probset9

# probset9 - \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 9} \\ \multicolumn{2}{c} {Due Thursday, November 12} \end{tabular*} \end{center} {\bfseries 1. Natural Resource Depletion} Consider a continuous time version of the natural resource depletion problem that you solved previously, in discrete time, using the Kuhn-Tucker theorem in problem set 6. Let $c(t)$ denote society's consumption of an exhaustible resource during each period $t \in [0,\infty)$, and suppose that a representative consumer gets utility from this resource as described by \int_{0}^{\infty} e^{-\rho t} \ln(c(t)) \, \mathrm{d}t, \tag{1} where the discount rate satisfies $\rho > 0$. Let $s(t)$ denote the stock of the resource that remains at each date $t \in [0,\infty)$. Since the resource is nonrenewable, this stock evolves according to -c(t) \geq \dot{s}(t) \tag{2} for $t \in [0,\infty)$, which just indicates that consumption during period $t$ subtracts from the stock that remains to be consumed from that period forward. Now the social planner's problem can be stated as: choose continuously differentiable functions $c(t)$ and $s(t)$ for $t \in [0,\infty)$ to maximize (1) subject to $s(0)=s_{0}$ given and (2) for all $t \in [0,\infty)$. \begin{description} \item a. Define the Hamiltonian for this problem, using $\pi(t)$ to denote the multiplier corresponding to the constraint (2) for period $t$. \item b. Next, write down the first-order condition for $c(t)$ and the pair of

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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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probset9 - \documentclass[12pt]cfw_article...

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