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Unformatted text preview: Solutions to Problem Set 10 EC720.01  Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Tuesday, November 24 1. LinearQuadratic Dynamic Programming The problem is to choose sequences { z t } t =0 and { y t } t =1 to maximize the objective function X t =0 t ( Ry 2 t + Qz 2 t ) , subject to the constraints y given and Ay t + Bz t y t +1 for all t = 0 , 1 , 2 ,... , where , R , Q , A , and B are constant, known parameters. a. The Bellman equation for this problem is v ( y t ; t ) = max z t Ry 2 t + Qz 2 t + v ( Ay t + Bz t ; t + 1) . b. Now guess that the value function takes the quadratic, timeinvariant form v ( y t ; t ) = v ( y t ) = Py 2 t , where P is an unknown constant, allowing the Bellman equation to be specialized to read Py 2 t = max z t Ry 2 t + Qz 2 t + P ( Ay t + Bz t ) 2 . Using this guess, the firstorder condition is for z t is 2 Qz t + 2 BP ( Ay t + Bz t ) = 0 and the envelope condition for y t is 2 Py t...
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 Fall '09
 IRELAND
 Economics

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