Problem Set 4 1. Edgeworth Box and Pareto Efficiency Consider the island economy with Friday and Robinson. They have agreed to share their resources and they have also agreed that the weight that Friday receives in the economy is w/ c (0. 1) and the weight that Robinson receives is w = 1-w. Their preferences are different. Friday likes bananas more than coconuts, and Robinson likes coconuts more than bananas (i.e. you can assume that a E (0.5), 1)). The utility functions are given by: w(r/) =alne + (1-a) lor!. and u"(r") = (1 - a) Inr; + alar. The endowment of the island is given by Cb = 2. ce = 2c > 2. (a) For every weight wy E (0. 1), find the allocation which would max- imize the social surplus given the weights; in other words, we are interested in finding the allocation (x/, 15, 24, If ) which maximizes the sum wu(x,)+ ( 1- wj)u, (i,) subject to the resource constraints of the economy.= w(1- a)+ (1-wfja (4) (b) For every weight wy E (0. 1), can you find an initial endowment of bananas and coconuts among Robinson and Friday and a pair of prices that the Pareto efficient allocation actually constitutes an equilibrium of the market. (Here we decentralize the Pareto efficient allocation via a market equilibrium.) It is sufficient to discuss the case of c = 1.2. Edgeworth Box and Competitive Equilibrium. Consider again Robinson and Friday. Now, Robinson and Friday have the same endow- ment of bananas and coconuts: = 4 = 1 ef = but their preferences are different. (a) We normalize the price of bananas to be equal to one, or po = 1. Compute the demand function of Robinson and Friday as a function of the price for coconuts pe-For all of the questions below, assume that there are only two consumers, Ms. A and Mr. B; and there are only two goods, good 1 and good 2. Also, ry denotes an amount of good i for Ms. A, and r, denotes an amount of good i for Mr. B. And, wa denotes the initial endowment of good i for Ms. A, and we denotes the initial endowment of good i for Mr. B. 1. Ms. A's utility function is uA (r) = Inof try. Mr. B's utility function is us (x*) = of +8x, . Endowments are given by wi = (4, 2) and we = (3,5). Find the contract curve.2. Ms. A's utility function is uA (x1 , x4) = 2ri + 12. while Mr. B's utility function is P, x7 ) = x8. There is a total of Wj of good 1 and w2 of good 2 available in the economy. Find the set of all Pareto efficient allocations in the economy. (Hint: Try drawing the indifference curves on an Edgeworth Box and checking for possible Pareto improvements at the different allocations.)Graphically, this is the set of all point on the bottom and right boundaries of the Edgeworth Box. Drawing the linear indifference curves in this problem with the appropriate slopes should help you see that all other points have Pareto improvements. 3. For Ms. A, goods 1 and 2 are perfect complements. For Mr. B. goods 1 and 2 are perfect substitutes. Endowments are given by wi = (10,0) and we = (0, 10) . Let p2 = 1. Find the competitive equilibrium.So, Ms. A will sell half of her endowment good 1 to Mr. B, in exchange for half of his good 2, at a price ratio of 2 = 1. The price/budget line therefore cuts across the entire Edgeworth Box from the top left corner to the bottom right corner at a slope of (negative) one. 4. For both Ms. A and Mr. B, goods 1 and 2 are perfect complements. Endowments are given by wa = (10,0) and w = (0, 10) . Let p2 = 1. Find the competitive equilibrium.5. Ms. A's utility function is while Mr. B's utility function is UB (CT , 17) = 1+ (a) Find the competitive (Walrasian) equilibrium in this economy when wA = 2, wf = 0, WA = 1, 09 = 2 and p2 = 1
