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Unformatted text preview: Problem Set 12 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Not Collected or Graded 1. Stochastic Linear-Quadratic Dynamic Programming This problem asks you to use dynamic programming to characterize the solution to a stochas- tic version of the linear-quadratic problem that you studied previously, under conditions of perfect foresight, in problem set 10. The problem is to choose contingency plans for a flow variable z t for all t = 0 , 1 , 2 ,... and a stock variable y t for all t = 1 , 2 , 3 ,... to maximize the objective function E X t =0 t ( Ry 2 t + Qz 2 t ) , subject to the constraints y given and Ay t + Bz t + t +1 y t +1 , (1) where 0 < < 1, R < 0, Q < 0, A , and B are all constant, known parameters and t +1 is an independently and identically distributed random shock that satisfies E t ( t +1 ) = 0 and E t ( 2 t +1 ) = 2 , that is, which has zero mean and variance 2 . The flow variable z t must be chosen at time t before t +1 is known, hence the constraint (1) must hold for all t = 0 , 1 , 2 ,... and for all possible realizations of t +1 . a. Guess that the value function for this problem depends only on y t and not t and takes the specific form v ( y t , t ) = v ( y t ) = Py 2 t + d, where P and d are unknown constants. Using this guess, write down the Bellman equation for the problem. In doing this, it might be helpful to note that the assumptions that t +1 is iid and that y t and z t are known at time t imply that E t [( Ay t + Bz t...
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- Fall '09