9Arrow-Pratt Theorem

9Arrow-Pratt Theorem - The Arrow-Pratt Theorem For t = 0,...

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The Arrow-Pratt Theorem For t = 0 , 1, let u t be a C 2 vN-M utility with u 0 t > 0 on R ++ . And let L be the set of cumulative distribution functions on R ++ with F (0) = 0 and F ( z ) = 1 for some number z < . 1 t = 0 , 1, denote the Arrow-Pratt absolute risk aversion measure by r t ( z ) = - u 00 t ( z ) u 0 t ( z ) , the certainty equivalent by CE t ( F ) and the risk premium by RP t ( F ). Theorem. The following are equivalent. 1. r 1 ( z ) r 0 ( z ) for all z > 0 . 2. There is a concave, strictly increasing function T : Range ( u 0 ) R such that u 1 ( z ) = T ( u 0 ( z )) for all z > 0 . 3. 1 ( F ) 0 ( F ) for all F ∈ L . 4. RP 1 ( F ) RP 0 ( F ) for all F ∈ L . Corollary. Let u,F satisfy the hypotheses of the Theorem, and let ( F,a ) be the certainty equivalent and RP ( ) the risk premium for vN-M utility u when the positive number a is added to the risk z (so final wealth is z + a ): u ( ( )) = R u ( z + a ) dF ( z ) and RP ( ) = R ( z + a ) dF ( z ) - ( ) . The following are equivalent 1. r ( · ) is decreasing on R ++ . 2. ( ) - a ( F, 0) for all F ∈ L and a > 0 . 3. RP ( ) RP ( F, 0) for all F ∈ L and a > 0 . To prove the corollary, set u = u 1 , define u 0 ( z ) = u
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This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.

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9Arrow-Pratt Theorem - The Arrow-Pratt Theorem For t = 0,...

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