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Unformatted text preview: STATE UNIVERSITY OF NEW YORK AT BUFFALO DEPARTMENT OF ECONOMICS Economics 666, Microeconomic Theory II Spring 2010 Peter B. Morgan Assignment Six Question 1. In the games that are independentprivatevalues models of auctions of a single object Chance moves first by drawing at random n private valuations v 1 ,...,v n for the object from a probability distribution with a cumulative density function F that is defined on an interval [ v , v ]. These valuations are then assigned randomly to the bidders 1 ,...,n . Use v (1) ,...,v ( n ) to denote the rankordered private valuations with v ( i ) ≥ v ( i +1) for all i = 1 ,...,n 1. For the purposes of this question, suppose that the private valuations are Uni formly distributed on [0 , 1]. Assume that all of the bidders are riskneutral. (i) What is the cumulative probability function of v (1) ? What is the probability density function of v (1) ? (ii) What is the cumulative probability function of v (2) ? What is the probability density function of v (2) ? (iii) Compute the means E v (1) [ v (1)  n ] and E v (2) [ v (2)  n ] of the distributions of v (1) and v (2) . Show that E v (1) [ v (1)  n ] > E v (2) [ v (2)  n ] for all n ≥ 2 and that both of these means converge to v = 1 as n → ∞ . (iv) Compute the variances of the distributions of v (1) and v (2) . Show that both of these variances converge to zero as n → ∞ . Is one of these variances larger than the other?...
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This note was uploaded on 03/21/2010 for the course ECO 666 taught by Professor P.morgan during the Spring '10 term at SUNY Buffalo.
 Spring '10
 P.Morgan
 Economics

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