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probset10 - Problem Set 10 EC720.01 Math for Economists...

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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 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 guess-and-verify 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 0 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 “linear-quadratic,” 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 < 0 as well. a. Write down the Bellman equation for this problem. b. Now guess that the value function also takes the quadratic, time-invariant form v ( y t ; t ) = v ( y t ) = Py 2 t , where P is an unknown constant. Using this guess, derive the first-order condition for z t and the envelope condition for y t . c. Use your results from above to show that the unknown P must satisfy P = R + βA 2 QP Q + βB 2 P .
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