dominated, so that it will never be used in any mixed strategy equilibrium (or indeed any equilibrium.) Hence this is really just a 2 × 2 matrix we need to consider. Let α = P r(U ) and β = P r(L), so that P r(C) = 1 − α and P r(C) = 1−β. Then for player 1 to mix we require β+4(1−β) = 3β+2(1−β), hence 2 = 4β, and hence β = 0.5. So if player 2 mixes with this probability then player 1 is indifferent between his two strategies. Now look at player 2: For 2 to be indifferent between the two strategies L and C we require 4α + 2(1 − α) = 2α + 3(1 − α). Hence 3α = 1 and thus α = 1/3. Thus the
Answers 171 mixed strategy Nash equilibrium is ((1/3, 2/3), (1/2, 1/2)). Note that the game has no pure strategy Nash equilibria. Question 6: This is a two player game (the court is not a strategic player and does not receive any payoffs.) The most natural extensive form for such a situation is probably as in the game tree on the next page. railway e ¡@ ¡ @ ¡ @ ¡ @ low high ¡ @ ¡ @ ¡ @ ¡ @ @sNature Nature s¡ ¡@ ¡@ ¡ ¡ @ No Acc @ No Acc ¡ (1 − pl )@ ¡ (1 − ph@ ) @ @ @ @ Accident ¡pl Accident ¡ph ¡ ¡ (20, 10) (25, 10) ¡ ¡ ¡ ¡ ¡ ¡ s¡ town s¡ ¡@ ¡@ ¡ @ ¡ @ Sue ¡ Sue ¡ @ Not Sue @ Not Sue ¡ @ ¡ @ ¡ @ ¡ @ ¡ @ ¡ @ (10, 0)
(12, 2)
(5, 10)
(17, 2)
Here it is important to note that ph > pl , reflecting the fact that if low care is taken the accident probability is higher. I have arbitrarily assigned payoffs which satisfy the description. High level of care costs the railway 5, accidents impose a cost of 8 on both parties, legal costs are 2 for each party. Let us try and find the Nash equilibrium of this game. As an exercise let us first find the strategic form: RT high low
Sue
N otSue
(20 − 10pl , 10 − 10pl ) (20 − 8pl , 10 − 8pl ) (25 − 20ph , 10)
(25 − 8ph , 10 − 8ph )
Note that low strictly dominates high for the railway if 0.5 > (2ph − pl ) while high strictly dominates low if ph − pl > 5/8. In those cases the
172 L-A. Busch, Microeconomics
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Nash equilibria are (low, Sue) and (high, N otSue), respectively. Otherwise there will be a mixed strategy equilibrium. Let α be the probability with which the railway uses the high effort level. The town is indifferent iff its expected payoffs from the two strategies are the same, that is, if 10−10αpl = 10 − 8ph + 8α(ph − pl ). This is the case if α = 4ph /(4ph + pl ). For lower α it prefers to Sue, for higher α it prefers to N otSue. Letting β denote the probability with which the town sues, the railway expects to receive 20 − 8pl − 2βpl from high and 25 − 8ph − 12βph from low. It is indifferent if β = (5 − 8(ph − pl ))/(12ph − 2pl ). So the mixed strategy Nash equilibrium is ¶ µ 5 5 − 8(ph − pl ) 1 4ph if ph − > pl > 2ph − . , (α, β) = 4ph + pl 12ph − 2pl 8 2 Question 7: There are two ways to draw this game. We can have nature move first and then Romeo (who does not observe nature’s move.) Or we can have Romeo move first, and then nature determines if the move is seen. The game tree for the first case is as drawn below.
Nature e © © HH HH ©©
see ©© p
©©
s © Romeo © ¡@ @ M ¡ S @ ¡ @ ¡ J1¡ s @sJ 2 ¢A ¢A ¢ A ¢ A S¢ S¢ AM AM ¢ A ¢ A ¢ A ¢ A ¢ A ¢ A
(30,50)
(1,1)
(5,5)
(50,30)
1-p HH not see HH
HHs ¡@ @ M ¡ S¡ @ @ ¡ J3¡ s @s ¢A ¢A ¢ A ¢ A S¢ S¢ AM AM ¢ A ¢ A ¢ A ¢ A ¢ A ¢ A
(30,50)
(1,1)
(5,5)
(50,30)
A strategy vector in this game is (sR , (s1J , s2J , s3J )). Subgames start at information sets J 1 and J 2 , the only other subgame is the whole tree. In the subgame perfect equilibrium Juliet therefore is restricted to (S, M, ·). Let α denote Romeo’s probability of moving S, and β Juliet’s (in J 3 .) Romeo’s (expected) payoff from S is 30p + (1 − p)(30β + 1 − β) and his payoff from M is 50p + (1 − p)(5β + 50(1 − β)). The β for which he is indifferent is (49 − 29p)/(74(1 − p)). Note that this is increasing in p and that β = 1 if p = 5/9! Juliet has payoffs of 50α+5(1−α) and α+30(1−α) from moving S
Answers 173 and M , respectively, in J 3 . Hence she is indifferent if α = 25/74. Of course, pure strategy equilibria may also exist, and we get the SPE equilibria to be µ µ ¶¶ 25 49 − 29p 5 , S, M, , (S, (S, M, S)) , (M, (S, M, M )) if p < . 74 74(1 − p) 9 Note that (S, (S, M, S)) requires that 30 > 50p + 5(1 − p), or p < 5/9 also. (M, (S, M, M )) requires that 50 > 30p+(1−p), or p < 49/29, which is always true. What if p > 5/9? In that case the equilibrium in which the outcome is coordination on S (preferred by Juliet) does not exist, and neither does the mixed strategy equilibrium. Hence the unique equilibrium if p ≥ 5/9 is (M, (S, M, M )). Romeo can effectively insist on his preferred outcome. Question 8: Each firm will ( maxqi
(
X j6=i
qj + qi − 10)2 qi − 0qi
)
.
The FOC for this problem is X X 2( qj + qi − 10)qi + ( qj + qi − 10)2 = 0 j6=i
Hence if
P
j6=i qj
j6=i
+ qi − 10 6= 0 we require 2qi + (
X j6=i
qj + qi − 10) = 0
P and get the reaction function qi = (10 − j6=i qj )/3. P With identical firms we then know that in equilibrium qj = qi , so that j6=i qj = (n − 1)qi . Hence 3qi = 10 − (n − 1)qi and we get that qi ∗ = 10/(n + 2) ∀i. Total market output then is 10n/(n + 2). Note that total market output approaches 10 from below as n gets large. Market price for a given n is 400/(n + 2)2 , which approaches zero as n gets large. (Note that the marginal cost is zero and hence the perfectly competitive price is zero!) Question 9: In the first instance the sellers can only vary price. To clarify ideas, let us focus on two prices only (as would be needed for a separating equilibrium.) The game then is as depicted below. We are to show that no separating equilibrium exists. If it did, it would have to be the two prices as indicated, where one firm charges one price (presumably the high quality firm charging the higher price) the other another. But given that, the consumer knows (in equilibrium) which firm produced the product. It is easy to see that the low quality firm would deviate to the higher price (being then mistaken
174 L-A. Busch, Microeconomics (9 − p, p − 3) (0, −3) ´´
buy ´
´
´
s ´α
´ No
(6 − p, p − 2)
Q No Q Q
seller s
pˆ
β QQ
Root: eNature low
Q Qs
buy
p
seller
pˆ
Q
Q
(9 − pˆ, pˆ − 3)
No QQ
Q (0, −3)
buyer ´ No ´
(1/2) s
buy
s
high (1/2)
buyer (0, −2) Q Q
p
May2004
s´
´
(0, −2) ´´
´
buy
(6 − pˆ, pˆ − 2)
for the high quality firm so that the consumer buys) since costs are unaffected by such a move, but a higher price is received. At this point the remainder is non-trivial and left for summer study! The key is that the consumer now has an information set for each price-warranty pair, and that there are two nodes in it, one for each type of firm. Question 10: What was not stated in the question was the fact that each consumer buys either one or no units. Each buyer purchases a unit of the good if and only if the price is below the valuation of the buyer. Hence total market demand is given by the number of buyers with a valuation above p, or 1 − F (p). F (v) is the cumulative distribution for the uniform distribution on [0, 2].R Since the pdf for the uniform distribution on [0, 2] is 0.5, we have v F (v) = 0 0.5dt = 0.5v. Hence market demand is 1−0.5p and inverse market demand is 2(1 − Q). A Cournot equilibrium is nothing but a Nash equilibrium in the game in which firms simultaneously choose output levels. Hence firm 1 solves maxq1 {2(1 − q1 − q2 )q1 − q1 /10} which leads to FOC 2(1 − q1 − q2 ) − 2q1 − 1/10 = 0 and the reaction function q1 (q2 ) = 19/40 − q2 /2. Firm 2 solves ª © maxq2 2(1 − q1 − q2 )q2 − q22
which leads to FOC 2(1 − q1 − q2 ) − 2q2 − 2q2 = 0 and the reaction function q2 (q1 ) = 1/3 − q1 /3. The Nash equilibrium then is (q1 , q2 ) = (37/100, 21/100). Market price is 42/50. Profits for the two firms are 74 × 37/10000 for firm 1 and (82 × 21 − 212 )/10000 for firm 2, so that joint profit is (74 × 37 + 82 × 21 − 212 )/10000.
Answers 175 In the Stackelberg leader case we consider the SPE of the game in which firm 1 chooses output first and firm 2, after observing firm 1’s output choice, picks its output level. Firm 1, the Stackelberg leader, therefore takes firm 2’s reaction function as given. Thus firm 1 solves ½ µ µ ¶¶ ¾ 1 q1 − maxq1 2 1 − q1 − q1 − q1 /10 3 3 The FOC for this is 4(1 − 2q1 )/3 − 1/10 = 0 and hence q1 = 37/80. Thus q2 = 43/240. Market price is 86/240. Profits for the two firms are (62 × 37)/(240×80) and 43×43/2402 . Joint profit thus is (186×37+43×43)/2402 . Joint profit maximization would require that the firms solve ª © maxq1 ,q2 2(1 − q1 − q2 )(q1 + q2 ) − q1 /10 − q22 .
This has FOCs
2(1 − q1 − q2 ) − 2(q1 + q2 ) − 1/10 = 0 2(1 − q1 − q2 ) − 2(q1 + q2 ) − 2q2 = 0 so that we know that 2q2 = 1/10 or q2 = 1/20. Hence 2(1 − q1 − 1/20) − 2(q1 + 1/20) − 1/10 = 0 and 2 − 4q1 − 3/10 = 0 and q1 = 7/40. Market price then is 2(31/40). Joint profits are 2(31/40)(9/40) − 8/400. This cannot be attained as a Nash equilibrium because neither output level is on the firm’s reaction function, and only output levels on the reaction function are, by design, a best response.
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Course Notes by Lutz-Alexander Busch Dept.of Economics University of Waterloo
Revised 2004
c °Lutz-Alexander Busch, 1994,1995,1999,2001,2002,2004 Do not quote or redistribute without permission
Contents
1 Preliminaries
1
1.1
About Economists . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
What is (Micro—)Economics? . . . . . . . . . . . . . . .
2
1.3
Economics and Sex . . . . . . . . . . . . . . . . . . . . . . . .
5
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Review 2.1
2.2
9
Modelling Choice Behaviour
1
. . . . . . . . . . . . . . . . . .
9
2.1.1
Choice Rules . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2
Preferences . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3
What gives? . . . . . . . . . . . . . . . . . . . . . . . . 13
Consumer Theory . . . . . . . . . . . . . . . . . . . . . . . . . 15 Special Utility Functions . . . . . . . . . . . . . . . . . . . . . 18 2.2.1
Utility Maximization . . . . . . . . . . . . . . . . . . . 20
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 2.3 1
Expenditure Minimization and the Slutsky equation . . 25
General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 27
This material is based on Mas-Colell, Whinston, Green, chapter 1
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2.4
2.3.1
Pure Exchange . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2
A simple production economy . . . . . . . . . . . . . . 33
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 35
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39
The consumer’s problem . . . . . . . . . . . . . . . . . . . . . 39 3.1.1
Deriving the budget set
. . . . . . . . . . . . . . . . . 40
No Storage, No Investment, No Markets . . . . . . . . . . . . 40 Storage, No Investment, No Markets . . . . . . . . . . . . . . 40 Storage, Investment, No Markets . . . . . . . . . . . . . . . . 41 Storage, No Investment, Full Markets . . . . . . . . . . . . . . 42 Storage, Investment, Full Markets . . . . . . . . . . . . . . . . 43 3.1.2
Utility maximization . . . . . . . . . . . . . . . . . . . 44
3.2
Real Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3
Risk-free Assets . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4
3.3.1
Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2
More on Rates of Return . . . . . . . . . . . . . . . . . 49
3.3.3
Resource Depletion . . . . . . . . . . . . . . . . . . . . 51
3.3.4
A Short Digression into Financial Economics . . . . . . 52
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Uncertainty 4.1
57
Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.1
Comparing degrees of risk aversion . . . . . . . . . . . 65
Contents iii 4.2
Comparing gambles with respect to risk
4.3
A first look at Insurance . . . . . . . . . . . . . . . . . . . . . 69
4.4
The State-Preference Approach . . . . . . . . . . . . . . . . . 72
4.5
4.6
. . . . . . . . . . . . 67
4.4.1
Insurance in a State Model . . . . . . . . . . . . . . . . 74
4.4.2
Risk Aversion Again . . . . . . . . . . . . . . . . . . . 76
Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.5.1
Diversification . . . . . . . . . . . . . . . . . . . . . . . 77
4.5.2
Risk spreading . . . . . . . . . . . . . . . . . . . . . . 78
4.5.3
Back to Asset Pricing . . . . . . . . . . . . . . . . . . . 79
4.5.4
Mean-Variance Utility . . . . . . . . . . . . . . . . . . 81
4.5.5
CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Information
91
5.1
Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2
Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3
Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4
The Principal Agent Problem . . . . . . . . . . . . . . . . . . 105 5.4.1
The Abstract P-A Relationship . . . . . . . . . . . . . 105
6 Game Theory 6.1
113
Descriptions of Strategic Decision Problems . . . . . . . . . . 117 6.1.1
The Extensive Form . . . . . . . . . . . . . . . . . . . 117
6.1.2
Strategies and the Strategic Form . . . . . . . . . . . . 121
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Solution Concepts for Strategic Decision Problems . . . . . . . 126 6.2.1
Equilibrium Concepts for the Strategic Form . . . . . . 127
6.2.2
Equilibrium Refinements for the Strategic Form . . . . 131
6.2.3
Equilibrium Concepts and Refinements for the Extensive Form . . . . . . . . . . . . . . . . . . . . . . . . . 134
Signalling Games . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3
Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Review Question Answers
145
7.1
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
List of Figures 2.1
Consumer Optimum: The tangency condition . . . . . . . . . 22
2.2
Corner Solutions: Lack of tangency . . . . . . . . . . . . . . . 23
2.3
An Edgeworth Box . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4
Pareto Optimal Allocations . . . . . . . . . . . . . . . . . . . 30
3.1
Storage without and with markets, no (physical) investment . 44
4.1
Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2
Risk Neutral and Risk Loving . . . . . . . . . . . . . . . . . . 63
4.3
The certainty equivalent to a gamble . . . . . . . . . . . . . . 64
4.4
Comparing two gambles with equal expected value . . . . . . . 67
4.5
An Insurance Problem in State-Consumption space . . . . . . 75
4.6
Risk aversion in the State Model . . . . . . . . . . . . . . . . 76
4.7
Efficient Portfolios and the Market Portfolio . . . . . . . . . . 83
5.1
Insurance for Two types . . . . . . . . . . . . . . . . . . . . . 99
5.2
Impossibility of Pooling Equilibria . . . . . . . . . . . . . . . . 101
5.3
Separating Contracts could be possible . . . . . . . . . . . . . 102
5.4
Separating Contracts definitely impossible . . . . . . . . . . . 103
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6.1
Examples of valid and invalid trees . . . . . . . . . . . . . . . 118
6.2
A Game of Perfect Recall and a Counter-example . . . . . . . 120
6.3
From Incomplete to Imperfect Information . . . . . . . . . . . 121
6.4
A Matrix game — game in strategic form . . . . . . . . . . . . 124
6.5
A simple Bargaining Game . . . . . . . . . . . . . . . . . . . . 125
6.6
The 4 standard games . . . . . . . . . . . . . . . . . . . . . . 126
6.7
Valid and Invalid Subgames . . . . . . . . . . . . . . . . . . . 135
6.8
The “Horse” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.9
A Signalling Game . . . . . . . . . . . . . . . . . . . . . . . . 139
6.10 A Minor Perturbation? . . . . . . . . . . . . . . . . . . . . . . 140
Chapter 1 Preliminaries These notes have been written for Econ 401 as taught by me at the University of Waterloo. As such the list of topics reflects the course material for that particular course. It is assumed that the student has mastered the prerequisites and little or no time is spent on them, aside from a review of standard consumer economics and general equilibrium in chapter 1. I assume that the student has access to a standard undergraduate micro theory text book. Any of the books commonly used will do and will give introductions to the topics covered here, as well as allowing for a review, if necessary, of the material from the pre-requisites. These notes will not give references. The material covered is by now fairly standard and can be found in one form or another in most micro texts. I wish to acknowledge two books, however, which have served as references: the most excellent book by Mas-Colell, Whinston, and Green, Microeconomic Theory, as well as the much more concise Jehle and Reny, Advanced Microeconomic Theory. I also would like to acknowledge my teachers Don Ferguson and Glen MacDonald, who have done much to bring microeconomics alive for me. This preliminary chapter contains extensive quotes which I have found informative, amusing, interesting, and thought provoking. Their sources have been indicated.
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About Economists
By now you may have heard many jokes about economists and noticed that modern economics has a bad reputation in some circles. If you mention your field of study in the bar, you are quite possibly forced to defend yourself against various stereotypical charges (focusing on assumptions, mostly.) The most eloquent quotes about economists that I know of are the following two quotes reproduced from the Economist, Sept.4, 1993, p.25: No real Englishman, in his secret soul, was ever sorry for the death of a political economist, he is much more likely to be sorry for his life. You might as well cry at the death of a cormorant. Indeed how he can die is very odd. You would think a man who could digest all that arid matter; who really preferred ‘sawdust without butter’; who liked the tough subsistence of rigid formulae, might defy by intensity of internal constitution all stomachic or lesser diseases. However they do die, and people say that the dryness of the Sahara is caused by a deposit of similar bones. (Walter Bagehot (1855)) Are economists human? By overwhelming majority vote, the answer would undoubtedly be No. This is a matter of sorrow for them, for there is no body of men whose professional labours are more conscientiously, or consciously, directed to promoting the wealth and welfare of mankind. That they tend to be regarded as blue-nosed kill-joys must be the result of a great misunderstanding. (Geoffrey Crowther (1952))
1.2
What is (Micro—)Economics?
In Introductory Economics the question of what economics is has received some attention. Since then, however, this question may have received no further coverage, and so I thought to collect here some material which I to use to start a course. It is meant to provide a background for the field as well as a defense, of sorts, of the way in which micro economics is practiced. Malinvaud sees economics as follows:
Preliminaries 3 Here we propose the alternative, more explicit definition: economics is the science which studies how scarce resources are employed for the satisfaction of the needs of men living in society: on the one hand, it is interested in the essential operations of production, distribution and consumption of goods, and on the other hand, in the institutions and activities whose object it is to facilitate these operations. [..] The main object of the theory in which we are interested is the analysis of the simultaneous determination of prices and the quantities produced, exchanged and consumed. It is called microeconomics because, in its abstract formulations, it respects the individuality of each good and each agent. This seems a necessary condition a priori for logical investigation of the phenomena in question. By contrast, the rest of economic theory is in most cases macroeconomic, reasoning directly on the basis of aggregates of goods and agents. [E. Malinvaud, Lectures on Microeconomic Theory, revised, N-H, 1985, p.1-2.] This gives us a nice description of what economics is, and in particular what micro theory entails. In following the agenda laid out by Malinvaud a certain amount of theoretical abstraction and rigor have been found necessary, and one key critique heard often is the “attempt at overblown rigor” and the “unrealistic assumptions” which micro theory employs. Takayama and Hildenbrand both address these criticisms in the opening pages of their respective books. First Takayama: The essential feature of modern economic theory is that it is analytical and mathematical. Mathematics is a language that facilitates the honest presentation of a theory by making the assumptions explicit and by making each step of the logical deduction clear. Thus it provides a basis for further developments and extensions. Moreover, it provides the possibility for more accurate empirical testing. Not only are some assumptions hidden and obscured in the theories of the verbal and “curve-bending” economic schools, but their approaches provide no scope for accurate empirical testing, simply because such testing requires explicit and mathematical representations of the propositions of the theories to be tested. [..] But yet, economics is a complex subject and involves many things that cannot be expressed readily in terms of mathematics.
4 L-A. Busch, Microeconomics Commenting on Max Planck’s decision not to study economics, J.M. Keynes remarked that economics involves the “amalgam of logic and intuition and wide knowledge of facts, most of which are not precise.” In other words, economics is a combination of poetry and hard-boiled analysis accompanied by institutional facts. This does not imply, contrary to what many poets and institutionalists feel, that hard-boiled analysis is useless. Rather, it is the best way to express oneself honestly without being buried in the millions of institutional facts. [..] Mathematical economics is a field that is concerned with complete and hard-boiled analysis. The essence here is the method of analysis and not the resulting collection of theorems, for actual economies are far too complex to allow the ready application of these theorems. J.M. Keynes once remarked that “the theory of economics does not furnish a body of settled conclusions immediately applicable to policy. It is a method rather than a doctrine, an apparatus of the mind, a technique of thinking, which helps its possessor to draw conclusions.” An immediate corollary of this is that the theorems are useless without explicit recognition of the assumptions and complete understanding of the logic involved. It is important to get an intuitive understanding of the theorems (by means of diagrams and so on, if necessary), but this understanding is useless without a thorough knowledge of the assumptions and proofs. [Akira Takayama, Mathematical Economics, 2nd ed., Cambridge, 1985, p. xv.] Hildenbrand offers the following: I cannot refrain from repeating here the quotation from Bertrand Russell cited by F. Hahn in his inaugural lecture in Cambridge: “Many people have a passionate hatred of abstraction, chiefly, I think, because of its intellectual difficulty; but as they do not wish to give this reason they invent all sorts of other that sound grand. They say that all abstraction is falsification, and that as soon as you have left out any aspect of something actual you have exposed yourself to the risk of fallacy in arguing from its remaining aspects alone. Those who argue that way are in fact concerned with matters quite other than those that concern science.” (footnote 2, p.2, with reference)
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Preliminaries 5 Let me briefly recall the main characteristics of an axiomatic theory of a certain economic phenomenon as formulated by Debreu: First, the primitive concepts of the economic analysis are selected, and then, each one of these primitive concepts is represented by a mathematical object. Second, assumptions on the mathematical representations of the primitive concepts are made explicit and are fully specified. Mathematical analysis then establishes the consequences of these assumptions in the form of theorems. [Werner Hildenbrand, Twenty Papers of Gerard Debreu, Econometric Society Monograph 4, Cambridge, 1983, page 4, quoted with omissions.]
1.3
Economics and Sex
I close this chapter with the following thought provoking excerpt from Mark Perlman and Charles R. McCann, Jr., “Varieties of uncertainty,” in Uncertainty in Economic Thought, ed. Christian Schmidt, Edward Elgar 1996, p 9-10. The problem as perceived As this is an opening paper, let us begin with what was once an established cultural necessity, namely a reference to our religious heritage. What we have in mind is the Biblical story of the Fall of Man, the details of which we shall not bore you with. Rather, we open consideration of this difficult question by asking what was the point of that Book of Genesis story about the inadequacy of Man. We are told that apparently whatever were God’s expectations, He became disappointed with Man. Mankind and particularly Womankind1 did not live up to His expectations.2 In any case, Adam and Eve were informed 1 Much has been made of the failure of women, perhaps that is because men wrote up the history. We should add, in order to avoid deleterious political correctness (and thereby cut off provocation and discussion), that since Eve was the proximate cause of the Fall, and Eve represents sexual attraction or desire, some (particularly St Paul, whose opinion of womankind was problematic) have considered that sexual attraction was in some way even more responsible for the Fall than anything else. Put crudely, even if economics is not a sexy subject, its origins were sexual. 2 What that says about His omniscience and/or omnipotence is, at the very least, para-
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that they had ‘fallen’ from Grace, and all of us have been made to suffer ever since. From our analytical standpoint there are two crucial questions: 1. What was the sin; and 2. What was the punishment? The sin seems to have been something combining (1) an inability to follow precise directions; (2) a willingness to be tempted, particularly when one could assert that ‘one was only doing what everyone else (sic) was doing;3 (3) a greed involving things (something forbidden) and time (instant gratification); (4) an inability to leave well enough alone; and (5) an excessive Faustian curiosity. Naturally, as academic intellectuals, we fancy the fifth reason as best. But what interests us directly is the second question. It is ‘What was God’s punishment for Adam and Eve’s vicarious sin, for which all mankind suffers?’ Purportedly a distinction has been made between what happened to Man and Woman, but, the one clear answer, particularly as seen by Aquinas and by most economists ever since, was that man is condemned to live with the paradigm of scarcity of goods and services and with a schedule of appetites and incentives which are, at best, confusing. In the more modern terms of William Stanly Jevons, ours is a world of considerable pain and a few costly pleasures. We are driven to produce so that we can consume, and production is done mostly by the ‘sweat of the brow’ and the strength of the back. The study of economics — of the production, distribution and even the consumption of goods and services — it follows, is the result of the Original Sin. When Carlyle called Economics the ‘Dismal Science’, he was, if anything, writing in euphemisms; Economics per se, is the Punishment for Sin. doxical. 3 Cf. Genesis, 3:9-12,16,17. [9] But the Lord God called to the man and said to him, ‘ Where are you?’ [10] He replied, ‘I heard the sound as you were walking the garden , and I was afraid because I was naked, and I hid myself.’ [11] God answered, ‘ Who told you that you were naked? Have you eaten from the tree which I forbade you?’ [12] The man said, ‘The woman you gave me for a companion, she gave me fruit from the tree and I ate.’ [Note: The story, as recalled, suggests that Adam was dependent upon Eve (for what?), and the price of that dependency was to be agreeable to Eve (‘It was really all her fault — I only did what You [God] had laid out for me.’)] ([16] and [17] omitted) [Again, for those civil libertarians amongst us, kindly note that God forced Adam to testify against himself. Who says that the Bill of Rights is an inherent aspect of divine justice? Far from it, in the Last Judgment, pleading the Fifth won’t do at all.]
Preliminaries 7 But, it is another line of analysis, perhaps novel, which we put to you. Scarcity, as the paradigm, may not have been the greatest punishment, because scarcity, as such, can usually be overcome. Scarcity simply means that one has to allocate between one’s preferences, and the thinking man ought to be able to handle the situation. We use our reasoning power, surely tied up with Free Will, to allocate priorities and thereby overcome the greater disasters of scarcity. What was the greater punishment, indeed the greatest punishment, is more basic. Insofar as we are aware, it was identified early on by another Aristotelian, one writing shortly before Aquinas, Moses Maimonides. Maimonides suggested that God’s real punishment was to push man admittedly beyond the limits of his reasoning power. Maimonides held that prior to the Fall, Adam and Eve (and presumably mankind, generally) knew everything concerning them; after the Fall they only had opinions.4 Requisite to the wise use of power is understanding and full specification; what was lost was any such claim previously held by man to complete knowledge and the full comprehension of his surroundings. In other words, what truly underlies the misery of scarcity is neither hunger nor thirst, but the lack of knowledge of what one’s preference schedule will do to one’s happiness. For if one had complete knowledge (including foreknowledge) one could compensate accordingly. If one pursues Maimonides’ line of inquiry, it seems that uncertainty (which is based not on ignorance of what can be known with study of data collection, but also on ignorance tied to the unknowable) is the real punishment.
4
[omitted]
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Notation: Blackboard versions of these symbols may differ slightly.
I will not distinguish between vectors and scalars by notation. Generally all small variables (x, y, p) are column vectors (even if written in the notes as row vectors to save space.) The context should clarify the usage. Capital letters most often denote sets, as in the consumption set X, or budget set B. Sets of sets are denoted by p capital P 2script letters, such as X = {X1 , X2 , . . . , Xk }, where Xi = {x ∈ 1 + r. Then what I should do is to borrow money at the interest rate r, say I dollars, and use those funds to buy the good in question, i.e., purchase I/p0 units. Then wait and sell the goods in the next period. That will yield Ip1 /p0 dollars. I also have to pay back my loan, at
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(1 + r)I dollars, and thus I have a profit of p1 /p0 − (1 + r) per dollar of loan. Note that the optimal loan size would be infinite. However, the resulting large demand would certainly drive up current prices (while also lowering future prices, since everybody expects a flood of the stuff tomorrow), and this serves to reduce the profitability of the exercise. In a zero-arbitrage equilibrium we p1 therefore must have (1 + r) = p1 /p0 , or, more tellingly, p0 = . The 1+r correct market price for an asset is its discounted future value! This discussion has an application to the debate about pricing during supply or demand shocks. For example, gasoline prices during the Gulf war, or the alleged price-gouging in the ice-storm areas of Quebec and Ontario: What should the price of an item be which is already in stock? Many people argue that it is unfair to charge a higher price for in-stock items. Only the replacement items, procured at higher cost, should be sold at the higher cost. While this may be “ethical” according to some, it is easily demonstrated to violate the above rule: The price of the good tomorrow will be determined by demand and supply tomorrow, and apparently all are agreed that that price might well be higher due to a large shift out in the demand and/or reduction in supply. Currently I own that good, and have therefore the choice of selling it tomorrow or selling it today. I would want to obtain the appropriate rate of return on the asset, which has to be equal between the two options. Thus I am only willing to part with it now if I am offered a higher price which foreshadows tomorrows higher price. Should I be forced not to do so I am forced to give money away against my will and better judgment. This would normally be considered unethical by most (just try and force them to give you money.) Of course, assets are not usually all the same, and we will see this later when we introduce uncertainty. For example, a house worth $100,000 and $100,000 cash are not equivalent, since the cash is immediately usable, while the house may take a while to sell — it is less “liquid.” The same is true for thinly traded stocks. Such assets may carry a liquidity premium — an illiquidity punishment, really — and will have a higher rate of return in order to compensate for the potential costs and problems in unloading them. This can, of course, be treated in terms of risk, since the realization of the house’s value is a random variable, at least in time, if not in the amount. Of course, there are other kinds of risk as well, and in general the future price of the asset is not known. (Note that bonds are an exception to some degree. If you choose to hold the bond all the way to the maturity date you do know the precise stream of payments. If you sell early, you face the uncertain sale price which depends on the interest rate at that point in time.)
Inter-temporal 51 Assets may also yield consumption returns while you hold them: a car or house are examples, as are dividend payments of stocks or interest payments of bonds. For one period this is still simple to deal with: The asset will generate benefit (say rent saved, or train tickets saved) of b and we thus p1 − p 0 + b compute the rate of return as . If the consumer holds multiple p0 assets in equilibrium, then we again require that this be equal to the rate of return on other assets. Complicating things in the real world is the fact that assets often differ in their tax treatment. For example, if the house is a principal residence any capital gains (the tax man’s term for p1 − p0 , and to add insult to injury they ignore inflation) are tax free. For another asset, say a painting, this is not true. Equilibrium requires, of course, that the rates of return as perceived by the consumer are equalized, and thus we may have to use an after tax rate for one asset and set it equal to an untaxed rate for another.
3.3.3
Resource Depletion
The simple discounting rules above can also be applied to gain some first insights into resource economics. We can analyse the question of simple resource depletion: at what rate should we use up a non-renewable resource. We can also analyse when a tree (or forest) should to be cut down. Assume a non-renewable resource currently available at quantity S. For simplicity, first assume a fixed annual demand D. It follows that there are S/D years left, after which we assume that an alternative has to be used which costs C. Thus the price in the last year should be pS/D = C. Arbitrage implies that pt+1 = (1 + r)pt , so that p0 = C/(1 + r)S/D . Note that additional discoveries of supplies lower the price since they increase the time to depletion, as do reductions in demand. Lowering the price of the alternative also lowers the current price. Finally, increases in the discount rate lower the price. This approach has a major flaw, however. It assumes demand and supply to be independent of price. So instead, let us assume some current price p0 as a starting value and let us focus on supply. When will the owner of the resource be willing to sell? If the market rate of return on other assets is r then the resource, which is just another asset, will also have to generate that rate of return. Therefore p1 = (1 + r)p0 , and in general we’d have to expect pt = (1 + r)t p0 . Note that the price of the resource is therefore increasing with time, which, in general equilibrium, means two things: demand will
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fall as customers switch to alternatives, and substitutes will become more competitive. Furthermore we might expect more substitutes to be developed. We will ultimately run out of the resource, but it is nearly always wrong to simply use a linear projection of current use patterns. This fact has been established over and over with various natural resources such as oil, tin, copper, titanium, etc. What about renewable resources? Consider first the ‘European’ model of privately owned land for timber production as an example. Here we have a company who owns an asset — a forest — which it intends to manage in order to maximize the present value of current and future profits. When should it harvest the trees? Each year there is the decision to harvest the tree or not. If it is cut it generates revenue right away. If it continues to grow it will not generate this revenue but instead generate more revenue tomorrow (since it is growing and there will be more timber tomorrow.) It follows that the two rates of return should be equalized, that is, the tree should be cut once its growth rate divided by its current size has slowed to the market interest rate. This fact has a few implications for forestry: Faster growing trees are a better investment, and thus we see mostly fast growing species replanted, instead of, say, oaks, which grow only slowly. (This discussion is ceteris paribus — ignoring general equilibrium effects.) Furthermore, what if you don’t own the trees? What if you are the James Bond of forestry, with a (time-limited) license to kill? In that case you will simply cut the trees down either immediately or before the end of your license, depending on the growth rate. Of course, in Canada most licenses are for mature forests, which nearly by definition have slow or no growth — thus the thing to do is to clear cut and get out of there. The Europeans, critical of clear-cutting, forget that they have long ago cut nearly all of their mature forests and are now in a harvesting model with mostly high growth forests. As a final note, notice that lack of ownership will also impact the replanting decision. As we will see later in the course, if we treat the logger as an agent of the state, the state has serious incentive problems to overcome within this principal agent framework.
3.3.4
A Short Digression into Financial Economics
I thought it might be useful to provide you with a short refresher or introduction to multi-period present value and compound interest computations. For starters, assume you put $1 in the bank at 5% interest, computed yearly, and that all interest income is also reinvested at this 5% rate. How much
Inter-temporal 53 money will you have in each of the following years? The answer is 1.05, 1.052 , 1.053 , . . . 1.05t . The important fact about this is that a simple interest rate and a compounded interest rate are not the same, since with compounding there is interest on interest. For example, if you get a loan at 12%, it matters how often this is compounded. Let us assume it is just simple interest; You then owe $1.12 for every dollar you borrowed at the end of one year. What you will quickly find out is that banks don’t normally do that. They at least compound semiannually, and normally monthly. Monthly compounding would mean that )12 = 1.1268. On a million dollar loan this would be you will owe (1 + .12 12 a difference of $6825.03. In other words, you are really paying not a 12% interest rate but a 12.6825% simple interest rate. It is therefore very important to be sure to know what interest rate applies and how compounding is applied (semi-annual, monthly, etc.?) Here is a handy little device used in many circles: the rule of 72, sometimes also referred to as the rule of 69. It is used to find out how long it will take to double your money at any given interest rate. The idea is that it will approximately take 72/r periods to double your money at an interest rate of r percent. The proof is simple: we want to solve for the t for which ¶t ¶ µ µ r% r% = ln2. = 2 ⇒ tln 1 + 1+ 100 100 However, for small x we know that ln(1 + x) ∼ x, thus t
100ln2 69.3147 r% ∼ ln2 ⇒ t ∼ = 100 r% r%
but of course 72 has more divisors and is much easier to work with. The power of compounding also comes into play with mortgages or other installment loans. A mortgage is a promise to pay every period for a specified length (typically 25 years, i.e., 300 months) a certain payment p. This is also known as a simple annuity. What is the value of such a promise, i.e., its present value? We need to compute the value of the following sum: δp + δ 2 p + δ 3 p + . . . + δ n p. Here δ = 1/(1 + r), where r is the interest rate we use per period. Thus ¡ ¢ PV = δ p + δp + δ 2 p + . . . + δ n−1 p ¡ ¢ = δp 1 + δ + δ 2 + . . . + δ n−1
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1 − δn 1−δ 1 − (1 + r)−n PV = p r = δp
(Recall in the above derivation that for δ < 1 we have
P∞
i=1
δ i = 1/(1 − δ).)
The above equation relates four variables: the principal amount, the payment amount, the payment periods, and the interest rate. If you fix any three this allows you to derive the fourth after only a little bit of manipulation. A final note: In Canada a mortgage can be at most compounded semi-annually. Thus the effective interest rate per month is derived by solving (1 + r/2)2 = (1 + rm )12 . If you are quoted a 12% interest rate per year the monthly rate is therefore (1.06)1/6 − 1 = 0.975879418%. The effective yearly interest rate in turn is (1.06)2 − 1 = 12.36%, and by law the bank is supposed to tell you about that too. Given the above, and the fact that nearly all mortgages are computed for a 25 year term (but seldom run longer than 5 years, these days), the monthly payment at a 10% yearly interest rate for an additional $1000 on the mortgage is $8.95. Before you engage in mortgages it would be a good idea to program your spreadsheet with these formulas and convince yourself how bi-weekly payments reduce the total interest you pay, how important a couple of percentage points off the interest rate are to your monthly budget, etc.
3.4
Review Problems
Question 1: There are three time periods and one consumption good. The consumer’s endowments are 4 units in the first period, 20 units in the second, and 1 unit in the third. The money price for the consumption good is known to be p = 1 in all periods (no inflation.) Let rij denote the (simple, nominal) interest rate from period i to j. a) State the restrictions on r12 , r23 and r13 implied by zero arbitrage. b) Write down the consumer’s budget constraint assuming the restriction in (a) holds. Explain why it is useful to have this condition hold (i.e., point out what would cause a potential problem in how you’ve written the budget if the condition in (a) fails. c) Draw a diagrammatic representation of the budget constraint in periods 2 and 3, being careful to note how period 1 consumption influences this diagram. Question 2: There are two goods, consumption today and tomorrow. Joe
Inter-temporal 55 has an initial endowment of (100, 100). There exists a credit market which allows him to borrow or lend against his initial endowment at market interest rates of 0%. A borrowing constraint exists which prevents him from borrowing against more than 60% of his period 2 endowment. Joe also possesses an investment technology which is characterized by a production function √ x2 = 10 x1 . That is, an investment of x1 units in period 1 will lead to x2 units in period 2. a) What is Joe’s budget constraint? A very clearly drawn and well labelled diagram suffices, or you can give it mathematically. Also give a short explanatory paragraph how the set is derived. b) Suppose that Joe’s preferences can be represented by the function U (c1 , c2 ) = exp(c41 c62 ). (Here exp() denotes the exponential function.) What is Joe’s final consumption bundle, how much does he invest, and what are his transactions in the credit market. Question 3: Anna has preferences over her consumption levels in two periods which can be represented by the utility function ¶ ¾ ½ µ 13 23 12 c1 + c 2 , c1 + c 2 . u(c1 , c2 ) = min 22 10 10 a) Draw a carefully labelled representation of her indifference curve map. b) What is her utility maximizing consumption bundle if her initial endowment is (9, 8) and the interest rate is 25%. c) What is her utility maximizing consumption bundle if her initial endowment is (5, 12) and the interest rate is 25%. d) Assume she can lend money at 22% and borrow at 28%. What would her endowment have to be for her to be a lender, a borrower? e) Assume she can lend money at 18% and borrow at 32%. Would Anna ever trade at all? (Explain.) Question 4: Alice has preferences over consumption in two periods represented by the utility function uA (c1 , c2 ) = lnc1 + αlnc2 , and an endowment of (12, 6). Bob has preferences over consumption in two periods represented by the utility function uB (c1 , c2 ) = c1 + βc2 , and an endowment of (8, 4). a) Draw an appropriately labelled representation of this exchange economy in order to “prime” your intuition. (Indicate the indifference maps and the Contract Curve.) b) Assuming, of course, that both α and β lie strictly between zero and one, what is the equilibrium interest rate and allocation?
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Chapter 4 Uncertainty So far, it has been assumed that consumers would know precisely what they were buying and getting. In real life, however, it is often the case that an action does not lead to a definite outcome, but instead to one of many possible outcomes. Which of these occurs is outside the control of the decision maker. It is determined by what is referred to as “nature.” These situations are ones of uncertainty — it is uncertain what happens. Often, however, the probabilities of the different possibilities are known from past experience, or can be estimated in some other way, or indeed are assumed based on some personal (subjective) judgment. Economists then speak of risk. Note that our “normal” model is already handling such cases if we take it at its most general level: commodities in the model were supposed to be fully specified, and could, in principle, be state contingent. We will develop that interpretation further later on in this chapter. First, however, we will develop a more simple model which is designed to bring the role of probabilities to the fore. One of the key facts about situations involving risk/uncertainty is that the consumer’s wellbeing does not only depend on the various possible outcomes, and which occurs in the end, but also on how likely each outcome is. The standard model of chapter 2 does not allow an explicit role for such probabilities. They are somehow embedded in the utility function and prices. In order to compare situations which differ only in the probabilities, for example, it would be nice to have probabilities explicitly in the model formulation. A particularly simple model that does this holds the outcomes fixed, they will all be assumed to lie in some set of alternatives X, and focuses on the different probabilities with which they occur. We call such a list 57
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of the probabilities for each outcome a lottery. Definition 1 A simple lottery is a list L = (p1P , p2 , . . . , pN ) of probabilities for the N different outcomes in X, with pi ≥ 0, N i=1 pi = 1. If we have a suitably defined continuous space of outcomes, for example 0. We are allowed to scale the utility index and to change its slope, but we are not allowed to change its curvature. The ˆ reason for this should be clear. Suppose we compare two lotteries, L and L,
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which differ only in that probability is shifted between outcomes k and j and ˆ so: outcomes m and n. Suppose U (L) > U (L), U (L) = n X i=1
pi u(xi ) −
n X
i=1 n X
ˆ = pi u(xi ) > U (L)
n X i=1
pˆi u(xi ) > 0
i=1
(pk − pˆk )u(xk ) + (pj − pˆj )u(xj )(pm − pˆm )u(xm ) + (pn − pˆn )u(xn ) > 0 (pk − pˆk )(u(xk ) − u(xj )) + (pm − pˆm )(u(xm ) − u(xn )) > 0 This comparison clearly depends on both, the differences in probabilities as well as the differences in the utility indices of the outcomes. If we multiply u(·) by a constant, it will factor out of the last line above. If, however, we were to transform the function u(·), even by a monotonic transformation, we would change the difference between outcome utilities, and this could change the above comparison. In fact, as we shall see later, the curvature of the Bernoulli utility index u(·) is crucial in determining the consumer’s behaviour with respect to risk, and will be used to measure the consumer’s risk aversion. Before we proceed to that, some famous paradoxes relating to uncertainty and our assumptions. Allais Paradox: The Allais paradox shows that consumers may not satisfy the axioms we had assumed. It considers the following case: Consider a space of outcomes for a lottery given by C = (25, 5, 0) in hundred thousands of dollars. Subjects are then asked which of two lotteries they would prefer, LA = (0, 1, 0) or LB = (.1, .89, .01). Often consumers will indicate a preference for LA , probably because they foresee that they would regret to have been greedy if they end up with nothing under lottery B. On the other hand, if they are asked to choose between LC = (0, .11, .89) or LD = (.1, 0, .9) the same consumers often indicate a preference for lottery D. Note that there is little regret possible here, you simply get a lot larger winning in exchange for a slightly lower probability of winning under D. These choices, however, violate our assumptions. This is easily checked by assuming the existence of some u(·): The preference for A over B then indicates that u(5) > .1u(25) + .89u(5) + .01u(0) .11u(5) > .1u(25) + .01u(0) .11u(5) + .89u(0) > .1u(25) + .9u(0)
pˆi u(xi )
Uncertainty 61 and the last line indicates that lottery C is preferred to D! Ellsberg Paradox: This paradox shows that consumers may not be consistent in their assessment of uncertainty. Consider an urn with 300 balls in it, of which precisely 100 are known to be red. The other 200 are blue or green in an unknown proportion (note that this is uncertainty: there is no information as to the proportion available.) The consumer is again offered the choice between two pairs of gambles: ½ LA : $1000 if a drawn ball is red. Choice 1 = LB : $1000 if a drawn ball is blue. ½ LC : $1000 if a drawn ball is NOT red. Choice 2 : = LD : $1000 if a drawn ball is NOT blue. Often consumers faced with these two choices will choose A over B and will choose C over D. However, letting u(0) be zero for simplicity, this means that p(R)u(1000) > p(B)u(1000) ⇒ p(R) > p(B) ⇒ (1 − p(R)) < (1 − p(B)) ⇒ p(¬R) < p(¬B) ⇒ p(¬R)u(1000) < p(¬B)u(1000). Thus the consumer should prefer D to C if choice were consistent. Other problems with expected utility also exist. One is the intimate relation of risk aversion and time preference which is imposed by these preferences. There consequently is a fairly active literature which attempts to find a superior model for choice under uncertainty. These attempts mostly come at the expense of much higher mathematical requirements, and many still only address one or the other specific problem, so that they too are easily ‘refuted’ by a properly chosen experiment.
4.1
Risk Aversion
We will now restrict the outcome space X to be one-dimensional. In particular, assume that X is simply the wealth/total consumption of the consumer in each outcome. With this simplification, the basic attitudes of a consumer concerning risk can be obtained by comparing two different lotteries: one that gives an outcome for certain (a degenerate lottery), and another that has the same expected value, but is non-degenerate. So, let L be a lottery ¯ given by the probability density f (x). It generates an expected on [0, X] R X¯ value of wealth of 0 xf (x)dx = C. We can now compare the consumer’s utility from obtaining C for certain, and that from the lottery L (which has
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expected wealth C.) Compare U (L) =
Z
¯ X
u(x)f (x)dx to u 0
ÃZ
¯ X
!
xf (x)dx . 0
Definition 2 A risk-averse consumer is one for whom the expected utility of any lottery is lower than the utility of the expected value of that lottery: ! ÃZ ¯ Z ¯ X
X
xf (x)dx .
u(x)f (x)dx < u
0
0
Utility u(w2)
3
♦
2.5
u(E(w)) 2 ♦ E(u()) ♦ 1.5 u(w1)
u(w)
♦ ♦ ♦
1 0.5
♦
0 0
w1
♦
♦
5 E(w)
10
w2
15
20 wealth
Figure 4.1: Risk Aversion The astute reader may notice that this is Jensen’s inequality, which is one way to define a concave function, in this case u(·) (see Fig. 4.1.) This is also the reason why only affine transformations were allowed for expected utility functions. Any other transformation would affect the curvature of the Bernoulli utility function u(·), and thus would change the risk-aversion of the consumer. Clearly, consumers with different risk aversion do not have the same preferences, however.2 Note that a concave u(·) has a diminishing marginal utility of wealth, an assumption which is quite familiar from introductory courses. Risk aversion therefore implies (and is implied by) the fact 2
To belabour the point, consider preferences over wealth represented by u(w) = w. In the standard framework√of chapter 1 positive monotonic transformations are ok, so that the functions w 2 and w both represent identical preferences. It is easy to verify that these two functions lead to a quite different relationship between the expected utility and the utility of the expected wealth than the initial one, however. Thus they cannot represent the same preferences in a setting of uncertainty/risk.
Uncertainty 63 U 14 12 10 E(u())= 8 u(E(w)) 6 ♦ 4 u(w1) 2 ♦ 0 ♦ 0 2 4 w1 u(w2)
♦ ♦ ♦
♦
6 8 10 12 14 E(w) w2
U 120 100 80 u(w2) 60 E(u()) 40 ♦ u(E(w)) 20 ♦ u(w1) 0 w 0
u(w)
♦ ♦ ♦ ♦ 1 w1
♦ 2
♦
3 4 E(w) w2
5w
Figure 4.2: Risk Neutral and Risk Loving that additional units of wealth provide additional utility, but at a decreasing rate. Of course, consumers do not have to be risk-averse. Risk neutral and risk loving are defined in the obvious way: The first requires that ¶ µZ Z u(x)f (x)dx = u
xf (x)dx .
while the second requires ¶ µZ Z xf (x)dx . u(x)f (x)dx > u
There is a nice diagrammatic representation of these available if we consider only two possible outcomes (Fig. 4.2). There are two other ways in which we might define risk aversion, and both reveal interesting facts about the consumer’s economic behaviour. The first is by using the concept of a certainty equivalent. It is the answer to the question “how much wealth, received for certain, is equivalent (in the consumer’s eyes, according to preferences) to a given gamble/lottery?” In other words: Definition 3 The certainty equivalent C(f, u) for a lottery with probability distribution f (·) under the (Bernoulli) utility function u(·) is defined by the equation Z u (C(f, u)) =
u(x)f (x)dx.
Again a diagram for the two-outcome case might help (Fig. 4.3).
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3
♦
2.5
u(w)
2 E(u()) u(w1)
1.5
♦
♦
♦
♦
♦
♦
1 0.5
♦
0 0
w1 C()5 E(w)
♦ 10
w2
15
20 wealth
Figure 4.3: The certainty equivalent to a gamble A risk averse consumer is one for whom the certainty equivalent of any gamble is less than the expected value of that gamble. One useful economic interpretation of this fact is that the consumer is willing to pay (give up expected wealth) in order to avoid having to face the gamble. InRdeed, the maximum amount which the consumer would pay is the difference wf (w)dw − C(f, u). This observation basically underlies the whole insurance industry: risk-averse consumers are willing to pay in order to avoid risk. A well diversified insurance company will be risk neutral, however, and therefore is willing to provide insurance (assume the risk) as long as it guarantees the consumer not more than the expected value of the gamble: Thus there is room to trade, and insurance will be offered. (More on that later.) Another way to look at risk aversion is to ask the following question: If I were to offer a gamble to the consumer which would lead either to a win of ² or a loss of ², how much more than fair odds do I have to offer so that the consumer will take the bet? Note that a fair gamble would have an expected value of zero (i.e., 50/50 odds), and thus would be rejected by the (risk averse) consumer for sure. This idea leads to the concept of a probability premium. Definition 4 The probability premium π(u, ², w) is defined by u(w) = (0.5 + π(·)) u(w + ²) + (0.5 − π(·)) u(w − ²). A risk-averse consumer has a positive probability premium, indicating that the consumer requires more than fair odds in order to accept a gamble.
Uncertainty 65 It can be shown that all three concepts are equivalent, that is, a consumer with preferences that have a positive probability premium will be one for whom the certainty equivalent is less than the expected value of wealth and for whom the expected utility is less than the utility of the expectation. This is reassuring, since the certainty equivalent basically considers a consumer with a property right to a gamble, and asks what it would take for him to trade to a certain wealth level, while the probability premium considers a consumer with a property right to a fixed wealth, and asks what it would take for a gamble to be accepted.
4.1.1
Comparing degrees of risk aversion
One question we can now try to address is to see which consumer is more risk averse. Since risk aversion apparently had to do with the concavity of the (Bernoulli) utility function it would appear logical to attempt to measure its concavity. This is indeed what Arrow and Pratt have done. However, simply using the second derivative of u(·), which after all measures curvature, will not be such a good idea. The reason is that the second derivative will depend on the units in which wealth and utility are measured.3 Arrow and Pratt have proposed two measures which largely avoid this problem: Definition 5 The Arrow-Pratt measure of (absolute) risk aversion is u00 (w) rA = − 0 . u (w) The Arrow-Pratt measure of relative risk aversion is u00 (w)w . rR = − 0 u (w) Note that the first of these in effect measures risk aversion with respect to a fixed amount of gamble (say, $1). The latter, in contrast, measures risk aversion for a gamble over a fixed percentage of wealth. These points can be demonstrated as follows: Consider a consumer with initial wealth w who is faced with a small fair bet, i.e., a gain or loss of some small amount ² with equal probability. 3
You can easily verify this by thinking of the units attached to the second derivative. If the first derivative measures change in utility for change in wealth, then its units must be u/w, while the second derivative is like a rate of acceleration. Its units are u/w 2 .
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How much would the consumer be willing to pay in order to avoid this bet? Denoting this payment by I we need to consider (note that w − I is the certainty equivalent) 0.5u(w + ²) + 0.5u(w − ²) = u(w − I). Use a Taylor series expansion in order to approximate both sides: 0.5(u(w) + ²u0 (w) + 0.5²2 u00 (w)) + 0.5(u(w) − ²u0 (w) + 0.5²2 u00 (w)) ∼ u(w) − Iu0 (w) . Collecting terms and simplifying gives us 0.5²2 u00 (w)) ∼ −Iu0 (w)
⇒
I∼
²2 −u00 (w) × 0 . 2 u (w)
Thus the required payment is proportional to the absolute coefficient of risk aversion (and the dollar amount of the gamble.) On the other hand, u00 w du0 w du0 /u0 %∆u0 = = ∼ . u0 dw u0 dw/w %∆w Thus the relative coefficient of risk-aversion is nothing but the elasticity of marginal utility with respect to wealth. That is, it measures the responsiveness of the marginal utility to wealth changes. Comparing across consumers, a consumer is said to be more risk averse than another if (either) Arrow-Pratt coefficient of risk aversion is larger. This is equivalent to saying that he has a lower certainty equivalent for any given gamble, or requires a higher probability premium. We can also compare the risk aversion of a given consumer for different wealth levels. That is, we can compute these measures for the same u(·) but different initial wealth. After all, rA is a function of w. It is commonly assumed that consumers have (absolute) risk aversion which is decreasing with wealth. Sometimes the stronger assumption of decreasing relative risk aversion is made, however. Note that a constant absolute risk aversion implies increasing relative risk aversion. Finally, note also that the only functional form for u(·) which has constant absolute risk aversion is u(w) = −e(−aw) . You may wish to verify that a consumer exhibiting decreasing absolute risk aversion will have a decreasing difference between initial wealth and the certainty equivalent (a declining maximum price paid for insurance) on the one hand, and a decreasing probability premium on the other.
Uncertainty 67 Utility
3 2.5
♦
2 E(u(3,4)) ♦ E(u(1,2)) ♦ 1.5
♦
♦
♦ ♦
♦
♦
♦
♦
♦
w1
5 w3 E(w)
10 w4
w2
u(w)
♦
1 0.5 0 0
15
20 wealth
Figure 4.4: Comparing two gambles with equal expected value
4.2
Comparing gambles with respect to risk
Another type of comparison of interest is not across consumers or wealth levels, as above, but across different gambles. Faced with two gambles, when do we want to say that one is riskier than the other? We could try to approach this question with purely statistical measures, such as comparisons of the various moments of the two lotteries’ distributions. This has the major problem, however, that the consumer may in general be expected to be willing to trade off a higher expected return for higher variance, say. Because of this, a definition based directly on consumer preferences is preferable. Two such measures are commonly employed in economics, first and second order stochastic dominance. Let us first focus on lotteries with the same expected value. For example, consider the two gambles depicted in Fig. 4.4. The first is a gamble over w1 and w2 . The second is a gamble over w3 and w4 . Both have an identical expected value of E(w). Nevertheless a risk averse consumer clearly will prefer the second to the first, as inspection of the diagram verifies. Note that in Fig. 4.4 E(w) − w1 > E(w) − w3 and w2 − E(w) > w4 − E(w). This clearly indicates that the second lottery has a lower variance, and thus that a risk averse consumer prefers to have less variability for a given mean. With multiple possible outcomes the question is not so simple anymore, however. One could construct an example with two lotteries that have the same mean and variance, but which differ in higher moments. What are the “obvious” preferences of a risk averse consumer about skurtosis, say?
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This has lead to a more general definition for comparing distributions which have the same mean: Definition 6 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). Let F (·) have the same mean as G(·). F (·) is said to dominate G(·) according to second order stochastic dominance if for every non-decreasing concave u(x): Z Z u(x)dF (x) ≥ u(x)dG(x) In words, a distribution second order stochastically dominates another if they have the same mean and if the first is preferred by all risk-averse consumers. This definition has economic appeal in its simplicity, but is one of those definitions that are problematic to work with due to the condition that for all possible concave functions something is true. In order to apply this definition easily we need to find other tests. Lemma 1 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). F (·) dominates G(·) according to second order stochastic dominance if Z Z Z x Z x tg(t)dt = tf (t)dt, and G(t)dt ≥ F (t)dt, ∀x. 0
0
I.e., if they have the same mean and there is more area under the cdf G(·) than under the cdf F (·) at any point of the distribution.4 A concept related to second order stochastic dominance is that of a mean preserving spread. Indeed it can be shown that the two are equivalent. Definition 7 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). G(·) is a mean preserving spread compared to F (·) if x is distributed according to F (·) and G(·) is the distribution of the R random variable x + z, where z is distributed according to some H(·) with zdH(z) = 0. 4
Note that the condition of identical means also implies a restriction on the total Rx Rx areas below the cumulative distributions. After all, x tdF (t) = [tF (t)]xx − x F (t)dt = Rx x − x F (t)dt.
Uncertainty 69 The above gives us an easy way to construct a second order stochastically dominated distribution: Simply add a zero mean random variable to the given one. While it is nice to be able to rank distributions in this manner, the condition of equal means is restrictive. Furthermore, it does not allow us to address the economically interesting question of what the trade off between mean and risk may be. The following concept is frequently employed in economics to deal with such situations. Definition 8 Let F (x) and G(x) be two cumulative distribution functions for a one-dimensional random variable (wealth). F (·) is said to dominate G(·) according to first order stochastic dominance if for every nondecreasing u(x): Z Z u(x)dF (x) ≥
u(x)dG(x)
This is equivalent to the requirement that F (x) ≤ G(x), ∀x.
Note that this requires that any consumer, risk averse or not, would prefer F to G. It is often useful to realize two facts: One, a first order stochastically dominating distribution F can be obtained form a distribution G by shifting up outcomes randomly. Two, first order stochastic dominance implies a higher mean, but is stronger than just a requirement on the mean. The other moments of the distribution get involved too. In other words, just because the mean is higher for one distribution than another does not mean that the first dominates the second according to first order stochastic dominance!
4.3
A first look at Insurance
Let us use the above model to investigate a simple model of insurance. To be concrete, assume an individual with current wealth of $100,000 who faces a 25% probability to loose his $20,000 car through theft. Assume the individual has vN-M expected utility. The individual’s expected utility then is U (·) = 0.75u(100, 000) + .25u(80, 000). Now assume that the individual has access to an insurance plan. Insurance works as follows: The individual decides on an amount of coverage, C. This
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coverage carries a premium of π per dollar. The contract specifies that the amount C will be paid out if the car has been stolen. (Assume that this is all verifiable.) How would our individual choose the amount of coverage? Simple: maximize expected utility. Thus maxC {0.75u(100, 000 − πC) + 0.25u(80, 000 − πC + C)}. The first order condition for this problem is (−π)0.75u0 (100, 000 − πC) + (1 − π)0.25u0 (80, 000 − πC + C) = 0. Before we further investigate this equation let us verify the second order condition. It requires (−π)2 0.75u00 (100, 000 − πC) + (1 − π)2 0.25u00 (80, 000 − πC + C) < 0. Clearly this is only satisfied if u(·) is concave, in other words, if the consumer is risk averse. So, what does the first order condition tell us? Manipulation yields the condition that u0 (100, 000 − πC) (1 − π) = 0 u (80, 000 − πC + C) 3π which gives us a familiar looking equation in that the LHS is a ratio of marginal utilities. It follows that total consumption under each circumstance is set so as to set the ratio of marginal utility of wealth equal to some fraction which depends on price and the probabilities. Even without knowing the precise function we can say something about the insurance behaviour, however. To do so, let us compute the actuarially fair premium. The expected loss is $5,000, so that an insurance premium which collects that amount for the $20,000 insured value would lead to zero expected profits for the insurance firm: 0.75πC + 0.25(πC − C) = 0 ⇒ π = 0.25. An actuarially fair premium simply charges the odds (there is a 1 in 4 chance of a loss, after all.) If we use this fair premium in the above first order condition we obtain u0 (100, 000 − πC) = 1. u0 (80, 000 − πC + C)
Since the utility function is strictly concave it can have the same slope only at the same point, and we conclude that5 (100, 000 − πC) = (80, 000 − πC + C) ⇒ C = 20, 000. 5
Ok, read that sentence again. Do you understand the usage of the word ‘Since’ ? I am not “cancelling” the u0 terms, because those indicate a function. Instead the equation tells us that numerator and denominator must be the same. But for what values of the independent variable wealth does the function u(·) have the same derivative? For none, if u(·) is strictly concave. Therefore the function must be evaluated at the same level of the independent variable.
Uncertainty 71 This is one of the key results in the analysis of insurance: at actuarially fair premiums a risk averse consumer will fully insure. Note that the consumer will not bear any risk in this case: wealth will be $95,000 independent of if the car is stolen, since a $5,000 premium is due in either case, and if the car is actually stolen it will be replaced. As we have seen before, this will make the consumer much better off than if he is actually bearing the gamble with this same expected wealth level. If you draw the appropriate diagram you can verify that the consumer does not have to pay any of the amount he would be willing to pay (the difference between the expected value and the certainty equivalent.) If we had a particular utility function we could now also compute the maximal amount the consumer would be willing to pay. We have to be careful, however, how we set up this problem, since simply increasing π will reduce the amount of coverage purchased! So instead, let us approach the question as follows: What fee would the consumer be willing to pay in order to have access to actuarially fair insurance? Let F denote the fee. Then we have the consumer choose between u(95, 000 − F ) and 0.25u(80, 000) + 0.75u(100, 000). (Note that I have skipped a step by assuming full insurance. The left term is the expected utility of a fully insured consumer who pays the fee, the right term is the expected utility of an uninsured consumer. You should verify that the lump sum fee does not stop the consumer from fully insuring at a fair premium.) For example, if u(·) = ln(·) then simple manipulation yields F ∼ 426. It is important to note why we have set up the problem this way. Consider the alternative (based on these numbers and the logarithmic function) and assume that the total payment of $5,426 which is made in the above case of a fair premium plus fee, were expressed as a premium. Then we get that π = 5426/20000 = 0.2713. The first order condition for the choice of C then requires that (recall that ∂ln(x)/∂x = 1/x) (80, 000 + 0.7287C) 0.7287 = = 0.895318835 (100, 000 − 0.2713C) 0.8139
⇒ C = 9, 810.50.
As you can see, if the additional price is understood as a per dollar charge for insured value, the consumer will not insure fully. Of course this is an implication of the previous result — the consumer now faces a premium which is not actuarially fair. Indeed, we could also compute the premium for which the consumer will cease to purchase any insurance. For logarithmic utility like this we would want to compute (remember, we are trying to find
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when C = 0 is optimal) 1−π 80, 000 = 100, 000 3π
⇒ π = 0.2941.
As indicated before, there is room to trade between insurance providers and risk averse consumers. Indeed, as you can verify in one of the questions at the end of the chapter, there is room for trade between two risk averse consumers if they face different risk or if they differ in their attitudes towards risk (degree of risk aversion.)
4.4
The State-Preference Approach
While the above approach lets us focus quite well on the role of probabilities in consumer choice, it is different in character to the ‘maximize utility subject to a budget constraint’ approach we have so much intuition about. In the first order condition for the insurance problem, for example, we had a ratio of marginal utilities on the one side — but was that the slope of an indifference curve? As mentioned previously, we can actually treat consumption as involving contingent commodities, and will do so now. Let us start by assuming that the outcomes of any random event can be categorized as something we will refer to as the states of the world. That is, there exists a set of mutually exclusive states which are adequate to describe all randomness in the world. In our insurance example above, for example, there were only two states of the world which mattered: Either the car was stolen or it was not. Of course, in more general settings we could think of many more states (such as the car is stolen and not recovered, the car is stolen but recovered as a write off, the car is stolen and recovered with minor damage, etc.) In accordance with this view of the world we now will have to develop the idea of contingent commodities. In the case of our concrete example with just two states, a contingent commodity would be delivered only if a particular state (on which the commodity’s delivery is contingent) occurs. So, if there are two states, good and bad, then there could be two commodities, one which promises consumption in the good state, and one which promises consumption in the bad state. Notice that you would have to buy both of these commodities if you wanted to consume in both states. Notice also that nothing requires that the consumer purchase them in equal amounts. They are, after all, different commodities now, even if the underlying good which gets delivered in each state is the same. Finally, note that if one of these commodities were missing
Uncertainty 73 you could not assure consumption in both states (which is why economists make such a fuss about “complete markets” — which essentially means that everything which is relevant can be traded. It does not have to be traded, of course, that is up to people’s choices, but it should be available should someone want to trade.) Of course, after the fact (ex post in the lingo) only one of these states does occur, and thus only the set of commodities contingent on that state are actually consumed. Before the fact (before the uncertainty is resolved, called ex ante) there are two different commodities available, however. Once we have this setting we can proceed pretty much as before in our analysis. To be concrete let there be just two states, good and bad. We will now index goods by a subscript b or g to indicate the state in which they are delivered. We will further simplify things by having just one good, consumption (or wealth). Given that there are two states, that means that there are two distinct (contingent) commodities, cg and cb . We may now assume that the consumer has our usual vN-M expected utility.6 If the individual assessed a probability of π to the good state occurring, then we would obtain an expected utility of consumption of U (cg , cb ) = πu(cg ) + (1 − π)u(cb ). This expression gives us the expected utility of the consumer. The consumers’ objective is to maximize expected utility, as before. It might be useful at this point to assess the properties of this function. As long as the utility index applied to consumption in each state, u(·), is concave, this is a concave function. It will be increasing in each commodity, but at a decreasing rate. We can also ask what the marginal rate of substitution between the commodities will be. This is easily derived by taking the total derivative along an indifference curve and rearranging: πu0 (cg )dcg + (1 − π)u0 (cb )dcb = 0,
πu0 (cg ) dcb =− . dcg (1 − π)u0 (cb )
Note the fact that the MRS now depends not only on the marginal utility of wealth but also on the (subjective) probabilities the consumer assesses for each state! Even more importantly, we can consider what is known as the certainty line, that is, the locus of points where cg = cb . Since the marginal utility of consumption then is equal in both states (we have state independent utility here, after all, which means that the same u(·) applies in each state), it follows that the slope of an indifference curve on the certainty 6
Note that this is somewhat more onerous than before now: imagine the states are indexed by good health and poor health. It is easy to imagine that an individual would evaluate material wealth differently in these two cases.
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line only depends on the probability the consumer assesses for each state. In this case, it is π/(1 − π). The other ingredient is the budget line, of course. Since we have two commodities, each might be expected to have a price, and we denote these by pg , pb respectively. The consumer who has a total initial wealth of W may therefore consume any combination which lies on the budget line pg cg + pb cb = W , while a consumer who has an endowment of consumption given by (wg , wb ) may consume anything on the budget line pg cg + pb cg = pg wg + pb wb . Where do these prices come from? As before, they will be determined by general equilibrium conditions. But if contingent markets are well developed and competitive, and there is general agreement on the likelihood of the states, then we might expect that a dollar’s worth of consumption in a state will cost its expected value, which is just the dollar times the probability that it needs to be delivered. (I.e., a kind of zero profit condition for state pricing.) Thus we might expect that pg = π and pb = (1 − π). The budget line also has a slope, of course, which is the rate at which consumption in one state can be transferred into consumption in the other state. Taking total derivatives of the budget we obtain that the budget slope is dcb /dcg = pg /pb . Combining this with our condition on “fair” pricing in the previous paragraph, we obtain that the budget allows transformation of consumption in one state to the other according to the odds.
4.4.1
Insurance in a State Model
So let us reconsider our consumer who was in need of insurance in this framework. In order to make this problem somewhat neater, we will reformulate the insurance premium into what is known as a net premium, which is a payment which only accrues in the case there is no loss. Since the normal insurance contract specifies that a premium be paid in either case, we usually have a payment of premium × Amount in order to obtain a net benefit of Amount − premium × Amount. One dollar of consumption added in the state in which an accident occurs will therefore cost premium/(1−premium) dollars in the no accident state. Thus, let pb = 1 and let pg = P , the net premium. The consumer will then solve maxcb ,cg {πu(cg ) + (1 − π)u(cb )} s.t. P cg + cb = P (100, 000) + 80, 000. The two first order conditions for the consumption levels in this problem are πu0 (cg ) − λP = 0 and (1 − π)u0 (cb ) − λ = 0.
Uncertainty 75 loss
20 cert. 15
10
5 @ endowment 0 0
5
10
15
20 no loss
Figure 4.5: An Insurance Problem in State-Consumption space Combining them in the usual way we obtain πu0 (cg ) = P. (1 − π)u0 (cb ) Now, as we have just seen the LHS of this is the slope of an Indifference curve. The RHS is the slope of the budget, and so this says nothing but the familiar “there must be a tangency”. We have also derived P = π/(1 − π) for a fair net premium before. Thus we get that πu0 (cg ) π = , 0 (1 − π)u (cb ) 1−π
which requires that
u0 (cg ) =1 u0 (cb )
⇒
cg = 1. cb
Thus this model shows us, just as the previous one, that a risk averse consumer faced with a fair premium will choose to fully insure, that is, choose to equalize consumption levels across the states. A diagrammatic representation of this can be found in diagram 3.5, which is standard for insurance problems. The consumer has an endowment which is off the certainty (45-degree) line. The fair premium defines a budget line along which the consumer can reallocate consumption from the good (no loss) state to the bad (loss) state. Optimum occurs where there is a tangency, which must occur on the certainty line since then the slopes are equalized. The picture looks perfectly “normal”, that is, just as we are used from introductory economics.
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4.4.2
May2004
Risk Aversion Again
Given the amount of time spent previously on risk-aversion, it is interesting to see how risk-aversion manifests itself in this setting. Intuitively it might be apparent that a more risk averse consumer will have indifference curves which are more curved, that is, exhibit less substitutability (recall that a straight line indifference curve means that the goods are perfect substitutes, while a kinked Leontief indifference curve means perfect complements.) It therefore stands to reason that we might be interested in the rate at which the MRS is falling. It is, however, much easier to think along the lines of certainty equivalents: Consider two consumers with different risk aversion, that is, curvature of indifference curves. For simplicity, let us consider a point on the certainty line and the two indifference curves for our consumers through that common point (see Fig. 4.6). loss
20 cert. B A
15
10
5
0 0
5
10
15
20 no loss
Figure 4.6: Risk aversion in the State Model Assume further that consumer B’s indifference curve lies everywhere else above consumer A’s. We can now ask how much consumption we have to add for each consumer in order to keep the consumer indifferent between the certain point and a consumption bundle with some given amount less in the bad state. Clearly, consumer B will need more compensation in order to accept the bad state reduction. Looked at it the other way around, this means that consumer B is willing to give up more consumption in the good state in order to increase bad state consumption. Note that both assess the same probabilities on the certainty line, since the slopes of their ICs are the same. How does this relate to certainty equivalents? Well, a budget line at π fair odds will have the slope − 1−π . Consider three such budget lines which are all parallel and go through the certain consumption point and the two
Uncertainty 77 gambles which are equivalent for the consumer to the certain point. Clearly (from the picture) consumer B’s budget is furthest out, followed by consumer A’s, and furthest in is the budget through the certain point. But we know that parallel budgets differ only in the income/wealth they embody. Thus there is a larger reduction in wealth possible for B without reducing his welfare, compared to A. The wealth reduction embodied in the lower budget is the equivalent of the certainty equivalent idea before. (The expected value of a given gamble on such a budget line is given by the point on the certainty line and that budget, after all.)
4.5
Asset Pricing
Any discussion of models of uncertainty would be incomplete without some coverage of the main area in which all of this is used, which is the pricing of assets. As we have seen before, if there is only time to contend with but returns or future prices are known, then asset pricing reduces to a condition which says that the current price of an asset must relate to the future price through discounting. In the “real world” most assets do not have a future price which is known, or may otherwise have returns which are uncertain — stocks are a good example, where dividends are announced each year and their price certainly seems to fluctuate. Our discussion so far has focused on the avoidance of risk. Of course, even a risk averse consumer will accept some risk in exchange for a higher return, as we will see shortly. First, however, let us define two terms which often occur in the context of investments.
4.5.1
Diversification
Diversification refers to the idea that risk can be reduced by spreading one’s investments across multiple assets. Contrary to popular misconceptions it is not necessary that their price movements be negatively correlated (although that certainly helps.) Let us consider these issues via a simple example. Assume that there exists a project A which requires an investment of $9,000 and which will either pay back $12,000 or $8,000, each with equal probability. The expected value of this project is therefore $10,000. Now assume that a second project exists which is just like this one, but (and this is important) which is completely independent of the first. How much each pays back in no way depends on the other. Two investors now could each invest $4,500 in each project. Each investor then has again a total
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investment of $9,000. How much do the projects pay back? Well, each will pay an investor either $6,000 or $4,000, each with equal probability. Thus an investor can receive either $12,000, $10,000, or $8,000. $12,000 or $8,000 are received one quarter of the time, and half the time it is $10,000. The total expected return thus is the same. BUT, there is less risk, since we know that for a risk-averse consumer 0.5u(12)+0.5u(8) < 0.25u(12)+0.25u(8)+0.5u(10) since 2(0.25u(12) + 0.25u(8)) < 2(0.5u(10)). Should the investor have access to investments which have negatively correlated returns (if one is up the other is down) risk may be able to be eliminated completely. All that is needed is to assume that the second project above will pay $8,000 when the first pays $12,000, and that it will pay $12,000 if the first pays $8,000. In that case an investor who invests half in each will obtain either $6,000 and $4,000 or $4,000 and $6,000: $10,000 in either case. The expected return has not increased, but there is no risk at all now, a situation which a risk-averse consumer would clearly prefer.
4.5.2
Risk spreading
Risk spreading refers to the activity which lies at the root of insurance. Assume that there are 1000 individuals with wealth of $35,000 and a 1% probability of suffering a $10,000 loss. If the losses are independent of one another then there will be an average of 10 losses per period, for a total $100,000 loss for all of them. The expected loss of each individual is $100, so that all individuals have an expected wealth of $34,900. A mutual insurance would now collect $100 from each, and everybody would be reimbursed in full for their loss. Thus we can guarantee the consumers their expected wealth for certain. Note that there is a new risk introduced now: in any given year more (or less) than 10 losses may occur. We can get rid of some of this by charging the $100 in all years and retaining any money which was not collected in order to cover higher expenses in years in which more than 10 losses occur. However, there may be a string of bad luck which might threaten the solvency of the plan: but help is on the way! We could buy insurance for the insurance company, in effect insuring against the unlikely event that significantly more than the average number of losses occurs. This is called re-insurance. Since an insurance company has a well diversified portfolio of (independent) risks, the aggregate risk it faces itself is low and it will thus be able to get fairly cheap insurance.
Uncertainty 79 These kind of considerations are also able to show why there may not be any insurance offered for certain losses. You may recall the lament on the radio about the fact that homeowners in the Red River basin were not able to purchase flood insurance. Similarly, you can’t get earth-quake insurance in Vancouver, and certain other natural disasters (and man-made ones, such as wars) are excluded from coverage. Why? The answer lies in the fact that all insured individuals would have either a loss or no loss at the same time. That would mean that our mutual insurance above would either require no money (no losses) or $10,000,000. But the latter requires each participant to pay $10,000, in which case you might as well not insure! (A note aside: often the statement that no insurance is available is not literally correct: there may well be insurance available, but only at such high rates that nobody would buy it anyways. Even at low rates many people do not carry insurance, often hoping that the government will bail them out after the fact, a ploy which often works.)
4.5.3
Back to Asset Pricing
Before we look at a more general model of asset pricing, it may be useful to verify that a risk-averse consumer will indeed hold non-negative amounts of risky assets if they offer positive returns. To do so, let us assume the simple most case, that of a consumer with a given wealth w who has access to a risky asset which has a return of rg or rb < 0 < rg . Let x denote the amount invested in the risky asset. Wealth then is a random variable and will be either wg = (w − x) + x(1 + rg ) or wb = (w − x) + x(1 + rb ). Suppose the good outcome occurs with probability π. What will be the choice of x? max0≤x≤w {πu(w + rg x) + (1 − π)u(w + rb x)} . The first and second order conditions are rg πu0 (w + rg x) + rb (1 − π)u0 (w + rb x) = 0 rg2 πu00 (w + rg x) + rb2 (1 − π)u00 (w + rb x) < 0 The second order condition is satisfied trivially if the consumer is risk averse. To show under what circumstances it is not optimal to have a zero investment consider the FOC at x = 0: rg πu0 (w) + rb (1 − π)u0 (w) ? 0. The LHS is only positive if πrg + (1 − π)rb > 0, that is, if expected returns are positive. Notice also that in that case there will be some investment! Of
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course this is driven by the fact that not investing guarantees a zero rate of return. Investing is a gamble, which the consumer dislikes, but also increases returns. Even a risk-averse consumer will take some risk for that higher return! Now let us consider a more general model with many assets. Assume that there is a risk-free asset (one which yields a certain return) and many risky ones. Let the return for the risk-free asset be denoted by R0 and the ˜ i , each of which is a random returns for the risky assets be denoted by R variable with some distribution. Initial wealth of the consumer is w. Finally, we can let xi denote the fraction of wealth allocated to asset i = 0, . . . , n. In the second period (we will ignore time discounting for simplicity and clarity) wealth will be a random variable the distribution of which on how P depends ˜ i , with the much is invested in eachPasset. In particular, w˜ = w0 ni=0 xi R budget constraint that ni=0 xi = 1. We can transform this expression as follows: ' # ' # n n n X X X ˜ i = w R0 + ˜ i − R0 ) . w˜ = w (1 − xi )R0 + xi R xi ( R i=1
i=1
i=1
The consumer’s goal, of course, is to maximize expected utility from this wealth by choice of the investment fractions. That is, ( Ã ' #!) n X ˜ i − R0 ) max{x} {Eu (w)} ˜ = max{x} Eu w R0 + xi ( R . i
i
i=1
Differentiation yields the first order conditions ˜ i − R0 ) = 0, Eu0 (w)( ˜ R
∀i.
Now we will do some manipulation of this to make it look more presentable and informative. You may recall that the covariance of two random variables, ˜i = X, Y , is defined as COV(X, Y ) = EXY −EXEY. It follows that Eu0 (w) ˜ R 0 ˜ 0 ˜ i . Using this fact and distributing the subtraction in COV(u , Ri )+Eu (w)E ˜ R the FOC across the equal sign, we obtain for each risky asset i the following equation: ˜ i + COV(u0 (w), ˜ i ). Eu0 (w)R ˜ 0 = Eu0 (w)E ˜ R ˜ R From this it follows that in equilibrium the expected return of asset i must satisfy 0 ˜i) ˜ R ˜ i = R0 − COV(u (w), . ER Eu0 (w) ˜
Uncertainty 81 This equation has a nice interpretation. The first term is clearly the riskfree rate of return. The second part therefore must be the risk-premium which the asset must garner in order to be held by the consumer in a utility maximizing portfolio. Note that if a return is positively correlated with wealth — that is, if an asset will return much if the consumer is already rich — then it is negatively correlated with the marginal utility of wealth, since that is decreasing in wealth. Thus the expected return of such an asset must exceed the risk free return if it is to be held. Of course, assets which pay off when wealth otherwise would be low can have a lower return than the risk-free rate since they, in a sense, provide insurance.
4.5.4
Mean-Variance Utility
The above pricing model required us to know the covariance and expectation of marginal utility, since, as we have seen before, it is the fact that marginal utility differs across outcomes which in some sense causes risk-aversion. A nice simplification of the model is possible if we specify at the outset that our consumer likes the mean but dislikes the variance of random returns, i.e., the mean is a good, the variance is a bad. We can then specify a utility function directly on those two characteristics of the distribution. (The normal distribution, for example is completely described by these two moments. If distributions differ in higher moments, this formulation would not be able to pick that up, however.) Recall that for a set of outcomes (w1 , w2 , . . . , wn ) with probabilities (π1 , π2 , . . . , πn ) n X The mean is µw = π i wi , i=1
and the variance is
σw2
=
n X i=1
πi (wi − µw )2 .
We now define utility directly on these: u(µw , σw ), although it is standard practice to actually use the standard deviation as I just have done. Risk aversion is now expressed through the fact that we assume that ∂u(·) = u1 (·) > 0 while ∂µw
∂u(·) = u2 (·) < 0. ∂σw
We will now focus on two portfolios only (the validity of this approach will be shown in a while.) The risk free asset has a return of rf , the risky
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asset (the pPhas a return of ms with probability πs . Let P “market portfolio”) rm = πs ms and σm = πs (ms − rm )2 . Assume that a fraction x of wealth is to be invested in the risky asset (the market). The expected return for a fraction x invested will be X X rx = πs (xms + (1 − x)rf ) = (1 − x)rf + x πs ms .
The variance of this portfolio is X X 2 σx2 = πs (xms + (1 − x)rf − rx )2 = πs (xms − xrm )2 = x2 σm . The investor/consumer will maximize utility by choice of x:
maxx {u(xrm + (1 − x)rf , xσm )} FOC u1 (·)[rm − rf ] + u2 (·)σm = 0 2 + 2σm [rm − rf ]u12 (·) ≤ 0 SOC u11 (·)[rm − rf ]2 + u22 (·)σm Assuming that the second order condition holds, we note that we will require [rm − rf ] > 0 since u2 (·) is negative by assumption. We may also note that we can rewrite the FOC as rm − r f −u2 (·) = . u1 (·) σm The LHS of this expression is the MRS between the mean and the standard deviation, that is, the slope of an indifference curve. The RHS can be seen to be the slope of the budget line since the budget is a mix of two points, (rf , 0) and (rm , σm ), which implies that the tradeoff of mean for standard deviation is rise over run: (rm − rf )/σm . In a general equilibrium everybody has access to the same market and the same risk free asset. Thus, everybody who does hold any of the market will have the same MRS — a result analogous to the fact that in our usual general equilibria everybody will have the same MRS. Of course, this is just a requirement of Pareto Optimality. In this discussion we had but one risky asset. In reality there are many. As promised, we derive here the justification for considering only the so-called market portfolio. The basic idea is simple. Assume a set of risky assets. Since we are operating in a two dimensional space of mean versus standard deviation, one can certainly ask what combination of assets (also known as a portfolio) will yield the highest mean for a given standard deviation, or, which is often easier to compute, the lowest standard deviation for a given mean.
Uncertainty 83 mean
20 budget efficient risk
15 market
10
5 risk free 0 0
5
10
15
20 standard dev.
Figure 4.7: Efficient Portfolios and the Market Portfolio P Let x denote the vector of shares in each of the assets so that I x = 1. Define the mean and standard deviation of the portfolio x as µ(x) and σ(x). Then it is possible to solve maxx {µ(x) + λ(s − σ(x))} or minx {σ(x) + λ(m − µ(x)} . For each value of the constraint there will be a portfolio (an allocation of wealth across the different risky assets) which achieves the optimum. It turns out (for reasons which I do not want to get into here: take a finance course or do the math) that this will lead to a frontier which is concave to the standard variation axis. Now, by derivation, any proportion of wealth which is held in risky assets will be held according to one of these portfolios. But which one? Well, the consumer can combine any one of these assets with the risk free asset in order to arrive at the final portfolio. Since this is just a linear combination, the resulting “budget line” will be a straight line and have a positive slope of (µx − µf )/σx , where µx , σx are drawn from the efficient frontier. A simple diagram suffices to convince us that the highest budget will be that which is just tangent to the efficient frontier. The point of tangency defines the market portfolio we used above! Note a couple more things from this diagram. First of all it is clear from the indifference curves drawn that the consumer gains from the availability of risky assets on the one hand, and of the risk-free asset on the other. Second, the market portfolio with which the risk free asset will be mixed will depend on the return from the risk free asset. Imagine sliding the risk free return
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up in the diagram. The budget line would have to become flatter and this means a tangency to the efficiency locus further to the right. Finally, the precise mix between market and risk free will depend on preferences. Indeed, there might be people who would want to have more than their entire wealth in the market portfolio, that is, the tangency to the budget would occur to the right of the market portfolio. This requires a “leveraged” portfolio in which the consumer is allowed to hold negative amounts of certain assets (short-selling.)
4.5.5
CAPM
The above shows that all investors face the same price of risk (in terms of the variance increase for a given increase in the mean.) It does not tell us anything about the risk-return trade-off for any given asset, however. Any risk unrelated to the market risk can be diversified away in this setting, however, so that any unsystematic risk will not attract any excess returns. An asset must, however, earn higher returns to the extent that it contributes to the market risk, since if it did not it would not be held in the market portfolio. Consideration of this problem in more detail (assume a portfolio with a small amount of this asset held and the rest in the market, compute the influence of the asset on the portfolio return and variance, rearrange) will yield the famous CAPM equation involving the asset’s ‘beta’ a number which is published in the financial press: σX,M µx = µf + (µm − µf ) 2 . σM Here x denotes asset x, m the market, f the risk free asset, and σX,M is the σX,M covariance between asset x and the market. The ratio 2 is referred to as σM the asset’s beta.
4.6
Review Problems
Question 1: Demonstrate that all risk-averse consumers would prefer an investment yielding wealth levels 24, 20, 16 with equal probability to one with wealth levels 24 or 16 with equal probability. Question 2: Compute the certainty equivalent for an expected utility maxi√ mizing consumer with (Bernoulli) utility function u(w) = w facing a gamble over $3600 with probability α and $6400 with probability 1 − α.
Uncertainty 85 Question 3: Determine at what wealth levels a consumer with (Bernoulli) utility function u(w) = lnw has the same absolute risk aversion as a consumer √ with (Bernoulli) utility function u(w) = 2 w. How do their relative risk aversions compare at that level. What does that mean? Question 4: Consider an environment with two states — call them rain, R, and shine, S, — with the probability of state R occurring known to be π. Assume that there exist two consumers who are both risk-averse, vN-M expected utility maximizers. Assume further that the endowment of consumer A is (10, 5) — denoting 10 units of the consumption good in the case of state R and 5 units in the case of S — and that the endowment of consumer B is (5, 10). What are the equilibrium allocation and price? (Provide a well labelled diagram and supporting arguments for any assertions you make.) Question 5: Anna has $10,000 to invest and wants to invest it all in one or both of the following assets: Asset G is gene-technology stock, while asset B is stock in a bible-printing business. There are only two states of nature to worry about, both of which occur with equal probability. One is that gene-technology is approved and flourishes, in which case the return of asset B is 0% while the return of asset G is 80%. The other is that religious fundamentalism takes hold, and gene-technology is severely restricted. In that case the return of asset B will be 40% but asset G has a return of (40%). Anna is a risk-averse expected-utility maximizer with preferences over wealth represented by u(w). a) State Anna’s choice problem mathematically. b) What proportion of the $10,000 will she invest in the gene-technology stock? (I.e. solve the above maximization problem.) Question 6∗ : Assume that consumers can be described by the following preferences: they are risk averse over wealth levels, but they enjoy gambling for its consumption attributes (i.e., while they dislike the risk on wealth which gambling implies, they get some utility out of partaking in the excitement (say, they get utility out of daydreaming about what they could do if they won.)) Let us further assume that consumers only differ with respect to their initial wealth level and their consumption utility from gambling, but that all consumers have the same preferences over wealth. In order to simplify these preferences further, assume that wealth and gambling are separable. We can represent these preferences over wealth, w, and gambling, g ∈ {0, 1} by some u(w, g) = v(w) + µi g, where v(w) is strictly concave, and µi is an individual parameter for each consumer. Finally, they are assumed to be expected utility maximizers. a) Assume for now that the gambling is engaged in via a lottery, in
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which consumers pay a fixed price p for one lottery ticket, and the ticket is either a “Try Again” (no prize) or “Winner” (Prize won.) Also assume for simplicity that they can either gamble once or not at all (i.e., each consumer can only buy one ticket.) i) First verify that in such an environment the government can make positive profits. (I.e., verify that some consumers will buy a ticket even if the ticket price exceeds the expected value of the ticket. ii) If consumers have identical µi , how does the participation of consumers then depend on their initial wealth level if their preferences exhibit {constant| decreasing} {absolute |relative} risk aversion? (The above notation means: consider all (sensible) permutations.) iii) Assume now that preferences are characterized by decreasing absolute and constant relative √ risk aversion. Verify that the utility functions v(w) = ln w and v(w) = w satisfy this assumption. Also assume that the consumption enjoyment of gambling is decreasing in wealth, that is, consumers with high initial wealth have low µi . (They know what pain it is to be rich and don’t daydream as much about it.) Who would gamble then? Question 7∗ : Assume that a worker only cares about the income he generates from working and the effort level he expends at work. Also assume that the worker is risk averse over income generated and dislikes effort. His √ preferences can be represented by u(w, e) = w − e2 . The worker generates income by accepting a contract which specifies an effort level e and the associated wage rate w(e). (Since leisure time does not enter in his preferences he will work full time (supply work inelastically) and we can normalize the wage rate to be per period income.) The effort level of the worker is not observed directly by the firm, and thus the worker has an incentive to expend as little effort as possible. The firm, however, cares about the effort expended, since it affects the marginal product it gets from employing the worker. It can conduct random tests of the worker’s effort level with some probability α. These tests reveal the true effort level employed by the worker. If the worker is not tested, then it is assumed that he did indeed work at the specified effort level and will receive the contracted wage. If he is tested and found to have shirked (not supplied the correct effort level) then a penalty p is assessed and deducted from the wage of the worker. a) What relationship has to hold between w(e), e, α and p in order for a worker to provide the correct effort level if p is a constant (i.e., not dependent on either the contracted nor the observed effort level)? b) If we can make p depend on the actual deviation in effort which we observe, what relationship has to be satisfied then? c) Is there any economic insight hidden in this? Think about how these problems would change if the probability of detection somehow depended on
Uncertainty 87 the effort level (i.e., the more I deviate from the correct effort level, the more likely I might be caught.) Question 8: A consumer is risk averse. She is faced with an uncertain consumption in the future, since she faces the possibility of an accident. Accidents occur with probability π. If an accident occurs, it is either really bad, in which case she looses B, or minor, in which case her loss is M < B. Bad accidents represent only 1/5 of all accidents, all other accidents are minor. Without the accident her consumption would be W > B. a) Derive her optimal insurance purchases if the magnitude of the loss is publicly observable and verifiable and insurance companies make zero profits. More explicitly, if the type of accident is verifiable then a contract can be written contingent on the type of loss. The problem thus is equivalent to a problem where there are two types of accident, each of which occurs with a different probability and can be insured for separately at fair premiums. b) Now consider the case if the amount of loss is private information. In this case only the fact that there was an accident can be verified (and hence contracted on), but the insurer cannot verify if the loss was minor or major, and hence pays only one fixed amount for any kind of accident. Assume zero profits for the insurer, as before, and show that the consumer now overinsures for the minor loss but under-insures for the bad loss, and that her utility thus is lowered. (Note that the informational distortion therefore leads to a welfare loss.) Question 9: Assume a mean-variance utility model, and let µ denote the expected level of wealth, and σ its variance. √ Take the boundary of the efficient risky asset portfolios to be given by µ = σ − 16. Assume further that there exists a risk-free asset which has mean zero and standard deviation zero (if this bothers you, you can imagine that this is actually measuring the increase in wealth above current levels.) Let there be two consumers who have meanσ2 σ2 variance utility given by u(µ, σ) = µ− and u(µ, σ) = 3µ− respectively. 64 64 Derive their optimal portfolio choice and contrast their decisions. Question 10: Fact 1: The asset pricing formula derived in class states that ˜ i = R0 − ER
˜i) Cov(U 0 (w), ˜ R . EU 0 (w) ˜
Fact 2: A disability insurance contract can be viewed as an asset which pays some amount of money in the case the insured is unable to generate income from work. Use Facts 1 and 2 above to explain why disability insurance can have
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a negative return, (that is, why the price of the contract may exceed the expected value of the payment) if viewed as an asset in this way. Question 11: TRUE/FALSE/UNCERTAIN: Provide justification for your answers via proof, counter-example or argument. 1) The optimal amount of insurance a risk-loving consumer purchases is characterized by the First Order Condition of his utility maximization problem, which indicates a tangency between an indifference curve and a budget line. 2) For a consumer who is initially a borrower, the utility level will definitely fall if interest rates increase substantially. 3) The utility functions (3000 ∗ lnx1 + 6000 ∗ lnx2 ) 1/3 2/3 + 2462 and Exp(x1 x2 ) 12 represent the same consumer preferences. 4) A risk-averse consumer will never pay more than the expected value of a gamble for the right to participate in the gamble. A risk-lover would, on the other hand. 5) The market rate of return is 15%. The stock of Gargleblaster Inc. is known to increase to $117 next period, and is currently trading for $90. This market (and the current stock price of Gargleblaster Inc.) is in equilibrium. 6) Under Risk-Variance utility functions, all consumers who actually hold both the risk-free asset and the risky asset will have the same Marginal Rate of Substitution between the mean and the variance, but may not have the same investment allocation. Question 12∗ : Suppose workers have identical preferences√ over wealth only, which can be represented by the utility function u(w) = 2 w. Workers are also known to be expected utility maximizers. There are three kinds of jobs in the economy. One is a government desk job paying $40,000.00 a year. This job has no risk of accidents associated with it. The second is a bus-driver. In this job there is a risk of accidents. The wage is $44,100.00 and if there is an accident the monetary loss is $11,700.00. Finally a worker could work on an oil rig. These jobs pay $122,500.00 and have a 50% accident probability. These are all market wages, that is, all these jobs are actually performed in equilibrium. a) What is the probability of accidents in the bus driver occupation? b) What is the loss suffered by an oil rig worker if an accident occurs there? c) Suppose now that the government institutes a workers’ compensation scheme. This is essentially an insurance scheme where each job pays a fair premium for its accident risk. Suppose that workers can buy this insurance
Uncertainty 89 in arbitrary amounts at these premiums. What will the new equilibrium wages for bus-drivers and oil rig workers be? Who gains from the workers’ compensation? d) Now suppose instead that the government decides to charge only one premium for everybody. Suppose that of the workers in risky jobs 40% are oil rig workers and 60% are bus drivers. Suppose that they can buy as much or little insurance as they wish. How much insurance do the two groups buy? Who is better off, who is worse off in this case (at the old wages)? Can we say what the new equilibrium wages would have to be? Question 13∗ : Prove that state-independent expected utility is homothetic if the consumer exhibits constant relative risk aversion. (This question arises since indifference curves do have the same slope along the certainty line. So could they have the same slope along any ray from the origin? In that case they would be homothetic.)
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Chapter 5 Information In the standard model of consumer choice discussed in chapter 1, as well as the model of uncertainty developed in chapter 3, it was assumed that the decision maker knows all relevant information. In chapter 1 this meant that the consumer knows the price of all goods, as well as the precise features of each good (all characteristics relevant to the consumer.) In chapter 3 this in particular implied that the consumer has information about the probabilities of states or outcomes. Not only that, this information is symmetric, so that all parties to a transaction have the same information. Hence in chapter 3 the explicit assumption that the insurance provider has the same knowledge of the probabilities as the consumer. What if these assumptions fail? What if there is no complete and symmetric information? Fundamentally, one of the key problems is asymmetric information — when one party to a transaction knows something relevant to the transaction which the other party does not know. This quite clearly will lead to problems, since Pareto efficiency necessitates that all available information is properly incorporated. Consider, for example, the famous “Lemon’s Problem”:1 Suppose a seller knows the quality of her used car, which is either high or low. The seller attaches values of $5000 or $1000 to the two types of car, respectively. Buyers do not know the quality of a used car and have no way to determine it before purchase. Buyers value good used cars at $6000 and bad used cars at $2000. Note that it is Pareto efficient in either case for the car to be sold. Will the market mechanism work in this case? Suppose that it is known by everybody that half of all cars are good, and half are bad. To keep it simple, suppose buyers and sellers are risk 1
This kind of example is due to Akerlof (1970) Quarterly Journal of Economics, a paper which has changed economics.
91
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neutral. Buyers would then be willing to pay at most $4000 for a gamble with even odds in which they either receive a good or a bad used car. Now suppose that this were the market price for used cars. At this price only bad cars are actually offered for sale, since good car buyers rather keep theirs, since their valuation of their own car exceeds the price. It follows that this price cannot be an equilibrium price. It is fairly easy to verify that the only equilibrium would have bad cars trade at a price somewhere between $1000 and $2000 while good cars are not traded at all. This is not Pareto efficient. Of course, it is not necessary that there be asymmetric information. Suppose that there exist many restaurants, each offering slightly different combinations of food and ambiance. Consumers have tastes over the characteristics of the meals (how spicy, what kind of meat, if any meat at all; Italian, French, eastern European, Japanese, Egyptian, etc.) as well as the kind of restaurant (formal, romantic, authentic, etc.) as well as the general quality of the cooking within each category. In a Pareto efficient general equilibrium each consumer must frequent (subject to capacity constraints) the most preferred restaurant, or if that is full the next preferred one. Can we expect this to be the equilibrium?2 To see why the general answer may be “No” consider a risk averse consumer in an environment where a dining experience is necessary in order to find out all relevant characteristics of a restaurant. Every visit to a new place carries with it the risk of a really unpleasant experience. If the expected benefit of finding a better place than the one currently known does not exceed the expected cost of having a bad experience, the consumer will not try a new restaurant, and hence will not find the best match! What seems to be important then, are two aspects of information: One, can all relevant information be acquired before the transactions is completed?; Two, is the information symmetric or asymmetric? Aside from a classification of problems into asymmetric or symmetric information, it is common to distinguish between three classes of goods, based on the kind of informational problems they present: search goods, experience goods, and (less frequently) faith goods. A search good is one for which the consumer is lacking some information, be it price or some attribute of the good, which can be fully determined before purchase of the good. Anytime 2 This is a question similar to one very important in labour economics: are workers matched to the correct jobs, that is, are the characteristics of workers properly matched to the required characteristics of the job? These kind of problems are analysed in the large matching literature. In this literature you will find interesting papers on stable marriages — is everybody married to their most preferred partner, or could a small coalition swap partners and increase their welfare?
Information 93 the consumer can discover all relevant aspects before purchasing we speak of a search good. Supposing that search is costly, which seems reasonable, we can then model the optimal search behaviour of consumers by considering the (marginal) benefits and (marginal) costs of search. Applying such thinking to labour markets we can study the efficiency effects of unemployment insurance; or we can apply it to advertising, which for such goods focuses on supplying the missing information, and is therefore possibly efficiency enhancing. For some goods such a determination of all relevant characteristics may not be possible. Nobody can explain to you how something tastes, for example; you will have to consume the good to find out. Similarly for issues of quality. Inasmuch as this refers to how long a durable good lasts, this can only be determined by consuming the good and seeing when (and if) it breaks. Such goods are called experience goods. The consumer needs to purchase the good, but the purchase and consumption of the good (or service!) will fully inform the consumer. This situation naturally leads to consumers trying a good, but maybe not necessarily finding the best match for them. Advertising will be designed to make the consumer try the product — free samples could be used.3 Why would the consumer not necessarily find the best product? If there is a cost to unsatisfactory consumption experiences this will naturally arise. As in the restaurant example above. Similar examples can be constructed for hair cuts, and many other services. What if the consumer never finds out if the product performs its function? This is the natural situation for many medical and religious services. The consumer will not discover the full implications until it is (presumably) too late. Such goods are termed faith goods and present tremendous problems to markets. In our current society the spiritual health of consumers is not judged to be important, and so the market failure for religious services is not addressed.4 Health, in contrast, is judged important — since consumers value it, for one, and since there are large externalities in a society with a social safety net — and thus there is extensive regulation for health care services in most societies.5 Since education also has certain attributes of a 3
It used to be legal for cigarette manufacturers to distribute “sample packs” for free, allowing consumers to experience the good without cost. The fact that nicotine is addictive to some is only helpful in this regard, as any local drug pusher knows: they also hand out free sample packs in schools. 4 A convincing argument can be made that societies which prescribe one state religion do so not in an attempt to maximize consumer welfare but tax revenue and control. The Inquisition, for example, probably had little to do with a concern for the welfare of heretics. Note also that I am speaking especially with respect to religions in which the “afterlife” plays a large role. We will encounter them again in the chapter on game theory. 5 Interesting issues arise when the provision of health care services is combined with the
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faith good — in the sense that it is very costly to unlearn, or to repeat the education — we also see strong regulation of education in most societies.6 Note that religion and health care probably differ in another dimension: in health care there is information asymmetry between provider and consumer; it may be argued that in religion both parties are equally uninformed.7 Presumably the fact that information is asymmetric makes it easier to exploit the uninformed party on the one hand, and to regulate on the other.8 The government attempts to combat this informational asymmetry by certifying the supply. Aside from all the above, there are additional problems with information. These days everybody speaks of the “information economy”. Clearly information is supposed to have some kind of value and therefore should have a price. However, while that is clear conceptually, it is far from easy to incorporate information into our models. Information is not a good like most others. For one, it is hard to measure. There are, of course, measurements which have been derived in communications theory — but they often measure the amount of meaningful signals versus some noise (as in the transmission of messages.) These measurements measure if messages have been sent, and how many. Economists, in contrast, are concerned with what kind of message actually contains relevant information and what kind may be vacuous. Much care is therefore taken to define the informational context of any given decision problem (we will encounter this again in the game theory part, where we will use the term information set to denote all situations in which the same information has been gathered, loosely speaking.) Aside from the problem of defining and measuring information, it is also a special good since it is not rivalrous (a concept you may have encountered in ECON 301): the fact that I possess some information in no way impedes your ability to have the same information. Furthermore, the fact that I “consume” the information somehow (let’s say by acting on it) does not stop you from consuming it or me from using it again later. There are therefore provision of spiritual services. 6 This is regulation for economically justifiable reasons. Because education is so costly to undo or repeat it is also frequently meddled with for “societal engineering” reasons. 7 I am not trying to belittle beliefs here, just pointing to the fact that while a medical doctor may actually know if a prescribed treatment works (while the patient does not), neither the religious official or the follower of a faith know if the actions prescribed by the faith will “work”. 8 Note the weight loss and aphrodisiac markets, or cosmetics, for example. Little difference between these and the “snake oil cures” of the past seems to exist. Regulation can take the simple form that only “verifiable” claims may be made.
Information 95 large public good aspects to information which require special consideration. 9 The long and short of this is that standard demand-supply models often don’t work, and that markets will in general misallocate if information is involved, which makes it even more important to have a good working model. A complete study of this topic is outside the scope of these notes, however. In what follows we will only outline some specific issues in further detail.
5.1
Search
One of the key breakthroughs in the economics of information was a simple model of information acquisition. The basic idea is a simple one (as all good ideas are.) A consumer lacks information — say about the price at which a good may be bought. Clearly it is in the consumer’s interest not to be “fleeced,” which requires him to have some idea about what the market price is. In general the consumer will not know what the lowest price for the product is, but can go to different stores and find out what their price is. The more stores the consumer visits the better his idea about what the correct price might be — the better his information — but of course the higher his cost, since he has to visit all these stores. An optimizing consumer may be expected to continue to find new (hopefully) lower prices as long as the marginal benefit of doing so exceeds the marginal cost of doing so. Therefore we “just” need to define benefits and costs and can then apply our standard answer that the optimum is achieved if the marginal cost equals the marginal benefit! The problem with this is the fact that we will have to get into sampling distributions and other such details (most of which will not concern us here) to do this right. The reason for this is that the best way to model this kind of problem is as the consumer purchasing messages (normally: prices) which are drawn from some distribution. For example: let us say that the consumer knows that prices Rare distributed according to some cumulative z distribution function F (z) = 0 f (p)dp, where f (p) is the (known) density. If the consumer where to obtain n price samples (draws) from this distribution 9
This is the problem in the Patent protection fight: Once somebody has invented (created the information for) a new drug the knowledge should be freely available — but this ignores the general equilibrium question of where the information came from. The incentives for its creation will depend on what property rights are enforceable later on. For while two firms both may use the information, a firm may only really profit from it if it is the sole user.
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then the probability that any given sample (say p0 ) is the lowest will be [1 − F (p0 )]n−1 f (p0 ). (This formula should make sense to you from Econometrics or Statistics: we are dealing with n independent random variables.) From this it follows that the expected value of the lowest price after having taken a sample of n prices is Z ∞ n plow = p[1 − F (p)]n−1 f (p)dp. 0
Note that this expected lowest sample decreases as n increases, but at a decreasing rate: the difference between sampling n times and n − 1 times is Z ∞ n−1 n plow − plow = − pF (p)[1 − F (p)]n−2 f (p)dp < 0. 0
So additional sampling leads to a benefit (lower expected price) but with a diminishing margin. What about cost? Even with constant (marginal) cost of sampling we would have a well defined problem and it is easy to see that individuals with higher search costs will have lower sample sizes and thus pay higher expected prices. Also note that the lowest price paid is still a random variable, and hence consumers do not buy at the same price (which is inefficient, in general!) Computing the variance you would observe that dispersion of the lowest price is decreasing in n — that means that the lower the search costs the ‘better’ the market can be expected to work. Indeed, competition policy is concerned with this fact in some places and attempts to generate rules which require that prices be posted (lowering the cost of search). Any discussion of search would be lacking if we did not point out that search is normally sequential. In the above approach n messages where bought, with n predetermined. This is the equivalent of visiting n stores for a quote irrespective of what the received quotes are. The dynamic problem is the much more interesting one and has been quite well studied. We will attempt to distill the most important point and demonstrate it by way of a fairly simple example. The key insight into this kind of problem is that it is often optimal to employ a stopping rule.10 That is, to continue sampling until a price has been obtained which is below some preset limit, at which point search is abandoned and the transaction occurs (a sort of “good enough” attitude.) The price at which one decides to stop is the reservation 10
It does depend on the distributional assumptions we make on the samples — independence makes what follows true.
Information 97 price — the highest price one is willing to pay! In order to derive this price we will have to specify if one can ‘go back’ to previous prices, or if the trade will have fallen through if one walks away. The latter is the typical setup in the matching literature in labour economics, the former is a bit easier and we will consider it first. Suppose the cost of another sample (search effort) is c. The outcome of the sample is a new price p, which will only be of benefit if it is lower than the currently known minimum price. Evaluated at the optimal reservation price pR , the expected gain from an additional sample is therefore the savings pR − p, “expected over” all prices p < pR . If these expected savings are equal to the cost of the additional sample, then the consumer is just indifferent between buying another sample or not, and thus the reservation price is found: Z pR pR satisfies (pr − p)f (p)dp = c. 0
Next, consider the labour market, where unemployed workers are searching for jobs. This is the slightly more complex case where the consumer cannot return to a past offer. Also, the objective is to find the highest price. First determine the value of search, V , which is composed of the cost, the expected gain above the wage currently on the table, and the fact that a lower wage might arise which would indicate that another search is needed. Thus, assuming a linear utility function for simplicity (no risk aversion), and letting p stand for wages R∞ Z pR Z ∞ −c + pR pf (p)dp f (p)dp =⇒ V = . pf (p)dp + V V = −c + 1 − F (pR ) 0 pR Note that I assume stationarity here and the fact that one pR will do (i.e., the fact that the reservation wage is independent of how long the search has been going on.) All of these things ought to be shown for a formal model. We now will ask what choice of pR will maximize the value above (which is the expected utility from searching.) Taking a derivative we get R∞ −pR f (pR )(1 − F (pR )) + f (pR )(−c + pR pf (p)dp) = 0. (1 − F (pR ))2 Simplifying and rearranging we obtain Z ∞ pf (p)dp − pR = c − pR F (pR ). pR
The LHS of this expression is the expected increase in the wage, the RHS is the cost of search, which consists of the actual cost of search and the fact
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that the current wage is foregone if a lower value is obtained. What now will be the effect of a decrease in the search cost c, for example if the government decides to subsidize non-working (searching) workers? This would lower c, and the LHS would have to fall to compensate, which will occur only if a higher pR is chosen. Of course, a higher pR lowers the RHS further. In mathematical terms it is easy to compute that −1 dpR = . dc 1 − F (pR ) Unemployment insurance will increase the reservation wage (and thus unemployment — note the pun: it ensures unemployment!) The reason is that our workers can be more choosy and search for the “right” job. They become more discriminating in their job search. Note that this is not necessarily bad. If the quality of the match (suitability of worker and firm with each other) is reflected in a higher wage, then this leads to better matches (fewer Ph.D. cab drivers). This may well be desirable for the general equilibrium efficiency properties of the model. Now, this model is quite simplistic. More advanced models might take into account eligibility rules. In those models unemployment insurance can be shown to cause some workers to take bad jobs (because that way they can qualify for more insurance later.) Similar models can also be used to analyse other matching markets. The market for medical interns comes to mind, or the marriage market. In closing let us note that certain forms of advertising will lower search costs (since consumers now can determine cheaply who has what for sale at which price) and thus are efficiency enhancing (less search, less resources spent on search, and lower price dispersion in the market.) Other forms of advertising (image advertising) do not have this function, however, and will have to be looked at in a different framework. This is where the distinction between search goods and experience goods comes in.
5.2
Adverse Selection
Most problems which will concern us in this course are actually of a different nature than the search for information above. What we are interested in most are the problems which arise because information is asymmetric. This means that two parties to a transaction do not have the same information, as is the case if the seller knows more about the quality (or lack thereof) of
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Figure 5.1: Insurance for Two types his good than the buyer, or if the buyer knows more about the value of the good to himself than the seller. In these types of environments we run into two well known problems, that of adverse selection (which you may think of as “hidden property”) and moral hazard (“hidden action.”) We will deal with the former in this section. Adverse selection lies at the heart of Akerlof’s Lemons Problem. These kind of markets lead naturally to the question if the informed party can send some sort of signal to reveal information. But how could they do so? Simply stating the fact that, say, the car is of good quality will not do, since such a statement is free and would also be made by sellers of bad cars (it is not incentive compatible.) Sometimes this problem can be fixed via the provision of a warranty, since a warranty makes the statement that the car is good more costly to sellers of bad cars than of cars which are, in fact, good. Let us examine these issues in our insurance model. Assume two states, good and bad, and assume two individuals who both have wealth endowment (wg , wb ); wg > wb . Suppose that these individuals are indistinguishable to the insurance company, but that one individual is a good risk type who has a probability of the good state occurring of πH , while the other is a bad risk type with probability of the good state of only πL < πH . To be stereotypical and simplify the presentation below, assume that the good risk is female, the bad risk male, so that grammatical pronouns can distinguish types in what follows. As we have seen before, the individuals’ indifference curves will have a slope of −πi /(1 − πi ) on the certainty line.
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If there were full information both types could buy insurance at fair premiums and would choose to be on the certainty line on their respective budget lines (with the high risk type on a lower budget line and with a lower consumption in each state.) However, with asymmetric information this is not a possible outcome. The bad risk cannot be distinguished from the good risk a priori and therefore can buy at the good risk premium. Now, if he were to maximize at this premium we know that he would over insure — and this action would then distinguish him from the good risk. The best he can do without giving himself away is to buy the same insurance coverage that our good risk would buy, in other words to mimic her. Thus both would attempt to buy full insurance for a good type. We now have to ask if this is a possible equilibrium outcome. The answer is NO, since the insurance company now would make zero (expected) profits on her insurance contract, but would loose money on his insurance contract. Consider the profit function (and recall that pi = πi ): (1 − πH )(wg − wb ) − (1 − πL )(wg − wb ) < 0. Foreseeing this fact, the insurance company would refuse to sell insurance at these prices. Well then, what is the equilibrium in such a market? There seem to be two options: either both types buy the same insurance (this is called pooling behaviour) or they buy different contracts (this is called separating behaviour.) Does there exist a pooling equilibrium in this market? Consider a larger market with many individuals of each of the two types. Let fH denote the proportion of the good types (πH ) in the market. An insurer would be making zero expected profits from selling a common policy for coverage of I at a premium of p to all types if and only if fH (πH pI − (1 − πH )(1 − p)I) + (1 − fH )(πL pI − (1 − πL )(1 − p)I) = 0. This requires a premium p = (1 − πL ) − fH (πH − πL ). Note that at this premium the good types subsidize the bad types, since the former pay too high a premium, the latter too low a premium. In Figure 5.2, this premium is indicated by a zero profit locus (identical to the consumers’ budget) at an intermediate slope (labelled ‘market’.) Any proposed equilibrium with pooling would have to lie on this line and be better than no insurance for both types. Such a point might be point M in the figure. However, this cannot be an equilibrium. In order for it to be an equilibrium
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15
20 no loss
Figure 5.2: Impossibility of Pooling Equilibria nobody must have any actions available which they could carry out and prefer to what is proposed. Consider, then, any point in the area to the right of the zero profit line, below the bad type’s indifference curve, and above the good type’s indifference curve: a point such as D. This contract, if proposed by an insurance company, offers less insurance but at a better price. Only the good type would be willing to take it (she would end up on a higher indifference curve). Since it is not all the way over on the good type zero profit line, the insurance company would make strictly positive profits. Of course, all bad types would remain at the old contract M , and since this is above the zero profit line for bad types whoever sold them this insurance would make losses. Notice that the same arguments hold whatever the initial point. It follows that a pooling equilibrium cannot exist. Well then, does a separating equilibrium exist? We now would need two contracts, one of which is taken by all bad types and the other by all good types. Insurance offerers would have to make zero expected profits. Of course, since each type takes a different contract the insurer will know who is who from their behaviour. This suggests that we look at contracts which insure the bad risks fully at fair prices. For the bad risk types to accept this type of contract the contract offered to the good risk type must be on a lower indifference curve. It must also be on the fair odds line for good types for there to be zero profits. Finally it must be acceptable to the good types. This suggests a point such as A in the Figure 5.3. There now are two potential problems with this. One is that the insurer and insured of good type would like to move to a different point, such as B, after the type
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20 cert.
loss
good market 15
10 bad * B
5 * A endowment * 0 0
5
10
15
20 no loss
Figure 5.3: Separating Contracts could be possible is revealed. But if that would indeed happen then it can’t, in equilibrium, because the bad types would foresee it and pretend to be good in order then to be mistaken for good. The other problem is that the stability of such separating equilibria depends on the mix of types in the market. If, as in the Figure 5.3, the market has a lot of bad types this kind of equilibrium works. But what if there are only a few bad types? In that case an insurer could deviate from our proposed contract and offer a pooling contract which makes strictly positive profits and is accepted by all in favour over the separating contract. This is point C in Figure 5.4. Of course, while this deviation destroys our proposed equilibrium it is itself not an equilibrium (we already know that no pooling contract can be an equilibrium.) This shows that a small proportion of bad risks who can’t be identified can destroy the market completely! Notice the implications of these findings on life or health insurance markets when there are people with terminal diseases such as AIDS or Hepatitis C, or various forms of cancer. The patient may well know that he has this problem, but the insurance company sure does not. Of course, it could require testing in order to determine the risk it will be exposed to. Our model shows that doing so would make sure that everybody has access to insurance at fair rates and that there is no cross-subsidization. This is clearly efficient. However, it will reduce the welfare of consumers with these diseases. Indeed, given that AIDS, for example, means a very high probability of seriously expensive treatment, the insurance premiums would be very high (justifiably so, by the way, since the risk is high.) What ought society do
Information 103 20 cert.
loss
good market 15
10 bad C * 5 * A endowment * 0 0
5
10
15
20 no loss
Figure 5.4: Separating Contracts definitely impossible about this? Economics is not able to answer that question, but it is able to show the implications of various policies. For example, banning testing and forcing insurance companies to bear the risk of insuring such customers might make the market collapse, in the worst scenario, or will at least lead to inefficiency for the majority of consumers. It would also most likely tend to lead to “alternative” screening devices — the insurance company might start estimating consumers’ “lifestyle choices” and then discriminate based on that information. Is that an improvement? An alternative would be to let insurance operate at fair rates but to directly subsidize the income of the affected minority. (This may cause “moral” objections by certain segments of the population.)
5.3
Moral Hazard
Another problem which can arise in insurance markets, and indeed in any situation with asymmetric information, is that of hidden actions being taken. In our insurance example it is often possible for the insured to influence the probability of a loss: is the car locked? Are there anti-theft devices installed? Are there theft and fire alarms, sprinklers? Do the tires have enough tread depth, and are they the correct tire for the season (in the summer a dedicated summer tire provides superior breaking and road holding to a M&S tire, while in the winter a proper winter tire is superior.) Some of these actions are observable and will therefore be included in the conditions of the insurance
104 L-A. Busch, Microeconomics
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contract. For example, in Germany your insurance will refuse to pay if you do not have “approved” tires, or if the car was not locked and this can be established. Indeed, insurance for cars without anti-theft devices is now basically unaffordable since theft rates are so high since the ‘iron curtain’ opened.11 If you insure your car as parked in a closed parking garage and then leave it out over night your insurance may similarly refuse to pay! Other actions are not so easily verified, however. How aggressively do you drive? How many risks do you decide to take on a given day? This is often not observable but nevertheless in your control. If the avoidance of accidents is costly to the insured in any way, then he can be expected to pick the optimal level of the (unobservable) action — optimal for himself, not the insurance company, that is. As a benchmark, let us consider an individual who faces the risk of loss L. The probability of the loss occurring depends on the amount of preventive action, A, taken and is denoted π(A). It would appear logical to assume that π 0 (A) < 0, π 00 (A) > 0. The activity costs money, and cost is c(A) with c0 (A) > 0, c00 (A) > 0. In the absence of insurance the individual would choose A so as to maxA {π(A)u(w − L − c(A)) + (1 − π(A))u(w − c(A))}. The first order condition for this problem is π 0 (A)u(w − L − c(A)) − c0 (A)π(A)u0 (w − L − c(A)) − π 0 (A)u(w − c(A)) − c0 (A)(1 − π(A))u0 (w − c(A)) = 0. Thus the optimal A∗ satisfies c0 (A∗ ) =
π 0 (A∗ )(u(w − L − c(A)) − u(w − c(A∗ ))) . π(A∗ )u0 (w − L − c(A∗ )) − (1 − π(A∗ ))u0 (w − c(A∗ ))
Now consider the consumer’s choice when insurance is available. To keep it simple we will assume that the only available contract has a premium of π(A∗ ) and is for the total loss L. Note that this would be a fair premium if the consumer continues to choose the level A∗ of abatement activity. However, the maximization problem the consumer now faces becomes (the consumer will assume the premium fixed) maxA {π(A)u(w − pL − c(A)) + (1 − π(A))u(w − pL − c(A))}. The first order condition for this problem is π 0 (A)u(·) − c0 (A)π(A)u0 (·) − π 0 (A)u(·) − c0 (A)(1 − π(A))u0 (·) = 0. 11
Yes, I am suggesting that most stolen cars end up in former eastern block countries — it is a fact.
Information 105 Thus the optimal A∗ satisfies ˆ = 0. c0 (A) Clearly (and expectedly) the consumer will now not engage in the activity at all. It follows that the probability of a loss is higher and that the insurance company would loose money. If we were to recompute the level of A at a fair premium for π(0) we would find that the same is true: no effort is taken. The only way to elicit effort is to expose the consumer to the right incentives: there must be a reward for the hidden action. A deductible will accomplish this to some extent and will lead to at least some effort being taken. This is due to the fact that the consumer continues to face risk (which is costly.) The amount of effort will in general not be the optimal, however.
5.4
The Principal Agent Problem
To conclude the chapter on information here is an outline of the leading paradigm for analysing asymmetric information problems. Most times one (uninformed) party wishes to influence the behavior of another (informed) party, the problem can be considered a principal-agent problem. This kind of problem is so frequent that it deserved its own name and there are books which nearly exclusively deal with it. We will obviously have only a much shorter introduction to the key issues in what follows. First note that this is a frequent problem in economics, to say the least. Second, note that at the root of the problem lies asymmetric information. We are, generically, concerned with situations in which one party — the principal — has to rely on some other party — the agent — to do something which affects the payoff of the first party. This in itself is not the problem, of course. What makes it interesting is that the principal cannot tell if the agent did what he was asked to do, that is, there is a lack of information, and that the agent has incentives not to do as asked. In short, there is a moral-hazard problem. The principal-agent literature explores this problem in detail.
5.4.1
The Abstract P-A Relationship
We will frame this analysis in its usual setting, which is that of a risk-neutral principal who tries to maximize (expected) profits, and one agent, who will
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have control over (unobservable) actions which influence profits. Call the principal the owner and the agent the manager. The agent/manager will be assumed to have choice of a single item, what we shall call his effort level e ∈ [e, e]. This effort level will be assumed to influence the level of profits before manager compensation, called gross-profits, which you may think of as revenue if there are no other inputs. More general models could have multiple dimensions at this stage. The manager’s preferences will be over monetary reward (wages/income) and effort: U (w, e). We assume that the manager is an expected utility maximizer. Also assume that the manager is risk-averse (that is u1 (·, ·) > 0, u11 (·, ·) < 0) and that effort is a bad (u2 (·, ·) < 0.) The owner can not observe the manager’s effort choice. Instead, only the realization of gross profits is observed. Note at this stage that this in itself does not yet cause a moral-hazard problem, since if the relationship between effort and profit is known we can simply invert the profit function to deduce the effort level. Therefore we need to introduce some randomness into the model. Let ρ be a random variable which also influences profits and which is not directly observable by the owner either. It could be known to the manager, but we will have to assume that it will become known only after the fact, so that the manager cannot condition his effort choice on the realization of ρ. Let the relationship between effort, profits and the random variable be denoted by Π(e, ρ). Note that Π(e, ρ) will be a random variable itself. All expectations in what follows will be taken with respect to the distribution of ρ. EΠ(e, ρ), for example, will be the expected profits for effort level e. Since the owner can only observe profits, the most general wage contract he can offer the manager will be a function w(Π). This formulation includes a fixed wage as well as all wage plus bonus schemes, or share contracts (where the manager gets a fixed share of profits.) Let us first, as a benchmark, determine the efficient level of effort, which would be provided under full information. In that case we would have to solve the Pareto problem, that is, solve maxe,w {EΠ(e, ρ) − w
s.t. U (w, e) = u} .
Here u is a level of utility sufficient to make the manager accept the contract. Note also that in formulating this problem like this I have already used the knowledge that a risk-neutral owner should fully insure a risk-averse manager by offering a constant wage. Assuming no corner solutions, it is easy to see that we would want to set the effort level such that the marginal benefit of effort (in terms of higher expected profits) is related to the marginal cost (in terms of higher disutility
Information 107 of effort, which will have to be compensated via the wages). In particular, we need that w EΠe (·, ·) = Ue (·, ·). Uw (·, ·) What if effort is not observable? In that case we will solve the model “backwards”: given any particular wage contract faced by the manager, we will determine the manager’s choice. Then we will solve for the optimal contract, “foreseeing” those choices. Assume in what follows that the owner wants to support some effort level eˆ (which is derived from this process.) So, our manager is faced with two decisions. One is to determine how much effort to provide given he accepts the contract: maxe {EU (w(Π(e, ρ)), e)}. This leads to FOC E [Uw (·, ·)w 0 (·)Πe (·, ·) + Ue (·, ·)] = 0, which determines the optimal e∗ . The other is the question if to accept the contract at all, which requires that maxe {EU (w(Π(e, ρ)), e)} = EU (w(Π(e∗ , ρ)), e∗ ) ≥ U0 . Here U0 is the level of utility the manager can obtain if he does not accept the contract but instead engages in the next best alternative activity (in other words, it is the manager’s opportunity cost.) Both of these have special names and roles in the principal-agent literature. The latter one is called the participation constraint, or individual rationality constraint. That is, any contract is constrained by the fact that the manager must willingly participate in it. Thus, the contract w ∗ (Π) must satisfy IR : maxe {EU (w(Π(e, ρ)), e)} ≥ U0 . The other constraint is that the manager’s optimal choice should, in equilibrium, correspond to what the owner wanted the manager to do. That is, if the owner wants to elicit effort level eˆ, then it should be true that the manager, in equilibrium, actually supplies that level. This is called the incentive compatibility constraint. Mathematically it says: IC : eˆ = argmaxe {EU (w(Π(e, ρ)), e)}. Here ‘argmax’ indicates the argument which maximizes. In other words, we want eˆ = e∗ .
108 L-A. Busch, Microeconomics
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Now we can write down the principal’s problem. The principal ‘simply’ wishes to maximize his own payoff subject to both, the participation and incentive compatibility constraints. This problem is easily stated maxe,w(Π) {E [Π(e, ρ) − w(Π(e, ρ)) +λP (U0 − U (w(Π(e, ρ)), e)) + λI (Uw (·, ·)w 0 (·)Πe (·, ·) + Ue (·, ·))]} , but hard to solve in general cases. We will, therefore, only look at three special cases in which some answers are possible. First, let us simplify by assuming a risk neutral manager. In that case we can have U (w, e) = w − v(e), where v(e) is the disutility of effort. Now note that we would not have to insure the manager in this case since he evaluates expected wealth the same as some fixed payment. Also note that there is no reason to over pay the manager compared to the outside option, that is, he needs to be given only U0 . Before we charge ahead and solve this brute force, let us think for a moment longer about the situation. We will not be insuring the manager. He will furthermore have the correct incentive to expend effort if he cares as much about profits as the owner. Thus it would stand to reason that if we were to sell him the profits he would be the owner and thus take the optimal action! But this is equivalent to paying him a wage which is equal to the profits less some fixed return for the owners. So, let us propose a wage contract of w = Π − p, and assume for the moment that the IR constraint can be satisfied (by choice of the correct p.) With such a contract the manager will choose an effort level such that Πe (e, ρ) − v 0 (e) = 0. Note that this is the efficient effort level under full information. The owner’s problem now simply is to choose the largest p such that the manager still accepts the contract! The other special case one can consider is that of an infinitely risk-averse manager. We can model this as a manager who has lexicographic preferences over his minimum wealth and effort. Independent of the effort levels the manager will prefer a higher minimum wealth, and for equal minimum wealth the manager will prefer the lower effort. For simplicity also assume that the lowest possible profit level is independent of effort (that is, only the probabilities of profits depend on the effort, not the level). In that case a manager will always face the same lowest wage for any wage function and thus will not be able to be enticed to provide more than the minimum level of effort.
Information 109 As you can see, we might expect anything between efficient outcomes and completely inefficient outcomes, largely depending on the precise situation. Let us return for a moment to the general setting above. We will restrict it somewhat by assuming that the manager’s preferences are separable: U (w, e) = u(w) − v(e) with the obvious assumptions of u0 (·) > 0, u00 (·) < 0, v 0 (·) ≥ 0, v 00 (·) > 0, v 0 (e) = 0, v 0 (e) = ∞. The last two are typical “Inada conditions” designed to ensure an interior solution. We will also assume that the cumulative distribution of profits exists and depends on effort, F (Π, e), and that this distribution is over Π ∈ [Π, Π], has a density f (Π, e) > 0, and satisfies first-order stochastic dominance. In this formulation the manager will solve (Z Π
maxe
Π
)
u(w(Π))f (Π, e)dΠ − v(e) .
This has first order condition Z Π u(w(Π))fe (Π, e)dΠ − v 0 (e) = 0. Π
We will ignore the SOC for a while. The participation constraint will be Z
Π Π
u(w(Π))f (Π, e)dΠ − v(e) ≥ U0 .
The owner will want to find a wage structure and effort level to (Z Π
maxw(·),e
Π
[(Π − w(Π))f (Π, e)
+λP (u(w(Π)) − v(e) − U0 )f (Π, e)
o + λC (u(w(Π))fe (Π, e) − v 0 (e)f (Π, e))] dΠ .
This will have to be differentiated, which is quite messy and, as pointed out before, hard to solve. However, consider the differentiation with respect to the wage function: −f (Π, e) + λP f (Π, e)u0 (w(Π)) + λC fe (Π, e)u0 (w(Π)) = 0. Rewriting we get something which is informative: 1 u0 (w(Π))
= λP + λC
fe (Π, e) . f (Π, e)
110 L-A. Busch, Microeconomics
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The left hand side of this is the inverse of the slope of the wealth utility function. It is therefore increasing in the wage. The right hand side consists of the constraint multipliers (which will both be positive since the constraints are binding) multiplied by some factors. Of particular interest is the last term, the ratio of the derivative of the density to the density. Let us further specialize the problem and suppose that there are only two effort levels, and that the owner wants to get the high effort level. In that case it is easy to see that the above equation will become ¶ µ fH (Π) − fL (Π) fL (Π) 1 . = λP + λC = λP + λC 1 − u0 (w(Π)) fH (Π) fH (Π) The term ffHL (Π) is called a likelihood ratio. It follows that the higher the (Π) relative probability that the effort was high for a given realized profit level, the higher the manager’s wage. Indeed, if the likelihood ratio is decreasing then then the wage function must be increasing with realized profits. What is going on here is that higher profits are a correct signal of higher effort, and thus we will make wages increase with the signal. Finally, assume that there are only two profit levels, with the high level of profits more likely under high effort. In that case it can be shown that 0
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formalism=' extension=' left=' advanced=' courses=' texts.br=' 118=' microeconomics=' acbr=' £b=' £=' bbr=' bcbr=' s£=' s=' bsbr=' ccbr=' bs=' ¡@=' @s=' s¡=' ¢a=' ¢=' aa=' ££=' s¢=' ¢sbr=' d=' cbr=' sbr=' s¡br=' ecbr=' ¤=' s¤=' cs¤=' £a=' c£=' s¢¢=' 6.1:=' valid=' invalid=' trees=' now=' ready=' defining=' bunch=' objects=' apply.=' capture=' need.=' information?=' play,=' i.e.,=' plays=' together=' player(s),=' knows=' making=' move,=' stage,=' exogenous=' exist,=' probability=' distribution=' formally,=' following:br=' 3=' n-player=' comprises:=' a;=' 2.=' associating=' vector=' length=' n=' γ;=' 3.=' partition=' {s0=' ,=' s1=' sn=' nonterminal=' nodes=' (the=' sets;)=' 4.=' s0=' immediate=' followers;=' 5.=' ∈=' {1,=' 2,=' n}=' si=' subsets=' sij=' (information=' sets),=' ∀b,=' 6.=' index=' iij=' 1-1=' map=' followers=' 119=' exhaustive=' list.=' notice=' (1)=' 1)=' sets.=' “nature”=' gets=' too.=' nature=' very=' concept.=' non-strategic=' choice=' environment,=' randomization=' (as=' accident=' choosing=' this.)=' (2)=' nature,=' non-strategic,=' (3)=' sets=' idea=' distinguish=' within=' every=' labels=' same.)=' restrictions=' implied,=' still=' draw=' important=' issues=' assumptions=' sets,=' words.=' restriction=' forget=' learn=' n-person=' recall=' never=' known,=' own=' previous=' moves:=' x,=' x0=' neither=' x=' nor=' predecessor=' one,=' xˆ=' xˆ,=' ),=' x˜=' leading=' x.=' bears=' close=' reading,=' quite=' intuitive=' practice:=' player’s=' set,=' follow=' forgets=' he=' himself=' moved=' previously=' furthermore,=' predecessors:=' true=' player,=' “the=' action”=' chosen=' nodes.=' otherwise,=' remember=' chose=' indistinguishable.=' always=' assume=' recall.=' words,=' moves,=' learned=' something=' opponents=' (such=' opponent=' took=' “up”=' moved)=' practice,=' sense.=' allow=' possibility=' know-br=' 120=' 1b=' ©=' hh=' hr=' ©©=' h=' rp=' ph=' r=' @=' ¡=' r¡=' @r=' 1,=' `=' @rbr=' 0,=' 0br=' 2br=' ³³=' ³=' ³br=' r³=' r¡br=' bppbr=' pp=' pbr=' pr2=' @pr=' 1p=' 6.2:=' counter-example=' ing=' simultaneously,=' his=' time.=' imperfect=' opposite=' singletons.=' crucial=' linguistic=' difference=' here:=' known=' more)=' players.=' order.=' all!=' non-trivial=' theorem=' harsanyi=' shows=' transform=' bayesian.6=' uncertainty=' 6br=' bayes’=' formula=' update=' beliefs.=' formulabr=' 121=' eb=' r©=' hrrm=' ight=' accom=' 10=' −2,=' r=' 4,=' ©br=' wolf=' lamb=' ep=' outbr=' m=' ac=' 10br=' 5br=' @rm=' 4br=' 6.3:=' outcomes,=' etc.,=' transformed=' only.=' that,=' construct=' information,=' equilibria=' coincide.=' entrant=' monopolist=' enjoys=' fighting=' entrant,=' types=' fight,=' playing,=' coincide=' entrant’s=' priors.7=' transformation=' specified=' 6.3.=' turns=' powerful=' results,=' makes=' tools=' lots=' nearly=' know.br=' 6.1.2br=' strategies=' analyze=' modelled=' employ=' concept=' strategies.=' complete,=' contingent=' plans=' just=' “move,”=' set.=' strategy=' plan,=' referee=' computer,=' according=' instructions,=' says=' event=' occurring=' times=' happening=' does,=' divided=' occurring:=' (a|b)=' 7br'="" (b|a)p=' (a)=' ∩=' b)=' .'="" (b)=' (b)br=' prior=' ex-ante=' belief=' ex-post=' posterior.br=' 122=' watch=' happens=' change=' mind=' point.=' submit=' plan=' before=' cover=' eventualities=' translates=' saying=' specify=' those=' rule=' out!=' σ=' associates=' element=' σi=' :=' 7→=' alternatively,=' randomize.8=' occur=' levels,=' strategies:=' 7=' behavioural=' βi=' elements=' 8=' mixed=' µi=' σi=' concepts,=' general.=' find=' corresponding=' strategy,=' (which=' properly=' associated=' introduce=' shortly.)=' decidedly=' easier=' with.=' are.=' only,=' now.=' giving=' σ2=' σn=' σ−i=' ).=' general,=' fully,=' nature.=' 8br=' somewhat=' controversial=' issue.=' people=' flip=' coins=' decisions?=' nevertheless=' generally=' accepted.=' circumstances=' seen=' equivalent=' bluffing:=' poker=' example.=' sometimes=' fold=' pair,=' stand,=' raise=' people.=' coin=' flip.=' instances=' randomizing=' explained=' person=' definite=' population=' probabilities=' correspond=' proportion=' strategy.=' worry=' necessary=' see!)br=' 123=' von=' neumann-morgenstern=' expected=' evaluate=' things.=' combination=' (vector)=' end,=' will,=' arrive=' precisely=' whatever=' vertex.=' played,=' implies=' evaluated=' using=' 9=' πi=' (σ).=' π(σ)='(π1' (σ),=' πn=' (σ)).=' γ.=' (it=' “normal=' form,”=' coming=' use.)=' fashion=' abbreviated=' representation=' cumbersome=' prevalent=' view=' nowadays,=' interpretation=' stressed=' “strategic=' form.”=' slightly=' viewpoint,=' socalled=' matrix-games=' analyzed=' first.=' thus,=' definition:=' 11=' g=' (normal)=' 3-tuple=' (n,=' s,=' u=' n},=' ×s2=' .×sn=' i’s=' rn=' ;=' implicit=' above.=' abstract=' suppressed,=' represented=' matrix=' (hence=' games.)=' row,=' column,=' third=' matrices.=' three=' clearly=' looses=' appeal.)=' 6.4=' related=' concept,=' abstract,=' here,=' outcomes=' specified,=' functions=' players.br=' 124=' 12br=' ←−=' 12=' rbr=' ubr=' (1,=' (2,=' 2)br=' 3,=' 3)br=' 1)br=' 0)br=' 4)br=' dbr=' (3,=' 0)=' 5)br=' 6.4:=' 12=' o)=' defined=' o=' couple=' things=' closely=' occur.=' principle,=' realizing=' indices=' arbitrary,=' relabel=' everything=' loss=' generality=' (does=' “option=' 1”?),=' with,=' say,=' eight=' entries=' each.=' four=' two.=' forms=' widely=' different,=' indeed.=' nevertheless,=' matrix.=' matter?=' say=' main=' don’t.=' point=' concerns=' convenient=' representation.=' 6.5=' gives=' announces=' 50=' 100=' dollars,=' same.=' add=' $100=' asked=' for,=' pay=' simple,=' ×=' 27=' matrix!9=' 9br=' why?=' announcement=' 3-tuple.=' 33=' vectors=' constructed.br=' 125=' (0,=' ..br=' 100)br=' 50br=' (50,=' 100br=' (100,=' (−1,=' −1)br=' 0»=' 2s»=' »»»=' 0=' ¡50=' @br=' 50)=' 100,=' »»=' »»»br=' »=' cxxbr=' 2sbr=' xxxbr=' ¡@br=' @100br=' ¡br=' −1=' −1br=' xx=' xxx=' xs=' @100=' 6.5:=' go=' on,=' 6.6=' below=' page=' listed=' beginning=' forms.=' am=' usual=' convention=' belongs=' row=' index.=' pennies=' t=' tbr=' sexes=' −1)=' 1bbr=' h¡¡=' @@t=' pr¡=' 2p=' p@=' −1,=' 1−1,=' mbr=' 30)br=' (5,=' (30,=' 50)br=' m¡=' @@s=' 50,=' 30br=' dilemma=' 5,=' 30,=' 3)=' 4)=' (4,=' c¡¡=' @@d=' 126=' “education=' game”=' b©©br=' r©©=' −3,=' cbbr=' ©hbr=' cbr=' hhn=' hrp=' (p,=' i)=' (i,=' )br=' nbbr=' −3)br=' i)br=' 6.6:=' gamesbr=' 6.2br=' problemsbr=' developed=' (“games”.)=' prediction=' “likely”=' this?10=' “solve”=' impose=' competition=' as,=' “markets=' clear”)=' planned=' buy=' sell=' desire=' price),=' (their=' game).=' equilibrium.=' impose,=' qualified=' name,=' “these=' constitute=' (bayes=' equilibrium,=' subgame=' cho-kreps=' criterion,..)=' nature.)=' general=' try=' “refine=' away”=' (lingo=' “discard”)=' appear=' sensible.=' 280=' select=' few=' gained=' wide=' acceptance=' easy=' (some=' others=' difficult=' apply=' game.)=' clear=' trying=' do:=' tell=' play.=' philosophical=' stake=' inclination!=' noted=' here=' manual=' interested=' iota=' modeled.br=' 127br=' 6.2.1br=' persuasive=' idea,=' old:=' why=' eliminate=' strictly=' worse=' do?=' elimination=' (strictly)=' dominated=' variants=' (iterated=' weakly=' strategies),=' principle=' do,=' 13=' dominates=' larger=' independent=' (a,=' s−i='> πi (b, s−i ) ∀s−i ∈ S−i . A similar definition can be made for weakly dominates if the strict inequality is replaced by a weak inequality. Other authors use the notion of a dominated strategy instead: Definition 14 Strategy a is is weakly (strictly) dominated if there exists a mixed strategy α such that πi (α, s−i ) ≥ (>)πi (a, s−i ), ∀s−i ∈ Σi and πi (α, s−i ) > πi (a, s−i ) for some s−i ∈ Σi . If we have a 2 × 2 game, then elimination of dominated strategies may narrow down our outcomes to one point. Consider the “Prisoners’ Dilemma” game, for instance. ‘Defect’ strictly dominates ‘Cooperate’ for both players, so we would expect both to defect. On the other hand, in “Battle of the Sexes” there is no dominated (dominating) strategy, and we would still not know what to predict. If a player has more than two strategies, we also do not narrow down the field much, even if there are dominated strategies. In that case, we can use Successive Elimination of Dominated Strategies, where we start with one player, then go to the other player, back to the first, and so on, until we can’t eliminate anything. For example, in the following game 12
(l, l)
(r, r)
(l, r)
(r, l)
L
(2, 0)
(2, 0)
R
(1, 0)
(2, −1)
(2, −1)
(3, 1)
(3, 1)
(1, 0)
128 L-A. Busch, Microeconomics
May2004
player 1 does not have a dominated strategy. Player 2 does, however, since (r, l) is strictly dominated by (l, r). If we also eliminate weakly dominated strategies, we can throw out (l, l) and (r, r) too, and then player 1 has a dominated strategy in L. So we would predict, after successive elimination of weakly dominated strategies, that the outcome of this game is (R, (l, r)). There are some criticisms about this equilibrium concept, apart from the fact that it may not allow any predictions. These are particularly strong if one eliminates weakly dominated strategies, for which the argument that a player should never choose those appears weak. For example you might know that the opponent will play that strategy for which you are indifferent between two strategies. Why then would you eliminate one of these strategies just because somewhere else in the game (where you will not be) one is worse than the other? Next, we will discuss the probably most widely used equilibrium concept ever, Nash equilibrium.11 This is the most universally accepted concept, but it is also quite weak. All other concepts we will see are refinements of Nash, imposing additional constraints to those imposed by Nash equilibrium. Definition 15 A Nash equilibrium in pure strategies is a set of strategies, one for each player, such that each player’s strategy maximizes that player’s payoff, taking the other players’ strategies as given: σ∗
is Nash iff
∀i, ∀σi ∈ Σi ,
∗ ∗ πi (σi∗ , σ−i ) ≥ πi (σi , σ−i ).
Note that the crucial feature of this equilibrium concept: each player takes the others’ actions as given and plays a best response to them. This is the mutual best response property we first saw in the Cournot equilibrium, which we can now recognize as a Nash equilibrium.12 Put differently, we only check against deviations by one player at a time. We do not consider mutual deviations! So in the Prisoners’ Dilemma game we see that one player alone cannot gain from a deviation from the Nash equilibrium strategies 11
Nash received the Nobel price for economics in 1994 for this contribution. He extended the idea of mutual best responses proposed by von Neumann and Morgenstern to n players. He did this in his Ph.D. thesis. von Neumann and Morgenstern had thought this problem too hard when they proposed it in their book Games and Economic Behaviour. 12 Formally we now have 2 players. Their strategies are qi ∈ [0, P −1 (0)]. Restricting attention to pure strategies, their payoff functions are πi (q1 , q2 ), so the strategic form is (π1 (q), π2 (q)). Denote by bi (q−i ) the best response function we derived in footnote 1 of this chapter. The Nash equilibrium for this game is the strategy vector (q∗ 1 , q∗2 ) = (b1 (q∗2 ), b2 (q∗1 )). This, of course, is just the computation performed in footnote 2.
Game Theory 129 (Defect,Defect). We do not allow or consider agreements by both players to defect to (Cooperate,Cooperate), which would be better! A Nash equilibrium in pure strategies may not exist, however. Consider, for example, the “Matching Pennies” game: If player 2 plays ‘H’ player 1 wants to play ‘H’, but given that, player 2 would like ‘T’, but given that 1 would like ‘T’, .. We may need mixed strategies to be able to have a Nash equilibrium. The definition for a mixed strategy Nash equilibrium is analogous to the one above and will not be repeated. All that changes is the definition of the strategy space. Since an equilibrium concept which may fail to give an answer is not that useful (hence the general disregard for elimination of dominated strategies) we will consider the question of existence next. Theorem 1 A Nash equilibrium in pure strategies exists for perfect information games. Theorem 2 For finite games a Nash equilibrium exists (possibly in mixed strategies.) Theorem 3 For (N, S, U ) with S ∈ Rn compact and convex and Ui : S 7→ R continuous and strictly quasi concave in si , a Nash equilibrium exists. Remarks: 1. Nash equilibrium is a form of rational expectations equilibrium (actually, a rational expectations equilibrium is a Nash equilibrium, formally.) As in a rational expectations equilibrium, the players can be seen to “expect” their opponent(s) to play certain strategies, and in equilibrium the opponents actually do, so that the expectation was justified. 2. There is an apparent contradiction between the first existence theorem and the fact that Nash equilibrium is defined on the strategic form. However, you may want to think about the way in which assuming perfect information restricts the strategic form so that matrices like the one for matching pennies can not occur. 3. If a player is to mix over some set of pure strategies {σi1 , σi2 , . . . , σik } in Nash equilibrium, then all the pure strategies in the set must lead
130 L-A. Busch, Microeconomics
May2004
to the same expected payoff (else the player could increase his payoff from the mixed strategy by changing the distribution.) This in turn implies that the fact that a player is to mix in equilibrium will impose a restriction on the other players’ strategies! For example, consider the matching pennies game: 12
H
T
H
(1, −1)
(−1, 1)
T
(−1, 1)
(1, −1)
For player 1 to mix we will need that π1 (H, µ2 ) = π1 (T, µ2 ). If β denotes the probability of player 2 playing H, then we need that β − (1 − β) = −β + (1 − β), or 2β − 1 = 1 − 2β, in other words, β = 1/2. For player 1 to mix, player 2 must mix at a ratio of 1/2 : 1/2. Otherwise, player 1 will play a pure strategy. But now player 2 must mix. For him to mix (the game is symmetric) we need that player 1 mixes also at a ratio of 1/2 : 1/2. We have, by the way, just found the unique Nash equilibrium of this game. There is no pure strategy Nash, and if there is to be a mixed strategy Nash, then it must be this. (Notice that we know there is a mixed strategy Nash, since this is a finite game!) The next equilibrium concept we mention is Bayesian Nash Equilibrium (BNE). This will be for completeness sake only, since we will in practice be able to use Nash Equilibrium. BNE concerns games of incomplete information, which, as we have seen already, can be modelled as games of imperfect information. The way this is done is by introducing “types” of one (or more) player(s). The type of a player summarizes all information which is not public (common) knowledge. It is assumed that each type actually knows which type he is. It is common knowledge what distribution the types are drawn from. In other words, the player in question knows who he is and what his payoffs are, but opponents only know the distribution over the various types which are possible, and do not observe the actual type of their opponents (that is, do not know the actual payoff vectors, but only their own payoffs.) Nature is assumed to choose types. In such a game, players’ expected payoffs will be contingent on the actual types who play the game, i.e., we need to consider π(σi , σ−i |ti , t−i ), where t is the vector of type realizations (potentially one for each player.) This implies that each player type will have a strategy, so that player i of type ti will have strategy σi (ti ). We then get the following:
Game Theory 131 Definition 16 A Bayesian Nash Equilibrium is a set of type contingent strategies σ ∗ (t) = (σi∗ (t1 ), . . . , σn∗ (tn )) such that each player maximizes his expected utility contingent on his type, taking other players’ strategies as given, and using the priors in computing the expectation: ∗ ∗ πi (σi∗ (ti ), σ−i |ti ) ≥ πi (σi (ti ), σ−i |ti ),
∀σi (ti ) 6= σi∗ (ti ), ∀i, ∀ti ∈ Ti .
What is the difference to Nash Equilibrium? The strategies in a Nash equilibrium are not conditional on type: each player formulates a plan of action before he knows his own type. In the Bayesian equilibrium, in contrast, each player knows his type when choosing a strategy. Luckily the following is true: Theorem 4 Let G be an incomplete information game and let G∗ be the complete information game of imperfect information that is Bayes equivalent: Then σ ∗ is a Bayes Nash equilibrium of the normal form of G if and only if it is a Nash equilibrium of the normal form of G∗ . The reason for this result is straight forward: If I am to optimize the expected value of something given the probability distribution over my types and I can condition on my types, then I must be choosing the same as if I wait for my type to be realized and maximize then. After all, the expected value is just a weighted sum (hence linear) of the conditional on type payoffs, which I maximize in the second case.
6.2.2
Equilibrium Refinements for the Strategic Form
So how does Nash equilibrium do in giving predictions? The good news is that, as we have seen, the existence of a Nash equilibrium is assured for a wide variety of games.13 The bad news is that we may get too many equilibria, and that some of the strategies or outcomes make little sense from a “common sense” perspective. We will deal with the first issue first. Consider the following game, which is a variant of the Battle of the Sexes game: 13
One important game for which there is no Nash equilibrium is Bertrand competition between 2 firms with different marginal costs. The payoff function for firm 1, say, is ( (p1 − c1 )Q(p1 ) if p1 < p2 π1 (p1 , p2 ) = α(p1 − c1 )Q(p1 ) if p1 = p2 0 otherwise which is not continuous in p2 and hence Theorem 3 does not apply.
132 L-A. Busch, Microeconomics
May2004
12
M
S
M
(6, 2)
(0, 0)
S
(0, 0)
(2, 6)
This game has three Nash equilibria. Two are in pure strategies — (M, M ) and (S, S) — and one is a mixed strategy equilibrium where µ1 (S) = 1/4 and µ2 (S) = 3/4. So what will happen? (Notice another interesting point about mixed strategies here: The expected payoff vector in the mixed strategy equilibrium is (3/2, 3/2), but any of the four possible outcomes can occur in the end, and the actual payoff vector can be any of the three vectors in the game.) The problem of too many equilibria gave rise to refinements, which basically refers to additional conditions which will be imposed on top of standard Nash. Most of these refinements are actually applied to the extensive form (since one can then impose restrictions on how information must be consistent, and so on.) However, there is one common refinement on the strategic form which is sometimes useful. Definition 17 A dominant strategy equilibrium is a Nash equilibrium in which each player’s strategy choice (weakly) dominates any other strategy of that player. You may notice a small problem with this: It may not exist! For example, in the game above there are no dominating strategies, so that the set of dominant strategy equilibria is empty. If such an equilibrium does exist, it may be quite compelling, however. There is another commonly used concept, that of normal form perfect equilibrium. We will not use this much, since a similar perfection criterion on the extensive form is more useful for what we want to do later. However, it is included here for completeness. Basically, normal form perfect will refine away some equilibria which are “knife edge cases.” The problem with Nash is that one takes strategies of the opponents as given, and can then be indifferent between one’s own strategies. Normal form perfect eliminates this by forcing one to consider completely mixed strategies, and only allowing pure strategies that survive after the limit of these completely mixed strategies is taken. This eliminates many of the equilibria which are only brought about by indifference. We first define an “approximate” equilibrium for completely mixed games, then take the limit:
Game Theory 133 Definition 18 A completely mixed strategy for player i is one that attaches positive probability to every pure strategy of player i: µi (si ) > 0 ∀si ∈ Si . Definition 19 A n-tuple µ(²) = (µ1 , . . . , µn ) is an ²-perfect equilibrium of the normal form game G if µi is completely mixed for all i ∈ {1, . . . , n}, and if µi (sj ) ≤ ² if πi (sj , µ−i ) ≤ πi (sk , µ−i ), sk 6= sj , ² > 0. Notice that this restriction implies that any strategies which are a poor choice, in the sense of having lower payoffs than other strategies, must be used very seldom. We can then take the limit as “seldom” becomes “never:” Definition 20 A Perfect Equilibrium is the limit point of an ²-perfect equilibrium as ² → 0. To see how this works, consider the following game: 12
T
B
t
(100, 0)
b
(100, 0)
(−50, −50) (100, 0)
The pure strategy Nash equilibria of this game are (t, T ), (b, B), and (b, T ). The unique normal form perfect equilibrium is (b, T ). This can easily be seen from the following considerations. Let α denote the probability with which player 1 plays t, and let β denote the probability with which player 2 plays T . 2’s payoff from T is zero independent of α. 2’s payoff from B is −50α, which is less than zero as long as α > 0. So, in the ²-perfect equilibrium we have to set (1 − β) < ², that is β > 1 − ² in any ²-perfect equilibrium. Now consider player 1. His payoff from t will be 150β − 50, while his payoff from b is 100. His payoff from t is therefore less than from b for all β, and we require that α < ². As ² → 0, both α and (1 − β) thus approach zero, and we have (b, T ) as the unique perfect equilibrium. While the payoffs are the same in the perfect equilibrium and all the Nash equilibria, the perfect equilibrium is in some sense more stable. Notice in particular that a very small probability of making mistakes in announcing or carrying out strategies will not affect the nPE, but it would lead to a potentially very bad payoff in the other two Nash equilibria.14 14
Note that an nPE is Nash, but not vice versa.
134 L-A. Busch, Microeconomics
6.2.3
May2004
Equilibrium Concepts and Refinements for the Extensive Form
Next, we will discus equilibrium concepts and refinements for the extensive form of a game. First of all, it should be clear that a Nash equilibrium of the strategic form corresponds one-to-one with a Nash equilibrium of the extensive form. The definition we gave applies to both, indeed. Since our extensive form game, as we have defined it so far, is a finite game, we are also assured existence of a Nash equilibrium as before. Consider the following game, for example, here given in both its extensive and strategic forms: 1b 21 U D ©H © HHD U© © HH (u, u) (6, 4) (1, 2) 2 r©© Hr2 u ¡@ d u ¡@ d (u, d) (6, 4) (8, 1) ¡ r¡
4, 6
@ @r
0, 4
¡ r¡
2, 1
@ @r
1, 8
(d, u)
(4, 0)
(1, 2)
(d, d)
(4, 0)
(8, 1)
This game has 3 pure strategy Nash equilibria: (D, (d, d)), (U, (u, u)), and (U, (u, d)).15 What is wrong with this? Consider the equilibrium (D, (d, d)). Player 1 moves first, and his move is observed by player 2. Would player 1 really believe that player 2 will play d if player 1 were to choose U , given that player 2’s payoff from going to u instead is higher? Probably not. This is called an incredible threat. By threatening to play ‘down’ following an ‘Up’, player 2 makes his preferred outcome, D followed by d, possible, and obtains his highest possible payoff. Player 1, even though he moves first, ends up with one of his worst payoffs.16 However, player 2, if asked to follow his strategy, would rather not, and play u instead of d if he finds himself after a move of U . The move d in this information set is only part of a best reply because under the proposed strategy for 1, which is taken as given in a Nash equilibrium, this information set is never reached, and thus it does not matter (to player 2’s payoff) which action is specified. This is a type of behaviour which we may want to rule out. This is done most easily by requiring all moves to be best replies for their part of the game, a concept we will now make more formal. (See also Figure 6.7) 15
There are also some mixed strategy equilibria, namely (U, (P21 (u) = 1, P22 (u) = α)) for any α ∈ [0, 1], and (D, (P21 (u) ≤ 1/4, P22 (u) = 0)). 16 It is sometimes not clear if the term ‘incredible threat’ should be used only if there is some actual threat, as for example in the education game when the parents threaten to punish. The more general idea is that of an action that is not a best reply at an information set. In this sense the action is not credible at that point in the game.
Game Theory 135
Bs©©©
¡ s E ¡ £B £ B £ B
¡A
©©
A As F £B £ B £ B
Ac ©H
HH
HH D Hs ¢A £B ¢ A £ B As ¢ £ Bs G¢ A ¡A H A ¡ ¢ A As ¡ A ¢ s I¢ A ¢ AK ¢ A ¢ A ¢ A ¢ A
C s
Subgames start at: A,E,F,G,D,H
No Subgames at: B,C,I,K
Figure 6.7: Valid and Invalid Subgames Definition 21 Let V be a non-terminal node in Γ, and let ΓV be the game tree comprising V as root and all its followers. If all information sets in Γ are either completely contained in ΓV or disjoint from ΓV , then ΓV is called a subgame. We can now define a subgame perfect equilibrium, which tries to exclude incredible threats by assuring that all strategies are best replies in all proper subgames, not only along the equilibrium path.17 Definition 22 A strategy combination is a subgame perfect equilibrium (SPE) if its restriction to every proper subgame is a subgame perfect equilibrium. In the example above, only (U, (u, d)) is a SPE. There are three proper subgames, one starting at player 2’s first information set, one starting at his second information set, and one which is the whole game tree. Only u is a best reply in the first, only d in the second, and thus only U in the last. Remarks: 1. Subgame Perfect Equilibria exist and are a strict subset of Nash Equilibria. 2. Subgame Perfect equilibrium goes hand in hand with the famous “backward induction” procedure for finding equilibria. Start at the end of the game, with the last information sets before the terminal nodes, 17
The equilibrium path is, basically, the sequence of actions implied by the equilibrium strategies, in other words the implied path through the game tree (along some set of arcs.)
136 L-A. Busch, Microeconomics
May2004
and determine the optimal action there. Then back up one level in the tree, and consider the information sets leading up to these last decisions. Since the optimal action in the last moves is now known, they can be replaced by the resulting payoffs, and the second last level can be determined in a similar fashion. This procedure is repeated until the root node is reached. The resulting strategies are Subgame Perfect. 3. Incredible Threats are only eliminated if all information sets are singletons, in other words, in games of perfect information. As a counterexample consider the following game: a³³³ (0,0) ³ 2³ (0,1) s ³ c A©©© a ÃÃ (1/2,0) 1c©©©B ÃÃÃ Ã s ` ` HH `c`` ` (-1,-1) H 2 H a C HHsP (1,0) PPc PP P (-1,-1) In this game there is no subgame starting with player 2’s information set after 1 chose B or C, and therefore the equilibrium concept reverts to Nash, and we get that (A, (c, c)) is a SPE, even though c is strictly dominated by a in the non-trivial information set. 4. Notwithstanding the above, Subgame Perfection is a useful concept in repeated games, where a simultaneous move game is repeated over and over. In that setting a proper subgame starts in every period, and thus at least incredible threats with regard to future retaliations are eliminated. 5. Subgame Perfection and normal Form Perfect lead to different equilibria. Consider the game we used before when we analyzed nPE: 1b ©H HHb © t© H © r© p © p p p p p p p p p 2p p p p p p p p H pH p r T ¡@ B T ¡@ B ¡ @ ¡ @ r¡ @r r¡ @r
100, 0 −50, −50
100, 0
100, 0
12
T
B
t
(100, 0)
b
(100, 0)
(−50, −50) (100, 0)
As we had seen before, the nPE is (b, T ), but since there are no subgames, the SPE are all the Nash equilibria, i.e., (b, T ), (t, T ) and (b, B).
Game Theory 137 1c
2s
A
D
l
s J
(3, 2, 2)
(1, 1, 1)
d 3
J
a
l
Jr JJ
(0, 0, 0)
(4, 4, 0)
s J
J
Jr JJ
(0, 0, 1)
Figure 6.8: The “Horse” As we can see, SPE does nothing to prevent incredible threats if there are no proper subgames. In order to deal with this aspect, the following equilibrium concept has been developed. For games of imperfect information we cannot use the idea of best replies in all subgames, since a player may not know at which node in an information set he is. We would like to ensure that the actions taken at an information set are best responses nevertheless. In order to do so, we have to introduce what the player believes about his situation at that information set. By introducing a belief system — which specifies a probability distribution over all the nodes in each of the player’s information sets — we can then require all actions to be best responses given the belief system. Consider the example in Figure 6.8, which is commonly called “The Horse.” This game has two pure strategy Nash equilibria, (A, a, r) and (D, a, l). Both are subgame perfect since there are no proper subgames at all. The second one, (D, a, l), is “stupid” however, since player 2 could, if he is actually asked to move, play d, which would improve his payoff from 1 to 4. In other words, a is not a best reply for player 2 if he actually gets to move. Definition 23 A system of beliefs φ is a vector of beliefs for each player, φi , where φi is a vector of probability distributions, φji , over the nodes in each of player i’s information sets Sij : φji
:
xjk
7→ [0, 1],
K X k=1
φji (xjk ) = 1; ∀xjk ∈ Sij .
Definition 24 An assessment is a system of beliefs and a set of strategies, (σ ∗ , φ∗ ).
138 L-A. Busch, Microeconomics
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Definition 25 An assessment (σ, φ) is sequentially rational if ¤ ¤ ∗ £ ∗ £ ∗ ∗ |Sij ) , ∀i, ∀σi ∈ Σi , ∀Sij . |Sij ) ≥ E φ πi (σi , σ−i E φ πi (σi∗ , σ−i Definition 26 An assessment (σ ∗ , φ∗ ) is consistent if (σ ∗ , φ∗ ) = limn→∞ (σn , φn ), where σn is a sequence of completely mixed behavioural strategies and φn are beliefs consistent with σn being played (i.e., obtained by Bayesian updating.) Definition 27 An assessment (σ ∗ , φ∗ ) is a sequential equilibrium if it is consistent and sequentially rational. As you can see, some work will be required in using this concept! Reconsider the horse in Figure 6.8. The strategy combination (A, a, r) is a sequential equilibrium with beliefs α = 0, where α denotes player 3’s probability assessment of being at the left node. You can see this by considering the following sequence of strategies: 1 plays D with (1/n)2 , which converges to zero, as required. 2 plays d with (1/n), also converging to zero. The consistent belief for three thus is given by (from Bayes’ Rule) α(n) =
(1/n)2 n 1 = = , (1/n)2 + (1 − (1/n)2 )(1/n) n2 + n − 1 n + 1 − 1/n
which converges to zero as n goes to infinity. As usual, the tougher part is to show that there is no sequence which can be constructed that will lead to (D, a, l). Here is a short outline of what is necessary: In order for 3 to play l we need beliefs which put at least a probability of 1/3 on being at the left node. We thus need that 1 plays down almost surely, since player 2 will play d any time 3 plays l with more than a 1/4 probability. But as weight shifts to l for 3, and 2 plays d, sequential rationality for 1 requires him to play A (4 > 3). This destroys our proposed setup.
Signalling Games This is a type of game used in the analysis of quality choice, advertising, warranties, or education and hiring. The general setup is that an informed party tries to convey information to an uninformed party. For example, the fact that I spend money advertising should convey the information that my product is of high quality to consumers who are not informed about the quality of my product. There are other sellers of genuinely low quality,
Game Theory 139 (3, 0)
l ´
(−4, −1) ´´
´
´
s ´α
L
1s
R
β QQ
l
s
Q
t¯ (1/2)
´r
(1, 2)
Q
r QQ
Q (−2, 1)
Root: eNature (−3, −2) Q Q (−2, −2)
Qr Q
l
Q
t (1/2) Q Qs
L
s
1
R
s´
´
´
r´
(−1, −1) ´´
´
l
(2, 3)
Figure 6.9: A Signalling Game however, and they will try to mimic my actions. In order to be credible my action should therefore be hard (costly) to mimic. The equilibrium concept used for this type of game will be sequential equilibrium, since we have to model the beliefs of the uninformed party. Consider the game in Figure 6.9. Nature determines if player 1 is a high type or a low type. Player 1 moves left or right, knowing his type. Player 2 then, without knowing 1’s true type, moves left or right also. The payoffs are as indicated. This type of game is called a game of asymmetric information, since one of the parties is completely informed while the other is not. The usual question is if the informed party can convey its private information or not. In the above game, the Nash equilibria are the following: Let player 1’s strategy vector (S1 , S2 ) indicate 1’s action if he is the low and high type, respectively, while player 2’s strategy vector (s1 , s2 ) indicates 2’s response if he observes L and R, respectively. Then we get that the pure strategy Nash equilibria are ((R, R), (r, r)), ((R, L), (r, r)), and ((L, R), (l, l)). Now introduce player 2’s beliefs. Let 2’s belief of facing a low type player 1 be denoted by α if 2 observes L, and by β if 2 observes R. We then can get two sequential equilibria: ((L, R), (l, r), (α = 1, β = 0)) and ((R, R), (r, r), (α = 0, β = 0.5)) (It goes without saying that you should try to verify this claim!). The first of these is a separating equilibrium. The action of player 1 completely conveys the private information of player 1. Only low types move Left, only high types move Right, and the move thus reveals the type of player. The second equilibrium, in contrast, is a pooling equilibrium. Both types take the same move in equilibrium, and no information is transmitted. Notice, however, that the belief that α = 0 is somewhat stupid. If you find yourself, as player 2, inadvertently in your first information set, what should you believe? Your beliefs here say that
140 L-A. Busch, Microeconomics
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you think it must have been the high type who made the mistake. This is “stupid”, since the high type’s move L is strictly dominated by R, while the low type’s move L is not dominated by R. It would be more reasonable to assume, therefore, that if anything it was the low type who was trying to “tell you something” by deviating (you are not on the equilibrium path if you observe L, remember!) There are refinements that impose restrictions like this last argument on the beliefs out of the equilibrium path, but we will not go into them here. Look up the Cho-Kreps criterion in any good game theory book if you want to know the details. The basic idea is simple: of the equilibrium path you should only put weight on types for whom the continuation equilibria off the equilibrium path are actually better than if they had followed the proposed equilibrium. The details are, of course, messy. Finally, notice a last problem with sequential equilibrium. Minor perturbations of the extensive form change the equilibrium set. In particular, the two games in Figure 6.10 have different sequential equilibria, even though the games would appear to be quite closely related. 1e ©© ¡ C ©© ¡ C L ©© ¡ M C R © ¡ C ©© © ¡ C 2 © © s¡ Cs ¡@ ¡@ (2, 2) ¡u d@ ¡u d@ ¡ @ ¡ @ (3, 3)
(1.5, 0) (0, 0)
(1, 1)
1e
L ©©
©© ©
(2, 2)
©
©©
©©
© s© © ¡@ @d ¡u @ ¡
(3, 3)
©©
(1.5, 0)
©
©©
Not L 1s ©
© ©© © M ©
2 ©©
s ©©
u ©©
(0, 0)
R
d
(1, 1)
Figure 6.10: A Minor Perturbation?
6.3
Review Problems
Question 1: Provide the definition of a 3-player game in extensive form. Then draw a well labelled example of such a game in which you indicate all the elements of the definition. Question 2: Define “perfect recall” and provide two examples of games
Game Theory 141 which violate perfect recall for different reasons. Question 3: Assume that you are faced with some finite game. Will this game have a Nash equilibrium? Will it have a Subgame Perfect Equilibrium? Why can you come to these conclusions? Question 4: Consider the following 3 player game in strategic form: Lef t 12
L
R
Player 3 C
Right 12 L
R
C
U
(1, 1, 1) (2, 1, 2)
(1, 3, 2)
U
(2, 2, 2) (4, 2, 4)
(2, 6, 4)
C
(1, 2, 1) (1, 1, 1)
(2, 3, 3)
C
(5, 0, 1) (1, 1, 1)
(0, 1, 1)
D
(2, 1, 2) (1, 1, 3)
(3, 1, 1)
D
(3, 2, 3) (2, 2, 4)
(4, 2, 2)
Would elimination of weakly dominated strategies lead to a good prediction for this game? What are the pure strategy Nash equilibria of this game? Describe in words how you might find the mixed strategy Nash equilibria. Be clear and concise and do not actually attempt to solve for the mixed strategies. Question 5: Find the mixed strategy Nash equilibrium of this game: 12
L
R
C
U
(1, 4) (2, 1)
(4, 2)
C
(3, 2) (1, 1)
(2, 3)
Question 6: Consider the following situation and construct an extensive form game to capture it. A railway line passes through a town. Occasionally, accidents will happen on this railway line and cause damage and impose costs on the town. The frequency of these accidents depends on the effort and care taken by the railway — but these are unobservable by the town. The town may, if an accident has occurred, sue the railway for damages, but will only be successful in obtaining damages if it is found that the railway did not use a high level of care. For simplicity, assume that there are only two levels of effort/care (high and low) and that the courts can determine with certainty which level was in fact used. Also assume that going to court costs the railway and the town money, that effort is costly for the railway (high effort reduces profits), that accidents cost the railway and the town money and that this cost is independent of the effort level (i.e., there is a “standard accident”). Finally, assume that if the railway is “guilty” it has to pay the town’s damages and court costs.
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Question 7: Consider the following variation of the standard Battle of the Sexes game: with probability α Juliet gets informed which action Romeo has taken before she needs to choose (with probability (1 − α) the game is as usual.) a) What is the subgame perfect equilibrium of this game? b) In order for Romeo to be able to insist on his preferred outcome, what would α have to be? Question 8: (Cournot Duopoly) Assume that inverse market demand is given by P (Q) = (Q − 10)2 , where Q refers to market output by all firms. Assume further that there are n firms in the market and that they all have zero marginal cost of production. Finally, assume that all firms are Cournot competitors. This means that they take the other firms’ P outputs as given and consider their own inverse demand to be pi (qi ) = ( j6=i qj + qi − 10)2 . Derive the Nash equilibrium output and price. (That is, derive the multilateral best response strategies for the output choices: Given every other firm’s output, a given firm’s output is profit maximizing for that firm. This holds for all firms.) Show that market output converges to the competitive output level as n gets large. (HINT: Firms are symmetric. It is then enough for now to focus on symmetric equilibria. One can solve for the so-called reaction function of one firm (it’s best reply function) which gives the profit maximizing level of output for a given level of joint output by all others. Symmetry then suggests that each firm faces the same joint output by its (n − 1) competitors and produces the same output in equilibrium. So we can substitute out and solve.) Question 9∗ : Assume that a seller of an object knows its quality, which we will take to be the probability with which the object breaks during use. For simplicity assume that there are only two quality levels, high and low, with breakdown probabilities of 0.1 and 0.4, respectively. The buyer does not know the type of seller, and can only determine if a good breaks, but not its quality. The buyer knows that 1/2 of the sellers are of high quality, and 1/2 of low quality. Assume that the seller receives a utility of 10 from a working product and 0 from a non-working product, and that his utility is linear in money (so that the price of the good is deducted from the utility received from the good.) If the seller does not buy the object he is assumed to get a utility level of 0. The cost of the object to the sellers is assumed to be 2 for the low quality seller and 3 for the high quality seller. We want to investigate if signalling equilibria exist. We also want to train our understanding of sequential equilibrium, so use that as the equilibrium concept in what follows. a) Assume that sellers can only differ in the price they charge. Show that no separating equilibrium exist.
Game Theory 143 b) Now assume that the sellers can offer a warranty which will replace the good once if it is found to be defective. Does a separating equilibrium exist? Does a pooling equilibrium exist? Question 10∗ : Assume a uniform distribution of buyers over the range of possible valuations for a good, [0, 2]. a) Derive the market demand curve. b) There are 2 firms with cost functions C1 (q1 ) = q1 /10 and C2 (q2 ) = q22 . Find the Cournot Equilibrium and calculate equilibrium profits. c) Assume that firm 1 is a Stackelberg leader and compute the Stackelberg equilibrium. (This means that firm 1 moves first and firm 2 gets to observe firm 1’s output choice. The Stackelberg equilibrium is the SPE for this game.) d) What is the joint profit maximizing price and output level for each firm? Why could this not be attained in a Nash equilibrium?
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Chapter 7 Review Question Answers 7.1
Chapter 2
Question 1: a) There are multiple C(·) which satisfy the Weak Axiom. Note, however, that you have to check back and forth to make sure that the WA is indeed satisfied. (I.e., C({x, y, z}) = {x}, C({x, y}) = {x, y} does not satisfy the axiom since while the check for x seems to be ok, you also have to check for y, and there it fails.) One choice structure that does work is C({x, y, z}) = {x}, C({x, z, w}) = {x}, C({y, w, z}) = {w}, C({y, w}) = {w}, C({x, z}) = {x}, C({x, w}) = {x}, C({x}) = {x}. b) Yes (I thought of that first, actually, in deriving the above) it is x º w º y º z. c) Yes, it is transitive. d) I was aiming for an application of out Theorem: our set of budget sets B does not contain all 2 and 3 element subsets of X. Missing are {x, y, w}, {x, y}, {y, z}, {w, z}. e) The best way to go about this one is to determine where we can possibly get this to work. Examination of the sets B shows that the two choices y, x only appear in one of the sets and thus must be our key if we want to satisfy the WA without having rational preferences. Some fiddling reveals that the following works: C({x, y, z}) = {x}, C({x, z, w}) = {w}, C({y, w, z}) = {y}, C({y, w}) = {y}, C({x, z}) = {x}, C({x, w}) = {w}, C({x}) = {x}. The problem is intransitivity, since the above implies that y º w º x º z but we also have x º y! Question 2: Here you have to make sure to maximize income for any given 145
146 L-A. Busch, Microeconomics work hrs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Part a 1 then 2 112 124 136 148 160 172 188 204 212 220 234 248 262 276 290 304
2 then 1 108 116 130 144 158 172 186 200 212 224 236 248 260 272 288 304
Max 112 124 136 148 160 172 188 204 212 224 236 248 262 276 290 304
May2004
Partb 1 then 2 2 then 1 112 108 124 116 136 130 32 148 145 31 160 160 172 174 32 188 189 31 204 204 212 216 220 228 234 32 240 249 31 252 264 264 2 278 3 276 293 31 292 308 308
Max 112 124 136 148 160 174 23 189 13 204 216 228 240 252 264 278 23 293 13 308
Table 7.1: Table 1: Computing maximal income amount of work. In parts (a) and (b) you have to choose to work either job 1 then job 2 (after 8 hours in job 1) or job 2 then job 1. Simply plotting the two and then taking the outer hull (i.e., the highest frontier) for each leisure level gives you the frontier. In (a) they only cross twice (at 9 and 12 hours of work) while in part (b) they cross 4 times. You can best see this effect by considering a table in which you tabulate total hours worked against total income, computed by doing job 1 first, and by doing job 2 first. This is shown in Table 1. In neither part a) nor in part b) is the budget set convex. c) This is a possibly quite involved problem. The intuitive answer is that it will not matter since marginal and average pay is (weakly) increasing in both jobs. Here is a more general treatment of these questions: We really are faced with an maximization problem, to max income given the constraints, for any given total amount worked. Let h1 and h2 denote hours worked in jobs 1 and 2, respectively. Then the objective function is I(h1 , h2 ) = h1 w1 (h1 ) + h2 w2 (h2 ), where wi (hi ) are½ the wage schedules. w1 if h1 ≤ C1 and The wage schedules have the general form w1 (h1 ) = w1 if h1 ≥ C1 ½ w2 if h2 ≤ C2 , where w i < wi . I ignore here that no hours w2 (h2 ) = w2 if h2 ≥ C2 above 8 are possible for either job, choosing to put that information into the constraints later.
Answers 147 Consider now the iso-income curves in (h1 , h2 ) space which result. We will have four regions to consider, namely A = {(h1 , h2 )|h1 ≤ C1 , h2 ≤ C2 }, B = {(h1 , h2 )|h1 ≤ C1 , h2 ≥ C2 }, C = {(h1 , h2 )|h1 ≥ C1 , h2 ≤ C2 }, D = {(h1 , h2 )|h1 ≥ C1 , h2 ≥ C2 }. The slope of the iso-income curves for the regions is easily seen to be the negative of the ratio of wages, so we have S(A) = −w 1 /w2 , S(B) = −w 1 /w2 , S(C) = −w 1 /w2 , S(D) = −w 1 /w2 . It is obvious that S(C) < S(D) and S(A) < S(B), as well as that S(C) < S(A) and S(D) < S(B). This implies, of course, that S(C) < S(A) < S(B) as well as that S(C) < S(D) < S(B), with the comparison of S(A) to S(D) indeterminate. (But luckily not needed in any case.) The important fact which follows from all of this is that the iso-income curves are all concave to the origin and piece-wise linear. Now superimpose the choice sets onto this. Note that without any restrictions H = h1 + h2 , that is, for any given number of hours H the hours in each job are “perfect substitutes”. These iso-hour curves are all straight lines with a slope of −1. (For our parameters all of S(A), S(C), S(D) are less than −1, while S(B) > −1, but his is not important.) For parts (a) and (b) the feasible set consists of the boundaries of the 8 × 8 square of feasible hours, where either hi = 0, hj < 8, or where hi = 8, 0 ≤ hj ≤ 8. The choice set is thus given by the intersection of the iso-hour lines with the feasible set (the box boundary). In part (c) this restriction is removed and the whole interior of the box is feasible. Due to the concavity to the origin of the iso-income lines this is of no relevance, however. Note how I have used our usual techniques of iso-objective curves and constraint sets to approach this problem. Works pretty well, doesn’t it! d) Now we “clearly” take up jobs in decreasing order of pay, starting with the highest paid and progressing to the lower paid ones in order. The resulting budget set will be convex. Question 3: The consumer will © ª 0.6 maxx x0.3 1 x2 + λ (m − x1 p1 − x2 p2 ) which leads to the first order conditions 0.3x−0.7 x0.6 1 2 = λp1 ,
−0.4 0.6x0.3 = λp2 , 1 x2
x1 p1 + x2 p2 = m.
The utility function is quasi-concave (actually, strictly concave in this case) and the budget set convex, so the second order conditions will be satisfied. Combining the first two first order conditions we get p1 0.3x2 = 0.6x1 p2
=⇒
x2 =
2p1 x1 . p2
148 L-A. Busch, Microeconomics
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Substitute into the budget constraint and simplify: x1 p 1 +
2p1 x1 p 2 = m p2
=⇒
x1 =
m . 3p1
³ ´ 2m m 2m Now use this to solve for x2 : x2 = 3p . So (x (p, m), x (p, m)) = , . 1 2 3p1 3p2 2 To find the particular quantity demanded, simply plug in the numbers and simplify: x1 = x2 =
412 + 24 436 3 × 412 + 1 × 72 = = ; 3×3 3 3
2(3 × 412 + 1 × 72) 2(412 + 24) = = 872. 3×1 1
Question 4: The key is to realize that this utility function is piece-wise linear with line segments at slopes −5, −1, −1/5, from left to right. The segments join at rays from the origin with slopes 3 and 1/3. Properly speaking, neither the Hicksian nor the Marshallian demands are functions. The function has either a perfect substitute or Leontief character. In the former the substitution effects approach infinity, in the latter they are zero. Demands are easiest derived from the price offer curve, which is a nice zigzag line. It starts at the intercept of the budget with the vertical axis (point A). It follows the indifference curve segment with −5 slope to the ray with slope 3. Call this point B. From there it follows the ray with slope 3 until that ray intersects a budget drawn from A with a slope of 1 (point C). It then continues on this budget and the coinciding indifference curve segment to the ray with slope 1/3 (point D). Up along that ray to an intersection with a budget from A with slope 1/5 (point E), along that budget to the intercept with the horizontal axis (point F), and then along the horizontal axis off to infinity. x2 AX £ X ³ ³ X £ XXX @ C ³³ ³ X ³ C @£ C XX XXX E ³³³ C £@ XX ³ ³³XXX C£ @ XXX ³³ BC£ ³ @ XX ³ D ³ XXX £C @ ³³ XXX ³ £ C ³³ @ XX ³ XX F £³ C @
x1 Now we can solve for the demands along the different pieces of the offer curve and get the Marshallian demand. Note that demand is either a whole range, or a “proper demand”. The ranges can be computed from the endpoints (i.e., A to B, C to D, E to F.) Along the rays demand is solved as for
Answers 149 a Leontief consumer: we know the ratio of consumption, we know the budget. So for example on the first ray segment (B to C) we know that x2 = 3x1 . Also, p1 x1 + p2 x2 = w. Hence x1 (p1 + 3p2 ) = w, and x1 = w/(p1 + 3p2 ). (For the Hicksian demand we simply need to fix one indifference curve and compute the points along it. We then get either a segment (like A to B above), or we stay at a kink for a range of prices.) The demands for good 1 therefore are 0, if p1 /p2 > 5; [0, 5w/(8p )] , if 5 = p1 /p2 ; 1 if 5 > p1 /p2 > 1; w/(3p2 + p1 ), x1 (p, w) = [w/(4p1 ), 3w/(4p1 )] , if p1 /p2 = 1; 3w/(3p1 + p2 ), if 1 > p1 /p2 > 1/5; [3w/(8p1 ), w/p1 ] , if p1 /p2 = 1/5; w/p1 , if 1/5 > p1 /p2 . 0, if p1 /p2 > 5; [0, u/8] , if p1 /p2 = 5; if 5 > p1 /p2 > 1; u/8, h1 (p, u) = [u/8, 3u/16] , if p1 /p2 = 1; 3u/16, if 1 > p1 /p2 > 1/5; [3u/16, u] , if p1 /p2 = 1/5; u, if 1/5 > p1 /p2 .
The demands for good 2 are similar and left as exercise. The income expansion paths and Engel curves can be whole regions at price ratios 1,5,1/5, otherwise the income expansion paths are the axes or rays, and the Engel curves are straight increasing lines.
Question 5: The elasticity of substitution measures by how much the consumption ratio changes as the price ratio changes (both measured in percentages.) In other words, as the price ratio changes the slope of the budget changes and we know this will cause a change in the ratio of the quantity demanded of the goods. But by how much? The higher the value of the elasticity, the larger the response in demands. Question 6: First we need to realize that the utility index which each function assigns to a given consumption point does not have to be the same. Instead, as long as the MRS is identical at every point, two utility functions represent the same preferences. So instead of taking the limit of the utility function directly, we will take the limits of the MRS and compare those to the MRSs of the other functions. M RS =
x21−ρ (1/ρ)(xρ1 + xρ2 )(1−ρ)/ρ (ρxρ−1 xρ−1 u1 1 ) 1 = . = = u2 (1/ρ)(xρ1 + xρ2 )(1−ρ)/ρ (ρxρ−1 xρ−1 x11−ρ 2 2 )
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The MRSs for the other functions are CD:
x2 ; x1
Perfect Sub: 1;
Leon: 0, or ∞.
So, consider the Leontief function min{x1 , x2 }. Its MRS is 0 or ∞. But as ρ → −∞ we see that (x2 /x1 )1−ρ → (x2 /x1 )∞ . But if x2 > x1 the fraction is greater than 1 and an infinite power goes to infinity. If x2 < x1 the fraction is less than one and the power goes to zero. The Cobb-Douglas function x1 x2 has MRS x2 /x1 . But as ρ → 0 the MRS of our function is just that. The perfect substitute function x1 + x2 has a constant MRS of 1. But as ρ → 1 the MRS of our function is (x2 /x1 )0 = 1. Therefore the CES function “looks like” those three functions for those choices of ρ. The parameter ρ essentially controls the curvature of the IC’s. Question 7: Set up the consumer’s optimization problem: maxx1 ,x2 ,x3 {x1 + lnx2 + 2lnx3 + λ(m − p1 x1 − p2 x2 − p3 x3 )} . The FOCs are 1 − λp1 = 0;
1 − λp2 = 0; x2
2 − λp3 = 0 x3
and the budget. The first of these allows us to solve for λ = 1/p1 . Therefore the second and third give us x2 = p1 /p2 and x3 = 2p1 /p3 . Combining this with the budget we get x1 = m/p1 − 3. Of course, this is only sensible if m > 3p1 . If it is not we must be at a corner solution. In that case x1 = 0 and all money is spent on x2 and x3 . The second and third FOC above tell us that x3 /x2 = 2p2 /p3 . Hence (remember x1 = 0 now) m = p2 x2 + 2p2 x2 and x2 = m/(3p2 ) while x3 = 2m/(3p3 ). So we get ´ ³ m − 3, p1 , 2 p1 if m > 3p1 p1 p2 ´ p3 x(p, m) = ³ 0, m , 2m if m ≤ 3p1 3p2 3p3 Question 8: Here we have a pure exchange economy with 2 goods and 2 consumers. We can best represent this in an Edgeworth box.
Answers 151 20
5
11.25
@ SSS S · S SSS S@ · sˆ S SSS S@ω · S SSS · S@ S SS· S@ S S S @ ·SSS S · SSS S x∗ -S· SSS S ω S S · S S 11 SSS · S S SSS · S SSS · S SSS · S SSS · S SSS · S SSS · S SSS · SSS · S SSS · S ·
OA
OB 20 8 Suppose x1 is on the horizontal axis and x2 is on the vertical, and let consumer B have the lower left hand corner as origin, consumer A the upper right hand corner. (I made this choice because I like to have the “harder” consumer oriented the usual way.) The dimensions of the box are 20 by 20 units. The first thing to do is to find the Pareto Set (the contract curve), since we know that any equilibrium has to be Pareto efficient. The MRS for person A is 4/3, the MRS for person B is 3x2 /(4x1 ). Therefore the Pareto Set is defined by x2 /x1 = 16/9 (in person B’s coordinates.) This is a straight ray from B’s origin with a slope greater than 1, and therefore above the main diagonal. The Pareto set is this ray and the portion of the upper boundary of the box from the ray’s intersection point to the origin of A. There now are 2 possibilities for the equilibrium. Either it is on the ray, and therefore must have a price ratio of 4/3. Or it is on the upper boundary of the box, in which case the price ratio must be below 4/3, but we know that B’s consumption level for good 2 is 20. In the first case we have 2 equations defining equilibrium. The ray, x2 = 16x1 /9, and the budget line (x2 − 11) = 4(8 − x1 )/3. From this we get 16x1 /9 − 11 = 32/3 − 12x1 /9 and from that 28x1 /9 = 65/3 and thus x1 = 195/28 < 20. It follows that x2 = (16 × 195)/(9 × 28) = 780/63 = 260/21 < 20. Since both of B’s consumption points are strictly within the interior of the box, we are done. All that remains is to compute A’s allocation. The equilibrium is therefore µ µ ¶ µ ¶¶ 4 195 780 195 780 A B (p∗, (x ), (x )) = , 20 − , 20 − , , . 3 28 63 28 63 Question 9: Again we have a square Edgeworth box, 20×20. Again I choose to put consumer B on the bottom left origin. B’s preferences are quasi-linear with respect to x1 , A’s are piece-wise linear with slopes 4/3 and 3/4 which
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meet at the kink line x2 = x1 (in A’s coordinates!) which coincides with the main diagonal. The MRS for B’s preferences is 3x2 /4. We have Pareto optimality when 3x2 /4 = 4/3 → x2 = 16/9 and when 3x2 /4 = 3/4 → x2 = 1. So, the Pareto Set is the vertical axis from B’s origin to xB 2 = 1, the horizontal line xB = 1 to the main diagonal (the point (1, 1) in other words), up the 2 B B main diagonal to the point (x1 , x2 ) = (16/9, 16/9), from there along the horizontal line xB 2 = 16/9 to the right hand edge of the Box, and then up that border to A’s origin. OA 20 Z ZS Z SZ ZZ Z ZSZ Z ZS Z ¡ Z Z SZZ Z S Z Z Z Z ¡ ZS Z Z ZS ZZZ Z Z¡ Z Z S 11 ωSS Z S S SS S¡ S S SS S S SS S S SS S S SS S S SSS S S S S S S S 16/9 ¡ x∗ S S SSS 1 SS S SSS
OB 8 20 By inspection, the most likely candidate for an equilibrium is a price ratio of 4/3 with an allocation on the second horizontal line segment. Let us attempt to solve for it. First, the budget equation (in B’s coordinates) is 3(x2 − 11) = 4(8 − x1 ). Second, we are presuming that x2 = 16/9. So we get 11 . Since this is less than 20 we have found 16/3 − 33 = 32 − 4x1 , or x1 = 14 12 an interior point and are done. The equilibrium is ¶ µ ¶¶ µ µ 4 1 164 11 16 A B , 5 , (p∗, (x ), (x )) = , 14 , . 3 12 9 12 9 Question 10: To prove this we need to show the implication in both directions: (⇐) : Suppose x  y. Then ∃B, x, y ∈ B with the property that x ∈ C(B), y ∈ / C(B). Consider all other B 0 ∈ B with the property that x, y ∈ B 0 . By the Weak Axiom ∃ 6 B 0 with y ∈ C(B 0 ) since otherwise the set B would violate the weak axiom (applied to the choice y with the initial set B 0 .) Therefore x Â∗ y. (⇒) : Let x Â∗ y. The first part of the definition requires ∃B, x, y, ∈ B, x ∈ C(B). By the weak axiom there are two possibilities: either all B 0 ∈ B with x, y ∈ B 0 have {x, y} ∈ C(B 0 ) or none have y ∈ C(B 0 ). The second part of
Answers 153 the definition requires us to be in the second case, but then y ∈ / C(B), and so x  y. If the WA fails a counter example suffices: Let X = {x, y, z}, B = {{x, y}, {x, y, z}}, C({x, y}) = {x}, C({x, y, z}) = {x, y}. This violates the WA. C({x, y}) = {x} demonstrates that x  y by definition. On the other hand it is not true that x Â∗ y (let B = {x, y} and B 0 = {x, y, z} in the definition of Â∗ . Question 11: a) This is another 20 by 20 box, with the endowment in the centre. Suppose B’s origin on the bottom left, A’s the top right. As in question 8, A’s indifference curves have a constant MRS of α and are perfect substitute type. B’s ICs have a MRS of βx2 /x1 and are Cobb-Douglas. The contract curve in the interior must have the MRSs equated, so it occurs where (in B’s coordinates) x2 /x1 = α/β. This is a straight ray from B’s origin and depending on the values of α and β it lies above or below the main diagonal. Since these cases are (sort of) symmetric we pick one, and assume that α/β > 1. The contract curve is this ray and then the part of the upper edge of the box to A’s origin. As in question 8 there are two cases for the competitive equilibrium. Either it occurs on the part of the Contract curve interior to the box, or it occurs on the boundary of the box. In the first case the slope of the budget and hence the equilibrium price must be α, since both MRSs have that slope along the ray and in equilibrium the price must equal the MRS. Note that the budget now coincides with A’s indifference curve through the endowment point. The equilibrium allocation is determined by the intersection of the contract curve and this budget/IC. So we have two equations in two unknowns: α=
x2 − 10 10 − x1
and x2 =
α x1 . β
Hence α(10 − x1 ) = αx1 /β or αβ10 = x1 (α + αβ) and thus x1 = β10/(1 + β) and x2 = α10/(1 + β). These are the consumption levels for B. A gets the B B B rest. The equilibrium thus would be (p∗ , (xA 1 , x2 ), (x1 , x2 )) = ¶ µ ¶¶ µ µ 2(1 + β) − α β10 α10 2+β , 10 , , α, 10 1+β 1+β 1+β 1+β which only makes sense if the allocation indeed is interior, that is, as long as 10α/(1 + β) < 20, or (α − β) < (2 + β). If that is not true we find ourselves in the other case. In that case we know that we are looking for an equilibrium on the upper boundary of the box and A thus know that xB 2 = 20 while x2 = 0. It remains to determine p and the allocations for good 1. At the equilibrium point the budget must be flatter
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than A’s IC (so that A chooses to only consume good 1). The allocation must also be the optimal choice for B and hence the budget must be tangent to B’s IC, since for B this is an interior consumption bundle (interior to B’s consumption set, that is.) So we again have to solve two equations in 2 unknowns: c2 − 10 βc2 = p and p = c1 10 − c1
while c2 = 20.
It follows that β20(10 − c1 ) = 10c1 and therefore c1 = 20β/(1 + 2β). A gets the rest. The equilibrium is therefore µ µ ¶ µ ¶¶ 1+β β20 ∗ A A B B (p , (x1 , x2 ), (x1 , x2 )) = 1 + 2β, 20 ,0 , , 20 . 1 + 2β 1 + 2β b) All endowments above and to the right of the line x2 = 40 − 2x1 in B’s coordinates will lead to a boundary equilibrium. All those on this line and below will lead to an interior equilibrium with p = 2. Question 12: a) The social planner’s problem is ¾ ½ √ 1 maxl ln(4 16 − l) + ln(l) 2 which has first order condition 1 2 1 √ − √ + = 0. 4 16 − l 16 − l 2l √ Hence 16 − l = l and so l∗ = 8, x∗ = 8, c∗ = 8 2. b) Since the profits, we solve for the firm √ consumer’s problem requires √ first. maxx {p4 x − wx} has FOC 2p/ x = w and leads to firm labour demand of x(p, w) = 4p2 /w2 , consumption good supply of c(p, w) = 8p/w, and profits of π(p, w) = 4p2 /w. The consumer will ½ ¾ 1 maxc,l lnc + lnl + λ (16w + π(p, w) − pc − wl) 2 which has first order conditions 1/c − λp = 0; 1/(2l) − λw = 0; 16w + π(p, w) = pc + wl. The first 2 imply that pc = 2lw. Substituting into the third and using the profits computed above yields demand of c(p, w) = 32w/(3p) + 8p/(3w) and leisure demand of l(p, w) = 16/3 + 4p2 /(3w2 ). We can now solve for the equilibrium price ratio. Take any one market and set demand equal to supply. For the goods market this implies 32w/(3p) +
Answers 155 8p/(3w) = 8p/w, and hence p2 /w2 = 2. Substituting into the demands and supplies this gives l∗ = 8, and hence all values are the same as in the social planner’s problem in part a). You may want to verify that you could have solved for the price ratio from the labour market. The complete statement of the general equilibrium is: The equilibrium price √ √ ratio is p/w = √ 2, the consumer’s allocation is (c, l) = (8 2, 8), and the firm produces 8 2 units consumption good from 8 units input. Note that we cannot state profits without fixing one √ of the prices. So let w = 1 (so that we use labour as numeraire), then p = 2 and profits are 8.
7.2
Chapter 3
Question 1: a) Zero arbitrage means that whichever way I move between periods, I get the same final answer. In particular I could lend in period 1 to collect in period three, or I could lend in period 1 to period 2, and then lend the proceeds to period 3. Hence the condition is (1 + r12 )(1 + r23 ) = (1 + r13 ). Note that if we where to treat r13 not as a simple interest rate but as a compounding one, we’d get (1 + r12 )(1 + r23 ) = (1 + r13 )2 instead. b) You have to adopt one period as your viewpoint and then put all other values in terms of that period (by discounting or applying interest). With period 3 as the viewpoint I use period 3 future values for everything: B = {(c1 , c2 , c3 )|(1 + r13 )c1 + (1 + r23 )c2 + c3 = (1 + r13 )m1 + (1 + r23 )m2 + m3 } Note that any other viewpoint is equally valid. The restriction in (a) means that it does not matter which interest rate I use to compute the forward value of c1 , say. Indeed, without that restriction I would get an infinite budget if it is possible to borrow infinite amounts. With some borrowing constraints in place I would have to compute the highest possible arbitrage profits for the various periods and compute the resulting budget. c) This is a standard downward sloping budget line in (c2 , c3 ) space with a slope of −(1 + r23 ). It does not necessarily have to go through the endowment point (m2 , m3 ), however. It will be below that point if c1 > m1 and above that point if c1 < m1 .
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c3
S{m1 > c1 } S S S ¡ {m1 < c1 } S S¡ S S¡ S S S m m3 S S SS SS
m2
c2
Question 2: a) The easy way to get this is to first ignore the technology. The market budget is a straight line with a slope of −1 through the point (100, 100), which is truncated at the point (160, 40), where the budget becomes vertical. Note that the gross rate of return is 1 since the interest rate is 0. Now consider the technology and the implications of zero arbitrage: Joe can move consumption from period 1 to period 2 in two ways, via the financial market, or via “planting”. Both must yield the same gross rate of return at the optimum (why? we know that at the optimum of a maximization problem the last dollar allocated to each option must yield the same marginal benefit.) The gross rate of return at the margin is nothing but the marginal product of the √ technology, however. So, compute the MP (5/ x1 ) and find the investment level at which the MP is 1. 5 √ =1 x1
−→
5=
√
x1
−→
x1 = 25.
At optimal use at an interior optimum Joe invests 25 units (and collects 50 in the next period.) This means that from any point on the financial market budget Joe can move left 25 and up 50. So that gives a straight line with slope −1 which starts at (0, 225) and goes to (135, 90). After this point there is a corner solution in technology choice: Joe cannot use the market any more. The technology therefore may give a higher return than the market. So the budget follows the (flipped over to the left) technology frontier down to the point (160, 40), and down to (160, 0) from there. b) First simplify the preferences (this step is not necessary!). Applying a natural logarithm gives the function Uˆ (c1 , c2 ) = c41 c62 which represents the .6 same preferences. Applying the 10th root gives U˜ (c1 , c2 ) = c.4 1 c2 which also represents the same preferences and is recognized as a Cobb-Douglas. Now you can either compute the MRS (2c2 /3c1 ) and set that equal to 1 (since most of the budget has a slope of −1 and we know that M RS = Slope at the optimum.) That gives you two equations in two unknowns, and we can solve: c2 = 225 − c1 , c2 = 3c1 /2
→
450 = 5c1
→
c1 = 90, c2 = 135.
Answers 157 We then double check that the assumption that we are on the −1 sloped portion of the budget was correct, which it is (by inspection.) Or you could use the demand function for C-D, so you know µ ¶ ¶ µ .4M .6M .4 × 225 .6 × 225 , , (c1 , c2 ) = = (90, 135). = p1 p2 1+0 1 Now this is his final consumption bundle. In order to get there he invested 25 units, so on the “market budget” line he must have started at (115, 85), and that required him to borrow 15 units. In summary, he borrows 15, giving him 115, of which he invests 25, so he has 90 left to consume. In the next period he gets 100 from his endowment, 50 from the investment, for a total of 150, of which he has to use 15 to pay back the loan, so he can consume 135! Question 3: I will not draw the diagram but describe it. You should refer to a rough diagram while reading these solutions to make sense of them. a) The indifference curves have two segments with a slope of −1.3 and −1.2 respectively. The switch (kink) occurs where µ ¶ µ ¶ 12 13 23 → c2 = (22×13−23×12)c1 /10 = c1 . c1 + c2 = 22 c1 + c 2 10 10 b) Note that the budget has a slope of 1.25 which is less than 1.3 and more than 1.2, so she consumes at the kink. Thus she is on the kink line and the budget: c1 = c2 and − 1.25 =
c2 − 8 c1 − 9
→
c1 = c2 = 77/9.
→
c1 = c2 = 73/9.
c) Again she consumes at the kink, so c1 = c2 and − 1.25 =
c2 − 12 c1 − 5
d) Here we need to work back. Note that at the slopes implied by the interest rates she continues to consume at her kink line. The reason is that both 1.25 and 1.28 are bigger than 1.2, the slope of her lower segment, but less than 1.3, the slope of the steep segment. Hence optimal consumption is at the kink and she borrows if she has less period 1 endowment than period 2 endowment. She lends money if she has larger period 1 endowment than period 2 endowment. So for all endowments above the main diagonal she is
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a borrower, for all endowments below a lender. e) Now she never trades. To lend money the budget slope is 1.18 which is less than either of her IC segment slopes. She would not want to lend ever at this rate no matter what her endowment. On the other hand, suppose she were to borrow. The budget slope is 1.32 which is steeper than even her steepest IC segment. She would not borrow. Thus she remains at the kink in her budget (the endowment point) no matter where it is. Question 4: Again I will not draw the diagram but describe it. You should refer to a rough diagram while reading these solutions to make sense of them. a) This is a 20 by 10 box. Suppose A’s origin on the bottom left, B’s the top right. A’s indifference curves have a MRS of c2 /(αc1 ) and are nice C-D type curves. B’s ICs have a MRS of 1/β and are straight lines. The contract curve in the interior must have the MRSs equated (from Econ 301: for differentiable utility functions an interior Pareto optimum has a tangency), so it occurs where c2 /c1 = α/β. This is a straight ray from A’s origin and depending on the values of α and β it lies above or below the main diagonal. Since these cases are (sort of) symmetric we pick one, and assume that α/β > 1/2. The contract curve is this ray and then the part of the upper edge of the box to B’s origin. b) There are two cases, either the Contract curve ray is shallow enough that the equilibrium occurs on it, or it is so steep that the equilibrium occurs on the top boundary of the box. In the first case the slope of the budget and hence the equilibrium price must be 1/β, since both MRSs have that slope along the ray and in equilibrium the price must equal the MRS. So the equilibrium interest rate is (1 − β)/β. Note that the budget now coincides with player B’s indifference curve through the endowment point. Hence the ray of the contract curve must intersect that, and it does so only if it intersects the top boundary to the right of the intersection of B’s IC with the boundary. The latter occurs at (4(3 − β), 10). The former occurs at (10β/α, 10). So the interior solution obtains if 10β/α > 4(3 − β), or if 10β > 12α − 4αβ. In that case the equilibrium allocations are derived by solving the intersection of the budget and the ray: c2 = αc2 /β and −
c2 − 6 1 = β c1 − 12
→
cA 1 =
6(2 + β) A α6(2 + β) , c2 = . 1+α β(1 + α)
B gets the remainder.
In the other case, when the ray fails to intersect B’s IC, we know that we are looking for an equilibrium on the upper boundary of the box (so B cA 2 = 10 and c2 = 0.) At this point we must have a budget flatter than B’s IC (so that B chooses to only consume good 1). It must also be tangent to
Answers 159 A’s IC, since for player A this is an interior consumption bundle (interior to his consumption set, that is.) So we require 1 + r = 10/(αc1 ) to have the tangency, and we require 1 + r = (10 − 6)/(12 − c1 ) in order to be on the budget line. These are two equations in two unknowns again, so we solve: cA 1 = 60/(5 + 2α) and r = (5 − 4α)/(6α). B gets the rest of good 1, of course.
7.3
Chapter 4
Question 1: We wish to show that for any concave u(x) 1 1 1 1 1 u(24) + u(20) + u(16) ≥ u(24) + u(16). 3 3 3 2 2 We can do the following: first bring the u(24) and u(16) to the RHS: 1 1 2 u(20) ≥ u(24) + u(16). 6 6 6 Then multiply both sides by 3: 1 1 u(20) ≥ u(24) + u(16). 2 2 The LHS of this represents a certain outcome of 20, the RHS a lottery with 2 equally likely outcomes. Now note that 1 1 24 + 16 = 20. 2 2 That is, the expected value of the lottery on the RHS of the last inequality above is equal to the expected value of the degenerate lottery on the LHS. Therefore this penultimate inequality must be true, since it coincides with the definition of a risk averse consumer. (utility of expectation greater than expectation of utility.) Question 2: The certainty equivalent is defined by Z X U (CE) = pi u(xi ) = u(x)dF (x). Using the particular function we are given: √ √ √ CE = α 3600 + (1 − α) 6400
CE = (α60 + (1 − α)80)2 = (80 − 20α)2 .
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Note that the expected value of the gamble is E(w) = α3600 + (1 − α)6400 = 6400 − α2800 and thus the maximal fee this consumer would pay for access to fair insurance would be the difference E(w) − CE = 400α(1 − α). Question 3: The coefficient of absolute risk aversion is defined as rA = −u00 (w)/u0 (w). Computing this for both functions we get u(w) = lnw −→ rA =
1 ; w
√ 1 u(w) = 2 w −→ rA = . 2w
Therefore the two consumers exhibit equal risk aversion if the second consumer has half the wealth of the first. Their relative risk aversion coefficients (defined as −u00 (w)w/u0 (w) are 1 and 1/2, respectively. That means that while, if the logarithm consumer has twice the wealth as the root consumer, he will have the same attitude towards a fixed dollar amount gamble, he will be more risk averse with respect to a gamble over a given proportion of wealth. (Note that the two statements don’t contradict one another: a $1 gamble represents half the percentage of wealth for a consumer with twice the wealth!) Question 4: Here we need an Edgeworth Box diagram, which is a square, 15 units a side. Suppose we have consumer A on the bottom left origin (B then goes top right). Suppose also that we put state R on the horizontal. Note that the certainty line is the main diagonal of the box! This observation is crucial, since it means that there is no aggregate risk! General equilibrium requires that demand is equal to supply for each good, but we can’t find those here (not knowing the consumers’ tastes), so it is not useful information. But we also know that in general equilibrium the price ratio must equal each consumer’s MRS (since GE is Pareto optimal and that requires MRSs to be equalized, at least for interior allocations.) Note that the two MRSs here are M RSA =
πu0A (cA R) 0 (1 − π)uA (cA S)
M RSB =
πu0B (cB R) 0 (1 − π)uB (cB S)
B B A On the certainty line (the main diagonal) cA S = cR and cS = cR , so M RSA = M RSB = π/(1 − π). In other words, the certainty line for each consumer coincides and together they are the set of Pareto optimal points.
Hence the equilibrium price ratio must be p∗ = π/(1 − π). The allocation is now easily computed: we know the price ratio and the endowment, hence the budget line for the consumers. We also know that
Answers 161 consumption is equal in both states. So π cA − 5 cA − 5 A → cA = S = S = cR = 5(1 + π) A 1−π 10 − c 10 − cA R B B A and since cB i = 15 − ci we get cS = cR = 5(2 − π).
Question 5:
a)
maxx {0.5u(10000(1 + 0.8x)) + 0.5u(10000(1.4 − 0.8x))}
b) The FOC for this is 0.5 × 0.8u0 (100000(1 + 0.8x)) − 0.5 × 0.8u0 (10000(1.4 − 0.8x)) = 0 implies : implies :
u0 (10000(1 + 0.8x)) = u0 (10000(1.4 − 0.8x)) 10000(1 + 0.8x) = 10000(1.4 − 0.8x)
since she is risk averse. It follows that 1 + .8x = 1.4 − .8x, and therefore that 1.6x = 0.4, so that x = 0.25. One quarter, or 25% are invested in gene technology. Question 6: i) Denote the probability with which a ticket wins by π and the prize by P . A fair price for this lottery ticket would have to be a fraction p per dollar of prize such that π(P − pP ) − (1 − π)pP = 0, or p = π. Let us start with this as a benchmark case (we know that normally such a lottery would not be accepted.) Utility maximization requires that for a gambling consumer v(w0 ) ≤ πv(w0 + (1 − p)P ) + (1 − π)v(w0 − pP ) + µi . Thus all consumers for whom µi ≥ v(w0 ) − πv(w0 + (1 − p)P ) − (1 − π)v(w0 − pP ) purchase a ticket. At a fair gamble this is µi ≥ v(w0 ) − πv(w0 + (1 − π)P ) − (1 − π)v(w0 − πP ) > v(w0 ) − v(π(w0 + (1 − π)P ) + (1 − π)(w0 − πP )) = v(w0 ) − v(w0 ) (the second strict inequality follows from the definition of risk aversion). Clearly a strictly positive µ is required. Can the government make money on this? Well, assume that the price p above is fair (p = π) and let there be an additional charge of q. Now all consumers gamble for whom µi ≥ v(w0 ) − πv(w0 + (1 − π)P − q) − (1 − π)v(w0 − πP − q). While such a µi is larger than before, it exists (for small q in any case) as long as things are sufficiently smooth and the µi go that high. Note that those who gamble have a high utility for it (a high taste parameter µi ) in this setting. Note that this implies that even though they lose money on average they have a higher
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welfare. (The anti-gambling arguments in public policy debates therefore come in two flavours: (i) your gambling is against my (religious) beliefs, and thus it ought to be banned, (ii) there are externalities: your lost money is really not yours but should have bought a lunch for your child/spouse/dog. Since your child/spouse/dog can’t make you stop, we will on their behalf.) ii) Now µ is fixed. Of course, the decision to gamble will still depend on the same inequality, namely µ > v(w0 ) − πv(w0 + (1 − π)P − q) − (1 − π)v(w0 − πP − q). We thus can translate this question into the question of how the right hand side depends on w0 and how this dependency relates to the different behaviours of risk aversion with wealth. So, is the right hand side increasing or decreasing with wealth, and is this a monotonic relationship? The right hand side is related, of course, to the utility loss from going to the expected utility from the expected value (ignoring q for a minute.) Intuitively, we would expect the difference to be declining in wealth for constant absolute risk aversion: Constant absolute risk aversion implies a constant difference between the expected value and the certainty equivalent.1 Let this difference be the base of a right triangle. Orthogonal to that we have the side which is the required distance between the two utilities. The third side must have a declining slope as wealth increases since it is related to the marginal utility of wealth at the certainty equivalent, which is declining in wealth by assumption. There you go, I’d expect the utility difference must fall with wealth. More formally, consider the original inequality again and approximate the RHS by its second order Taylor series expansion (that way we get first and second derivatives, which we want in order to form rA : v(w0 ) − πv(w0⊕ ) − (1 − π)v(w0 − πP − q) ≈ v(w0 ) − π(v(w0 ) + ⊕v 0 (w0 ) + ⊕2 v 00 (w0 )/2) − (1 − π)(v(w0 ) − ªv 0 (w0 ) + ª2 v 00 (w0 )/2) = −π ⊕ v 0 (w0 ) − π ⊕2 v 00 (w0 )/2(1 − π)(ªv 0 (w0 ) − ª2 v 00 (w0 )/2) = v 0 (w0 ) [(1 − π) ª (1 − ªrA /2) − π ⊕ (1 − ⊕rA /2)] . This looks more like it! Now note that we use ª and ⊕ as positive quantities (which are not equal: ª is larger!) Furthermore we know that (a) this quantity must be positive and (b) that π is probably a very small number. Now, if rA is constant then the term in brackets is constant, but of course 1
Is there a general proof for that? Note that constant rA has for example the functional form u(w) = −e−aw , for which the above is certainly true.
Answers 163 v 0 (w) falls with w and thus the right hand side of our initial inequality (way above) falls. Any given µ is therefore more likely to be larger than it. Thus rich consumers participate, poor consumers don’t if we have constant absolute risk aversion. If we have decreasing absolute risk aversion this effect is strengthened. Now, since relative risk aversion is just rA w, it follows that constant relative risk aversion requires a decreasing absolute risk aversion, and that decreasing relative risk aversion requires an even more decreasing absolute risk aversion. Thus in all cases the rich gamble and the poor don’t. (Note here that they are initially rich. Since they loose money on average they will become poor and stop gambling.) 2 iii) If v(w) = ln w then v 0 (w) = 1/w and v 00 (w) . Therefore √ √ = −1/w 0 rA = 1/w, with ∂rA /∂w < 0, and rR = 1. If v(w) = w then v (w) = 1/2 w and v 00 (w) = −w 3/2 /4. Therefore rA = 1/(2w) and rR = 1/2. We now know two pieces of information: the consumers’ risk aversion to a given size gamble is declining with wealth. This would, ceteris paribus make them more likely to purchase the gamble for a constant µ (see above). But µ now is also declining with wealth. The final outcome therefore depends on what declines faster, and we can’t make a definite statement. (As an aside note the following. Suppose we are talking stock market participation here. Then it might be reasonable to assume that the utility of participating in it is increasing in wealth, on average, and so we get higher participation by wealthier people. Now, if the stock market on average is a bad bet we get mean reversion in wealth, while if the stock market is on average more profitable than savings accounts etc we get the rich getting richer. If you now run a voting model where the mean voter wins, you get the desire to redistribute (i.e., tax the investing and profiting rich and give the cash to those who have a too high marginal utility of wealth to invest themselves.) Note also that progressive taxes reduce the returns of a given investment proportional to wealth, counteracting the above effect of more participation by wealthy individuals. . . . See how much fun you can have with these simple models and a willingness to extrapolate wildly?) Question 7: This question forms part of a typical incomplete information contracting environment. Here we focus only on the consumer’s behaviour. a) Assume that the worker has a contractual obligation to provide an effort level of E. Once he has signed the contract, however, he knows that his actual effort is not observable and thus would try to shirk. Expected utility is maximized for p p e∗ = argmax{α w(E) − p + (1 − α) w(E) − e2 }.
The first order condition for this problem is −2e = 0 if e 6= E. I.e., given the worker shirks he will go all the way (after all, the punishment does not
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depend on the severity of the crime in any way.) Thus wepneed to ensure that is the casep if w(E) − E 2 ≥ p the worker will notpshirk at all, 2which p α w(E) − p + (1 − α) w(E), or E ≤ α( w(E) − w(E) − p). If the wage function satisfies this inequality for all E, it will elicit the correct effort levels in all cases. b) Now we have a potentially variable punishment. Given some job with contractual obligation E, the worker now will maximize expected utility and set p p e∗ = argmax{α w(E) − p(E − e) + (1 − α) w(E) − e2 }. p The FOC for this problem is αp0 (·)(2 w(E) − p(E − e))−1 − 2e = 0. (There are also second order conditions which need to hold!) This implies that the worker will play off the cost of shirking against the gains from doing so. We need to make sure that this equation is only satisfied for e∗ = E, in which case he “voluntarily” chooses the contracted level. This clearly requires a p −1 0 0 positive p (). In particular, αp (0)(2 w(E)) − 2E = 0. Note: We could also vary the detection/supervision p probability and makepα depend on E. Then we get e∗ = argmax{α(E) w(E) − p + (1 − α(E)) w(E) − e2 }. As in (a), if the worker deviates he will go all the way here. So the problem is similar to (a), only the wage schedule is now different since α(E) can also vary now. What this shows us is that we tend to want a punishment and a detection probability which both depend on the deviation from the correct level. (This is going to be a question about the technology available: some technologies may be able to detect flagrant shirking more readily than slight shirking.) c) What this seems to indicate is that we would like to make punishments fit the crime. (So for example, if the punishment for a hold-up with a weapon is as severe as if somebody actually gets shot during it, then I might as well shoot people when I’m at it and I think that helps (and if it does not increase the effort the police put into finding me.)) Furthermore, if detection is a function of the actual effort level (the more you fudge the books the more likely will you be detected) then we need lower punishments, ceteris paribus, since the increasing risk will provide some disincentive to cheat anyways. Question 8: a) Let CB denote the coverage purchased for bad losses, and CM the coverage for minor losses. Zero profits imply that the premiums pB and pM for bad and minor losses, respectively, are pB = π/5 and pM = 4π/5. Hence the consumer’s expected utility maximization problem becomes ½ maxCB ,CM (1 − π)u(W − pM CM − pB CB )+ 1 π( u(W − pM CM − pB CB + CB − B)+ 5
Answers 165 4 u(W − pM CM − pB CB + CM − M )) 5
¾
The first order conditions for this problem are 4π π −pM (1 − π)u0 (n) − pM u0 (b) + (1 − pM ) u0 (m) = 0 5 5 π 4π −pB (1 − π)u0 (n) + (1 − pB ) u0 (b) − pB u0 (m) = 0 5 5 Using the fair premiums this simplifies to µ ¶ π 0 4π 0 −(1 − π)u (n) − u (b) + 1 − u0 (m) = 0 5 5 ³ 4π 0 π´ 0 u (b) − u (m) = 0 −(1 − π)u0 (n) + 1 − 5 5
Hence
¶ µ ³ π´ 0 π 4π 0 4π u (m) u (b) − u0 (m) − u0 (b) = 1 − 1− 5 5 5 5
and thus u0 (b) = u0 (m), which finally implies that u0 (n) = u0 (b) = u0 (m) and therefore that 0 = CB − B = CM − M. As expected, the consumer buys full insurance for each accident type separately. b) Now only one coverage can be purchased, denote it by C, and will be paid in case of either accident. Zero profits imply that the premium p is p = π. Hence the consumer’s expected utility maximization problem becomes ½ maxC (1 − π)u(W − pC)+ 1 π( u(W − pC + C − B)+ 5 ¾ 4 u(W − pC + C − M )) 5
The first order condition for this problem is π 4π −p(1 − π)u0 (n) + (1 − p) u0 (b) + (1 − p) u0 (m) = 0 5 5 Using the fair premium this simplifies to 1 4 u0 (n) = u0 (b) + u0 (m) 5 5
166 L-A. Busch, Microeconomics
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and hence either u0 (b) > u0 (n) > u0 (m) or u0 (b) < u0 (n) < u0 (m). Thus either W − B + (1 − π)C < W − πC < W − M + (1 − π)C or W − B + (1 − π)C > W − πC > W − M + (1 − π)C, but this implies either −B + 1C < 0 < −M + 1C or −B + 1C > 0 > −M + 1C. Since B > M by definition we obtain that B > C > M , the consumer over insures against minor losses, and is under insured against big losses. Question 9: From Figure 3.7 in the text, we can take the consumers’ budget line to be the line from the risk free asset point (the origin in this case) to a tangency with the efficient portfolio frontier. where √ Now this tangency √ occurs −1 the margin is equal to the average, so that σ − 16/σ = (2 σ − 16) . That means that the market portfolio has 2(σ − 16) = σ or σ = 32. Therefore µ = 4. The slope of the portfolio line thus is 4/32. For an optimal solution the consumer’s MRS must equal the slope of the portfolio line. For the two consumers given the MRS is σ/32 and σ/96. Thus the optima are σ = 4, µ = 1/2 and σ = 12, µ = 3/2. As expected, the consumer with the higher marginal utility for the mean will have a higher mean at the same prices (and given that both have the same disutility from variance.) Question 10: The asset pricing formula implies that the expected return of the insurance equals the expected risk-free return less a covariance term. If insurance has a lower expected return than the risk-free asset, this covariance term must be positive. In the denominator we have the expected marginal utility, guaranteed to be positive. Thus the numerator must be positive. This means that Cov(u0 (w), Ri ) > 0. But since u00 (w) < 0 this implies that the covariance between w and Ri is negative, that is, if wealth is low the return to the policy is high, if wealth is high, the return to the policy is low. That of course is precisely the feature of disability insurance which replaces income from work if and only if the consumer is unable to work. Question 11: 1) False. The second order condition would indicate a minimum as demonstrated here: maxC {πu(w − L − pC + C) + (1 − π)u(w − pC)} has FOC π(1−p)u0 (w−L+(1−p)C)−p(1−π)u0 (w−pC) = 0. The second order condition for a maximum is π(1−p)2 u00 (w−L+(1−p)C)+p2 (1−π)u00 (w−pC) ≤ 0. Note that 1 ≥ π, p ≥ 0, so that the SOC requires u00 (·) to be negative for at least one of the terms. A risk-lover has, by definition, u00 (·) > 0. 2) Uncertain. We can draw 2 diagrams to demonstrate. In both we have two intersecting budget lines, one steeper, one flatter. The flatter one corresponds to the initial situation. They intersect at the consumer’s endowment. Since the consumer is a borrower, the initial consumption point is below and
Answers 167 to the right of the endowment on the initial budget. The indifference curve through this point is tangent to this budget. It may, however, cut the new budget (so that the IC tangent to the new budget represents a higher level of utility) or lie everywhere above it (in which case utility falls.) 3) True. Apply the following positive monotonic transformations to the first function: −2462, ×12, collect terms in one logarithm, take exponential, take the 9000th root. What you get is the second function. R 4) True. A risk averse consumer is defined as having u( xg(x)dx) > R u(x)g(x)dx. Let the consumer have initial wealth w and suppose he could participate in a lottery which leads to a change in his initial wealth by x, distributed as f (x). Suppose the payment for this lottery is p. If this payment is equal to the expected value of the lottery then the consumer will not have a change in expected wealth, but will face risk. Thus by definition he would not buy this lottery. If the payment is less, then the expected value of wealth from participating in the lottery exceeds the initial wealth. Depending on by how much, the consumer may purchase. A risk loving consumer, of course, would already buy at when the expected net gain is zero. (This argument could be made more precise, and you should try to put it into equations!) 5) False. The market rate of return is 15%. Gargleblaster stock has a rate of return of (117 − 90)/90 = 30%. This violates zero arbitrage. 6) True. All consumers face the same budget line in mean-variance space. At an interior optimum (and assuming their MRS is defined) they all consume on this line where the tangency to their indifference curve occurs. This may be anywhere along the line, depending on tastes, but the slope is dictated by the market price for risk. Question 12: a) Since workers work as bus driver and at a desk job we require √ √ √ 2 40000 = α2 44100 − 11700 + (1 − α)2 44100 Therefore
√ √ 1 44100 − 40000 210 − 200 √ = . α= √ = 210 − 180 3 44100 − 32400
b) Since workers work on oil rigs and at a desk job we require √ √ √ 2 40000 = 0.5 × 2 122500 − Loss + 0.5 × 2 122500. √ √ Thus 400 = 122500 − Loss + 350 and hence 50 = 122500 − Loss or Loss = 120000. c) At fair premiums the workers will fully insure. That is, they suffer their expected loss for certain. For a bus driver the √ expected loss √ is 11700/3 = 3900. Thus the bus driver wage must satisfy 2 40000 = 2 w − 3900 and
168 L-A. Busch, Microeconomics
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hence it is $43900. For the oil rig worker the expected loss is $60000, and their wages will fall to $100000 under workers compensation. Note the the condition that workers take all jobs together with a fixed desk job wage fixes the utility level in equilibrium for workers. However, the wage premium for risky jobs will not have to be paid: the wages of the risky occupations fall which benefits the firms in those industries by lowering their wage costs. (This is why industries are in favour of workers’ compensation.) d) The average probability of an accident now is 0.4 × 0.5 + 0.6 × 1/3 = 0.4. If we were to use this as a fair premium (but see below!) this premium is too high for bus drivers, who will under insure, and too low for oil rig workers, who will over √ bus drivers will choose to buy insurance Cb √ insure. Indeed, the such that 6 44100 √ − 0.4Cb = 8 32400 +√.6Cb (Take the first order condition √ from for maxCb {(1/3) 32400 + 0.6Cb + (2/3) 44100 − 0.4Cb }, bring the the denominator into the numerators and loose the 1/30 on both sides.) Thus we require 9(44100−0.4Cb ) = 16(32400+0.6Cb ), or Cb = 10(9×44100−16× 32400)/(6×16+4×9) = −9204. What does this mean? It means that the bus drivers would like to bet on themselves having an accident buying negative amounts of insurance! (The ultimate in under insurance!) Note that the governments expected profit from bus drivers is −0.4×9204/3+1.2×9204/3 = 2454.40 > 0. The√oil rig workers would need to solve √ + 122500 − 0.4Co }, which leads to max √ Co { 2500 + 0.6Co √ 3 122500 − 0.4Co = 2 2500 + 0.6C and thus Co = 182083.33. Note that the govt looses money on them, since (0.5 × 0.4 − 0.5 × 0.6) × 182083.33 = −18208.30. Overall then the govt makes losses of 5810.68N , where N is the number of workers in risky occupations. At the old wages both groups are better off (and thus there would be an influx of desk workers and a reallocation towards oil rigs.) In order to break even the insurance rates would have to be changed, in particular raised. It also seems that the govt would ban the purchase of negative insurance amounts. In which case the bus drivers would find it optimal to buy no insurance, and then premiums would have to be 0.5 for the govt to break even. This would be deemed unjust by all involved, and so in practice the govt forces all workers to buy a fixed amount of insurance! In principle we could compute equilibrium wages if we treat the insurance purchase as a function of the wage. So, for example we know from the above√that Cb (w) = 10(9×w−16×(w−11700))/(6×16+4×9). We then solve p p for s 40000 = (2/3) w + (1 − 0.4)Cb (w) + (4/3) w − 11700 − 0.4Cb (w). The details are left to the reader.
Answers 169 The important point here is that it is important to charge the correct premiums. If that is not done things will work out funny. That in turn leads to real life plans which do not allow a choice — workers have to insure, the amount is dictated (often capped, that is, the insured amount is a function of the wage up to a maximum.) You can see that such plans can be quite complicated and that it can be quite complicated to figure out who would want to do what, what the distributional implications are, etc. Question 13: Let us translate the question into notation: We are to show u0 (c1 ) c2 u00 (w)w that 0 = k if = λ if the function u(·) satisfies 0 = a, ∀w. u (c2 ) c1 u (w) c2 u0 (c1 ) = k and =λ 0 u (c2 ) c1
=⇒
u0 (c1 ) = k(λ)u0 (λc1 ).
If the left and right hand side of that last expression are identical functions, then their derivatives must equal: u00 (c1 ) = k(λ)λu00 (λc1 ), but we know that k(λ) = u0 (c1 )/u0 (λc1 ), so that u00 (c1 ) =
u0 (c1 ) λu00 (λc1 ) u0 (λc1 )
=⇒
u00 (c1 ) u00 (λc1 ) = λ u0 (c1 ) u0 (λc1 )
Thus the MRS is constant for any consumption ratio λ if u00 (c1 )c1 u00 (λc1 )λc1 = u0 (c1 ) u0 (λc1 )
∀λ,
which is constant relative risk aversion.
7.4
Chapter 6
Question 1: A 3-player game in extensive form comprises a game tree, Γ, a payoff vector of length three for each terminal node, a partition of the set of non-terminal nodes into player sets S0 , S1 , S2 , S3 , a partition of the player sets S1 , S2 , S3 into information sets. Further, a probability distribution for each node in S0 over the set of immediate followers and for each SiJ an index set IiJ and a 1-1 mapping from Iij to the set of immediate followers of the nodes in Sij . Any carefully labelled game tree diagram will do. It does not even have to have nature (i.e., S0 could be empty.) Question 2: Perfect recall is when each player never forgets any of his own previous moves (so that for any two nodes within an information set one
170 L-A. Busch, Microeconomics
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may not be a predecessor of the other and any two nodes may not have a common predecessor in another information set of that player such that the arc leading to the nodes differs) and never forgets information once known (so that any two nodes in a player’s information set may not have predecessors in distinct previous information sets of this player.) Counter examples as in the text, or any game which violates these requirements. Question 3: Yes, any finite game has a Nash equilibrium, possibly in mixed strategies. This follows from the Theorem we have in the text. The game therefore will also have a SPE (they are a subset of the Nash equilibria, but keep in mind that the condition of subgame perfection may have no ‘bite’, in which case we revert to Nash.) Question 4: Player 1 has no weakly dominated strategies since u1 (D, L, Lef t) > u1 (U, L, Lef t) but u1 (D, R, Lef t) < u1 (U, R, Lef t), while u1 (C, L, Right) > u1 (U, L, Right). Player 3 does also not have a weakly dominated strategy. Depending on the opponents’ moves he gets a higher payoff sometimes in the left and sometimes in the right matrix. Player 2 does have weakly dominated strategies: Both L and R are weakly dominated by C. This does not leave us with a good prediction yet, aside from the fact that 2 can be argued to play C. However, if we now consider repeated elimination we can narrow down the answer to what is also the unique Nash equilibrium in pure strategies in this case, (D, C, Right). To find mixed strategy Nash we assign probabilities to the strategies for players, so let µ1 = P r(U ), µ2 = P r(C), γ1 = P r(L), γ2 = P r(R), and α = P r(Lef t). We can then compute the payoffs for players for each of their pure strategies. So for example u1 (U, γ, α) = α(γ1 + 2γ2 + (1 − γ1 − γ2 ) + (1 − α)(2γ1 + 4γ2 + 2(1 − γ1 − γ2 )). We then can ask, when is player 1, say, actually willing to mix? Only if the payoff from the pure strategies in the support of the mixed strategy are equal, so that the player does not care. Question 5: This one is made easier by the fact that strategy R is (strictly) dominated, so that it will never be used in any mixed strategy equilibrium (or indeed any equilibrium.) Hence this is really just a 2 × 2 matrix we need to consider. Let α = P r(U ) and β = P r(L), so that P r(C) = 1 − α and P r(C) = 1−β. Then for player 1 to mix we require β+4(1−β) = 3β+2(1−β), hence 2 = 4β, and hence β = 0.5. So if player 2 mixes with this probability then player 1 is indifferent between his two strategies. Now look at player 2: For 2 to be indifferent between the two strategies L and C we require 4α + 2(1 − α) = 2α + 3(1 − α). Hence 3α = 1 and thus α = 1/3. Thus the
Answers 171 mixed strategy Nash equilibrium is ((1/3, 2/3), (1/2, 1/2)). Note that the game has no pure strategy Nash equilibria. Question 6: This is a two player game (the court is not a strategic player and does not receive any payoffs.) The most natural extensive form for such a situation is probably as in the game tree on the next page. railway e ¡@ ¡ @ ¡ @ ¡ @ low high ¡ @ ¡ @ ¡ @ ¡ @ @sNature Nature s¡ ¡@ ¡@ ¡ ¡ @ No Acc @ No Acc ¡ (1 − pl )@ ¡ (1 − ph@ ) @ @ @ @ Accident ¡pl Accident ¡ph ¡ ¡ (20, 10) (25, 10) ¡ ¡ ¡ ¡ ¡ ¡ s¡ town s¡ ¡@ ¡@ ¡ @ ¡ @ Sue ¡ Sue ¡ @ Not Sue @ Not Sue ¡ @ ¡ @ ¡ @ ¡ @ ¡ @ ¡ @ (10, 0)
(12, 2)
(5, 10)
(17, 2)
Here it is important to note that ph > pl , reflecting the fact that if low care is taken the accident probability is higher. I have arbitrarily assigned payoffs which satisfy the description. High level of care costs the railway 5, accidents impose a cost of 8 on both parties, legal costs are 2 for each party. Let us try and find the Nash equilibrium of this game. As an exercise let us first find the strategic form: RT high low
Sue
N otSue
(20 − 10pl , 10 − 10pl ) (20 − 8pl , 10 − 8pl ) (25 − 20ph , 10)
(25 − 8ph , 10 − 8ph )
Note that low strictly dominates high for the railway if 0.5 > (2ph − pl ) while high strictly dominates low if ph − pl > 5/8. In those cases the
172 L-A. Busch, Microeconomics
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Nash equilibria are (low, Sue) and (high, N otSue), respectively. Otherwise there will be a mixed strategy equilibrium. Let α be the probability with which the railway uses the high effort level. The town is indifferent iff its expected payoffs from the two strategies are the same, that is, if 10−10αpl = 10 − 8ph + 8α(ph − pl ). This is the case if α = 4ph /(4ph + pl ). For lower α it prefers to Sue, for higher α it prefers to N otSue. Letting β denote the probability with which the town sues, the railway expects to receive 20 − 8pl − 2βpl from high and 25 − 8ph − 12βph from low. It is indifferent if β = (5 − 8(ph − pl ))/(12ph − 2pl ). So the mixed strategy Nash equilibrium is ¶ µ 5 5 − 8(ph − pl ) 1 4ph if ph − > pl > 2ph − . , (α, β) = 4ph + pl 12ph − 2pl 8 2 Question 7: There are two ways to draw this game. We can have nature move first and then Romeo (who does not observe nature’s move.) Or we can have Romeo move first, and then nature determines if the move is seen. The game tree for the first case is as drawn below.
Nature e © © HH HH ©©
see ©© p
©©
s © Romeo © ¡@ @ M ¡ S @ ¡ @ ¡ J1¡ s @sJ 2 ¢A ¢A ¢ A ¢ A S¢ S¢ AM AM ¢ A ¢ A ¢ A ¢ A ¢ A ¢ A
(30,50)
(1,1)
(5,5)
(50,30)
1-p HH not see HH
HHs ¡@ @ M ¡ S¡ @ @ ¡ J3¡ s @s ¢A ¢A ¢ A ¢ A S¢ S¢ AM AM ¢ A ¢ A ¢ A ¢ A ¢ A ¢ A
(30,50)
(1,1)
(5,5)
(50,30)
A strategy vector in this game is (sR , (s1J , s2J , s3J )). Subgames start at information sets J 1 and J 2 , the only other subgame is the whole tree. In the subgame perfect equilibrium Juliet therefore is restricted to (S, M, ·). Let α denote Romeo’s probability of moving S, and β Juliet’s (in J 3 .) Romeo’s (expected) payoff from S is 30p + (1 − p)(30β + 1 − β) and his payoff from M is 50p + (1 − p)(5β + 50(1 − β)). The β for which he is indifferent is (49 − 29p)/(74(1 − p)). Note that this is increasing in p and that β = 1 if p = 5/9! Juliet has payoffs of 50α+5(1−α) and α+30(1−α) from moving S
Answers 173 and M , respectively, in J 3 . Hence she is indifferent if α = 25/74. Of course, pure strategy equilibria may also exist, and we get the SPE equilibria to be µ µ ¶¶ 25 49 − 29p 5 , S, M, , (S, (S, M, S)) , (M, (S, M, M )) if p < . 74 74(1 − p) 9 Note that (S, (S, M, S)) requires that 30 > 50p + 5(1 − p), or p < 5/9 also. (M, (S, M, M )) requires that 50 > 30p+(1−p), or p < 49/29, which is always true. What if p > 5/9? In that case the equilibrium in which the outcome is coordination on S (preferred by Juliet) does not exist, and neither does the mixed strategy equilibrium. Hence the unique equilibrium if p ≥ 5/9 is (M, (S, M, M )). Romeo can effectively insist on his preferred outcome. Question 8: Each firm will ( maxqi
(
X j6=i
qj + qi − 10)2 qi − 0qi
)
.
The FOC for this problem is X X 2( qj + qi − 10)qi + ( qj + qi − 10)2 = 0 j6=i
Hence if
P
j6=i qj
j6=i
+ qi − 10 6= 0 we require 2qi + (
X j6=i
qj + qi − 10) = 0
P and get the reaction function qi = (10 − j6=i qj )/3. P With identical firms we then know that in equilibrium qj = qi , so that j6=i qj = (n − 1)qi . Hence 3qi = 10 − (n − 1)qi and we get that qi ∗ = 10/(n + 2) ∀i. Total market output then is 10n/(n + 2). Note that total market output approaches 10 from below as n gets large. Market price for a given n is 400/(n + 2)2 , which approaches zero as n gets large. (Note that the marginal cost is zero and hence the perfectly competitive price is zero!) Question 9: In the first instance the sellers can only vary price. To clarify ideas, let us focus on two prices only (as would be needed for a separating equilibrium.) The game then is as depicted below. We are to show that no separating equilibrium exists. If it did, it would have to be the two prices as indicated, where one firm charges one price (presumably the high quality firm charging the higher price) the other another. But given that, the consumer knows (in equilibrium) which firm produced the product. It is easy to see that the low quality firm would deviate to the higher price (being then mistaken
174 L-A. Busch, Microeconomics (9 − p, p − 3) (0, −3) ´´
buy ´
´
´
s ´α
´ No
(6 − p, p − 2)
Q No Q Q
seller s
pˆ
β QQ
Root: eNature low
Q Qs
buy
p
seller
pˆ
Q
Q
(9 − pˆ, pˆ − 3)
No QQ
Q (0, −3)
buyer ´ No ´
(1/2) s
buy
s
high (1/2)
buyer (0, −2) Q Q
p
May2004
s´
´
(0, −2) ´´
´
buy
(6 − pˆ, pˆ − 2)
for the high quality firm so that the consumer buys) since costs are unaffected by such a move, but a higher price is received. At this point the remainder is non-trivial and left for summer study! The key is that the consumer now has an information set for each price-warranty pair, and that there are two nodes in it, one for each type of firm. Question 10: What was not stated in the question was the fact that each consumer buys either one or no units. Each buyer purchases a unit of the good if and only if the price is below the valuation of the buyer. Hence total market demand is given by the number of buyers with a valuation above p, or 1 − F (p). F (v) is the cumulative distribution for the uniform distribution on [0, 2].R Since the pdf for the uniform distribution on [0, 2] is 0.5, we have v F (v) = 0 0.5dt = 0.5v. Hence market demand is 1−0.5p and inverse market demand is 2(1 − Q). A Cournot equilibrium is nothing but a Nash equilibrium in the game in which firms simultaneously choose output levels. Hence firm 1 solves maxq1 {2(1 − q1 − q2 )q1 − q1 /10} which leads to FOC 2(1 − q1 − q2 ) − 2q1 − 1/10 = 0 and the reaction function q1 (q2 ) = 19/40 − q2 /2. Firm 2 solves ª © maxq2 2(1 − q1 − q2 )q2 − q22
which leads to FOC 2(1 − q1 − q2 ) − 2q2 − 2q2 = 0 and the reaction function q2 (q1 ) = 1/3 − q1 /3. The Nash equilibrium then is (q1 , q2 ) = (37/100, 21/100). Market price is 42/50. Profits for the two firms are 74 × 37/10000 for firm 1 and (82 × 21 − 212 )/10000 for firm 2, so that joint profit is (74 × 37 + 82 × 21 − 212 )/10000.
Answers 175 In the Stackelberg leader case we consider the SPE of the game in which firm 1 chooses output first and firm 2, after observing firm 1’s output choice, picks its output level. Firm 1, the Stackelberg leader, therefore takes firm 2’s reaction function as given. Thus firm 1 solves ½ µ µ ¶¶ ¾ 1 q1 − maxq1 2 1 − q1 − q1 − q1 /10 3 3 The FOC for this is 4(1 − 2q1 )/3 − 1/10 = 0 and hence q1 = 37/80. Thus q2 = 43/240. Market price is 86/240. Profits for the two firms are (62 × 37)/(240×80) and 43×43/2402 . Joint profit thus is (186×37+43×43)/2402 . Joint profit maximization would require that the firms solve ª © maxq1 ,q2 2(1 − q1 − q2 )(q1 + q2 ) − q1 /10 − q22 .
This has FOCs
2(1 − q1 − q2 ) − 2(q1 + q2 ) − 1/10 = 0 2(1 − q1 − q2 ) − 2(q1 + q2 ) − 2q2 = 0 so that we know that 2q2 = 1/10 or q2 = 1/20. Hence 2(1 − q1 − 1/20) − 2(q1 + 1/20) − 1/10 = 0 and 2 − 4q1 − 3/10 = 0 and q1 = 7/40. Market price then is 2(31/40). Joint profits are 2(31/40)(9/40) − 8/400. This cannot be attained as a Nash equilibrium because neither output level is on the firm’s reaction function, and only output levels on the reaction function are, by design, a best response.
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