［美］Varian, Hal R.-Microeconomic Analysis (3rd) Answers

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1、AnswerstoExercisesMicroeconomicAnalysisThirdEditionHalR.VarianUniversityofCaliforniaatBerkeleyW.W.Norton&CompanyNewYorkLondonCopyrightc1992,1984,1978byW.W.Norton&Company,Inc.AllrightsreservedPrintedintheUnitedStatesofAmericaTHIRDEDITION0-393-96282-2W.W.Norton&Company,Inc.,500FifthAv

2、enue,NewYork,N.Y.10110W.W.NortonLtd.,10CopticStreet,LondonWC1A1PU234567890ANSWERSChapter1.Technology1.1False.Therearemanycounterexamples.Considerthetechnologygeneratedbyaproductionfunctionf(x)=x2.TheproductionsetisY=f(y;−x):yx2gwhichiscertainlynotconvex,buttheinputre-pquirementsetisV

3、(y)=fx:xygwhichisaconvexset.1.2Itdoesn'tchange.1.31=aand2=b.1.4Lety(t)=f(tx).ThenXndy@f(x)=xi;dt@xii=1sothatXn1dy1@f(x)=xi:ydtf(x)@xii=11.5Substitutetxifori=1;2toget11f(tx1;tx2)=[(tx1)+(tx2)]=t[x1+x2]=tf(x1;x2):ThisimpliesthattheCESfunctionexhibitsconstantreturnstoscaleandhen

4、cehasanelasticityofscaleof1.1.6Thisishalftrue:ifg0(x)>0,thenthefunctionmustbestrictlyincreasing,buttheconverseisnottrue.Consider,forexample,thefunctiong(x)=x3.Thisisstrictlyincreasing,butg0(0)=0.1.7Letf(x)=g(h(x))andsupposethatg(h(x))=g(h(x0)).Sincegismonotonic,itfollowsthath(x)=h(x0)

5、.Nowg(h(tx))=g(th(x))andg(h(tx0))=g(th(x0))whichgivesustherequiredresult.1.8Ahomotheticfunctioncanbewrittenasg(h(x))whereh(x)isho-mogeneousofdegree1.HencetheTRSofahomotheticfunctionhasthe2ANSWERSform0@h@hg(h(x))@x1@x1=:g0(h(x))@h@h@x2@x2Thatis,theTRSofahomotheticfunctionisjusttheTRSof

6、theun-derlyinghomogeneousfunction.ButwealreadyknowthattheTRSofahomogeneousfunctionhastherequiredproperty.1.9Notethatwecanwrite11a1a2(a1+a2)x1+x2:a1+a2a1+a21Nowsimplydeneb=a1=(a1+a2)andA=(a1+a2).1.10Toproveconvexity,wemustshowthatforallyandy0inYand0t1,wemusthavety+(1−t)y0inY.

7、Butdivisibilityimpliesthattyand(1−t)y0areinY,andadditivityimpliesthattheirsumisinY.Toshowconstantreturnstoscale,wemustshowthatifyisinY,ands>0,wemusthavesyinY.Givenanys>0,letnbeanonnegativeintegersuchthatnsn−1.Byadditivity,nyisinY;sinces=n1,divisibilityimplies(s=n)ny=syisinY.1.11.aT

8、hisis