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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 LinearQuadratic Dynamic Programming This problem asks you to use dynamic programming to characterize the solution to a stochas tic version of the linearquadratic 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 β t ( Ry 2 t + Qz 2 t ) , subject to the constraints y given and Ay t + Bz t + ε t +1 ≥ y t +1 , (1) where 0 < β < 1, R < 0, Q < 0, A , and B are all constant, known parameters and ε t +1 is an independently and identically distributed random shock that satisfies E t ( ε t +1 ) = 0 and E t ( ε 2 t +1 ) = σ 2 , that is, which has zero mean and variance σ 2 . The flow variable z t must be chosen at time t before ε t +1 is known, hence the constraint (1) must hold for all t = 0 , 1 , 2 ,... and for all possible realizations of ε t +1 . a. Guess that the value function for this problem depends only on y t and not ε t and takes the specific form v ( y t ,ε 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 ε 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
 IRELAND
 Economics, Dynamic Programming, Recursion, Optimization, Bellman equation, Rt+1

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