problem-set-4-answers

# problem-set-4-answers - Economics 703 Advanced...

This preview shows pages 1–2. Sign up to view the full content.

Economics 703 Advanced Microeconomics Prof. Peter Cramton Problem Set 4: Suggested Answers 1. & 2. Consider an n-bidder auction with valuations v i independently and identically distributed according to F(v i ) on support [v,v]. Let the highest bidder pay the price (1 - k)b f + kb s to the seller, where k [0,1], b f = the (first) highest of the n bids, and b s = the second-highest of the n bids. This is a k+1 st price auction: if k = 0 we have a first-price auction, and if k = 1 we have a second-price auction. The utility function for bidder i is then u (b ,b ,v ,v ) v [(1 k)b k maxb ] if b maxb , 0 otherwise. i i i i i i i j i j i j i j - - = - - + R S T So if the n - 1 other bidders play the strategy b(v j ), i's best response solves max v i (1 k)b i kb(y)}dG(y) b 0 b (b ) i 1 i - - + X Z Y - where y max j i v j and G(y) = F n-1 (y) is the distribution of the highest of n - 1 random variables independently distributed according to F( ). The first-order condition is 0 (1- k)dG(y) d db b (b ) {v [(1 k)b kb(b (b ))]}g[b (b )] i 1 i 0 b (b ) i i -1 i -1 i 1 i =- + L N M O Q P X Z Y - - + - - where g(y) = (n - 1)F n-2 (y)f(y) is the density of G and f is the density of F. Using the fact that d db b (b ) = 1 b [b b ) i 1 i -1 i - ( ] , The first-order-condition becomes (*) ( [ ( ) ( ] . 1 1 - - - - k)b b b )]= (v b )(n = f[b (b )] F[b b ) -1 i i i 1 i -1 i

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/21/2009 for the course ECON ECON703 taught by Professor Professorpetercramton during the Fall '09 term at Maryland.

### Page1 / 4

problem-set-4-answers - Economics 703 Advanced...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online