\documentclass{article} \usepackage{amsmath} \usepackage{amssymb} \begin{document} \textbf{Question A} Consider an household that has consumption set $\mathbb{R}_L^+$ and complete, transitive, continuous, and strictly convex preferences $\succ$ over $\mathbb{R}_L^+$. The household can use income $w \geq 0$ to purchase a commodity vector $x \in \mathbb{R}_L^+$ at prices $p \in \mathbb{R}_L^+$. In addition, the household can use technology $Y \subseteq \mathbb{R}_L^+$ to produce a vector $y$; $Y$ follows the standard notation for production possibility sets (negative numbers denote inputs and positive numbers denote outputs), and satisfies $0 \in Y$. This household ultimately consumes the sum of purchased and produced goods. \begin{enumerate} \item Write down in detail the consumer's problem of choosing an optimal bundle for preferences $\succ$ given the various constraints, and briefly explain intuitively how the production technology enters this problem. \item Assume $L = 2$, $w = 10$, $p = (1,2)$, the technology is described by $y_1 \leq -4y_2$, and $\succ$ is represented by the utility function $\min \{x_1 + y_1, x_2 + y_2\}$. Find the optimal consumption bundle