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 0 be a random variable representing a workers true ability. Consider two tests of worker ability, i = 1 , 2, with test scores t 1 and t 2 given by: t 1 = + and t 2 = + + , where and 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: i ( t ) = E [ | 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 + and E ( ) = E ( t 1 ) = E ( t 2 ). Further- more, t 2 is a garbling of t 1 . We have that 2 ( t ) = E ( | t 2 = t ) = E [ E ( | t 1 = t- )] = E ( 1 ( t- )) . Let u : R R be an arbitrary, concave function. Then: E t [ u ( 2 ( t ))] = E t [ u ( E [ | t 2 = t ])] = E t [ u ( E [ 1 ( t- )])] E t [ E [ u ( 1 ( t- ))]] = E t 1 [ u ( 1 ( t 1 ))] . Prepared by Omer Ozak (ozak@brown.edu), 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 1 (wages after seeing t 1 ) is riskier than the distribution of 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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