2. (30) lCons1der an economy with production. A representative consumer has Cobb Douglas utility function U(:c,y) = 5r: - y2 and 12 units of labor L. A consumer owns a representative rm X that produces good .1: with production function a: = 4J1: and one representative rm Y that produces y with production function y = 2m, where LE and L,ir are the labor inputs of the two rms. Without loss of generality, we can normalize the wage rate of labor to 1. Denote the prices of good .1: and good y as p,E and 1:3,, respectively. In this economy, there are three markets. The consumer supplies his labor to the two rms, receives his wage as workers and two rms' prots as the owners of the two rms, and consumes the products of the two rms. Questions (a) (c) guild you to nd the general equilibrium in this economy. (a) {10) Given pm, py, and the normalized wage 1 per unit of labor, nd two rms' demand functions for labors, supply functions for good .1: and good y, and their prot functions. (b) {10) Find the consumer's demand functions for good a: and good 3;. Note that a representative consumer owns one rm X and one rm Y. (c) {5) Provide all three marketclearing conditions and nd the general equilibrium prices pg and 39,, and equilibrium allocation (amounts of good :1: and good 9 pro duced and consumed, and two rms' labor inputs in the general equilibrium). (d) {5) Find the Pareto efcient allocation (amounts of good a: and good 3;, and how to divide the 12 units of labor in the two productions) by providing the optimization problem for the Pareto eicient allocation and solving the problem, and compare it to the general equilibrium outcome from part (c)