Reference no: EM13995088

Suppose that there are two consumers in an exchange economy with two goods. There are total of ten units of each good. The initial endowments for these consumers are (e1x, e1y) for consumer 1 and (e2x, e2y) for consumer 2, where e2x = 10 – e1x and e2y = 10 – e1y. Consumer 1 has utility function u(x1, y1) = x1 1/3 y1 2/3, while consumer 2 has utility function v(y1, y2) = x2 2/3 y2 1/3. (a) Normalize the price of good 1 to 1 and denote the price of good 2 by p. Write the budget constraints for each consumer. (b) Solve for each consumer's optimal consumption bundle as a function of price p. (c) Is each person's consumption of good x increasing or decreasing as a function of p? Explain the meaning of this result in words -- is this the result that you would expect? (d) What is the equilibrium price for the initial allocation of (6, 2) to consumer 1 and (4, 8) to consumer 2? What are the resulting equilibrium bundles? (e) What is the first-order condition for Pareto optimality? Check that the equilibrium bundles satisfy this condition. (f) We know that with Cobb-Douglas preferences of this form, consumer 1 spends 1/3 of her wealth on good 1, while consumer 2 spends 2/3 of her wealth on good 1. So why doesn't the equilibrium allocation leave consumer 1 with twice as much of good 2 as of good 1 - and similarly for consumer 2?