probset4

probset4 - \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 4 } \\ \multicolumn{2}{c} {Due Thursday, October 8} \end{tabular*} \end{center} The two welfare theorems of economics tell us that optimal and equilibrium resource allocations coincide -- but only under certain conditions. Sometimes when market failures prevent the theorems from holding, however, government policy can help improve equilibrium outcomes, as the questions below reveal. {\bfseries 1. Optimal Allocations} Consider an economy in which output is produced with capital $k$ and labor (``hours worked'') $h$ according to the Cobb-Douglas specification $k^{\alpha}h^{1-\alpha}$, where $0<\alpha<1$. In this static model, the capital stock $k$ is taken as given, but hours worked $h$ and consumption $c$ are chosen by a benevolent social planner in order to maximize the utility $\ln(c)-h$ of a representative consumer, where $\ln$ denotes the natural logarithm. Hence, an optimal resource allocation solves the problem $$ \max_{h,c} \ln(c)-h \text{ subject to } k^{\alpha}h^{1-\alpha} \geq c. $$ Find solutions for the optimal choices for $h$ and $c$ in terms of the parameters $k$ and $\alpha$. {\bfseries 2. Equilibrium Allocations} Now consider the same economy, but where perfectly competitive markets for inputs and outputs replace the social planner in allocating resources. \begin{description} \item a. Now, the representative consumer (standing in for a large number of identical consumers) is endowed with $k^{s}$ units of capital, and chooses labor supply $h^{s}$ and consumption $c$ to maximize utility subject to a budget constraint, that is, to solve $$ \max_{h^{s},c} \ln(c)-h^{s} \text{ subject to } rk^{s} + wh^{s} \geq c,
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probset4 - \documentclass[12pt]cfw_article...

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