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# Expected utility completeness reexivity and

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Unformatted text preview: d utility representation? Expected Utility  Completeness, Reﬂexivity and Transitivity  Independence Axiom. For all lotteries p; p 0 and p 00 and all ˛ 2 Œ0; 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 . 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 Œ0; 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 .35000 c1/ 1  /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 / D 1 . 1 .  Suppose insurance company breaks even, that is, makes zero expected proﬁt: K K .1 /  0 D 0 ∴ D ; i.e., “fair insurance.”  Then, optimal choice of K is such that u0.c1 / D u0.c2 /; i.e., “full insurance.”  A risk-averse expected ut...
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