Problem 1 (45 points): There are two goods X and Y and two consumers named Ann and Bob. UA(XA; YA) = XAYA and UB(XB; YB) = XBY B where > 0: MUA;X = YA; MUA;Y = XA; MUB;X = Y B ; MUB;Y = XBY 1 B : Total endowments of goods X and Y are respectively denoted wX and wY : a) (10 points) Find Pareto e¢ cient allocation in which XA = XB = wX=2 (that is, Önd Y A and Y B such that, XA = XB = wX=2 along with Y A and Y B constitutes PE allocation). Show your work. b) (3 points) Is Y A increasing or decreasing in wY ? c) (3 points) Is Y A increasing or decreasing in ? d) (3 points) Is Y B increasing or decreasing in wY ? e) (5 points) Is Y B increasing or decreasing in ? f) (2 points) Interpret your answers to parts c and e. Now for the rest of the parts suppose that wA;X = 1; wA;Y = 1; wB;X = 1; wB;Y = 1: g) (4 points) What is total endowment of good X in the society? h) (15 points) Normalize pX = 1 and assume = 4: Solve for the general equilibrium price and quantities consumed. That is, Önd pY ; X A; X B; Y A; Y B: Show your work. Problem 2 (30 points): Suppose UA(X; Y ) = minfX; Y g and UB(X; Y ) = minf2X; Y g: wX = 1; wY = 1: a) (15 points) Draw the edgeworth box, label it completely and depict the set of Pareto e¢ cient allocations. For the following part, suppose that initial endowments are wA;X = 0:3; wA;Y = 0:5; wB;X = 0:7; wB;Y = 0:5: b) (15 points) Which bundles are Pareto optimal if trade takes place and Ann obtains all the gains from trade? Problem 3 (25 points): Suppose Ann and Bob each have 40 units of good X and 40 units of good Y . UA(X; Y ) = 1:5X + 1:8Y; UB(X; Y ) = 1:5X + 1:8Y: MUA;X = MUB;X = 1:5 and MUA;Y = MUB;Y = 1:8: a) (10 points) Are there Pareto improving trades? If yes, what are they? Now suppose that Ann and Bob get contradictory information. Ann now thinks that good Y is totally worthless and Bob thinks that X is totally worthless. Hence, their utilities become UA(X; Y ) = X; UB(X; Y ) = Y: b) (10 points) Draw the dgeworth box with indi§erence curves at the initial endowments. Clearly label everything. 1 c) (5 points) Assume Ann and Bob exhaust all potential Pareto improving trades. Which allocation(s) could result?

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