production_function_properties.pdf

《production_function_properties.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、LectureNote:PropertiesofProductionFunctions¤U.MichaelBergmanInstituteofEconomics,UniversityofCopenhagenSeptember11,20041IntroductionThisnotediscussessomeimportantpropertiesofproductionfunctions.Insection2wede¯nehomogeneityanddiscussesitsrelationtoreturnstoscale.Asanexample,weshowthattheCobb{Dougla

2、sfunctionusedthroughoutthe¯rstthreechaptersinJonesisho-mogeneousofdegree1andexhibitsconstantreturnstoscale.Insection3wediscussthethreealternativewaysofintroducingtechnologicalprogressintotheproductionfunction.Inaddition,weshowthatunderaCobb{Douglastechnology,thesethreewaysofhandlingtechnologicalpr

3、ogressareequivalent.Finally,insection4weshowthatthereexistsasteady{statewithconstantgrowthratesintheneoclassicalgrowthmodelonlywhentech-nologicalprogressismodelledaslabor{augmenting.Thenoteshouldberegardedasacomplementtothemaintextandisnotrequiredreading.2HomogeneityandreturnstoscaleLetusstartbyde

4、¯ningtheproductiontechnologyY=F(K;L;¿)whereKiscapital,Lislaborand¿istechnology.Theproductionfunctionistwicecontinuouslydi®erentiable.ThemarginalproductofinputK,say,intheproductionofYistherateatwhichoutputYchangesinresponsetoKholdingallotherinputsconstant.Letustakethepartial¤°c2002byU.MichaelBergma

5、n.Thisdocumentmaybereproducedforeducationalpurposes,solongasthecopiescontainthisnoticeandareretainedforpersonaluseorisdistributedfree.ThislecturenoteistobeusedonlyinconjunctionwiththecourseMakro¿konomi1givenatUniversityofCopenhagen.ThenotedrawsheavilyonSyds½terandHammond(1995),Jones(1998),Barroand

6、Sala{i{Martin(1995)andalecturenotewrittenbyHenrikJensenatUniversityofCopenhagen.ManythankstoHenrikJensenforallowingmetousehislecturenotes.{1{derivativeswithrespecttoeachinputfactorsuchthat@FMPL==FL@L@FMPK==FK@K@FMPT==F¿.@¿Thesecond{orderpartialderivativesaretheratesatwhichthemarginalproductsofthei

7、nputschangeasoneoftheinputsisvaried.Thesecond{orderpartialderivativesare@2F=FLL@L2@2F=FKK@K2@2F=F¿¿.@¿2Thesignsofthesederivativesdeterminewhetherthemarginalproductsarediminishing(Fii<0),constant(Fii=0)orincreasin