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1、Chapter6DUALITYAPPROACHESTOMICROECONOMICTHEORY*W.E.Diewert1.IntroductionWhatdowemeanwhenwesaythatthereisa“duality”betweencostandproductionfunctions?SupposethataproductionfunctionFisgivenandthatu=F(x),whereuisthemaximumamountofoutputthatcanbeproducedbythetechnologyduringacertainperi
2、odifthevectorofinputquantitiesx≡(x1,x2,...,xN)isutilizedduringtheperiod.Thus,theproductionfunctionFdescribesthetechnologyofthegivenfirm.Ontheotherhand,thefirm’sminimumtotalcostofproducingatleasttheoutputlevelugiventheinputprices(p1,p2,...,pN)≡pisdefinedasC(u,p)anditisobviouslyafunctio
3、nofu,pandthegivenproductionfunctionF.Whatisnotsoobviousisthat(undercertainregularityconditions)thecostfunctionC(u,p)alsocompletelydescribesthetechnologyofthegivenfirm;i.e.,giventhefirm’scostfunctionC,itcanbeusedinordertodefinethefirm’sproductionfunctionF.Thus,thereisadualitybetweencost
4、andproductionfunctionsinthesensethateitherofthesefunctionscandescribethetechnologyofthefirmequallywellincertaincircumstances.Inthefirstpartofthischapter,wedevelopthisdualitybetweencostandproductionfunctionsinmoredetail.InSection2,wederivetheregularityconditionsthatacostfunctionCmusth
5、ave(irrespectiveofthefunctionalformorspecificregularitypropertiesfortheproductionfunctionF),andweshowhowaproductionfunctioncanbeconstructedfromagivencostfunction.InSection3,wedevelopthisdualitybetweencostandproductionfunctionsinamoreformalmanner.*ThispaperisanabridgedversionofIMSSST
6、echnicalReportNo.281,Stan-fordUniversity,1978.AnevenmoreabridgedversionwaspublishedinK.J.ArrowandM.D.Intriligator(eds.),HandbookofMathematicalEconomics,Am-sterdam:North-HollandPublishingCo.,1982,pp.535–599.TheauthorwouldliketothankC.BlackorbyandR.C.AllenforhelpfulcommentsandtheCana
7、daEssaysinIndexNumberTheory,VolumeICouncilforfinancialsupport.MythanksalsotoMayMcKeeandShelleyHeyW.E.DiewertandA.O.Nakamura(Editors)fortypingadifficultmanuscript.Apreliminaryversionofthispaperwaspre-c1993ElsevierSciencePublishersB.V.Allrightsreserved.sentedattheNewYorkmeetingsoftheEco
8、nometricSociety,December19
