probset2

# probset2 -...

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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}{[email protected]{\extracolsep{\fill}}r} \multicolumn{2}{c} {\bfseries Problem Set 2 } \\ & \\ EC720.01 - Math for Economists & Peter Ireland \\ Boston College, Department of Economics & Fall 2009 \\ & \\ \multicolumn{2}{c} {Due Thursday, September 24} \end{tabular*} \end{center} \ {\bfseries 1. Utility Maximization - Second-Order Conditions} { The following result specializes Theorem 19.8 from Simon and Blume's book to provide first and second-order conditions for a constrained optimization problem with two choice variables and a single constraint that is assumed to bind at the optimum. o {\bfseries Theorem} Let $F: \mathbf{R^{2}} \rightarrow \mathbf{R}$ and $G: \mathbf{R^{2}} \rightarrow \mathbf{R}$ be twice continuously differentiable functions, and consider the constrained optimization problem $$\max_{x_{1},x_{2}} F(x_{1},x_{2}) \text{ subject to } c \geq G(x_{1},x_{2}),$$ with parameter $c \in \mathbf{R}$. Associated with this problem, define the Lagrangian $$L(x_{1},x_{2},\lambda) = F(x_{1},x_{2}) + \lambda[c-G(x_{1},x_{2})].$$ Suppose there exist values $x_{1}^{*}$, $x_{2}^{*}$, and $\lambda^{*}$ of $x_{1}$, $x_{2}$, and $\lambda$ that satisfy the first-order conditions $$L_{1}(x_{1}^{*},x_{2}^{*},\lambda^{*}) = F_{1}(x_{1}^{*},x_{2}^{*}) - \lambda^{*}G_{1}(x_{1}^{*},x_{2}^{*}) = 0,$$ $$L_{2}(x_{1}^{*},x_{2}^{*},\lambda^{*}) = F_{2}(x_{1}^{*},x_{2}^{*}) - \lambda^{*}G_{2}(x_{1}^{*},x_{2}^{*}) = 0,$$ $$L_{3}(x_{1}^{*},x_{2}^{*},\lambda^{*}) = c - G[(x_{1}^{*},x_{2}^{*}) \geq 0,$$ $$\lambda^{*} \geq 0,$$ and $$This preview has intentionally blurred sections. Sign up to view the full version. View Full Document \lambda^{*}[c - G(x_{1}^{*},x_{2}^{*})] = 0.$$ Suppose also that $c-G(x_{1}^{*},x_{2}^{*})$, so that the constraint binds at the optimum, and that the bordered Hessian'' matrix $$H = \begin{bmatrix} 0 & G_{1}(x_{1}^{*},x_{2}^{*}) & G_{2}(x_{1}^{*},x_{2}^{*}) \\ G_{1}(x_{1}^{*},x_{2}^{*}) & L_{11}(x_{1}^{*},x_{2}^{*},\lambda^{*}) & L_{21} (x_{1}^{*},x_{2}^{*},\lambda^{*}) \\ G_{2}(x_{1}^{*},x_{2}^{*}) & L_{12}(x_{1}^{*},x_{2}^{*},\lambda^{*}) & L_{22} (x_{1}^{*},x_{2}^{*},\lambda^{*}) \end{bmatrix}$$ satisfies the second-order condition that $|H|>0$, so that the determinant of $H$ is strictly positive. Then $x_{1}^{*}$ and $x_{2}^{*}$ are local maximizers of $F(x_{1},x_{2})$ subject to $c \geq G(x_{1},x_{2})$. $Note that this result provides sufficient conditions for a solution to the problem: it says that if the first and second-order conditions are satisfied, then the values of$x_{1}^{*}$and$x_{2}^{*}$constitute at least a local maximum. v With this result in mind, return to the problem solved by a consumer who uses his or her income$I$to purchase$c_{1}$units of good 1 at the price of$p_{1}\$ per
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