高级微观经济学-范里安(第三版)答案

AnswerstoExercisesMicroeconomicAnalysisThirdEditionHalR.VarianUniversityofCaliforniaatBerkeleyW.W.Norton&CompanyNewYorkLondonCopyrightc1992,1984,1978byW.W.Norton&Company,Inc.AllrightsreservedPrintedintheUnitedStatesofAmericaTHIRDEDITION0-393-96282-2W.W.Norton&Company,Inc.,500FifthAvenue,NewYork,N.Y.10110W.W.NortonLtd.,10CopticStreet,LondonWC1A1PU234567890ANSWERSChapter1.Technology1.1False.Therearemanycounterexamples.Considerthetechnologygeneratedbyaproductionfunctionf(x)=x2.TheproductionsetisY=f(y;−x):yx2gwhichiscertainlynotconvex,buttheinputre-quirementsetisV(y)=fx:xpygwhichisaconvexset.1.2Itdoesn'tchange.1.31=aand2=b.1.4Lety(t)=f(tx).Thendydt=nXi=1@f(x)@xixi;sothat1ydydt=1f(x)nXi=1@f(x)@xixi:1.5Substitutetxifori=1;2togetf(tx1;tx2)=[(tx1)+(tx2)]1=t[x1+x2]1=tf(x1;x2):ThisimpliesthattheCESfunctionexhibitsconstantreturnstoscaleandhencehasanelasticityofscaleof1.1.6Thisishalftrue: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).Nowg(h(tx))=g(th(x))andg(h(tx0))=g(th(x0))whichgivesustherequiredresult.1.8Ahomotheticfunctioncanbewrittenasg(h(x))whereh(x)isho-mogeneousofdegree1.HencetheTRSofahomotheticfunctionhasthe2ANSWERSformg0(h(x))@h@x1g0(h(x))@h@x2=@h@x1@h@x2:Thatis,theTRSofahomotheticfunctionisjusttheTRSoftheun-derlyinghomogeneousfunction.ButwealreadyknowthattheTRSofahomogeneousfunctionhastherequiredproperty.1.9Notethatwecanwrite(a1+a2)1a1a1+a2x1+a2a1+a2x21:Nowsimplydeneb=a1=(a1+a2)andA=(a1+a2)1.1.10Toproveconvexity,wemustshowthatforallyandy0inYand0t1,wemusthavety+(1−t)y0inY.Butdivisibilityimpliesthattyand(1−t)y0areinY,andadditivityimpliesthattheirsumisinY.Toshowconstantreturnstoscale,wemustshowthatifyisinY,ands0,wemusthavesyinY.Givenanys0,letnbeanonnegativeintegersuchthatnsn−1.Byadditivity,nyisinY;sinces=n1,divisibilityimplies(s=n)ny=syisinY.1.11.aThisisclosedandnonemptyforally0(ifweallowinputstobenegative).TheisoquantslookjustliketheLeontieftechnologyexceptwearemeasuringoutputinunitsoflogyratherthany.Hence,theshapeoftheisoquantswillbethesame.Itfollowsthatthetechnologyismonotonicandconvex.1.11.bThisisnonemptybutnotclosed.Itismonotonicandconvex.1.11.cThisisregular.Thederivativesoff(x1;x2)arebothpositivesothetechnologyismonotonic.Fortheisoquanttobeconvextotheorigin,itissucient(butnotnecessary)thattheproductionfunctionisconcave.Tocheckthis,formamatrixusingthesecondderivativesoftheproductionfunction,andseeifitisnegativesemidenite.TherstprincipalminoroftheHessianmusthaveanegativedeterminant,andthesecondprincipalminormusthaveanonnegativedeterminant.@2f(x)@x21=−14x−321x122@2f(x)@x1@x2=14x−121x−122@2f(x)@x22=−14x121x−322Ch.2PROFITMAXIMIZATION3Hessian=−14x−3=21x1=2214x−1=21x−1=2214x−1=21x−1=22−14x1=21x−3=22#D1=−14x−3=21x1=220D2=116x−11x−12−116x−11x−12=0:Sotheinputrequirementsetisconvex.1.11.dThisisregular,monotonic,andconvex.1.11.eThisisnonempty,butthereisnowaytoproduceanyy1.Itismonotonicandweaklyconvex.1.11.fThisisregular.Tocheckmonotonicity,writedowntheproductionfunctionf(x)=ax1−px1x2+bx2andcompute@f(x)@x1=a−12x−1=21x1=22:Thisispositiveonlyifa12qx2x1,thustheinputrequirementsetisnotalwaysmonotonic.LookingattheHessianoff,itsdeterminantiszero,andthedeterminantoftherstprincipalminorispositive.Thereforefisnotconcave.Thisaloneisnotsucienttoshowthattheinputrequirementsetsarenotconvex.Butwecansayevenmore:fisconvex;therefore,allsetsoftheformfx1;x2:ax1−px1x2+bx2ygforallchoicesofyareconvex.Exceptfortheborderpointsthisisjustthecomplementoftheinputrequirementsetsweareinterestedin(theinequalitysigngoesinthewrongdirection).Ascomplementsofconvexsets(suchthattheborderlineisnotastraightline)ourinputrequirementsetscanthereforenotbethemselvesconvex.1.11.gThisfunctionisthesuccessiveapplicationofalinearandaLeontieffunction,soithasallofthepropertiespossessedbythesetwotypesoffunctions,includingbeingregular,monotonic,andconvex.Chapter2.ProfitMaximization4ANSWERS2.1Forprotmaximization,theKuhn-Tuckertheoremrequiresthefollow-ingthreeinequalitiestoholdp@f(x)@xj−wjxj=0;p@f(x)@xj−wj0;xj0:Notethatifxj0,thenwemusthavewj=p=@f(x)=@xj.2.2Supposethatx0isaprot-maximizingbundlewithpositiveprots(x0)0.Sincef(tx0)tf(x0);fort1,wehave(tx0)=pf(tx0)−twx0t(pf(x0)−wx0)t(x0)(x0):Therefore,x0couldnotpossiblybeaprot-maximizingbundle.2.3Inthetextthesupplyfunctionandthefactordemandswerecomputedforthistechnology.Usingthoseresults,theprotfunctionisgivenby(p;w)=pwapaa−1−wwap1a−1:Toprovehomogeneity,notethat(tp;tw)=tpwapaa−1−twwap1a−1=t(p;w);whichimpliesthat(p;w)isahomogeneousfunctionofdegree1.BeforecomputingtheHessianmatrix,factortheprotfunctioninthefollowingway:(p;w)=p11−awaa−1aa1−a−a11−a=p11−awaa−1(a);where(a)isstrictlypositivefor0a1.TheHessianmatrixcannowbewrittenasD2(p;!)=@2(p;w)@p2@2(p;w)@p@w@2(p;w)@w@p@2(p;w)@w2!=0BBB@a(1−a)2p2a−11−awaa−1−a(1−a)2pa1−aw1a−1−a(1−a)2pa1−aw1a−1a(1−a)2p11−aw2−aa−1