Dirk BergemannDepartment of EconomicsYale UniversitySolutions Prepared by Olga TimoshenkoEconomics 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 class. In contrast to the model in class, Robinson andFriday have the same endowment of bananas and coconuts:xbf= xbr= 1xcf= xcr= c > 1but their preferences are different. Friday likes bananas more than co-conuts, and Robinson likes coconuts more than bananas (i.e. you canassume that α ∈ (0.5, 1)). The utility function are given by:uf(xf) = α ln xbf+ (1 − α) ln xcf,andur(xr) = (1 − α) ln xbr+ α ln xcr.(a) We normalize the price of bananas to be equal to one, or pb= 1.Compute the demand function of Robinson and Friday as a functionof the price for coconuts pc.SolutionUtility maximization problem for Robinson ismaxxbr,xcr(1 − α) ln xbr+ α ln xcrs. t. xbr+ pcxcr= 1 + pccThe first order conditions are(1 − α)xcrαxbr=1pcxbr+ pcxcr= 1 + pcc1Solve the two equations for Robinson’s demand for coconuts andbananasxbr= (1 − α)(1 + pcc)xcr= α1 + pccpcBy the symmetry of the preferences Friday’s demand isxbf= α(1 + pcc)xcf= (1 − α)1 + pccpc(b) Find the equilibrium price of this island economy and determine thenet trading quantities.i. Start with the case of c = 1.SolutionConsider the market clearing condition for bananasxbf+ xbr= 1 + 1α(1 + pcc) + (1 − α)(1 + pcc) = 21 + pc= 2pc= 1At the equilibrium price the consumption of Robinson and Fridayisxbr= 2(1 − α)xcr= 2αxbf= 2αxcf= 2(1 − α)The net trading quantities areebr− xbr= 2α − 1ecr− xcr= 1 − 2αebf− xbf= 1 − 2αecf− xcf= 2α − 12ii. Continue to describe the equilibrium for general c > 1 and de-scribe the intuition.Consider the market clearing condition for bananasxbf+ xbr= 1 + 1α(1 + pcc) + (1 − α)(1 + pcc) = 21 + pcc = 2pc=1cObserve that the price of coconuts is declining in the in thenumber of coconuts available in the economy representing theidea that as a good becomes less scares its price declines.The net trading quantities areebr− xbr= 2α − 1ecr− xcr= c(1 − 2α)ebf− xbf= 1 − 2αecf− xcf= c(2α − 1)Notice that the trade and consumption of bananas is independentof the number of coconuts in the economy. On the one hand, asthe number of coconuts increases, the total income of a personshould increase and the consumption of bananas should increase.On the other hand, though, as coconuts become abundant in theeconomy, its price declines. Due to the summery of the prefer-ences and endowments here, the price of coconuts would declineexactly at the same rate as the number of coconuts increases,leaving the nominal value of individual’s income unchanged. Asa result, since a person spends a constant fraction of income onthe consumption of bananas and the price of bananas does notchange, the consumption of bananas remains independent of thenumber of coconuts in the economy.2. Pareto Efficiency. Consider the island economy again but this timeFriday and Robinson have agreed to share their resources and they havealso agreed that the weight that Friday receives in the economy is wfandthe weight that Robinson receives is wr= 1 − wf.(a) For every weight wf∈ (0, 1), find the allocation which would maxi-mize the social surplus given the weights; in other words, we are in-terested in finding the allocation (xbf, xbr, xcf,xcr) which maximizesthe sumwfuf(xf) + (1 − wf) ur(xr)3subject to the resource constraints of the economy.SolutionThe maximization problem ismaxxbf,xbr,xcf,xcrwf(α ln xbf+ (1 − α) ln xcf) + (1 − wf) ((1 − α) ln xbr+ α ln xcr)s. t. xbf+ xbr= 2xcf+ xcr= 2cThe first order conditions are (after eliminating the Lagrange multi-pliers)wfαxbr= (1 − wf)(1 − α)xbfwf(1 − α)xcr= (1 − wf)αxcfxbf+ xbr= 2xcf+ xcr= 2cSolve this system of equations to obtainxbr=2(1 − wf)(1 − α)wfα + (1 − wf)(1 − α)(1)xbf=2wfαwfα + (1 − wf)(1 − α)(2)xcr=2c(1 − wf)αwf(1 − α) + (1 − wf)α(3)xcf=2cwf(1 − α)wf(1 − α) + (1 − wf)α(4)(b) For every weight wf∈ (0, 1), can you find an initial endowment of ba-nanas and coconuts among Robinson and Friday and a pair of pricesthat the Pareto efficient allocation actually constitutes an equilibriumof the market. (Here we decentralize the Pareto efficient allocationvia a market equilibrium.) It is sufficient to discuss the case of c = 1.SolutionFor every weight wf∈ (0, 1), let the initial endowment be given by thesystem of pareto efficient allocations in (1)-(4). Normalize the priceof bananas to 1, i.e. pb= 1. We need to find the price of coconutsthat together with endowments given by (1)-(4) will constitute acompetitive equilibrium.Since (1)-(4) is a Pareto efficient allocation, the marginal rate of sub-stitution between coconuts and bananas are the same for Robinson4and Friday (verify by yourself that it is actually true). Next, recallthat at a competitive equilibrium MRS equals the price ratio. Wewill use that property to recover the price for coconuts.MRScb(x∗br, x∗cr) =pcpbαx∗br(1 − α)x∗cr= pcα2(1−wf)(1−α)wfα+(1−wf)(1−α)(1 − α)2c(1−wf)αwf(1−α)+(1−wf)α= pcwf(1 − α) + (1 − wf)αc(wfα + (1 − wf)(1 − α))= pcThus, prices (pb, pc) =(1,wf(1−α)+(1−wf)αc(wfα+(1−wf)(1−α)))and the allocationgiven by (1)-(4) constitute a competitive equilibrium in an economywhere initial endowments are given by (1)-(4). It will be just a simpleeconomy where everybody consumes their own endowment at theequilibrium prices.Reading Assignment: NS Chapter 9, 10, 11,