l061106riskpreferencesWEB

# l061106riskpreferencesWEB - S.Grant ECON501 2.3 Money...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 10

l061106riskpreferencesWEB - S.Grant ECON501 2.3 Money...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online