final09

final09 - \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 Final Exam} \\ \multicolumn{2}{c} {Sunday, December 20, 2009} \end{tabular*} \end{center} This exam has two questions on four pages; before you begin, please check to make sure your copy has all two questions and all four pages. The first question is worth 40 points and the second question is worth 20 points; as indicated at the beginning of the semester, this 60-point final will account for 60 percent of your course grade and the problem sets will account for the remaining 40 percent of your course grade. {\bfseries 1. The Maximum Principle in Continuous Time} This question asks you to use the maximum principle to solve a continuous-time, infinite-horizon model in which economic growth is driven by the accumulation of both physical and human capital. During each period $t \in [0,\infty)$, a representative consumer divides up a total stock of physical capital $k(t)$ into a fraction $u(t)$ used to produce goods for consumption and investment in physical capital and a fraction $1-u(t)$ used to produce additional human capital. Likewise, the consumer ``divides up'' a total stock of human capital $h(t)$ by spending a fraction $v(t)$ of his or her time producing goods for consumption and investment in physical capital and a fraction $1-v(t)$ of his or her time accumulating additional human capital. Hence, $u(t)k(t) $ and $v(t)h(t)$ measure the total amounts of physical and human capital used to produce goods and $[1-u(t)]k(t)$ and $[1-v(t)]h(t)$ measure the total amounts of physical and human capital used to accumulate more human capital. Suppose that goods and new human capital both get produced according to the same Cobb-Douglas production function in which the parameter $\alpha$, satisfying $0<\alpha<1$, measures the share of physical versus human capital in production. Suppose also that both capital stocks depreciate at the same rate, measured by the parameter $\delta$ satisfying $0<\delta<1$. Then in this economy, physical capital gets accumulated according to the constraint \begin{equation} [u(t)k(t)]^{\alpha}[v(t)h(t)]^{1-\alpha} - \delta k(t) - c(t) \geq \dot{k}(t), \tag{1} \end{equation} for all $t \in [0,\infty)$. In (1), $c(t)$ denotes the amount of goods consumed
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during period $t$. Thus, when this constraint holds as an equality, as it will when the consumer chooses quantities optimally, it says that the total amount $ [u(t)k(t)]^{\alpha}[v(t)h(t)]^{1-\alpha}$ of goods produced during each period $t$ gets used for three purposes: (1) to replace physical capital that depreciates away, (2) for consumption, and (3) to add, on net, to the stock of physical
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