HW2 - Microeconomic Theory II Assignment 2: Risk and...

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Unformatted text preview: Microeconomic Theory II Assignment 2: Risk and Information Solutions * February 15, 2007 Problem 1. Concerning Measures of Risk Aversion: Let u i : < -→ < ,i = 1 , 2 , be two strictly increasing, strictly concave, twice differentiable Bernoulli utility functions; and, let F : < -→ [0 , 1] be a cumulative distribution function for some real-valued random variable, ˜ y , which is the (positive or negative) reward from accruing some risky situation. So, agent i ’s preferences over gambles can be represented as follows: U i ( F ) = R u i ( y ) dF ( y ) ,i = 1 , 2. Define c ( F ; u i ) to be the certainty equivalent of the gamble represented by F for agent i : c ( F ; u i ) ≡ u- 1 i [ R u i ( y ) dF ( y )] . We say that agent 1 is “more risk averse than agent 2” if and only if: ∀ F : c ( F ; u 1 ) 5 c ( F ; u 2 ). Prove that the following three claims are equivalent: (i.) Agent 1 is more risk averse than agent 2. (ii.) There exists an increasing concave function v : < -→ < such that, ∀ y ∈ < : u 1 ( y ) = v ( u 2 ( y )) . (iii.) ∀ y ∈ < :- u 00 1 ( y ) u 1 ( y ) =- u 00 2 ( y ) u 2 ( y ) Solution . • (ii.) ⇐⇒ (iii.): Notice that since both u 1 ( y ) and u 2 ( y ) are increasing we can always find an increasing function v ( · ) such that u 1 ( y ) = v ( u 2 ( y )). Taking the derivative in in this equation and using the chain rule we have that u 1 ( y ) = v ( u 2 ( y )) u 2 ( y ) . * Prepared by ¨ Omer ¨ Ozak ([email protected]), Department of Economics, Brown University. If you find any typos or mistakes please let me know, so that it can be fixed. 1 Microeconomic Theory II Solutions Taking logarithms on both sides and differentiating again we get u 00 1 ( y ) u 1 ( y ) = u 00 2 ( y ) u 2 ( y ) + v 00 v so that v 00 ≤ 0 if, and only if, u 00 1 ( y ) u 1 ( y ) ≤ u 00 2 ( y ) u 2 ( y ) . • (ii.) ⇒ (i.): By definition u 2 ( c ( F ; u 2 )) = R u 2 ( y ) dF ( y ), the hypothesis implies that u 1 ( c ( F ; u 1 )) = R v ( u 2 ( y )) dF ( y ). Since v ( · ) is concave, then Z v ( u 2 ( y )) dF ( y ) ≤ v Z u 2 ( y ) dF ( y ) ¶ . Putting the pieces together we get u 1 ( c ( F ; u 1 )) = Z v ( u 2 ( y )) dF ( y ) ≤ v Z u 2 ( y ) dF ( y ) ¶ = v ( u 2 ( c ( F ; u 2 ))) = u 1 ( c ( F ; u 2 )) and since u i ( · ) is strictly increasing, we have that c ( F ; u 1 ) 5 c ( F ; u 2 ). • (i.) ⇒ (ii.): Let v ( · ) again be the increasing function that makes u 1 ( y ) = v ( u 2 ( y )). By hypothesis c ( F ; u 1 ) 5 c ( F ; u 2 ) so that u 1 ( c ( F ; u 1 )) ≤ u 1 ( c ( F ; u 2 )). But this implies that Z v ( u 2 ( y )) dF ( y ) = u 1 ( c ( F ; u 1 )) ≤ u 1 ( c ( F ; u 2 )) =...
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This note was uploaded on 07/06/2009 for the course ECON 206 taught by Professor G.loury during the Spring '07 term at Brown.

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HW2 - Microeconomic Theory II Assignment 2: Risk and...

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