Final_2010_soln - MS&E 246 Feryal Erhun Final Examination...

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MS&E 246 Final Examination Feryal Erhun March 15, 2010 Solutions Problem 1 (15 points). Answer each of the following questions TRUE or FALSE. If TRUE, provide a brief (1-2 sentence) justification for your answer; if FALSE, provide a brief counterexample. (1 point per correct answer; 4 points per correct justification/counterexample) (a) (5 points) Consider the following payoff table and assume g > h > 0 , w > 0 , v > 0 . Company Monitor No monitor Worker Work 0 , - h w , - w Shirk w - g , v - w - h w - g , v - w This game is dominance solvable when w < g . Solution: FALSE. We also need h 6 = w for the game to be dominance solvable. (b) (5 points) Consider the following game: Player 2 C D Player C 1 , 1 - 1 , 2 1 D 2 , - 1 0 , 0 Suppose that this game is played infinitely many times with discount fac- tor δ . Consider the following strategies: Play C if all previous outcomes have been (C,C), otherwise, play D forever. These strategies form a Nash equilibrium when δ 1 / 2 . Solution: TRUE. Recall that subgame perfection requires a Nash equilib- rium in every subgame , i.e., the game starting from any possible history. 1
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First consider any history up to stage t - 1 where both players collaborated. If a player deviates she will receive 2 in period t and 0 after that. If the player continues to collaborate, she is going to receive 1 1 - δ 2 . Therefore, this deviation is not profitable. If, instead, the previous history dictates that the players should be in punishment mode at stage t , then players cannot make a profitable deviation. Since there are no profitable deviations in any subgame, the given strategies are subgame perfect. (c) (5 points) Alice is a moderate drinker who plans to buy a bottle of wine. Her utility is U = θq - t where q is the quality of the wine, t is the price of the wine, θ is a positive parameter that indexes her taste for quality. If Alice decides not to buy any wine, her utility is 0. Alice may either be a “coarse” (in which case her θ is θ 1 = 5 ) or a “sophisticated” (in which case her θ is θ 2 = 10 ) drinker with equal probability. Bob is a local wine seller and he can produce wine of any quality q with a cost of C ( q ) = q 2 . Assume that he produces two types of wine (a high quality q 2 = 5 and a low quality q 1 = 2 wine) and charges them t 2 = 10 and t 1 = 2 . 5 , respectively. Alice will buy a high quality wine is she is of type “sophisticated” and she will buy a low quality wine otherwise. Solution: FALSE. The IR constraints for both types and the IC constraint for the sophisticated type are satisfied. However, the IC constraint for the coarse type is violated. Problem 2 (25 points). Two firms compete in a market. Let q 1 and q 2 be the production quantities of firm 1 and firm 2 respectively. Assume that firm 1 can only choose q 1 = 10 or q 1 = 5 , and firm 2 can only choose q 2 = 5 or q 2 = 2 . Production costs are zero for both firms, and the market price is set as a function of total quantity as follows: p ( q 1 , q 2 ) = a - q 1 - q 2 .
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