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exchange economy

The attached question is about 2 consumers with 3 goods in an exchange economy. The consumer's utility functions are given. The question asks to find demand functions for each consumer as well as finding Walrasian equilibrium under certain assumptions.


Question (1)

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.
(a) Explain why consumer 1's utility maximization implies that x11 = x21 = 3x31. Then find consumer 1's demand as a function p2 and p3.
(b) Derive consumer 2's demand.
(c) Find a Walrasian equilibrium in which consumer 2 buys strictly positive quantities of all goods. Describe both consumers' consumption in this equilibrium.

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Solution Summary

The following posting examines an exchange economy. The solution covers concepts including utility maximization, consumer demand and consumer consumption.