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1、Public Economics –Homework #1 Binzhen WuAssigned: Sep. 27, 2012Due in two weeks: 9:50am, Oct. 11/12 (Thursday/Friday)1.(10 Points) Pareto OptimalConsideracaseofpureexchange,withnoexternalities.AmyhastheutilityfunctionUA=0.5xA+0.5yAand0.50.5BettyhastheutilityfunctionUB=0.5xByB.Theresource
2、constraintisxA+xB=10andyA+yB=8.Considerthefollowingtwoallocations.OneisxA=0,yA=0andxB=10,yB=8;theotherisxA=5,yA=4andxB=5,yB=4.ArethesetwoallocationsParetoefficient?Explain.Iftheanswerisno,findaParetoimprovement.ANS:ThefirstoneisParetoefficient,thelatteroneisnot.Forthefirstallocation:AAAM
3、RSx,y=MUx/MUy=1/1=1BBB-0.50.50.5-0.5MRSx,y=MUx/MUy=0.5*0.5xByB/0.5*0.5xByB=yB/xB=8/10ABSoMRSx,y≠MRSx,y.However,thisallocationisParetooptimal,becausethisallocationisacornerallocation.WedonotrequireABMRSx,y=MRSx,yforcornersolutions.Forthisallocation,thereisnowaytomakeAbetteroff,withoutmaki
4、ngBworseoff.ThereforeitisParetooptimal.Secondallocation(innersolution):AAAMRSx,y=MUx/MUy=1/1=1BBB-0.50.50.5-0.5MRSx,y=MUx/MUy=0.5*0.5xByB/0.5*0.5xByB=yB/xB=4/5ABMRSx,y>MRSx,y,itisnotParetoefficient.Forthesecondallocation,Amygetsrelativelymoreutilityfromx,whileBettygetsrelativelymoreutili
5、tyfromy.Amy’spossibleParetoimprovementistogiveAmyonemoreunitofX,giveBettyonemoreunitofY.ThenewallocationisxA=6,yA=6andxB=4,yB=4.(ReceivefullcreditaslongasitisParetoimprovement.)Underthenewallocation,Amy’sutilityis0.5*6+0.5*6=6;Betty’sutilityis0.5*square(4*4)=2.Undertheoldallocation,Amy’s
6、utilityis0.5*5+0.5*7=6;Betty’sutilityis0.5*square(15)<2.Therefore,BettyisbetteroffwhileAmyisnotworseoff.ThisisaParetoimprovement.There seems to be some errors in this part of answer, a Pareto Improvement which is also the answer for most studentscould be xA=5.5,yA=3.5andxB=4.5,yB=4.52.(3
7、0Points)CompetitiveEquilibriumSupposeFredistheonlyconsumerintheeconomy.Fredderivesutilityfromtwothings:clothesandbooks.He3/41/4hasanincomeof$24andautilityfunctiongivenby:U(C,B)=CB.Thereisonefirmthatproduces1/41/4clothesusinglabor(L)andcapital(K)andatechnologygivenby:fC(L,
