This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 realvalued, 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 secondorder 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 riskaverse 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 secondorder 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...
View
Full
Document
 Spring '07
 G.LOURY
 Microeconomics

Click to edit the document details