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Unformatted text preview: Intermediate Microeconomics, Winter 2008 Problem Set No 20  Solutions due: Monday, April 7; Tuesday, April 8 Reading: Chaper 18, Asymmetric Information and Adverse Selection, pp635  646 Q1. Alternating Offer Bargaining . Suppose a buyer and a seller bargain over a price for a good. The value of the good is v for the buyer and c for the seller, with c < v (you might want to choose numbers at first, eg, v = 1 and c = 0). These values are known to both of them. They bargain according to the following protocol: First, the seller makes a price offer p s ∈ [0 , 1]. Then, the buyer accepts or rejects. If the buyer accepts the offer, the game is over and the payoffs are v p s for the buyer and p s c for the seller. If the buyer rejects the offer, the buyer can make a price offer p b himself in the next period and the seller can accept or reject p b . If they trade in the second period, payoffs are discounted by δ ∈ (0 , 1) so the payoffs are δ ( v p b ) and δ ( p b c ) respectively if the seller accepts. Payoffs are zero if the seller rejects. a) If the buyer has rejected the offer in the first period, what will hap pen in the second period, i.e., what will be p b ? The buyer knows that seller is willing to accept any offer at or above the seller’s valuation and will thus offer p b = c . As a result the buyer’s payoff will be δ ( v c ) and the seller’s payoff will be zero. b) Given your knowledge of p b , what will be p s ? The seller knows that by rejecting the buyer can guarantee himself a payoff of δ ( v c ) . In order to persuade the buyer not to reject, the seller must offer a price that makes the buyer no worse off, thus p s = v δ ( v c ) . The buyer’s payoff will be δ ( v c ) and the seller’s payoff will be v p s = (1 δ )( v c ) . Note that it is ambiguous whether it is the buyer or seller who is better off. For high values of δ , the buyer does not lose much by rejecting and waiting and as a result collects a higher portion of the gains from trade than the seller. For low values of δ , the opposite is true. 1 c) (Challenging): Suppose bargaining has now three stages, i.e., first the seller offers p s 1 , then, if p s 1 is rejected, the buyer offers p b , and finally, if p b is rejected, the seller can offer p s 2 . Of course, payoffs in the third period are discounted by δ 2 < δ < 1. Can you find the solution for each stage? We can solve this game in exactly the same fashion. If the third stage is ever reached, the seller will offer p s 2 = v and the payoffs will be δ 2 ( v c ) to the seller and zero to the buyer. In the second stage, the buyer knows that the seller can guarantee for himself a payoff of δ 2 ( v c ) and thus offers p b = c + δ ( v c ) . The seller’s payoff is then δ ( p b c ) = δ 2 ( v c ) and the buyer’s payoff is δ ( v p b ) = ( δ δ 2 )( v c ) . Finally, in the first period the seller knows that the buyer is guaranteed a payoff of ( δ δ 2 )( v c ) and thus offers p...
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 Winter '08
 Burbidge,John
 Microeconomics, Game Theory, seller, subgames, high types

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