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HW16_solutions - Intermediate Microeconomics Econ 401...

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Intermediate Microeconomics, Econ 401 Winter 2008 Problem Set No 16: Solutions (14.2*) Show the payoff matrix and explain the reasoning in the prisoners’ dilemma example where Larry and Duncan, possible criminals, will get one year in prison if neither talks; if one talks, one goes free and the other gets five years; and if both talk, both get two years. (Note: The payoffs are negative because they represent years in jail, which is a bad). Duncan talk don’t talk Larry talk -2 , -2 0 , -5 don’t talk -5, 0 -1, -1 (From answer, p. A-13) If Duncan plays “don’t talk”, Larry gets 0 if he talks and -1 (a year in jail) if he doesn’t talk. If Duncan talks, Larry gets -2 if he talks and -5 if he doesn’t talk. Thus Larry is better off by choosing to talk in either case, so “talk” is his dominant strategy. In the normal form game above, we underline these best responses. By the same reasoning (and identical payoffs), Duncan’s dominant strategy is to talk. As a result, the Nash equilibrium is for both to talk. (17.28 W) (Hint: This question is in fact very similar to our Cournot game, in which firms choose Quantities. Here, the players choose the hours of working. You need to find the best response functions and solve them.) Anna and Bess are assigned to write a joint paper within a 24-hour period about the Pareto optimal provision of public goods. Let t A and t B denote the number of hours that Anna and Bess contribute to the project, respectively. The grade that Anna and Bess earn is: 23 ln( t A + t B ) . Anna gets utility from her grade and from her leisure, and if she works t A hours on the project, she can enjoy 24 - t A hours of leisure. Anna’s utility function is U A = 23 ln( t A + t B ) + ln(24 - t A ) . Bess’s is identical. If they choose hours simultaneously and independently, what is the Nash equilibrium number of hours that each will work on the project? What is the number of hours each should contribute to the project, that maximizes the sum of their utilities? To find the Nash equilibrium (NE), let’s maximize each person’s individual utility function, to get their best response functions and then substitute one into the other, to solve for t * A and t * B . For Anna, max U A ∂U A ∂t A = 0 = 23 t A + t B - 1 24 - t A 23 t A + t B = 1 24 - t A 552 - 23 t A = t A + t B t A = 23 - 1 24 t B . 1
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Follow the same steps to find Bess’s best response function: t B = 23 - 1 24 t A . Set them equal to solve for t A = t B , and substitute in to either best response function to find that t A = t B = 22 . 08 in the Nash equilibrium. (Note that the web answer finds an incorrect value here, because of a calculation mistake.) To find the socially optimal amounts for t A and t B , we need to maximize their joint utility: max U A + U B = max t A ,t B [46 ln( t A + t B ) + ln(24 - t A ) + ln(24 - t B )] ∂U A + U B ∂t A = 0 = 46 t A + t B - 1 24 - t A ∂U A + U B ∂t B = 0 = 46 t A + t B - 1 24 - t B Again, solve for both best response functions and set them equal: 24 - t A = 24 - t B t A = t B .
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