9Arrow-Pratt Theorem - The Arrow-Pratt Theorem For t = 0 1...

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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 For 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. CE 1 ( F ) CE 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 CE ( F, a ) be the certainty equivalent and RP ( F, a ) 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 ( CE ( F, a )) = R u ( z + a ) dF ( z ) and RP ( F, a ) = R ( z + a ) dF ( z ) - CE ( F, a ) . The following are equivalent 1. r ( · ) is decreasing on R ++ . 2. CE ( F, a ) - a CE ( F, 0) for all F ∈ L and a > 0 . 3. RP ( F, a ) RP ( F, 0) for all F ∈ L and a > 0 . To prove the corollary, set
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