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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 independent-private-values 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 rank-ordered 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 risk-neutral. (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