Probset3 - \documentclass[12pt]cfw_article\usepackagecfw_amsmath\usepackagecfw_geometry\geometrycfw_top=1in,bottom=1in,left=1in,right=1in\setlengthc

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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 3 } \\ \multicolumn{2}{c} {Due Thursday, October 1} \end{tabular*} \end{center} Many famous results from microeconomic theory are now understood to be special cases of the envelope theorem. This problem set will ask you to invoke the envelope theorem repeatedly to ``prove'' some of these results. {\bfseries 1. Hotelling's Lemma} Consider a firm that produces output $y$ with capital $k$ and labor $l$ according to the technology described by $$ f(k,l) \geq y. $$ The firm sells each unit of output at the price $p$, rents each unit of capital at the rate $r$, and hires each unit of labor at the wage $w$. Hence it chooses $y$, $k$, and $l$ to maximize profits $$ py - rk - wl $$ subject to the technological constraint just shown above. \begin{description} \item a. Set up the Lagrangian for this problem, letting $\lambda$ denote the multiplier on the constraint. \item b. Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values $y^{*}$, $k^{*}$, and $l^{*}$ that solve the firm's problem, together with the associated value $\lambda^{*}$ for the multiplier. \item c. Assume that the output and input prices $p$, $r$, and $w$ and the production function $f$ are such that it is possible to solve uniquely for the values of $y^{*}$, $k^{*}$, $l^{*}$, and $\lambda^{*}$ in terms of the parameters $p$, $r$, and $w$. Then the function $y^{*}(p,r,w)$ describing the optimal level of output represents the firm's supply function, and functions $k^{*}(p,r,w)$ and $l^{*}(p,r,w)$ describing the optimal inputs are the firm's factor demand curves. Along with these functions, define the firm's profit function as
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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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Probset3 - \documentclass[12pt]cfw_article\usepackagecfw_amsmath\usepackagecfw_geometry\geometrycfw_top=1in,bottom=1in,left=1in,right=1in\setlengthc

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