probset12

# probset12 - \documentclass[12pt]cfw_article...

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\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{geometry} \geometry{top=1in,bottom=1in,left=1in,right=1in} \setlength{\parindent}{0in} \setlength{\parskip}{2ex} \begin{document} \begin{center} \begin{tabular*}{6.5in}{l@{\extracolsep{\fill}}r} \multicolumn{2}{c} {\bfseries Problem Set 12} \\ \multicolumn{2}{c} {Not Collected or Graded} \end{tabular*} \end{center} {\bfseries 1. Stochastic Linear-Quadratic Dynamic Programming} This problem asks you to use dynamic programming to characterize the solution to a stochastic version of the linear-quadratic problem that you studied previously, under conditions of perfect foresight, in problem set 10. 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 $$E_{0} \sum_{t=0}^{\infty} \beta^{t}(Ry_{t}^{2}+Qz_{t}^{2}),$$ subject to the constraints $y_{0}$ given and Ay_{t}+Bz_{t} + \varepsilon_{t+1} \geq y_{t+1}, \tag{1} where $0<\beta<1$, $R<0$, $Q<0$, $A$, and $B$ are all constant, known parameters and $\varepsilon_{t+1}$ is an independently and identically distributed random shock that satisfies $E_{t}(\varepsilon_{t+1})=0$ and $E_{t} (\varepsilon_{t+1}^{2})=\sigma^{2}$, that is, which has zero mean and variance $\sigma^{2}$. The flow variable $z_{t}$ must be chosen at time $t$ before $\varepsilon_{t+1}$ is known, hence the constraint (1) must hold for all $t=0,1,2,. ..$ and for all possible realizations of $\varepsilon_{t+1}$. \begin{description} \item a. Guess that the value function for this problem depends only on $y_{t}$ and not $\varepsilon_{t}$ and takes the specific form $$v(y_{t},\varepsilon_{t}) = v(y_{t}) = Py_{t}^{2} + d,$$ where $P$ and $d$ are unknown constants. Using this guess, write down the Bellman

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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.

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probset12 - \documentclass[12pt]cfw_article...

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