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# answer12 - Solutions to Problem Set 12 EC720.01 Math for...

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Solutions to Problem Set 12 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Not Collected or Graded 1. Stochastic Linear-Quadratic Dynamic Programming 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 0 X t =0 β t ( Ry 2 t + Qz 2 t ) , subject to the constraints y 0 given and Ay t + Bz t + ε t +1 y t +1 (1) for all t = 0 , 1 , 2 , ... and all possible realizations of ε t +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 . a. Using the 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, the Bellman equation for this problem becomes Py 2 t + d = max z t Ry 2 t + Qz 2 t + βPE t [( Ay t + Bz t + ε t +1 ) 2 ] + βd = max z t Ry 2 t + Qz 2 t + βP ( Ay t + Bz t ) 2 + βPσ 2 + βd b. Using the expression on the last line from part (a), the first-order condition for z t becomes 2 Qz t + 2 βBP ( Ay t + Bz t ) = 0 and the envelope condition for y t becomes 2 Py t = 2 Ry t + 2 βAP ( Ay t + Bz t ) . Notably, the introduction of uncertainty into the linear-quadratic problem does not alter the form of the optimality conditions for z t and y t . This special feature of the stochastic LQ model is often referred to as the property of “certainty equivalence,” and it does not apply more generally for problems that do not have the linear-quadratic structure.

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