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14.123 Spring 2001 PROBLEM SET #3 Due: Tuesday, February 27 1.

Peter Diamond Page 1 of 4

Consider a two-person (Ann and Bob) two-good (x and y) competitive exchange economy. Ann has utilityfunction UA = Min {x, 2y}. Bob has utility function UB = Min {3x, y}. Both have the nonnegative quadrant as a consumption possibility set. The aggregate endowment in this economy is (12,12). (a) (b) (c)(d) Contrast the sets of Pareto Optimal allocations under the two different definitions of Pareto Optimality given in class – with weak and strong inequalities on utilities. Assume that Ann has all ofgood x and Bob has all of good y. Describe the competitive equilibrium. Describe the set of individual endowments, consistent with the aggregate endowment, that result in a competitive equilibriumwith positive prices for both goods. Calculate the core of this economy if there is only one consumer of each type.

2. Consider a two-person (Ann and Bob) two-good (x and y) competitive exchangeeconomy. Ann has utility function UA = Min[xA, yA] Bob has utility function UB = 4xB + 4yB Both have the nonnegative quadrant as a consumption possibility set. Ann’s initial endowment is 12 units of x and12 units of y; and Bob's initial endowment is 12 units of y and no x. (a) Describe the prices and quantities in competitive equilibrium. Is there more than one price ratio that supports competitiveequilibrium? Explain.

14.123 Spring 2001

Peter Diamond Page 2 of 4

(b) For what other initial endowments are the competitive equilibrium quantities the same as in your answer to part a? Givethe equilibrium prices that support the equilibrium for each such initial endowment 3. Does an exchange economy with identical consumers with strictly monotonic, strictly convex preferences haveunique competitive equilibrium prices? Are prices unique if the quantities are interior to the consumption possibility set? Consider the following economy: There are three goods, legume, tillip and...