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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 expectedutility maximizer is riskaverse 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 riskaverse expected ut...
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 Spring '09
 SAMUELSON
 Microeconomics

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