probset3

# 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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