problem-set-4-answers

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

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

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