probset10

# probset10 - \documentclass[12pt]cfw_article...

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\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{geometry} \usepackage{url} \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 10} \\ \multicolumn{2}{c} {Due Tuesday, November 24} \end{tabular*} \end{center} {\bfseries 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 $\ {z_{t}\}_{t=0}^{\infty}$ for a flow variable and $\{y_{t}\}_{t=1}^{\infty}$ for a stock variable to maximize the objective function $$\sum_{t=0}^{\infty} \beta^{t} (Ry_{t}^2+Qz_{t}^{2}),$$ subject to the constraints $y_{0}$ given and $$Ay_{t}+Bz_{t} \geq y_{t+1}$$ for all $t=0,1,2,. ..$, where $\beta$, $R$, $Q$, $A$, 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, $0<\beta<1$, and to make the objective function concave, it is helpful to assume that $R<0$ and $Q<0$ as well. \begin{description} \item a. Write down the Bellman equation for this problem. \item b. Now guess that the value function also takes the quadratic, time-invariant

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## probset10 - \documentclass[12pt]cfw_article...

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