PS_20_ asym_infos_and_repeated_games_2_

PS_20_ asym_infos_and_repeated_games_2_ - Intermediate...

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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 (0 , 1) so the payoffs are ( v- p b ) and ( 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 sellers valuation and will thus offer p b = c . As a result the buyers payoff will be ( v- c ) and the sellers 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 ( 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- ( v- c ) . The buyers payoff will be ( v- c ) and the sellers payoff will be v- p s = (1- )( v- c ) . Note that it is ambiguous whether it is the buyer or seller who is better off. For high values of , 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 , 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 2 < < 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 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 2 ( v- c ) and thus offers p b = c + ( v- c ) . The sellers payoff is then ( p b- c ) = 2 ( v- c ) and the buyers payoff is ( v- p b ) = ( - 2 )( v- c ) . Finally, in the first period the seller knows that the buyer is guaranteed a payoff of ( - 2 )( v- c ) and thus offers p...
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This note was uploaded on 12/10/2011 for the course ECON 401 taught by Professor Burbidge,john during the Winter '08 term at Waterloo.

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PS_20_ asym_infos_and_repeated_games_2_ - Intermediate...

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