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.
Consider an exchange economy with two consumers and three goods
Let be the amount of good ( )consumed by consumer . Each consumer consumption set is Suppose that each consumer has an endowment of one unit of each good. Suppose consumer 1's utility function is and consumer 2's utility function is . Suppose that the price of good 1, , is normalized to one, and that the prices of goods 2 and 3 are denoted by and , and are strictly positive.
(a) Explain why consumer 1's utility maximization implies that . Then find consumer 1's demand as a function of and .
(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.
This job examines an exchange economy.