l061106riskpreferencesWEB

l061106riskpreferencesWEB - S.Grant ECON501 2.3 Money...

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S.Grant ECON501 2.3 Money Outcomes and Risk Aversion Ref: MWG 6.C If individual is a subjective expected utility maximizer, then % over acts can be characterized by π , prob. measure on S representing beliefs and preference scaling utility function u : X R . So can identify act a =[ δ x 1 ,E 1 ; ... ; δ x n n ] with lottery L = [ x 1 ,p 1 ; ; x n n ] where p i = π ( E i ) . Focus on situation where outcomes are amounts of wealth. An act is now a random variable ˜ x : S X . Identify act with its cumulative distribution function (CDF) F ˜ x ( x )= π ( s S x ( s ) x ) prob. realized outcome no greater than x . 1 S.Grant ECON501 Axioms place no restrictions on preference scaling utility function for wealth, but economics does. 1. u is increasing (or u 0 ( x ) > 0 ). 2. u is concave (or u 00 ( x ) 0 ) 3. u 000 ( x ) > 0 (or u 0 ( . ) is convex) 1. “more is better” — local non-satiation 2. Risk aversion 3. Decreasing absolute risk aversion. 2
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S.Grant ECON501 De f nition 2.3.1: An individual is (weakly) risk averse if for any act ˜ x ,the act that yields E[˜ x ]= Z xdF ˜ x ( x ) Ã = X x X ( s S x ( s )= x ) ! with certainty is weakly preferred to ˜ x . Proposition 2.3.1: If U x Z u ( x ) dF ˜ x ( x ) Ã = X x X u ( x ) π ( s S x ( s x ) ! represents % ,t h e n % exhibits (weak) risk aversion if and only if the preference-scaling utility function u is concave. Proof. I. concave u risk aversion: By Jensen’s inequality, if u ( . ) is concave then Z u ( x ) dF ( x ) u μZ xdF ( x ) ,fo ra l l F ( . ) 3 S.Grant ECON501 II. risk aversion u is concave. (We will show u not concave % does not exhibit risk aversion.) Suppose u is not concave. That is, there exists y,z R + and α (0 , 1) satisfying u ( αy +(1 α ) z ) <αu ( y )+(1 α ) u ( z ) . Butitthenfo l lowsfo rtheevent E with π ( E α and the act ˜ x ,where ˜ x ( s ½ y if s E z if s/ E ,&so F ˜ x ( x 0 if x<y α if x [ ) 1 if x z , we have Z u ( x ) dF ˜ x ( x αu ( y α ) u ( z ) >u ( αy α ) z ) = u μZ xdF ˜ x ( x ) 4
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S.Grant ECON501 2.4 Measures of Risk Aversion Certainty Equivalent de f ned as c x, u )= u 1 μZ u ( x ) dF ˜ x ( x ) Obs: If risk averse, then risk premium
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l061106riskpreferencesWEB - S.Grant ECON501 2.3 Money...

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