A: Consider the special 2-good case where consumer 1 views the goods x1 and x2 as perfect complements— with utility equal to the lower of the quantities of x1 and x2 in her basket. Consumer 2, on the other hand, views the goods as perfect substitutes, with utility equal to the sum of the quantities of x1 and x2 in his basket.
(a) Illustrate the contract curve for these two consumers in the Edge worth box assuming the overall endowment of each of the two goods in the economy is e.
(b) What does the utility possibility frontier that derives from this contract curve look like? Carefully label intercepts and slopes.
(c) How would the utility possibility frontier be different if the utility of consumer 2 were given by half the sum of the quantities of x1 and x2 in his basket?
(d) Consider the original utility possibility frontier from part (b). Suppose the two individuals are currently endowed with the midpoint of the Edge worth box. Locate the point on the utility possibility frontier that corresponds to this allocation of goods.
(e) Suppose that the government does not have access to efficient taxes for the purpose of redistributing resources. Rather, the government uses distortion nary taxes, with the marginal cost of redistributing $1 increasing with the level of redistribution. What do you think the second best utility possibility frontier now looks like relative to the first best? Do the two shares any points in common?
(f) Suppose instead that the current endowment bundle lies off the contract curve on the diagonal that runs from the upper left to the lower right corner of the Edge worth box. If competitive markets are allowed to operate, do your first and second best utility possibility frontiers differ from those you derived so far?
(g) If markets were not allowed to operate in the case described in (e), where would the second best utility possibility frontier now lie relative to the first best?
B: Suppose we have an Edge worth economy in which both individuals have the utility function u(x1, x2) = xα1 x2 (1−α) and where the economy’s endowment of each of the two goods is e.
(a) Set up a maximization problem in which the utility of consumer 1 is maximized subject to the economy-wide endowment constraints and subject to keeping individual 2’s utility at u. (By defining individual 2’s consumption of each good as the residual left over from individual 1’s consumption, you can write this problem with just a single constraint.)
(b) Derive the contract curve for this economy.
(c) Use this to derive the utility possibility frontier. What shape does it have?
(d) How would your answers change i f we had specified the utility function as u(x1, x2) = x β 1 x (0.5−β) 2 with 0 < β < 0.5.
(e) Do the two different utility functions represent the same underlying (ordinal) preferences? If so, explain the difference in the two utility possibility frontiers.
(f) How could you keep the same (ordinal) preferences but transform the utility function in such a way as to cause the utility possibility set to be non-convex? Explain.