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Unformatted text preview: 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 This problem will give you more practice with dynamic programming under certainty; it presents another case in which an explicit solution for the value function can be found using the guessandverify method. The problem is to choose sequences { z t } ∞ t =0 for a flow variable and { y t } ∞ t =1 for a stock variable to maximize the objective function ∞ X t =0 β 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 β , R , Q , A , and B are constant, known parameters. This problem can be described as being “linearquadratic,” because the constraint is linear and the objective function is quadratic. The discount factor lies between zero and one, 0 < β < 1, and to make the objective function concave, it is helpful to assume that R < 0 and Q < as well. a. Write down the Bellman equation for this problem. b. Now guess that the value function also takes the quadratic, timeinvariant form v ( y t ; t ) = v ( y t ) = Py 2 t , where P is an unknown constant. Using this guess, derive the firstorder condition for z t and the envelope condition for y t ....
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
 Fall '09
 IRELAND
 Economics, The Land

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