3. Alice and Bill live in an exchange economy with two goods, x1 and x2. Alice owns 1 unit of good 1 and 9 units of good 2 whereas Bill owns 9 units of good 1 and 1 unit of good 2. Alice’s utility function is (x1, x2 ) = x1 x2 1− and Bill’s utility function is (x1, x2 ) = x1 x2 1− where ∈ (0,1) and ∈ (0,1). a. Suppose that Alice and Bill enter into a trading agreement. Show all feasible allocations of goods 1 and 2 for Alice and Bill in an Edgeworth box. Show the allocations that are Pareto improvements over the endowment allocation. b. Determine the competitive equilibrium price. c. Assume that = and that Alice and Bill trade at the price ratio that you have found in part b. Show that there are gains from trade. d. Assume again that = . Determine how the equilibrium price ratio and the consumption levels change as increases. Give an intuitive explanation for your answer. e. Suppose that initially Alice has 9 units of both goods and Bill has 1 unit of both goods, i.e, the endowments are very unequal. i. Assume that ≠ . Determine if there are still gains from trade. What can you infer about the rich and the poor? ii. Now assume that = . The government thinks that the desired level of consumption of both goods is 5 units for both Alice and Bill. Therefore, it levies a proportional tax ∗ on any goods purchased, that is, if the initial price ratio is P = P1 P2 , then the price ratio after the proportional tax becomes P ′ = P1 (1+ ∗) P2 (1+ ∗) = P. The government will use the tax revenue as a lump sum payment to reach the desired allocation. Explain if the government’s scheme complies with the Second Welfare Theorem.
