Chapter 08, Exact and Superlative Index Numbers

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1、Chapter8EXACTANDSUPERLATIVEINDEXNUMBERS*W.E.Diewert1.IntroductionOneofthemosttroublesomeproblemsfacingnationalincomeaccountantsandeconometricianswhoareforcedtoconstructsomedataseries,isthequestionofwhichfunctionalformforanindexnumbershouldbeused.Inthepresentpaper,weconsiderthisquestionandrela

2、tefunctionalformsfortheunder-lyingproductionorutilityfunction(oraggregatorfunction,touseaneutralterminology)tofunctionalformsforindexnumbers.First,defineaquantityindexbetweenperiods0and1,Q(p0,p1,x0,x1),asafunctionofthepricesinperiods0and1,p0>0andp1>0(where0NNNisanNdimensionalvectorofzeros),and

3、thecorrespondingquantityvectors,x0>0andx1>0.Apriceindexbetweenperiods0and1,P(p0,p1,x0,x1),NNisafunctionofthesamepriceandquantityvectors.Giveneitherapriceindexoraquantityindex,theotherfunctioncanbedefinedimplicitlybythefollowingequation(Fisher’s[1922]weakfactorreversaltest):010101011100(1.1)P(p

4、,p,x,x)Q(p,p,x,x)=p·x/p·x;i.e.,theproductofthepriceindextimesthequantityindexshouldyieldtheexpenditureratiobetweenthetwoperiods.(Weindicatetheinnerproductoftwovectorsasp·xorpTx.)ExamplesofpriceindexesareP(p0,p1,x0,x1)≡p1·x0/p0·x0(Laspeyrespriceindex),La01011101PPa(p,p,x,x)≡p·x/p·x(Paaschepric

5、eindex).*ThisarticlewasfirstpublishedintheJournalofEconometrics4(2),1976,pp.115–145.ApreliminaryversionwaspresentedatStanfordinAugust1973.TheauthorisindebtedtoL.J.Lau,D.Aigner,K.J.Arrow,E.R.Berndt,C.Black-orby,L.R.ChristensenandK.Lovellforhelpfulcomments.ThisresearchwasEssaysinIndexNumberTheor

6、y,VolumeIpartiallysupportedbyNationalScienceFoundationGrantGS-3269-A2attheW.E.DiewertandA.O.Nakamura(Editors)InstituteforMathematicalStudiesintheSocialSciencesatStanfordUniver-c1993ElsevierSciencePublishersB.V.Allrightsreserved.sity,andbytheCanadaCouncil.224EssaysinIndexNumberTheory8.Exactand

7、SuperlativeIndexNumbers225ThegeometricmeanofthePaascheandLaspeyresindexeshasbeensug-arbitrarytwicedifferentiablelinearlyhomogeneousfunction.ForaproofthatgestedasapriceindexbyBowley[1928]andPigou[1912],butitisIrvingthefunctionalform(xTAx)1/2can