A Simplified Proof for Preservation of the Ordering by Risk Aversion with Background Risks in the Expected Utility Theory

概要

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In this paper, I give a simplified proof of Nachman's cerebrated result on preserving the ordering by risk aversion over basic utility functions in the expected utility theory. At first, I prove a lemma that says, if a basic utility function u_1 is more risk averse than a basic utility function u_2 then, for any randomized initial wealth, the randomized marginal utility measure of u_2 stochastically dominates that of u_1 by the first order. The proof is a simple implication of the famous result that u_1 = g°u_2 for some increasing concave utility-transformation function g. Then this lemma is used to prove the main theorem that says randomizing initial wealth does not affect the ordering of basic utility function by risk aversion. The proof is a direct implication of the fact that the ARA of the basic utility function with a randomized initial wealth is equal to the expectation of the ARA of the original basic utility function with respect to the randomized marginal utility measure associated with this randomized initial wealth.
千葉大学の論文
2005-06-29