1、Chapter第七Chapter第七PRODUCTION1生產函Thefirmsproduction生產函Thefirmsproductionfunctionfora particular good (q) shows theamountofthetcanbeusingalternativecombinationsofcapitaland labor q = 2邊際實物產Tostudyvariationinasingleinput,邊際實物產Tostudyvariationinasingleinput,wedefinephysicaltheadditionaltcantinput produc

2、edbyemployingonemoreunitholdingotherinputsconstant們定義邊際實物產量一 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因為邊際生產力遞減，Bhemarginalproductivityoflaborove

4、ralsodependonchangesinotherch as weneedtoconsiderflk whichisoften需要考慮flk （通常5平均實物產Labor productivity is often measured by average productivity勞動生產力通常以平均生產力衡量平均實物產Labor productivity is often measured by average productivity勞動生產力通常以平均生產力衡量q f(k,lAP llabor lltAPl alsodependsontheamountcapital employed注

5、意，APl也依賴于所使用資本的數6ATwo-InputProductionetheproductionfunctionATwo-InputProductionetheproductionfunctioncanberepresentedq=f(k,l)=600k2l2 -kToconstructMPl andAPl,wemustmeavaluefor為了得到MPl和須假定k令k let k=-Theproductionq=7ATwo-InputProductionThemarginalproductivityfunctionisATwo-InputProductionThemarginalpro

6、ductivityfunctionisMPl = q/l=120,000l-whichdiminishes aslme l20上式隨著l增加遞減(定l20）,因為fll 120,000umvalue這意味著，qThistqhas120,000l- 3000l2 = 40l= l= Laborinputbeyondl=40output超過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，當APl達到其最大值時，APlMPl等Isoquant等產量sibleIsoquant等產量siblesubstitutionofoneinputanotherweuseanisoquantmapAnisoquantshowsthosecombinationsofkandtproduceagivenlevelofoutput可以生產給

8、定產出水平的k和lf(k,l)=IsoquantMap等產量Eachisoquant representsIsoquantMap等產量Eachisoquant representsadifferentlevelof outputrisesaswemovenortheast隨著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等產量線的斜率，表示了l可以被k替k rateoftechnicalsubstitution代率0andisdiminishingasinginputsof0kAl -slope=-RTSBq=MarginalRateofTechnicalSubstitutionMarginalRateofTechnicalSubstitutionThemarginalrateoftechnicalsubstitution(RTS)showsrateatwhichlaborcanbesubst

10、itutedforcapitalholdingonstantalongan邊際技術替代率(RTS)RTS(lfork) dk0RTSandMarginalTakethetotaldifferentialoftheproductiondqRTSandMarginalTakethetotaldifferentialoftheproductiondq f dl f dk MP dlMP dklkAlong anisoquant dq0so沿同一等產量線dq0MPl dlMdkRTS(lfork) dk0HencetheRTSisgivenbytheratiooftheinputs. RTSandMa

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

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

13、someranges但一些生產函數在某些投入區間內，有fkl whenwemediminishingRTSwearet MPl MPk diminishquicklyenoughtocompensatefor 假ADiminishingRTS遞eADiminishingRTS遞etheproductionfunctionisq= f(k,l)=600k2l2 -k3lForthisproductionfunctionMPl = fl =1200k2l- 3k3lMPk =fk =1200kl2 -3k2l thesemarginalproductivitieswillitiveforvalue

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

15、= 2400kl-9k2litiveonlyforkl僅在which 266ADiminishingRTS遞ADiminishingRTS遞Thus,forthisproductionfunction,RTSisdiminishing200kl266所以，對于這一生產函數，當2001),如果生產函數為qf(k,l，并且所有投入乘以同一正的常數(t 1)，那么Effect on 對產出Returns to 規f(tk,tl) = Constant不f(tk,tl) Increasing 遞ReturnstoScalesibleforaproductionfunctiontoexhibitRetu

16、rnstoScalesibleforaproductionfunctiontoexhibitItreturnstoscaleforsomelevelsofinputusageincreasingordecreasingreturnsforothereconomistsrefertothedegreeofreturnstoscalewiththetonlyafairlynarrowrangeofvariationininputandtherelated level ofoutputisbeingConstantReturnsto規不ConstantConstantReturnsto規不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次齊次的，它的導數是 ConstantReturnsto規不ThemarginalproductivityConstan

18、tReturnsto規不Themarginalproductivityofanyinputdependsonratioofcapitalandlabor(notontheevelstheseinputs)的比值（而非這些投入的絕對水平nkandldependsonlyontheratioof,TheRTSthescaleofConstantReturnsto規不ConstantReturnsto規不TheproductionfunctionwillbeGeometrically,alloftheisoquantsradialsofoneConstantReturnstoAlongarayCon

19、stantReturnstoAlongarayfromtheorigin(constantk/l),theRTSwillbetheonallisoquants在一條出發自原點的射線上(k/l，RTS在所k Theisoquantsareequallyasoutputexpands這些等產量線是q=q=q=l ReturnstoScaleReturnstoscalecanbeReturnstoScaleReturnstoscalecanbegeneralizedtoaproductionwithninputs可以擴展到有nq= Ifallinputsaredbyitiveconstantt,we

20、如果所有投入乘以一個正的常數f(tx1,tx2,txn)=Ifk1,wehaveconstantreturnstoscale如果k1Ifk1,wehavedecreasingreturnstoscale如果k1,Ifk1,wehaveincreasingreturnstoscale如果Ifk1,Elasticityof Substitution替代彈Theelasticityofsubstitution()measurestheElasticityof Substitution替代彈Theelasticityofsubstitution()measuresthechangeink/lrelat

21、ivetotheproportionatehe alonganisoquant 替代彈性()衡量的是沿著等產量線 %(k/l) d(k/l)RTS ln(k/l%RTSlnRTSdRTSk/Thevalueof will always itivebecause k/l RTShesamedirection因為k/l和RTS向運動，Elasticityof Substitution替代彈BothRTSand k/l will changeElasticityof Substitution替代彈BothRTSand k/l will changeaswemovefromAto B從點A沿等產量線移

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

23、alonganisoquantorastheItofproductionElasticityof Substitution替代彈GeneralizingtheelasticityElasticityof Substitution替代彈Generalizingtheelasticityofsubstitutiontothemany-caseraisesseveralcomplications將替代彈性擴展到多種ifwedefinetheelasticity ofsubstitutionntwoinputstotheproportionateheratioofthetwoinputstopropo

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

25、laborareTheLinearProductionCapitalandlaborareperfect資本和勞動是完全可替代k RTSisconstantask/l 隨著k/l變化，RTS是常 =slope=-l FixedProportionsttheproductionFixedProportionsttheproductionfunctioniseq = min a,b Capitalandlabormustalwaysbeusedinafixed thefirmwillalwaysoperatealongaraywherek/lisBecausek/lisconstant,0因為k/

26、l為常數，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)可以證明，彈性為1參考格拉斯生產函數的替Cobb-DouglasProduction柯TheCobb-Do

28、uglasProduction柯TheCobb-Douglasproductionfunctionisinlogarithmslnq = ln A+ a ln k +b ln aistheelasticityofoutputwithrespecttok bistheelasticityofoutputwithrespecttob是產出對l的彈生產函ttheproduction生產函ttheproductionfunctioneq = f(k,l) = k+ l/ 1, 0, 1increasingreturnsto1規遞 Differentiatingtheproductionfunction

29、withrespect to time we get生產函數對時間微分，得到dq dAf(k,lDifferentiatingtheproductionfunctionwithrespect to time we get生產函數對時間微分，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對于任何變量x，(dx/dt)/x是x在 denotethisby用Gx表Then,wecanwritetheermsof所以可以用增長率的形式重寫等 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如果除一種投入and外，其他投入都保持不變，可以推導出這個投入變量和產出的關系th

33、emarginalphysicalproductivityisthechangeoutputresultingfromaone-unithe oftheinput邊際實物生產力是指投入增加一amed todeclineasuseoftheinputincreases stoNoteImportantTheentireproductionstoNoteImportantTheentireproductionfunctioncanbebyanisoquantmap整個生產函數可以用等產量theslope ofansthemarginalratetechnicalsubstitutionRTSitshowshowoneanbesubstitutedforanother hol