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grasshopper - Relative and Absolute Risk Aversion Question...

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Unformatted text preview: Relative and Absolute Risk Aversion Question 1. A) Define the Arrow-Pratt measure of absolute risk aversion. Answer : Where u is the von Neumann-Morgenstern utility function, R A ( y ) =- u 00 ( y ) u ( y ) . B) Consider the following von Neumann Morgenstern utility function u ( x ) =- 1 α e- αx . For what values of α is a consumer with this utility function risk-averse? Does this consumer display increasing, decreasing, or constant absolute risk aversion? Explain. Answer : This consumer is risk averse if and only if α > 0. For this function, R A ( y ) = α . Since α does not change with y , this consumer has constant absolute risk aversion. C) Consider the following von Neumann Morgenstern utility function u ( x ) = 1 α x α . For what values of α is a consumer with this utility function risk-averse? Does this consumer display increasing, decreasing, or constant absolute risk aver- sion? Does this consumer display increasing, decreasing, or constant relative risk aversion? Answer : In this case, u 00 ( x ) = ( α- 1) x α- 2 . We see that u 00 ( x ) < 0 so long as α < 1, so he is risk averse if and only if α < 1. Taking derivatives and simplifying, we find that R A ( x ) = 1- α x Differentiating with respect to x , we see that he has decreasing absolute risk aversion if α < 1....
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grasshopper - Relative and Absolute Risk Aversion Question...

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