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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 7 Outline of the course
I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course
I. Introduction II. Individual choice Budget constraint (Ch. 2) Preferences (Ch. 3) Utility (Ch. 4) Consumer problem (Ch. 5) Revealed preference (Ch. 7) Slutsky equation (Ch. 8) Endowment income effect (Ch. 9) Intertemporal choice (Ch. 10) Choice under uncertainty (Ch. 12) III. Competitive markets IV. Market failure Demand for Insurance Suppose consumer initially has $35;000 in wealth. But his final wealth is uncertain and he may loose $10;000, say, because
his car may be stolen. An insurance contract the consumer can buy pays him $1 in case the
loss occurs. To buy insurance the consumer must pay upfront a premium of dollars per contract he buys. How much insurance will the consumer buy, if any? How do we even approach this problem? Demand for Insurance Consumer decides how many contracts to buy If he buys K insurance contracts his final wealth is: w D
0 ( 25;000 C K 35;000 K W car stolen K W car not stolen To simplify matters, assume there is only one consumption good so that
consumer cares directly about wealth. (In reality, with two or more goods, wealth matters indirectly because of its effect in the final decision of which bundle to consume.) Different choices of K yield different contingent consumption plans. How do we extend consumer theory to deal with choice under
uncertainty? In terms of standard consumer theory... There is a way to translate this problem into standard consumer theory. Two states of nature: state 1 (car stolen) and state 2 (car not stolen). A bundle is .x1; x2/, that is, consumption (of the same good!) in each
state. That is, we allow the consumer to choose different amounts of a good
contingent on the state. But the choice is made before the true state is realized, like in the
insurance example. In the insurance example, the choice of the amount K of insurance to buy is, effectively, the choice of a bundle .x1 ; x2/. Choice under uncertainty So, in principle, our theory could just postulate a utility function u.x1; x2/
and we would be done. However, choice under uncertainty seems to be special in at least two
ways. Perhaps it makes sense to impose more structure on consumer preferences. First, it sounds intuitive that the probability / likelihood that the car be
stolen should matter. So we would like a theory that takes the effect of these probability assessments into account explicitly. Second, in terms of actual consumption, either the consumer consumes x1 or he consumes x2 units of the good! Either the car is stolen or not!
Although this contingent consumption can be formally translated into standard consumer theory, behaviorally speaking things are different. Expected utility theory will take care of the two points above. In fact, we
will see that they are related. Model basics Outcomes, states or consequences: C D fc1; c2; : : : ; cng Lotteries are probability distributions over C , that is, numbers p1; p2; : : : ; pn (between 0 and 1) which sum to 1. Set of all lotteries over C denoted L. pi is the probability of outcome ci consumer has a preference relation over L This is similar to consumer theory, except that the choice space is
different. Example Three (monetary) outcomes: $2;500;000, 500;000 and 0. The lottery p D .0; 1; 0/ means $500;000 for sure The lottery p 0 D .1=3; 1=3; 1=3/ means ... The lottery p 00 D .p1; p2; p3/ means ... Do you prefer p or p 0 ? Expected Utility
Definition. A preference relation over lotteries has an expected utility representation if there exist utility numbers u.ci / such that for all lotteries p , p 0 2 L, p p 0 if and only if
0 0 0 p1u.c1/ C p2u.c2/ C : : : C pnu.cn/ p1u.c1/ C p2u.c2/ C : : : C pnu.cn/; that is, Expected utility under p D Ep u Ep 0 u D Expected utility under p 0: When does a preference relation have an expected utility representation? Expected Utility Completeness, Reflexivity and Transitivity Independence Axiom. For all lotteries p; p 0 and p 00 and all 2 OE0; 1, p p 0 if and only if p C .1 /p 00 p 0 C .1 /p 00: To understand IA need to understand notation p C .1 /p 00. This is just a lottery that assigns probability p.ci / C .1 /p 00.ci / to each outcome ci . The axiom makes sense when we assume that the decision maker
views compound lotteries as equivalent to simple lotteries. Expected Utility
Theorem. A preference relation satisfies Comp, Refl, Trans and Indep if and only if it has an expected utility representation. Proof idea: Compl., Refl. and Trans. imply that the preference has a utility representation. IA implies that indifference curves are straight lines, which means utility is linear in lotteries. Risk Aversion
Consider monetary lotteries only, that is, lotteries over amounts of money. For a lottery p let Ep denote the lottery that pays the expected value of lottery p with certainty. An individual is risk averse if for every monetary lottery p he prefers Ep than p . p over Ep Proposition. An expected-utility maximizer is risk-averse if and only if his utility function over deterministic monetary outcomes is concave. Recall that a function f W R ! R is concave if for all x , y 2 R, 2 OE0; 1, f .x C .1 /y/ f .x/ C .1 /f .y/: Demand for Insurance Consumer decides how many contracts to buy If he buys K insurance contracts his final wealth is: ( c1 D 35;000 K W w/ prob. 1 wD c2 D 25;000 C K K W w/ prob. Budget line: c2 D 25000 C Slope of the budget line D U.c1 ; c2/ D .1
1 1 .35000 c1/ /u.c1/ C u.c2/. and so MRS D .1 /u0.c1/ M U1 D M U2 u0.c2/ Demand for Insurance Choose K such that MRS D .1 /u0.c1/ u0 .c2 / 1 . D 1 . Suppose insurance company breaks even, that is, makes zero expected
profit: insurance premium K
i.e., "fair insurance." K .1 / 0 D 0 D ; cost of payoffs Then, optimal choice of K is such that u0.c1 / D u0.c2 /;
i.e., "full insurance."
The optimal choice of the number of insurance contracts equalizes the marginal utilities of consumption in state 1 and the MU of consumption in state 2 A risk-averse expected utility maximizer who is offered fair insurance
optimally chooses to be fully insured. Investment on a Risky Asset initial wealth w two states, "good" or "bad" risky assets returns rg > 0 or rb < 0, depending on the state If consumer invests x in risky asset he gets ( w C xrg W w/ prob. 0 w D w C xrb W w/ prob. 1
so his expected utility is EU.x/ D u.w C xrg / C .1 Marginal utility of x : EU 0.x/ D rg u0.w C xrg / C .1 /u.w C xrb / /rb u0.w C xrb / Investment on a Risky Asset Consider second derivative:
2 EU 00.x/ D rg u00.w C xrg / C .1 2 /rb u00.w C xrb /; which is negative if consumer is risk averse. Look at marginal utility at x D 0: EU 0.0/ D u0.w/ . rg C .1 /rb / ,, , ...
expected return A risk averse consumer does not invest in a risky asset with negative
expected return; A risk averse consumer always invests a positive amount in a risky asset
with positive expected return. Taxing a Risky Asset after tax returns: .1 t/rg in the good state and .1 t/rb in the bad state The FOC determines his optimal investment level: EU 0.x/ D 0;
that is, .1 t/rg u0.w C x.1 t/rg / C .1 /.1 t/rb u0.w C x.1 t/rb / D 0: Does imposing a tax encourage or discourage investment in the risky
asset? Taxing a Risky Asset The answer, surprisingly, is it encourages investment in the risky asset! Let x be the optimal investment level when t D 0, and x the optimal O investment under t > 0. Then we must have: xD O And the proof comes from the FOC: .1 t/rg u0.w C x.1 O t/rg / C .1 /.1 t/rb u0.w C x.1 O t/rb / D 0; 1 x t see blackboard. The tax reduces the expected return but it also reduces the risk,
because in the bad state the tax is actually a subsidy. By increasing investment, consumer can get the same consumption
pattern as before, offsetting the tax. ...
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This note was uploaded on 03/10/2012 for the course ECON 121 at Yale.