# HW3 - Microeconomic Theory II Assignment 3 Applications of...

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Unformatted text preview: Microeconomic Theory II Assignment 3: Applications of Stochastic Dominance Solutions * February 21, 2007 Problem 1. Let α ≥ 0 be a random variable representing a worker’s true ability. Consider two tests of worker ability, i = 1 , 2, with test scores t 1 and t 2 given by: t 1 = α + ε and t 2 = α + ε + η, where ε and η are real-valued, zero mean, statistically independent random variables represent- ing error in the testing process. Suppose that a worker with test score t is paid a wage equal to his conditional mean ability, given that test score. Thus: μ i ( t ) = E [ α | t i = t ] , i = 1 , 2 . gives the wage of a worker whose score on test i is t. Show that the population distribution of wages under test i = 1 is “more unequal” (i.e., “riskier” in the sense of second-order stochastic dominance) than is the population distribution of wages under test i = 2. Solution . From the definitions of t 1 and t 2 we have that t 2 = t 1 + η and E ( α ) = E ( t 1 ) = E ( t 2 ). Further- more, t 2 is a garbling of t 1 . We have that μ 2 ( t ) = E ( α | t 2 = t ) = E η [ E ( α | t 1 = t- η )] = E η ( μ 1 ( t- η )) . Let u : R → R be an arbitrary, concave function. Then: E t [ u ( μ 2 ( t ))] = E t [ u ( E [ α | t 2 = t ])] = E t [ u ( E η [ μ 1 ( t- η )])] ≥ E t [ E η [ u ( μ 1 ( t- η ))]] = E t 1 [ u ( μ 1 ( t 1 ))] . ∗ 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 Therefore, since the above inequality holds for all risk-averse utility functions, u ( · ), the distri- bution of μ 1 (wages after seeing t 1 ) is riskier than the distribution of μ 2 (wages after seeing t 2 ) in terms of second-order stochastic dominance. trianglesolid Problem 2. Consider an expected utility maximizing agent who does not know how much money is in his bank account. Let ˜ y ≥ 0 be this uncertain balance, with E [˜ y ] < ∞ , and let F ( y ) = Pr { ˜ y ≤ y } be the CDF of ˜...
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HW3 - Microeconomic Theory II Assignment 3 Applications of...

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