Problem:
Consider an exchange economy with two consumers (i = 1,2) and three goods (l = 1,2,3)
Let x_u (less than or equal to) 0 be the amount of good (l) consumed by consumer (i). Each consumer i's consumption set is Xi = R^3. Suppose that each consumer has an endowment of one unit of each good. Suppose consumer 1's utility function is u1=min{x11,x21, 2x31} and consumer 2's utility function is u2 = x12 + x22 + 2x32. Suppose that the price of good 1, p1, is normalized to one, and that the prices of goods 2 and 3 are denoted by p2 and p3, and are strictly positive.
Q1. Explain why consumer 1's utility maximization implies that x11 = x21 = 3x31. Then find consumer 1's demand as a function p2 and p3.
Q2. Derive consumer 2's demand.
Q3. Find a Walrasian equilibrium in which consumer 2 buys strictly positive quantities of all goods. Describe both consumers' consumption in this equilibrium.