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ps9macro2sp08ans - Econ 387L Macro II Spring 2008...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Answers Problem Set # 9 Problem 2 Consider a version of the environment studied by Stokey (1989) that was taught in class. Let ( x, X, y ) be the choice variables available to a representative agent, the market as a whole, and a benevolent government, respectively, where x, X X = { x L , x H } and y Y = { y L , y H } . Let per period payo ff s to the government be denoted u ( x i , X j , y k ) . In the case where x j = X j let payo ff s be given as in the table u ( x i , X j , y k ) X L X H y L 0* 20 y H 1 10* The values of u ( x i , X j , y k ) not reported in the table are such that the competitive equilibria (i.e. ones where x = X = h ( y ) ) are the outcome pairs denoted by the asterisk. 1 Objective : The objective of this problem is to help you understand how we may take a Ramsey problem (a sequential game) and imbed it in a dynamic simultaneous move game. Using a trigger strategy (i.e., a threat), the government can be induced to play the best outcome even though in a single period, its incentive is to deviate. We will also see that many strategies may be played that satisfy the trigger strategy and that there exists a tension between the agent’s discount factor and the punishment strategy - as the pain of the punishment increases, the discount required to support the optimal strategy decreases. Conversely, if I care more about the future then the punishment in fl icted upon me in the future matters more. (1) De fi ne a Ramsey plan and a Ramsey outcome for a one-period economy. Find the Ramsey outcome. Answer: In a Ramsey equilibrium, government goes fi rst and the public responds to that decision, playing a best response which is necessarily a competitive equilibrium. Assuming that government’s objective is to maximize public utility, a Ramsey equilibrium solves max y u ( h ( y ) , h ( y ) , y ) (1) which in our example is ( y, x ) = ( y h , x h ) . (2) De fi ne a Nash equilibrium (in pure strategies) for the one-period economy. Answer: A NE is a triplet ¡ x NE , X NE , y NE ¢ such that this set solves max x,y u ( x, X, y ) (2) where ¡ x NE , X NE , y NE ¢ is a competitive equilibrium. (3) Show that there exists no Nash equilibrium (in pure strategies) for the one period economy. Answer: 1 The problem imposes consistency ( x = X ) in the set up, so the solutions will also impose this consistency. A more complete set-up would include pay-o ff s that allow agents to play strategies di ff erent than the average agent. For completeness assume
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