中级微观经济学中微课件

1、Chapter第七Chapter第七PRODUCTION1生产函Thefirmsproduction生产函Thefirmsproductionfunctionfora particular good (q) shows theamountofthetcanbeusingalternativecombinationsofcapitaland labor q = 2边际实物产Tostudyvariationinasingleinput,边际实物产Tostudyvariationinasingleinput,wedefinephysicaltheadditionaltcantinput produc

2、edbyemployingonemoreunitholdingotherinputsconstant们定义边际实物产量一 marginal physical product of capital MPkkmarginalphysicalproduct of labor MP q ll3DiminishingMarginalThemarginalphysicalproductofaninputdependsontinputDiminishingMarginalThemarginalphysicalproductofaninputdependsontinputisusedmuchIngeneral

3、,wemediminishingmarginalMP2fMP2fkk f11 k4DiminishingMarginalBecauseofdiminishingmarginalproductivity,19thDiminishingMarginalBecauseofdiminishingmarginalproductivity,19theconomist Thomas Malthus worried about the effect of populationgrowthonlaborproductivity因为边际生产力递减，Bhemarginalproductivityoflaborove

4、ralsodependonchangesinotherch as weneedtoconsiderflk whichisoften需要考虑flk （通常5平均实物产Labor productivity is often measured by average productivity劳动生产力通常以平均生产力衡量平均实物产Labor productivity is often measured by average productivity劳动生产力通常以平均生产力衡量q f(k,lAP llabor lltAPl alsodependsontheamountcapital employed注

5、意，APl也依赖于所使用资本的数6ATwo-InputProductionetheproductionfunctionATwo-InputProductionetheproductionfunctioncanberepresentedq=f(k,l)=600k2l2 -kToconstructMPl andAPl,wemustmeavaluefor为了得到MPl和须假定k令k let k=-Theproductionq=7ATwo-InputProductionThemarginalproductivityfunctionisATwo-InputProductionThemarginalpro

6、ductivityfunctionisMPl = q/l=120,000l-whichdiminishes aslme l20上式随着l增加递减(定l20）,因为fll 120,000umvalue这意味着，qThistqhas120,000l- 3000l2 = 40l= l= Laborinputbeyondl=40output超过l408ATwo-InputProductionATwo-InputProductionTofindaverageproductivityweholdk=10andsolveAPl =q/l=60,000l-umwhere APlAPl/l=60,000-200

7、0l=l=APl reaches9ATwo-InputProductionInfact,ATwo-InputProductionInfact,whenl=30,Pl andMPl areequalThus,whenAPl isatum,APl andMPl 所以，当APl达到其最大值时，APlMPl等Isoquant等产量sibleIsoquant等产量siblesubstitutionofoneinputanotherweuseanisoquantmapAnisoquantshowsthosecombinationsofkandtproduceagivenlevelofoutput可以生产给

8、定产出水平的k和lf(k,l)=IsoquantMap等产量Eachisoquant representsIsoquantMap等产量Eachisoquant representsadifferentlevelof outputrisesaswemovenortheast随着k q=q=l MarginalRateofTechnicalSubstitutionThe slope of an isoquant shows the rate at which l can besubstitutedforMarginalRateofTechnicalSubstitutionThe slope of

9、an isoquant shows the rate at which l can besubstitutedfork等产量线的斜率，表示了l可以被k替k rateoftechnicalsubstitution代率0andisdiminishingasinginputsof0kAl -slope=-RTSBq=MarginalRateofTechnicalSubstitutionMarginalRateofTechnicalSubstitutionThemarginalrateoftechnicalsubstitution(RTS)showsrateatwhichlaborcanbesubst

10、itutedforcapitalholdingonstantalongan边际技术替代率(RTS)RTS(lfork) dk0RTSandMarginalTakethetotaldifferentialoftheproductiondqRTSandMarginalTakethetotaldifferentialoftheproductiondq f dl f dk MP dlMP dklkAlong anisoquant dq0so沿同一等产量线dq0MPl dlMdkRTS(lfork) dk0HencetheRTSisgivenbytheratiooftheinputs. RTSandMa

11、rginalBecauseRTSandMarginalBecauseMPl andMPk willbothbenonnegative,RTSwillitiveorzero因为MPl和MPk都是非负的，RTS（或者However,itisgenerallysibletoderiveaRTSfromthemptionofdiminishingmarginal RTSandMarginaltisoquantsareconvex,wewouldliketoRTSandMarginaltisoquantsareconvex,wewouldliketotd(RTS)/dl0d(RTS)/dl0,thede

12、nominator itive已假设fk 0Because fll andfkk are bo ratio will be negative if fkl ismedtobenegative,the itive 因为fll和fkkRTSandMarginaluitively,itseemstfklRTSandMarginaluitively,itseemstfkl=flk should上，fklflk ifworkershavemorecapital,theywouldhavehigherproductivitiesButsomeproductionfunctionshavefkl 0over

13、someranges但一些生产函数在某些投入区间内，有fkl whenwemediminishingRTSwearet MPl MPk diminishquicklyenoughtocompensatefor 假ADiminishingRTS递eADiminishingRTS递etheproductionfunctionisq= f(k,l)=600k2l2 -k3lForthisproductionfunctionMPl = fl =1200k2l- 3k3lMPk =fk =1200kl2 -3k2l thesemarginalproductivitieswillitiveforvalue

14、sofkandforwhichkl当k和l满足kl400ADiminishingRTS递ADiminishingRTS递Becausefll =1200k2-6k=1200l2 -6klthisproductionfunctionexhibitsdiminishingproductivities for sufficiently large values of fll and fkk ADiminishingRTS递CrossADiminishingRTS递Crossdifferentiationofeitherofmarginalproductivityfunctionsfkl = flk

15、= 2400kl-9k2litiveonlyforkl仅在which 266ADiminishingRTS递ADiminishingRTS递Thus,forthisproductionfunction,RTSisdiminishing200kl266所以，对于这一生产函数，当2001),如果生产函数为qf(k,l，并且所有投入乘以同一正的常数(t 1)，那么Effect on 对产出Returns to 规f(tk,tl) = Constant不f(tk,tl) Increasing 递ReturnstoScalesibleforaproductionfunctiontoexhibitRetu

16、rnstoScalesibleforaproductionfunctiontoexhibitItreturnstoscaleforsomelevelsofinputusageincreasingordecreasingreturnsforothereconomistsrefertothedegreeofreturnstoscalewiththetonlyafairlynarrowrangeofvariationininputandtherelated level ofoutputisbeingConstantReturnsto规不ConstantConstantReturnsto规不Const

17、antreturns-to-scaleproductionfunctionshomogeneousofdegreeoneininputsf(tk,tl)= t1f(k,l)= Thisstthemarginalproductivityfunctionshomogeneousofdegreezero if a function is homogeneous of degree k, its derivatives are homogeneousofdegreek-1如果方程是k次齐次的，它的导数是 ConstantReturnsto规不ThemarginalproductivityConstan

18、tReturnsto规不Themarginalproductivityofanyinputdependsonratioofcapitalandlabor(notontheevelstheseinputs)的比值（而非这些投入的绝对水平nkandldependsonlyontheratioof,TheRTSthescaleofConstantReturnsto规不ConstantReturnsto规不TheproductionfunctionwillbeGeometrically,alloftheisoquantsradialsofoneConstantReturnstoAlongarayCon

19、stantReturnstoAlongarayfromtheorigin(constantk/l),theRTSwillbetheonallisoquants在一条出发自原点的射线上(k/l，RTS在所k Theisoquantsareequallyasoutputexpands这些等产量线是q=q=q=l ReturnstoScaleReturnstoscalecanbeReturnstoScaleReturnstoscalecanbegeneralizedtoaproductionwithninputs可以扩展到有nq= Ifallinputsaredbyitiveconstantt,we

20、如果所有投入乘以一个正的常数f(tx1,tx2,txn)=Ifk1,wehaveconstantreturnstoscale如果k1Ifk1,wehavedecreasingreturnstoscale如果k1,Ifk1,wehaveincreasingreturnstoscale如果Ifk1,Elasticityof Substitution替代弹Theelasticityofsubstitution()measurestheElasticityof Substitution替代弹Theelasticityofsubstitution()measuresthechangeink/lrelat

21、ivetotheproportionatehe alonganisoquant 替代弹性()衡量的是沿着等产量线 %(k/l) d(k/l)RTS ln(k/l%RTSlnRTSdRTSk/Thevalueof will always itivebecause k/l RTShesamedirection因为k/l和RTS向运动，Elasticityof Substitution替代弹BothRTSand k/l will changeElasticityof Substitution替代弹BothRTSand k/l will changeaswemovefromAto B从点A沿等产量线移

22、向点B，RTS和k/lk istheratiooftheseproportionalmeasuresthecurvaturetheABq=Bl Elasticityof Substitution替代弹IfisElasticityof Substitution替代弹Ifishigh,theRTSwillnotchangemuchrelativeto theisoquantwillberelatively flatIfislow,theRTSwillchangebyasubstantialamountas theisoquantwillbesharplycurvedsiblefortochange

23、alonganisoquantorastheItofproductionElasticityof Substitution替代弹GeneralizingtheelasticityElasticityof Substitution替代弹Generalizingtheelasticityofsubstitutiontothemany-caseraisesseveralcomplications将替代弹性扩展到多种ifwedefinetheelasticity ofsubstitutionntwoinputstotheproportionateheratioofthetwoinputstopropo

24、rtionatechangeinRTS,weneedtoholdoutputandtheofotherinputsconstantTheLinearProductionttheproductionTheLinearProductionttheproductionfunctionisq = f(k,l)= ak + Thisproductionfunctionexhibitsconstantreturnstoef(tk,tl)= atk+btl=t(ak+bl)= Allisoquantsarestraightlines RTSis = TheLinearProductionCapitaland

25、laborareTheLinearProductionCapitalandlaborareperfect资本和劳动是完全可替代k RTSisconstantask/l 随着k/l变化，RTS是常 =slope=-l FixedProportionsttheproductionFixedProportionsttheproductionfunctioniseq = min a,b Capitalandlabormustalwaysbeusedinafixed thefirmwillalwaysoperatealongaraywherek/lisBecausek/lisconstant,0因为k/

26、l为常数，所以Fixed Proportions固定比Nosubstitution n labor andcapital k k/l isfixedatb/aFixed Proportions固定比Nosubstitution n labor andcapital k k/l isfixedatb/a =l Cobb-DouglasProduction柯ttheproductionfunctionisCobb-DouglasProduction柯ttheproductionfunctioniseq= f(k,l)=A,a,b Thisproductionfunctioncanexhibitan

27、yreturnstof(tk,tl)= )b =Ata+b=ifab1constantreturnstoscale如果ab1ifab1increasingreturns toscale 如果ab1ifab1decreasingreturnstoscale如果ab1Cobb-DouglasProduction柯ItcanbeCobb-DouglasProduction柯Itcanbesimplyttheelasticitysubstitutionis1forCobb-DouglasP197)可以证明，弹性为1参考格拉斯生产函数的替Cobb-DouglasProduction柯TheCobb-Do

28、uglasProduction柯TheCobb-Douglasproductionfunctionisinlogarithmslnq = ln A+ a ln k +b ln aistheelasticityofoutputwithrespecttok bistheelasticityofoutputwithrespecttob是产出对l的弹生产函ttheproduction生产函ttheproductionfunctioneq = f(k,l) = k+ l/ 1, 0, 1increasingreturnsto1规递 Differentiatingtheproductionfunction

29、withrespect to time we get生产函数对时间微分，得到dq dAf(k,lDifferentiatingtheproductionfunctionwithrespect to time we get生产函数对时间微分，得到dq dAf(k,l) Adf(k,ldtdtdtdq dAqf(k,l)kdtdtdtAdtDividingbyqgivesus除以q，得dq/dt dA/dt f Dividingbyqgivesus除以q，得dq/dt dA/dt f /k dk f /l qAf(k,ldtf(k,l) dtdq/dt dA/dt dk/dt kdl/dtlqAf

30、(k,lkf(k,lldq/dtdA/dt dk/dt kdl/dtlqAf(kdq/dtdA/dt dk/dt kdl/dtlqAf(k,lkf(k,llForanyvariablex,(dx/dt)/xisthegrowthrateinx对于任何变量x，(dx/dt)/x是x在 denotethisby用Gx表Then,wecanwritetheermsof所以可以用增长率的形式重写等 klqAklf(k,lf(k,l klqAklf(k,lf(k,lSince klqAklf(k,lf(k,lSince qkf(k,ll q f(k,lGqGAWeeq,kGk eqheCobb-Doug

31、las柯ttheproductionfunctionheCobb-Douglas柯ttheproductionfunctionisq= A(t)f(k,l)= A(t)kl1-eIfwettechnicalprogressexponentialthenA(t)= q=Aetkl1-heCobb-Douglas柯TakinglogarithmsanddifferentiatingrespectheCobb-Douglas柯Takinglogarithmsanddifferentiatingrespecttotgivesthegrowthequationlnq lnq q/qqheCobb-Dou

32、glas柯(lnAtheCobb-Douglas柯(lnAtlnk(1)lnl)Gqlnk(1)lnGk (1Herethetechnicalchangefactorisexplicitlyedthe asticitiesaregivenbythevaluesofheCobb-Douglas.stoNoteImportantIfallbutoneofstoNoteImportantIfallbutoneoftheinputsareheldconstant,nthesinglevariableanbederived如果除一种投入and外，其他投入都保持不变，可以推导出这个投入变量和产出的关系th

33、emarginalphysicalproductivityisthechangeoutputresultingfromaone-unithe oftheinput边际实物生产力是指投入增加一amed todeclineasuseoftheinputincreases stoNoteImportantTheentireproductionstoNoteImportantTheentireproductionfunctioncanbebyanisoquantmap整个生产函数可以用等产量theslope ofansthemarginalratetechnicalsubstitutionRTSitshowshowoneanbesubstitutedforanother hol