Centre for Efficiency and Productivity Analysis

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1 Centre for Efficiency and Productivity Analysis Working Paper Series No. WP06/2015 Exact Relationships between Fisher Indexes and Theoretical Indexes E. Grifell-Tatjé, C. A. K. Lovell Date: May 2015 School of Economics University of Queensland St. Lucia, Qld Australia ISSN No

2 ExactRelationshipsbetweenFisherIndexes andtheoreticalindexes E.Grifell*Tatjé,UniversitatAutònomadeBarcelona C.A.K.Lovell,UniversityofQueensland Abstract In this paper we develop exact relationships between empirical Fisher indexes and their theoretical Malmquist and Konüs counterparts. We begin by using implicit Malmquistpriceandpricerecoveryindexestoestablishexactrelationshipsbetween Malmquist quantity and productivity indexes and Fisher quantity and productivity indexes.wethenshowthatmalmquistquantityandproductivityindexesandfisher price and price recovery indexes almost satisfy the product test with the relevant valuechange,andwederiveaquantitymixfunctionthatensuressatisfactionofthe producttest.wenextuseimplicitkonüsquantityandproductivityindexestoestablish exactrelationshipsbetweenkonüspriceandpricerecoveryindexesandfisherprice and price recovery indexes. We then show that Konüs price and price recovery indexesandfisherquantityandproductivityindexes almost satisfytheproducttest withtherelevantvaluechange,wederiveapricemixfunctionthatensuressatisfaction oftheproducttest,andweshowthatthispricemixfunctiondiffersfundamentallyfrom thequantitymixfunctionrelatingmalmquistandfisherindexes. Keywords: implicit Malmquist indexes, implicit Konüs indexes, Fisher indexes, quantitymixandpricemixfunctions JELClassificationcodes:C43,D24,D61 0

3 ExactRelationshipsbetweenFisherIndexesandTheoreticalIndexes 1.Introduction Theoretical Malmquist quantity and productivity indexes differ from empirical Fisherquantityandproductivityindexes.ThismattersbecauseMalmquistindexescan beestimatedusingempiricaldata,andempiricalmalmquiststudiesareproliferating. OurfirstobjectiveistorelatetheoreticalMalmquistquantityandproductivityindexes to empirical Fisher quantity and productivity indexes, and to provide economically meaningful expressions for the relationships. These expressions also enable Malmquist quantity and productivity indexes and Fisher price and price recovery indexestosatisfytheproducttestwiththerelevantvaluechange.thekeyingredients inthisanalysisareimplicitmalmquistpriceandpricerecoveryindexes. Similarly, theoretical Konüs price and price recovery indexes differ from empiricalfisherpriceandpricerecoveryindexes.thisalsomatters,becausekonüs indexes also can be estimated using empirical data, although to date this has not becomeapopularexercise.nonethelessoursecondobjectiveistorelatetheoretical KonüspriceandpricerecoveryindexestoempiricalFisherpriceandpricerecovery indexes, and to provide (fundamentally different) economically meaningful expressions for the relationships. These expressions also enable Konüs price and price recovery indexes and Fisher quantity and productivity indexes to satisfy the producttestwiththerelevantvaluechange.thekeyingredientsinthisanalysisare implicitkonüsquantityandproductivityindexes. Theliteraturerelatingtheoreticalandempiricalindexnumbershastakentwo approaches. One approach seeks restrictions on the structure of production technology,inconjunctionwithaformofoptimizingbehavior,thatequateanempirical indexwithacorrespondingtheoreticalindex.diewert(1992)followsthisapproachto provide astrongeconomicjustification fortheuseoffisherquantityandproductivity indexes. A second approach imposes relatively weak regularity conditions on the structure of production technology, sufficient for duality to hold, augmented with Mahler inequalities, to establish approximate relationships between empirical and theoreticalindexes.balk(1998)makesextensiveuseofthisapproach. Ourpaperfitsintoneithercategory.Ouranalysisbeginswithimplicittheoretical priceandquantityindexes.weusetheseimplicitindexestoderivefunctionsthatlink FisherindexeswithMalmquistandKonüsindexes,andthatguaranteesatisfactionof the analogous product tests. We provide economic intuition behind the content of these functions, which characterize variation in the mix of choice variables, either quantitiesorprices. 1 Our paper unfolds as follows. In Section 2 we provide some background to motivateouranalysisrelatingempiricalandtheoreticalindexnumbers.insection3 weuseimplicitmalmquistpriceandpricerecoveryindexestorelatefisherquantity andproductivityindexestomalmquistquantityandproductivityindexes.wealsoshow thatmalmquistquantityandproductivityindexesandfisherpriceandpricerecovery indexes almost satisfytheproducttestwiththerelevantvaluechange,andwederive and provide economic interpretations of quantity mix functions that guarantee 1

4 satisfaction of the product test. In Section 4 we use implicit Konüs quantity and productivityindexestorelatefisherpriceandpricerecoveryindexestokonüsprice andpricerecoveryindexes.wealsoshowthatkonüspriceandpricerecoveryindexes andfisherquantityandproductivityindexes almost satisfytheproducttestwiththe relevantvaluechange,wederivepricemixfunctionsthatguaranteesatisfactionofthe producttest,andweshowthatthesefunctionsdifferfundamentallyfromtheanalogous functionsrelatingmalmquistandfisherindexes.section5concludes. 2.Background Lety t R # andx t R $ beoutputandinputquantityvectorswithcorresponding pricevectorsp t R # andw t R $,andletrevenuer t =p tt y t,costc t =w tt x t,and profitability(orcostrecovery)π t =R t /C t,allfortwotimeperiods,abaseperiodt=0and acomparisonperiodt=1.letthetechnologyt t ={(y,x):xcanproduceyinperiodt}, theconvexoutputsetp t (x)={y:(y,x) T t }withfrontierip t (x)={y:y P t (x),λy P t (x), λ>1},andtheconvexinputsetl t (y)={x:(x,y) T t }withfrontieril t (y)={x:x L t (y), λx L t (y),λ<1}.finallylettherevenuefrontierr t (x,p)=max y {p T y:y P t (x)} R t and thecostfrontierc t (y,w)=min x {w T x:x L t (y)} C t. WeknowfromBalk(1998)thatourbestempiricalandtheoreticalquantityand productivityindexesarerelatedby Y F Y M (x 1,x 0,y 1,y 0 ) X F X M (y 1,y 0,x 1,x 0 ) % % ((* +,* -,. +,. - ) ' ' ( (. +,. -,* +,* -, (1) ) where Y F, X F and Y F /X F are Fisher output quantity, input quantity and productivity indexes, and Y M (x 1,x 0,y 1,y 0 ), X M (y 1,y 0,x 1,x 0 ) and Y M (x 1,x 0,y 1,y 0 )/X M (y 1,y 0,x 1,x 0 ) are Malmquistoutputquantity,inputquantityandproductivityindexesingeometricmean form. 2 Itfollowsthat 3 P F Y M P F Y F = W F X M W F X F = % ( ' ( 2 3 % ' = + -. (2) Results (1) and (2) are based on Mahler inequalities, which use distance functionstoboundtheallocativeefficienciesofquantityvectors[r t (x,p) p T y/d 5 6 (x,y) 2

5 p,y,xandc t (y,w) w T x/d 7 6 (y,x) w,x,y],withanassumptionofwithin*periodallocative efficiency[p tt y t /D 5 6 (y t,x t )=r t (x t,p t )andw tt x t /D 7 6 (x t,y t )=c t (y t,w t ),t=0,1],whered 5 6 (x,y)= min{φ>0:y/φ P t (x)} 1 y P t (x)areoutputdistancefunctionsandd 7 6 (y,x)=max{θ>0: x/θ L t (y)} 1 x L t (y)areinputdistancefunctions. Wealsoknowthatourbestempiricalandtheoreticalpriceandpricerecovery indexesarerelatedby P F P K (x 1,x 0,p 1,p 0 ) W F W K (y 1,y 0,w 1,w 0 ) 2 2 8(* +,* -,9 +,9 - ) (. +,. -,: +,: -, (3) ) where P F, W F and P F /W F are Fisher output price, input price and price recovery indexes, and P K (x 1,x 0,p 1,p 0 ), W K (y 1,y 0,w 1,w 0 ) and P K (x 1,x 0,p 1,p 0 )/W K (y 1,y 0,w 1,w 0 ) are Konüsoutputprice,inputpriceandpricerecoveryindexes. 4 Itfollowsthat Y F P K Y F P F = X F W K X F W F = % ' % ' 2 3 = + -. (4) Results (3) and (4) are not based on Mahler inequalities. These results are basedoninequalitieshavingsimilarform[r t (x,p) p T y p,y,xandc t (y,w) w T x w,x,y], but they use revenue and cost frontiers to bound the overall efficiencies, and the efficienciesbeingboundedarethoseofpricevectorsratherthanquantityvectors. In Sections 3 and 4 we derive exact relationships between empirical and theoretical index numbers, and we provide economic interpretations of the mix functions that convert the approximations to equalities. We also show that the economiccontentofthemixfunctionsthatconverttheapproximationsin(1)and(2) toequalitiescoincide,andtheydifferfundamentallyfromtheeconomiccontentofthe mixfunctionsthatconverttheapproximationsin(3)and(4)toequalities,whichalso coincide.thequantitymixfunctionsinsection3,butnotthepricemixfunctionsin Section4,areratioanaloguestotheproductmixandresourcemixeffectsinGrifell* TatjéandLovell(1999o1182,1184). ThestartingpointsinouranalysesareimplicitMalmquistoutputandinputprice indexesinsection3,andimplicitkonüsoutputandinputquantityindexesinsection 4. Neither set of implicit indexes satisfies the fundamental homogeneity property in 3

6 pricesorquantities,respectively(diewert(1981o174,176)).howeverwedonottreat theseimplicitindexesaspriceorquantityindexesoweusethemforotherpurposes,to convert the economicapproximations in(1) and(3)toexact relationships, which in turn eliminates the product test gaps in (2) and (4), and to provide economic interpretationsofthegapstheyeliminate. 3. ImplicitMalmquistPriceandPriceRecoveryIndexes InthissectionweexploitimplicitMalmquistoutputprice,inputpriceandprice recovery indexes. These indexes enable us to derive exact relationships between Fisher and Malmquist output quantity, input quantity and productivity indexes, and exactdecompositionsofrevenuechange,costchangeandprofitabilitychange. 3.1TheOutputSide AbaseperiodimplicitMalmquistoutputpriceindexis PI = # (x 0,p 1,p 0,y 1,y /0 - ) % - ((* -,. +,. - ) = 9+?. + /@ Ā (* -,. + ) 9 -?. - /@ Ā (* -,. -, (5) ) inwhichy # = (x 0,y 1,y 0 )=D 5 = (x 0,y 1 )/D 5 = (x 0,y 0 )isabaseperiodmalmquistoutputquantity index.multiplyinganddividingbyp 0T y 1 /D 5 = (x 0,y 1 )yields PI # = (x 0,p 1,p 0,y 1,y 0 )=P P 9-? [. + /@ Ā (* -,. + )] 9 -? [. - /@ Ā (* -,. - )] % E =P P % - ((* -,. +,. - ) =P P YM # = (x 0,p 0,y 1,y 0 ), (6) in which P P = p 1T y 1 /p 0T y 1 is a Paasche output price index, Y L = p 0T y 1 /p 0T y 0 is a Laspeyres output quantity index, and YM # = (x 0,p 0,y 1,y 0 ) = [p 0T y 1 /D 5 = (x 0,y 1 )]/[p 0T y 0 /D 5 = (x 0,y 0 )] is a base period Malmquist output quantity mix function,sonamedbecauseitisbasedonoutputdistancefunctionsthatscaleoutput vectorsy 1 andy 0 tothebaseperiodfrontierip 0 (x 0 ),therebyeliminatinganymagnitude differencebetweenthem,leavingonlydifferenceintheirmix.thisfunctionistheratio oftherevenuegeneratedbyy 1 /D 5 = (x 0,y 1 )tothatgeneratedbyy 0 /D 5 = (x 0,y 0 )whenboth arevaluedatbaseperiodoutputprices.thesecondequalityin(6)providesanexact decomposition of a base period implicit Malmquist output price index. The third equalitydemonstratesthatthebaseperiodmalmquistoutputquantitymixfunctionis 4

7 the ratio of a Laspeyres output quantity index to a base period Malmquist output quantity index. In the presence of base period prices we expect normalized base period quantities y 0 /D 5 = (x 0,y 0 ) to generate at least as much revenue as normalized comparisonperiodquantitiesy 1 /D 5 = (x 0,y 1 ),andsoweexpectym # = (x 0,p 0,y 1,y 0 ) 1,and thusy L Y # = (x 0,y 1,y 0 ). Revenuechangeis Y # = (x 0,y 1,y 0 ) PI # = (x 0,p 1,p 0,y 1,y 0 ) =[Y # = (x 0,y 1,y 0 ) P P ] YM # = (x 0,p 0,y 1,y 0 ), (7) whichuses(6)toprovideanexactdecompositionofrevenuechange,showingthat theproductofabaseperiodmalmquistoutputquantityindex,apaascheoutputprice index,andabaseperiodmalmquistoutputquantitymixfunctionsatisfiestheproduct testwithrevenuechange. ThebaseperiodoutputquantitymixfunctionhasavalueofunityifM=1,orif M>1 and y 1 = λy 0, λ>0, which effectively converts the problem to a single output problem.ifym # = (x 0,p 0,y 1,y 0 )=1,PI # = (x 0,p 1,p 0,y 1,y 0 )=P P andy L =GY # = (x 0,y 1,y 0 )in(6)and R 1 /R 0 =Y # = (x 0,y 1,y 0 ) P P in(7),sothat,undereitherofthestipulatedconditions,a baseperiodimplicitmalmquistoutputpriceindexisequaltoapaascheoutputprice index,abaseperiodmalmquistoutputquantityindexisequaltoalaspeyresoutput quantityindex,andtheproductofabaseperiodmalmquistoutputquantityindexand apaascheoutputpriceindexsatisfiestheproducttestwithrevenuechange. Ifneitheroftheseconditionsholds,YM # = (x 0,p 0,y 1,y 0 )<1.Baseperiodoutput allocativeefficiency(butnotnecessarilytechnicalefficiency)ofy 0 relativetop 0 [i.e., p 0T y 0 /D 5 = (x 0,y 0 ) = r 0 (x 0,p 0 ) in (6)] is sufficient for YM # = (x 0,p 0,y 1,y 0 ) < 1, and thus for PI # = (x 0,p 1,p 0,y 1,y 0 ) < P P, Y L < Y # = (x 0,y 1,y 0 ), and R 1 /R 0 Y # = (x 0,y 1,y 0 ) P P. A less restrictivesufficientconditionforallthreeinequalitiesrequiresonlythaty 0 bemore allocatively efficient than y 1 relative to (x 0,p 0 ) on base period technology [i.e., p 0T y 1 /D 5 = (x 0,y 1 )<p 0T y 0 /D 5 = (x 0,y 0 ) r 0 (x 0,p 0 )in(6)].thisassumptionisweakerthanone ofbaseperiodoutputallocativeefficiency(e.g.,balk(1998))orofbaseperiodrevenue maximization(e.g.,diewert(1981)). AcomparisonperiodimplicitMalmquistoutputpriceindexis PI H # (x 1,p 1,p 0,y 1,y /0 - ) % + ((* +,. +,. - ) = 9+?. + /@ + A (* +,. + ) 9 -?. - /@ + A (* +,. -, (8) ) 5

8 in which Y # H (x 1,y 1,y 0 ) = D 5 H (x 1,y 1 )/D 5 H (x 1,y 0 ) is a comparison period Malmquist output quantityindex.multiplyinganddividingbyp 1T y 0 /D 5 H (x 1,y 0 )yields PI # H (x 1,p 1,p 0,y 1,y 0 )=P L 9+? [. + /@ A + (* +,. + )] 9 +? [. - /@ A + (* +,. - )] % I =P L % + ((* +,. +,. - ) =P L YM # H (x 1,p 1,y 1,y 0 ), (9) in which P L = p 1T y 0 /p 0T y 0 is a Laspeyres output price index, Y P = p 1T y 1 /p 1T y 0 is a Paasche output quantity index, and YM # H (x 1,p 1,y 1,y 0 ) = [p 1T y 1 /D 5 H (x 1,y 1 )]/[p 1T y 0 /D 5 H (x 1,y 0 )]isacomparisonperiodmalmquistoutputquantitymix functionthatistheratiooftherevenuegeneratedbyy 1 /D 5 H (x 1,y 1 )tothatgeneratedby y 0 /D 5 H (x 1,y 0 ) when both are valued at comparison period output prices. The second equality in (9) provides an exact decomposition of a comparison period implicit Malmquist output price index. The third equality shows that the comparison period MalmquistoutputquantitymixfunctionistheratioofaPaascheoutputquantityindex to a comparison period Malmquist output quantity index. In the presence of comparison period prices we expect normalized comparison period quantities y 1 /D 5 H (x 1,y 1 )togenerateatleastasmuchrevenueasnormalizedbaseperiodquantities y 0 /D 5 H (x 1,y 0 ),andsoweexpectym # H (x 1,p 1,y 1,y 0 ) 1,andthusY P Y # H (x 1,y 1,y 0 ). Revenuechangeis Y # H (x 1,y 1,y 0 ) PI # H (x 1,p 1,p 0,y 1,y 0 ) =[Y # H (x 1,y 1,y 0 ) P L ] YM # H (x 1,p 1,y 1,y 0 ), (10) whichprovidesasecondexactdecompositionofrevenuechange,inwhichtheproduct of a comparison period Malmquist output quantity index, a Laspeyres output price index,andacomparisonperiodmalmquistoutputquantitymixfunctionalsosatisfies theproducttestwithrevenuechange. ThecomparisonperiodoutputquantitymixfunctionhasavalueofunityifM=1, orifm>1andy 1 =λy 0,λ>0.Undereitheroftheseconditionsacomparisonperiod implicit Malmquist output price index is equal to a Laspeyres output price index, a comparison period Malmquist output quantity index is equal to a Paasche output quantity indexo andthe product of a comparison period Malmquist output quantity index and a Laspeyres output price index satisfies the product test with revenue change. 6

9 If neither of these conditions holds, comparison period output allocative efficiency of y 1 relative to p 1 [i.e., p 1T y 1 /D 5 H (x 1,y 1 ) = r 1 (x 1,p 1 ) in (9)] is sufficient for YM # H (x 1,p 1,y 1,y 0 )>1,andthusforPI # H (x 1,p 1,p 0,y 1,y 0 )>P L,Y P >Y # H (x 1,y 1,y 0 ), andr 1 /R 0 > Y # H (x 1,y 1,y 0 ) P L. A less restrictive sufficient condition for all three inequalities requires only that y 1 be more allocatively efficient than y 0 relative to (x 1,p 1 ) on comparisonperiodtechnology[i.e.,p 1T y 0 /D 5 H (x 1,y 0 )<p 1T y 1 /D 5 H (x 1,y 1 ) r 1 (x 1,p 1 )in(9)]. Figure1illustratesthebaseperiodandcomparisonperiodoutputquantitymix functions for M=2. Convexity of the output sets guarantees that YM # = [x 0,p 0,y 1 /D 5 = (x 0,y 1 ),y 0 /D 5 = (x 0,y 0 )] 1andthatYM # H [x 1,p 1,y 1 /D 5 H (x 1,y 1 ),y 0 /D 5 H (x 1,y 0 )] 1.Withinperiodallocativeefficiencyissufficientbutnotnecessaryoallthatisrequired isthaty 0 /D 5 = (x 0,y 0 )bemoreallocativelyefficientthany 1 /D 5 = (x 0,y 1 )relativetop 0,andthat y 1 /D 5 H (x 1,y 1 )bemoreallocativelyefficientthany 0 /D 5 H (x 1,y 0 )relativetop 1. InsertFigure1abouthere AnimplicitMalmquistoutputpriceindexisthegeometricmeanof(6)and(9), andso PI # (x 1,x 0,p 1,p 0,y 1,y 0 )=P F YM # (x 1,x 0,p 1,p 0,y 1,y 0 ) % =P F % ( (* +,* -,. +,. -, (11) ) inwhichp F = [P P P L ] 1/2 isafisheroutputpriceindex,y F =[Y L Y P ] 1/2 isafisheroutput quantityindex,y M (x 1,x 0,y 1,y 0 )= Y = # x =, y H, y = G GY H # (x H, y H, y = ) H/L isamalmquistoutput quantityindex,andthemalmquistoutputquantitymixfunctionym # (x 1,x 0,p 1,p 0,y 1,y 0 ) =[YM = # (x 0,p 0,y 1,y 0 ) YM H # (x 1,p 1,y 1,y 0 )] 1/2. Itfollowsfrom(11)that Y F =Y # (x 1,x 0,y 1,y 0 ) YM # (x 1,x 0,p 1,p 0,y 1,y 0 ),(12) which provides an exact relationship between the empirical Fisher output quantity indexandthetheoreticalmalmquistoutputquantityindex. Revenuechangeisthegeometricmeanof(7)and(10),andso =[Y #(x 1,x 0,y 1,y 0 ) P F ] YM # (x 1,x 0,p 1,p 0,y 1,y 0 ), (13) whichprovidesanexactdecompositionofrevenuechange. TheoutputquantitymixfunctionhasavalueofunityifM=1,orify 1 =λy 0,λ>0. UndereitheroftheseconditionsPI # (x 1,x 0,p 1,p 0,y 1,y 0 )=P F in(11),y F =Y # (x 1,x 0,y 1,y 0 ) 7

10 in(12),andr 1 /R 0 =Y # (x 1,x 0,y 1,y 0 ) P F in(13).ifneitheroftheseconditionsholds, YM # (x 1,x 0,p 1,p 0,y 1,y 0 ) 1providesaneconomicallymeaningfulcharacterizationofthe differencespi # (x 1,x 0,p 1,p 0,y 1,y 0 ) P F in(11),y F Y M (x 1,x 0,y 1,y 0 )in(12),andr 1 /R 0 Y # (x 1,x 0,y 1,y 0 ) P F in(13). 3.2TheInputSide WeexploittheimplicitMalmquistinputpriceindexinasimilarmanner,using the same strategies and the same quantity mix logic. The base period implicit Malmquist input price index is WI # = (y 0,w 1,w 0,x 1,x 0 ) (C 1 /C 0 )/X # = (y 0,x 1,x 0 ) and the comparison period implicit Malmquist input price index is WI # H (y 1,w 1,w 0,x 1,x 0 ) (C 1 /C 0 )/X # H (y 1,x 1,x 0 ).Weomitallintermediatestepsandarriveatthegeometricmean ofthetwo,theimplicitmalmquistinputpriceindex WI # (y 1,y 0,w 1,w 0,x 1,x 0 )=W F :-? [* + /@ Ō (. -,* + )] : -? [* - /@ Ō (. -,* - )] G G :+? [* + /@ O + (. +,* + )] : +? [* - /@ O + (. +,* - )] H/L ' =W F ' ( (. +,. -,* +,* - ) =W F XM # (y 1,y 0,w 1,w 0,x 1,x 0 ),(14) inwhichthefisherinputpriceindexw F = [W P W L ] 1/2,theFisherinputquantityindex X F =[X L X P ] 1/2,andtheMalmquistinputquantityindexX # (y 1,y 0,x 1,x 0 )=[X # = (y 0,x 1,x 0 ) X # H (y 1,x 1,x 0 )] 1/2.TheMalmquistinputquantitymixfunctionXM # (y 1,y 0,w 1,w 0,x 1,x 0 )is thegeometricmeanofabaseperiodmalmquistinputquantitymixfunctionthatisthe ratioofthecostincurredatx 1 /D 7 = (y 0,x 1 )tothatatx 0 /D 7 = (y 0,x 0 )whenbotharevaluedat base period input prices, and a comparison period Malmquist input quantity mix functionthatistheratioofthecostincurredatx 1 /D 7 H (y 1,x 1 )tothatatx 0 /D 7 H (y 1,x 0 )when both are valued at comparison period input prices. The second equality in (14) providesanexactdecompositionoftheimplicitmalmquistinputpriceindex.thethird equalityshowsthatthemalmquistinputquantitymixfunctionistheratioofafisher inputquantityindextoamalmquistinputquantityindex,fromwhichitfollowsthat X F =X # (y 1,y 0,x 1,x 0 ) XM # (y 1,y 0,w 1,w 0,x 1,x 0 ),(15) whichprovidesanexactrelationshipbetweenanempiricalfisherinputquantityindex andatheoreticalmalmquistinputquantityindex. that SincecostchangecanbeexpressedasC 1 /C 0 =X F W F,itfollowsfrom(15) =[X #(y 1,y 0,x 1,x 0 ) W F ] XM # (y 1,y 0,w 1,w 0,x 1,x 0 ),(16) whichprovidesanexactdecompositionofcostchange. 8

11 TheinputquantitymixfunctionhasavalueofunityifN=1,orifx 1 =µx 0,µ>0, whicheffectivelyconvertstheproblemtoasingleinputproblem.undereitherofthese conditionswi # (y 1,y 0,w 1,w 0,x 1,x 0 )=W F in(14),x F =X # (y 1,y 0,x 1,x 0 )in(15), andc 1 /C 0 = X # (y 1,y 0,x 1,x 0 ) W F in (16). If neither of these conditions holds, we exploit the expectationthatxm # (y 1,y 0,w 1,w 0,x 1,x 0 ) 1,evenintheabsenceofwithin*periodinput allocative efficiency, which generates WI # (y 1,y 0,w 1,w 0,x 1,x 0 ) W F in (14), X F X # (y 1,y 0,x 1,x 0 )in(15), andc 1 /C 0 X # (y 1,y 0,x 1,x 0 ) W F in(16). Figure2illustratesthebaseperiodandcomparisonperiodinputquantitymix functions with N=2. Convexity of the input sets guarantees that XM # = [y 0,w 1,w 0,x 1 /D 7 = (y 0,x 1 ),x 0 /D 7 = (y 0,x 0 )] 1 and that XM # H [y 1,w 1,w 0,x 1 /D 7 H (y 1,x 1 ),x 0 /D 7 H (y 1,x 0 )] 1.Aswithanoutputorientation,withinperiod allocativeefficiencyissufficient,butnotnecessary,forxm # (y 1,y 0,w 1,w 0,x 1,x 0 ) 1. InsertFigure2abouthere 3.3CombiningtheOutputSideandtheInputSide Weignorebaseperiodandcomparisonperiodindexesandproceeddirectlyto animplicitmalmquistpricerecoveryindex.theratioof(11)and(14)is 2P ( (* +,* -,9 +,9 -,. +,. - ) 3P ( (. +,. -,: +,: -,* +,* - ) = 2 3 M # (y H, y =, x H, x =, p H, p =, w H, w = ),(17) in which M # (y H, y =, x H, x =, p H, p =, w H, w = ) = YM # (x 1,x 0,p 1,p 0,y 1,y 0 )/XM # (y 1,y 0,w 1,w 0,x 1,x 0 )isamalmquistquantitymixfunctionthat provides an economic characterization of the gap, if any, between P F /W F and PI # (x 1,x 0,p 1,p 0,y 1,y 0 )/WI # (y 1,y 0,w 1,w 0,x 1,x 0 ). From (11) (13) we expect YM # (x 1,x 0,p 1,p 0,y 1,y 0 ) 1,andfrom(14) (16)weexpectXM # (y 1,y 0,w 1,w 0,x 1,x 0 ) 1. ConsequentlyweexpectM # (y H, y =, x H, x =, p H, p =, w H, w = ) 1,inwhichcaseaFisher price recovery index is approximately equal to an implicit Malmquist price recovery index.aunitaryratiowouldrequireequalitybetweentheoutputquantitymixfunction andtheinputquantitymixfunction,asufficientbutnotnecessaryconditionforwhich isy 1 =λy 0,λ>0andx 1 =µx 0,µ>0,whichconvertsamultipleoutput,multipleinput problemtoasingleoutput,singleinputproblemthatdoesnotrequireindexnumbers ofeithersort. is Anexpressionforproductivitychangeisgivenbytheratioof(12)and(15),and % ' = % ((* +,* -,. +,. - ) ' ( (. +,. -,* +,* - ) M #(y H, y =, x H, x =, p H, p =, w H, w = ),G (18) 9

12 which provides an exact relationship between a Fisher productivity index and a Malmquist productivity index, with the Malmquist quantity mix function providing an economicinterpretationofthe(presumablysmall)gapbetweenthetwo. is Anexpressionforprofitabilitychangeisgivenbytheratioof(13)and(16),and + -=[ % ((* +,* -,. +,. - ) ' ( (. +,. -,* +,* - ) 2 3 ] M # (y H, y =, x H, x =, p H, p =, w H, w = ),G (19) and if the Malmquist quantity mix function is approximately unity a Malmquist productivityindexandafisherpricerecoveryindexapproximatelysatisfytheproduct testwithprofitabilitychange. 5 InthissectionwehaveusedimplicitMalmquistpriceandpricerecoveryindexes to relate Malmquist quantity and productivity indexes to Fisher quantity and productivityindexes.theimportantfindingsarecontainedin(11),(14)and(17)o(12), (15)and(18)oand(13),(16)and(19).Thefirstsetofresultsrelatesimplicittheoretical price and price recovery indexes to their explicit empirical counterparts, and establishesthefoundationsforthesecondandthirdsetsofresults.(12),(15)and(18) clarify the sense in which Fisher quantity and productivity indexes and Malmquist quantityandproductivityindexesareapproximatelyequal.(13),(16)and(19)clarify thesenseinwhichmalmquistquantityandproductivityindexesapproximatelysatisfy therelevantproducttestwithfisherpriceandpricerecoveryindexes.eachofthese sets of results depends fundamentally on Malmquist output and input quantity mix functions,whichhavecleareconomicinterpretations.itisworthemphasizingthatthe quantitymixfunctionscomparetheallocativeefficienciesofpairsofquantityvectors, whicharethechoicevariablesintheexercises. (13), (16) and (19) warrant special emphasis from an empirical perspective, because of their decomposability properties. Y # (x 1,x 0,y 1,y 0 ), X # (y 1,y 0,x 1,x 0 ) and Y # (x 1,x 0,y 1,y 0 )/X # (y 1,y 0,x 1,x 0 ) decompose into the product of economic drivers of productivity change: technical change, technical efficiency change, mix efficiency changeandscaleefficiencychange(o Donnell(2012)).Incontrast,P F,W F andp F /W F decompose into contributions of individual output and input price changes (Balk (2004)).GThese two features enable a decomposition of value (revenue, cost and profitability)changeintotheeconomicdriversofquantitychangeandtheindividual pricedriversofpricechange. 4.ImplicitKonüsQuantityandProductivityIndexes In this section we exploit implicit Konüs output quantity, input quantity and productivityindexes.theseindexesleadustoexactrelationshipsbetweenfisherand Malmquist output price, input price and price recovery indexes, and to exact 10

13 decompositionsofrevenuechange,costchangeandprofitabilitychange.bothsetsof resultsdifferfromanalogousresultsinsection3. 4.1TheOutputSide WebeginwithabaseperiodimplicitKonüsoutputquantityindex YI = S (x 0,p 1,p 0,y 1,y 0 0 ) + / (* -,9 +,9 - ) = 9+?. + /T - (* -,9 + ) 9 -?. - /T - (* -,9 -, (20) ) in which P S = (x 0,p 1,p 0 ) = r 0 (x 0,p 1 )/r 0 (x 0,p 0 ) is a base period Konüs output price index. Multiplyinganddividingbyp 1T y 0 /r 0 (x 0,p 1 )yields YI S = (x 0,p 1,p 0,y 1,y 0 )=Y P 9+?. - /T - (* -,9 + ) 9 -?. - /T - (* -,9 - ) 2 E =Y P 2-8(* -,9 +,9 - ) =Y P PM S = (x 0,y 0,p 1,p 0 ), (21) in which Y P = p 1T y 1 /p 1T y 0 is a Paasche output quantity index, P L = p 1T y 0 /p 0T y 0 is a Laspeyresoutputpriceindex,andPM S = (x 0,y 0,p 1,p 0 )=[y 0T p 1 /r 0 (x 0,p 1 )]/[y 0T p 0 /r 0 (x 0,p 0 )]is abaseperiodkonüsoutputpricemixfunction,sonamedbecauseitisafunctionof revenuefunctionsthatcoincideapartfromtheiroutputpricevectors.thisfunctionis theratiooftherevenuegeneratedbyy 0 atnormalizedcomparisonperiodoutputprices p 1 /r 0 (x 0,p 1 )tothatgeneratedbyy 0 atnormalizedbaseperiodoutputpricesp 0 /r 0 (x 0,p 0 ). Thetwonormalizedpricevectorsdifferonlyintheiroutputpricemix.Analternative interpretationofthebaseperiodkonüsoutputpricemixfunctionisthatitistheratio oftworevenueefficiencies,bothwithbaseperiodtechnologyandquantityvectorsbut withdifferentoutputpricevectors. Thesecondequalityin(21)providesanexactdecompositionofabaseperiod implicit Konüs output quantity index. The third equality demonstrates that the base periodkonüsoutputmixfunctionistheratioofalaspeyresoutputpriceindexanda baseperiodkonüsoutputpriceindex.thismixfunctionisboundedabovebyunityif y 0 isrevenueefficientrelativeto(x 0,p 0 )onbaseperiodtechnology[i.e.,p 0T y 0 =r 0 (x 0,p 0 ) in(21)],orify 0 ismorerevenueefficientrelativeto(x 0,p 0 )thanto(x 0,p 1 )onbaseperiod technology[i.e.,r 0 (x 0,p 0 ) p 0T y 0 /r 0 (x 0,p 0 ) p 1T y 0 /r 0 (x 0,p 1 )in(21)].ineithercasep L P S = (x 0,p 1,p 0 )andyi S = (x 0,p 1,p 0,y 1,y 0 ) Y P.YI S = (x 0,p 1,p 0,y 1,y 0 )=Y P ifeitherm=1orp 1 = λp 0, λ>0, which essentially converts the problem to a single input problem. These 11

14 boundsdonotrequirebaseperiodrevenuemaximizingbehavior,orevenbaseperiod allocativeefficiency. Revenuechangeis =P S = (x 0,p 1,p 0 ) YI S = (x 0,p 1,p 0,y 1,y 0 ) =[P S = (x 0,p 1,p 0 ) Y P ] PM S = (x 0,y 0,p 1,p 0 ), (22) whichstatesthattheproductofabaseperiodkonüsoutputpriceindex,apaasche outputquantityindexandabaseperiodkonüsoutputpricemixfunctionsatisfiesthe producttestwithr 1 /R 0.AsaboveweexpectR 1 /R 0 P S = (x 0,p 1,p 0 ) Y P.Howeverif eitherm=1orp 1 =λp 0,λ>0,(21)and(22)collapsetoYI S = (x 0,p 1,p 0,y 1,y 0 )=Y P andr 1 /R 0 =P S = (x 0,p 1,p 0 ) Y P,inwhichcaseabaseperiodimplicitKonüsoutputquantityindexis equal to a Paasche output quantity index, and consequently a Konüs output price indexandapaascheoutputquantityindexsatisfytheproducttestwithr 1 /R 0. WenowsketchtheresultsofacomparisonperiodimplicitKonüsoutputquantity index. Following the same procedures as above, after multiplying and dividing by p 0T y 1 /r 1 (x 1,p 0 )wehave YI H S (x 1,p 1,p 0,y 1,y /0 - )= (* +,9 +,9 - ) =Y L.+? 9 + /T + (* +,9 + ). +? 9 - /T + (* +,9 - ) 2 I =Y L (* +,9 +,9 - ) =Y L PM S H (x 1,y 1,p 1,p 0 ), (23) inwhichy L =p 0T y 1 /p 0T y 0 isalaspeyresoutputquantityindex,p P =y 1T p 1 /y 1T p 0 isa Paascheoutputpriceindex,andP S H (x 1,p 1,p 0 )=r 1 (x 1,p 1 )/r 1 (x 1,p 0 )isacomparisonperiod Konüs output price index. The comparison period Konüs output price mix function PM S H (x 1,y 1,p 1,p 0 )istheratiooftherevenueefficiencyoftwooutputpricevectors,given comparison period technology and quantity vectors. If y 1 is more revenue efficient relative to (x 1,p 1 ) than to (x 1,p 0 ) on comparison period technology, then PM S H (x 1,y 1,p 1,p 0 ) 1,YI S H (x 1,y 1,y 0 ) Y L andp P P S H (x 1,p 1,p 0 ). Revenuechangeis =P S H (x 1,p 1,p 0 ) YI S H (x 1,p 1,p 0,y 1,y 0 ) 12

15 =[P S H (x 1,p 1,p 0 ) Y L ] PM S H (x 1,y 1,p 1,p 0 ), (24) which states that the product of a comparison period Konüs output price index, a Laspeyres output quantity index and a comparison period Konüs output price mix function satisfies the product test with R 1 /R 0. Under the conditions above, R 1 /R 0 P S H (x 1,p 1,p 0 ) Y L.IfeitherM=1orp 1 =λp 0,λ>0,R 1 /R 0 =P S H (x 1,p 1,p 0 ) Y L. Figure 3 illustrates the base period and comparison period output price mix functionsform=2.itisnotnecessarythaty 0 berevenueefficientrelativeto(x 0,p 0 )on baseperiodtechnologyoallthatisrequiredisthaty 0 bemorerevenueefficientrelative to(x 0,p 0 )thanto(x 0,p 1 )onbaseperiodtechnology.asimilarremarkappliestoy 1. InsertFigure3abouthere Thegeometricmeanof(21)and(23)isanimplicitKonüsoutputquantityindex YI S (x 1,x 0,p 1,p 0,y 1,y 0 )=Y F PM S = x =, y =, p H, p = G GPM S H x H, y H, p H, p = H/L 2 =Y F 2 8 (* +,* -,9 +,9 - ) =Y F PM S x H, x =, y H, y =, p H, p =, (25) which states that an implicit Konüs output quantity index is the product of a Fisher outputquantityindexandakonüsoutputpricemixfunction.becauseonecomponent oftheoutputpricemixfunctionisboundedabovebyunityandtheotherisbounded belowbyunityweexpectyi S (x 1,x 0,p 1,p 0,y 1,y 0 ) Y F. Itfollowsfromthesecondandthirdequalitiesin(25)that P F =P S (x 1,x 0,p 1,p 0 ) PM S (x H, x =, y H, y =, p H, p = ), (26) whichenablesustocalculatethegapbetweenthetheoreticalandempiricaloutput priceindexes. Thegeometricmeanof(22)and(24)yieldstheexpressionforrevenuechange =[P S(x 1,x 0,p 1,p 0 ) Y F ] PM S (x H, x =, y H, y =, p H, p = ), (27) which states that a Konüs output price index and a Fisher output quantity index approximately satisfy the product test with R 1 /R 0, the approximation becoming an equalityifeitherm=1orp 1 =λp 0,λ>0. 13

16 4.2TheInputSide We now consider the implicit Konüs input quantity index. The base period implicitkonüsinputquantityindexisxi S = (y 0,w 1,w 0,x 1,x 0 )=(C 1 /C 0 )/W S = (y 0,w 1,w 0 )andthe comparison period implicit Konüs input quantity index is XI S H (y 1,w 1,w 0,x 1,x 0 ) = (C 1 /C 0 )/W S H (y 1,w 1,w 0 ).Thegeometricmeanofthetwo,theimplicitKonüsinputquantity index,is XI S (y 1,y 0,w 1,w 0,x 1,x 0 )=X F WM S = y =, x =, w H, w = G GWM S H y H, x H, w H, w = H/L 3 =X F 3 8 (. +,. -,: +,: - ) =X F WM S y H, y =, x H, x =, w H, w =,(28) inwhichthekonüsinputpricemixfunctionwm S y H, y =, x H, x =, w H, w = measuresthe gapbetweenxi S (y 1,y 0,w 1,w 0,x 1,x 0 )andx F,andisdefinedanalogouslytotheoutput pricemixfunctionin(25). Fromthesecondandthirdequalitiesin(28) W F =W S (y 1,y 0,w 1,w 0 ) WM S y H, y =, x H, x =, w H, w =, (29) whichprovidesanexactrelationshipbetweenempiricalfisherandtheoreticalkonüs inputpriceindexes. Costchangeis =[W S(y 1,y 0,w 1,w 0 ) X F ] WM S y H, y =, x H, x =, w H, w =. (30) Thebaseperiodandcomparisonperiodinputpricemixfunctionsareillustrated infigure4,inwhichcostefficiencyofx 0 andx 1 isnotrequiredoallthatisrequiredis that x 0 be more cost efficient relative to (y 0,w 0 ) than to (y 0,w 1 ) on base period technology, and that x 1 be more cost efficient relative to (y 1,w 1 ) than to (y 1,w 0 ) on comparisonperiodtechnology. InsertFigure4abouthere 4.3CombiningtheOutputSideandtheInputSide WenowconstructanimplicitKonüsproductivityindex.Weignorebaseperiod andcomparisonindexesandproceeddirectlytoanimplicitkonüsproductivityindex. Theratioof(25)and(28)is 14

17 %P 8 (* +,* -,9 +,9 -,. +,. - ) 'P 8 (. +,. -,: +,: -,* +,* - ) =% ' M S (y H, y =, x H, x =, p H, p =, w H, w = ), (31) in which the Konüs price mix function M S (y H, y =, x H, x =, p H, p =, w H, w = ) = PM S x H, x =, y H, y =, p H, p = /WM S y H, y =, x H, x =, w H, w = measures the gap between YI S (x 1,x 0,p 1,p 0,y 1,y 0 )/XI S (y 1,y 0,w 1,w 0,x 1,x 0 ) and Y F /X F. Because we expect PM S x H, x =, y H, y =, p H, p = 1 and we expect WM S y H, y =, x H, x =, w H, w = 1 we also expectm S (y H, y =, x H, x =, p H, p =, w H, w = ) 1,inwhichcasetheimplicitKonüsproductivity indexisapproximatelyequaltoafisherproductivityindex.equalitywouldrequirethe output and input price mix functions to be equal, a sufficient but not necessary conditionforwhichisp 1 =λp 0,λ>0andw 1 =µw 0,µ>0,whichconvertstheproblemto asingleoutput,singleinputproblemforwhichindexnumbersareunnecessary. Theratioof(26)and(29) 2 = 2 8(* +,* -,9 +,9 - ) (. +,. -,: +,: - ) M S(y H, y =, x H, x =, p H, p =, w H, w = ),(32) providesanexactrelationshipbetweenanempiricalfisherpricerecoveryindexand atheoreticalkonüspricerecoveryindex. The ratio of(27) and(30) provides an implicit Konüs measure of profitability change + -= 0+ /0-1 + /1 - =[ 2 8(* +,* -,9 +,9 - ) 3 8 (. +,. -,: +,: - ) % ' ] M S (y H, y =, x H, x =, p H, p =, w H, w = ),(33) andifm S (y H, y =, x H, x =, p H, p =, w H, w = ) 1aKonüspricerecoveryindexandaFisher productivityindexapproximatelysatisfytheproducttestwithprofitabilitychange. 6 InthissectionwehaveusedimplicitKonüsquantityandproductivityindexesto relate Konüs price and price recovery indexes to Fisher price and price recovery indexes. The important findings are contained in(25),(28) and(31)o(26),(29) and (32)o and (27), (30) and (33). The first three relate implicit theoretical quantity and productivity indexes to their explicit empirical counterparts, and establish the foundationsforthesecondandthirdsetsofresults.(26),(29)and(32)clarifythesense inwhichkonüspriceandpricerecoveryindexesapproximatefisherpriceandprice recoveryindexes.(27),(30)and(33)clarifythesenseinwhichkonüspriceandprice recoveryindexesapproximatelysatisfytherelevantproducttestwithfisherquantity andproductivityindexes.inboththesecondandthirdsetsofresultsclarityisprovided bytherelevantkonüspricemixfunction. 15

18 In Section 3 the product test expressions (13), (16) and (19) have useful empiricalapplications,sincemalmquistquantityandproductivityindexesdecompose by economic driver and Fisher price and price recovery indexes decompose by individualprices.heretheproducttestexpressions(27),(30)and(33)areofpotential, butasyetunrealized,empiricalvalue.thefisherquantityandproductivityindexes have been decomposed by economic driver of productivity change, although agreementonapreferreddecompositionremainselusive. 7 TheKonüspriceandprice recoveryindexeshaveyettobedecomposedbyeconomicdriversofpricechange (ratherthan,ascommonlypracticed,byindividualprices),althoughresearchonthis issueisunderway. WeemphasizethattheKonüspricemixfunctionsdiffersignificantlyfromthe MalmquistquantitymixfunctionsinSection3,althoughtheyservethesamepurposes, toconvertapproximationstoexactrelationshipsandtocloseproducttestgaps.the Malmquist quantity mix functions are ratios of values generated by two normalized quantityvectorsweightedbyacommonpricevector.thekonüspricemixfunctions areratiosofvaluesgeneratedbyasinglequantityvectorweightedbytwonormalized pricevectors. 5.SummaryandConclusions WehaveexploitedimplicitMalmquistpriceandpricerecoveryindexestoderive exactrelationshipsbetweenmalmquistandfisherquantityandproductivityindexes, and to derive economically meaningful functions describing the ability of Malmquist quantityandproductivityindexestosatisfyproducttestswithfisherpriceandprice recoveryindexes.thekeytotheseexactrelationshipsistheconceptofmalmquist output and input quantity mix functions, in which quantities are allowed to vary betweenbaseandcomparisonperiodsbutpricesarefixedateitherbaseperiodvalues orcomparisonperiodvalues.thesmallerthevariationinthequantitymixbetween base and comparison periods, the smaller the gap between Fisher and Malmquist quantityandproductivityindexes. We also have exploited implicit Konüs quantity and productivity indexes to derive exact relationships between Konüs and Fisher price and price recovery indexes, and to derive fundamentally different, but nonetheless economically meaningfulfunctionsdescribingtheabilityofkonüspriceandpricerecoveryindexes tosatisfyproducttestswithfisherquantityandproductivityindexes.thekeytothese exact relationships is the concept of Konüs output and input pricemix functions, in whichpricesareallowedtovarybetweenbaseandcomparisonperiodsbutquantities arefixedateitherbaseperiodvaluesorcomparisonperiodvalues.thesmallerthe variationinthepricemixbetweenbaseandcomparisonperiods,thesmallerthegap betweenfisherandkonüspriceandpricerecoveryindexes. The exact relationships have clear economic interpretations, as allocative efficiency effects, although these effects differ between Sections 3 and 4. These allocative efficiency effects are easy to calculate, using data required to calculate 16

19 FisherindexesandestimateMalmquistandKonüsindexes,asBreaetal.(2011)have demonstratedforfisher/malmquistpairings. 17

20 References Balk,B.M.(1998),Industrial2Price,2Quantity2and2Productivity2Indexes.Boston:Kluwer AcademicPublishers. Balk, B. M. (2004), Decompositions of Fisher Indexes, Economics2 Letters 82:1 (January),107*13. Brea, H., E. Grifell*Tatjé and C. A. K. Lovell (2011), Testing the Product Test, Economics2Letters113:2(November),157*59. Diewert,W.E.(1981), TheEconomicTheoryofIndexNumbers:ASurvey, Chapter 7inA.Deaton,ed.,Essays2In2The2Theory2and2Measurement2of2Consumer2Behaviour2 In2Honour2of2Sir2Richard2Stone.Cambridge:CambridgeUniversityPress. Diewert,W.E.(1992), FisherIdealOutput,InputandProductivityIndexesRevisited, Journal2of2Productivity2Analysis3:3(September),211*48. Grifell*Tatjé,E.,andC.A.K.Lovell(1999), ProfitsandProductivity, Management2 Science45:9(September),1177*93. Grifell*Tatjé,E.,andC.A.K.Lovell(2015),Productivity2Accounting:2The2Economics2 of2business2performance.newyork:cambridgeuniversitypress. Kuosmanen,T.,andT.Sipiläinen(2009), ExactDecompositionoftheFisherIdeal TotalFactorProductivityIndex, Journal2of2Productivity2Analysis31:3(June),137*150. O Donnell, C. J. (2012), An Aggregate Quantity Framework for Measuring and DecomposingProductivityChange, Journal2of2Productivity2Analysis38:3(December), 255*72. Ray, S. C., and K. Mukherjee (1996), Decomposition of the Fisher Ideal Index of Productivity:ANonparametricDualAnalysisofU.S.AirlinesData, Economic2Journal 106:439(November),1659*

21 y 2 y 1 /U V H (W H, X H ) y 1 /U V = (W =, X H ) y 0 /U V H (W H, X = ) p 1 y 0 /U V = (W =, X = ) p 0 y 1 IP 0 (x 0 ) IP 1 (x 1 ) Figure1OutputQuantityMixFunctions 19

22 x 2 W = /U Y H (X H, W = ) W = /U = Y (X =, W = ) W H /U = Y (X =, W H ) W H /U Y H (X H, W H ) IL 1 (y 1 ) w 1 w 0 IL 0 (y 0 ) x 1 Figure2InputQuantityMixFunctions 20

23 y 2 y 1 p 0 p 1 y 0 p 1 p 0 IP 0 (x 0 ) IP 1 (x 1 ) y 1 Figure3OutputPriceMixFunctions 21

24 x 2 x 1 w 0 w 1 IL 1 (y 1 ) x 0 w 1 w 0 IL 0 (y 0 ) x 1 Figure4InputPriceMixFunctions 22

25 Endnotes 1 OuranalysisextendsresultsinGrifell*TatjéandLovell(2015oChapter3). 2 The Fisher quantity indexes are defined as Y F = ^-[ _ + ^+[ _ + ^-[ _ - ^+[ _ - `ā _ -,\ + `ā _ -,\ - `a + _ +,\ + `a+ _ +,\ - Z -[ \ + Z+[ \ + Z -[ \ - Z +[ \ - H/L and X F = H/L, and the Malmquist quantity indexes are defined as Y M = H/L andx M = `b- \ -,_ + `b- \ -,_ - `b + \ +,_ + `b+ \ +,_ - 3 The Fisher price indexes are defined as P F = _ -[^+ _ -[^- _+[^+ _ +[^- H/L. 4 The Konüs price indexes are defined as P K = d - (\ -,^+) (\ + H/L,^+) d - (\ -,^-) G Gd+. d + (\ +,^-) H/L. \ -[ Z + \+[ Z + \ -[ Z - \ +[ Z - H/L and W F = c - (_ -,Z + ) (_ +,Z + H/L ) c - (_ -,Z - ) G Gc+ and W c + (_ +,Z - K = ) 5 Allapproximationresultsinthissectionalsocanoccurifthetechnologiesallowinfiniteoutput substitutionpossibilitiesalongip 0 (x 0 )andip 1 (x 1 )betweenoutputraysdefinedbyy 1 andy 0 in Figure1,andinfiniteinputsubstitutionpossibilitiesalongIL 0 (y 0 )andil 1 (y 1 )betweeninputrays definedbyx 1 andx 0 infigure2. 6 Allapproximationresultsinthissectioncanalsooccurify 0 andy 1 infigure3andx 0 andx 1 infigure4areverticesofpiecewiselineartechnologiesthatallowp 1T y 0 =p 0T y 0,p 0T y 1 =p 1T y 1 andw 1T x 0 =w 0T x 0,w 0T x 1 =w 1T x 1,asmightoccurwithDEA. 7 Compare,forexample,thedecompositionsproposedbyRayandMukherjee(1996)andby KuosmanenandSipiläinen(2009). 23