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answmdtrmtwo - EC 206 Microeconomics II Second Midterm Exam...

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EC 206 Microeconomics II: Second Midterm Exam April 9, 2007 (Professor Glenn C. Loury) ANSWERS 1. {35 points} Three bidders participate in the auction sale of a single object which is worth v i to bidder i , where v i is uniformly distributed on the unit interval. Each bidder knows his own valuation, but not that of the other two bidders. (a) Suppose the auctioneer solicits sealed bids and sells the object to the highest bidder at a price equal to that bid. Find the symmetric, BNE bidding strategy for this game. Answer: Let b ( v ) be the symmetric BNE equilibrium strategy, and assume that b ( v ) is a strictly increasing, di ff erentiable function. In equilibrium it must be that agent i with valuation v i = v maximizes his expected payo ff by choosing b = b ( v ) . This implies the following maximization problem: b ( v ) ArgMax b 0 { [ v b ] Pr[ Max { b ( v j ) : j 6 = i } b ] } (1) The identity above says that for any agent i and any valuation v , the equilibrium strategy speci fi es a bid b ( v ) which maximizes the agent’s expected surplus, which equals the excess of his valuation over his bid — ( v b ) — times the probability of winning the acution with a bid of b , given that the other two players also follow the strategy b ( v ) . Now, since the equilibrium strategy is strictly increasing it must be invertible. De fi ne φ ( w ) b ∗− 1 ( w ) . So then, b ( φ ( w )) w. Di ff erentiating this identity with respect to w using the chain rule, gives: φ 0 ( w ) = 1 / [ db dv | v = φ ( w ) ] . Hence, the fi rst-order necessary condition for an interior optimum in (1) is: 0 = d db { [ v b ] · [ φ ( b )] 2 } = φ ( b ) 2 + 2( v b ) φ ( b ) φ 0 ( b ) . The condition above must hold at b = b ( v ) . Using this fact and rearranging terms, we have: d dv [ v 2 b ( v )] = v 2 db dv ( v ) + 2 vb ( v ) = 2 v 2 = 2 3 d dv [ v 3 ] .
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