Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 7: Competitive Equilibrium and Edgeworth Box2/24/10This problem set is not due, but a solution will be posted on the class website on Monday. This problem set contains excellent material for preparation ofthe midterm exam.1. Edgeworth Box and Competitive Equilibrium. Consider Robinsonand Friday from c lass. In c ontrast to the model in class, Robinson andFriday have the same endowment of bananas and coconuts:xbf= xbr= 1xcf= xcr= c > 1but their preferences are di¤erent. Friday likes bananas more than co-conuts, and Robinson likes coconuts more than bananas (i.e. you canassume that  2 (0:5; 1)). The utility function are given by:uf(xf) =  ln xbf+ (1  ) ln xcf;andur(xr) = (1  ) ln xbr+  ln xcr.(a) We normalize the price of bananas to be equal to one, or pb= 1.Compute the deman d function of Robinson and Friday as a functionof the price for co conuts pb.(b) Find the equilibrium price of this island economy and determine thenet trading quantities.i. Start with the case of c = 1.ii. Continue to describe the equilibrium for general c > 1 and de-scribe the intuition.2. Pareto E¢ ciency. Consider the island economy again but th is timeFriday and Robinson have agreed to share their resources and they h avealso agreed that the weight that Friday receives in the economy is wfandthe weight that Robinson receives is wr= 1  wf.(a) For every weight wf2 (0; 1), …nd the allocation which would maxi-mize the social surplus given the weights; in other words, we are in-terested in …nding the allo c ation (xbf; xbr; xcf;xcr) which maximizesthe sumwfuf(xf) + (1  wf) ur(xr)subject to the resource constraints of the economy.1(b) For every weight wf2 (0; 1), can you …nd an initial endowmentof bananas and coconuts among Robinson and Friday and a pairof prices that the Pareto e¢ cient allocation actually constitutes anequilibrium of the market. (Here we decentralize the Pareto e¢ cientallocation via a market equilibrium.) It is su¢ cient to discuss thecase of c = 1.Reading Assignment: NS Chapter 9, 10, 11,