In the insurance example the choice of the amount k

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Unformatted text preview: nce 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 expecte...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.

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