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answer10 - Solutions to Problem Set 10 EC720.01 - Math for...

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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. Linear-Quadratic 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, time-invariant 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 first-order 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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answer10 - Solutions to Problem Set 10 EC720.01 - Math for...

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