Index Number Concepts, Measures and Decompositions of Productivity Growth

Transcription

1 C Journal of Produciviy Analyi, 9, 27 59, Kluwer Academic Publiher. Manufacured in The Neherl. Inde Number Concep, Meaure Decompoiion of Produciviy Growh W. ERWIN DIEWERT Deparmen of Economic, Univeriy of Briih Columbia, 873 Ea Mall, Buchanan Tower, room 997, Vancouver, BC V6T Z ALICE O. NAKAMURA alice.nakamura@ualbera.ca School of Buine, Univeriy of Albera, 3-23 Buine Building, Edmonon, Albera T6G 2R6 Abrac Thi paper eplore he definiion properie of oal facor produciviy growh (TFPG) indee, focuing epecially on he Paache, Lapeyre, Fiher, Törnqvi, implici Törnqvi one. Thee indee can be evaluaed from obervable price quaniy daa, cerain of hee are hown o be meaure of TFPG concep heoreical indee ha have been propoed in he lieraure. The mahemaical relaionhip beween hee quaniy aggregae, financial meaure, price quaniy indee are eplored. Decompoiion of he produciviy growh indee are alo given. The paper conclude wih a brief overview of ome limiaion on our analyi. JEL claificaion: O3, O30; O4, O4.7; M Keyword: inde number, produciviy, TFP, TFPG, echnical change Produciviy A raio of oupu o inpu. (Akinon, Banker, Kaplan Young, 995, Managemen Accouning, p. 54) To profi from produciviy improvemen, managemen need meauremen procedure for monioring produciviy performance idenifying improvemen opporuniie. (David M. Miller, Harvard Buine Review, May June 984, pp ) Thi paper preen conider a number of concepual definiion of produciviy growh, meaure decompoiion of hee ha are ueful no only for economic analy hoe involved in he conrucion of official aiic bu alo for buine manager. Secion inroduce four concep meaure of oal facor produciviy growh (TFPG) for he impliic cae in which he inde number problem i aben: he one inpu, one oupu cae. In he - cae, i i eaily een ha he meaure are all equal. In he general N inpu, M oupu cae, he quaniie for he differen inpu oupu mu be aggregaed hi lead o he ue of inde number formula. If price weigh are ued, hen iue of price change mu be deal wih oo. In Secion 2, we inroduce Correponding auhor.

2 28 DIEWERT AND NAKAMURA he Paache, Lapeyre, Fiher Törnqvi TFPG formula. We preen hee afer fir defining aggregae quaniy price indee ha can be ued o define he TFPG inde formula. The meaure of he four concep of TFPG ha are defined for a general N inpu, M oupu producion iuaion are hown o be equal for appropriae choice of funcional form for aggregaing he quaniy or value price daa. Secion 3 give a way of decompoing any TFPG inde in erm of inpu-oupu coefficien. The reciprocal of he inpu-oupu coefficien he oupu-inpu coefficien are he mo widely ued ype of ingle facor produciviy meaure for hop floor operaion managemen. The decompoiion preened provide a framework for eamining he effec on overall TFPG of change in pecific ingle facor produciviy coefficien. Two main approache o chooing among he differen propoed funcional form for he TFPG inde are he aiomaic (or e) approach reviewed briefly in Secion 4, he eac inde number approach inroduced in Secion 5 hen ued in Secion 6. The aiomaic approach make ue of li of deired properie, ermed aiom or e, ha are formalizaion of common ene properie of good inde number or generalizaion of properie ha hold for all propoed inde number formula in he impliic - cae. The eac inde number approach i rooed in economic heory, i omeime referred o a an economic approach. Secion 7 i devoed o ye anoher meaure of produciviy performance propoed by Diewer Morrion decompoiion of ha meaure. Secion 8 conclude.. Four Concep of TFPG I i convenien o inroduce concep noaion in he implified cone of a one inpu, one oupu producion proce. For = 0,,...,T, he quaniy of he inpu ued in period i wih uni price w he quaniy of he oupu produced i y wih uni price p. The baic definiion of oal facor produciviy (TFP) i he rae of ranformaion of oal inpu ino oal oupu. Engineer oher inereed in phyical efficiency ofen focu on hi ranformaion rae, which i meaured by he raio of oupu produced o inpu: TFP ( y ) a. () The upercrip on TFP indicaing he ime period will only be hown hereafer when hi i no clear from he cone. The parameer a defined along wih TFP in () i an oupu-inpu coefficien. An oupu-inpu coefficien alway involve ju one oupu one inpu, alhough coefficien of hi or are ofen defined ued in he cone of muliple inpu, muliple oupu producion iuaion. Indeed, we how a new way of doing hi in Secion 3. In hi paper, we focu on oal facor produciviy growh raher han TFP. 2 A number of differen way have been propoed for concepualizing TFPG, four of which are conidered in hi paper. 3 A fir concep i imply he rae of growh for TFP. Meaure of hi fir concep will be denoed by TFPG(). For he - cae, he growh in TFP beween period i meaured by he raio of he oupu-inpu coefficien for period : 4 TFPG() ( y )( y ) = a a. (2)

3 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 29 A econd concep of oal facor produciviy growh i he raio of he rae of oupu growh o inpu growh. For he - cae, he naural meaure for hi i TFPG(2) ( y y )( ). (3) The hird fourh concep relae TFPG o he financial revenue co oal o margin. For he - cae, oal revenue co are given for =,...,T by R p y C w. (4) Dividing he revenue raio for period by he oupu price raio for he wo period yield (R R )(p p ) = ( p y p y )( p p ) = y y. (5) Similarly, when he co raio for period i divided by he inpu price raio, we ge (C C )(w w ) = ( w w )( w w ) =. (6) The hird concep i he growh rae for real revenue veru real co. For he - cae, hi i meaured by he rae of growh in he raio of revenue o co conrolling for price change: [ R R ][ C TFPG(3) C ] ( y )( = ). (7) p w p w Buine manager are inereed in enuring ha revenue eceed co, which lead o an inere in margin. The period margin, m, can be meaured by y + m R C, (8) he meaure for margin growh conrolling for price change i defined for he - cae a TFPG(4) [( + m )( + m )] [( w w )( p p )]. (9) If we inerpre he margin a a reward for managerial or enrepreneurial inpu, hen TFPG(4) can be inerpreed a he growh rae of he price of he inpu broadly defined o a o include managerial enrepreneurial inpu divided by he growh rae of he price of he oupu good. Uing (8) o eliminae he margin growh rae on he righ-h ide of (9) comparing hi wih he epreion in (2), (3) (7), we ee ha in he - cae, he meaure for he four concep of oal facor produciviy growh are all equal. 2. The N-M Cae Mo real producion procee yield muliple oupu virually all ue muliple inpu, leading o a need for inde number o aggregae hee oupu inpu. We begin by

4 30 DIEWERT AND NAKAMURA defining quaniy aggregae ha are componen of he Paache, Lapeyre Fiher Ideal quaniy, price TFPG indee, hen give he formula for hee indee. Törnqvi implici Törnqvi inde number are alo defined. 2.. Price Weighed Quaniy Aggregae For a general N-inpu, M-oupu producion proce, he period inpu oupu price vecor are denoed by w [w,...,w N ] p [p, p 2,...,p M ], while [,..., N ] y [y,...,y M ] denoe he period inpu oupu quaniy vecor. Nominal oal co C revenue R can be viewed a price weighed quaniy aggregae of he micro level daa for he individual ranacion, are defined a follow for period : C C wn n, R pm y m, (0) wn n R pm y m. () We alo define four hypoheical quaniy aggregae. 5 The fir wo reul from evaluaing period quaniie uing period price weigh: wn n pm y m (2) Thee aggregae are wha he co revenue would have been if he period inpu had been purchaed he period oupu had been old a period price. In conra, he hird fourh aggregae are um of period quaniie evaluaed uing period price: wn n pm y m. (3) Thee are wha he co revenue would have been if he period inpu had been purchaed he period oupu had been old a period price. The eigh aggregae given in (0) hrough (3) are all ha are needed o define he Paache, Lapeyre Fiher quaniy, price, TFPG indee. Tradiionally hee were defined a weighed average of quaniy price relaive. 6 The equivalen definiion preened here are more convenien for eablihing ha each of hee TFPG indee i a meaure of all four of he differen concep of TFPG inroduced in Secion.

5 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS The Paache, Lapeyre Fiher Quaniy Price Indee The Paache (874), Lapeyre (87), Fiher (922, p. 234) oupu quaniy indee can be defined a follow uing he quaniy aggregae given in (0) (3): Q P i= Q L p i y i M p j y j, (4) j= pi y i i= j= M p j y j, (5) Q F (Q P Q L ) (2). (6) Similarly, he Paache, Lapeyre Fiher inpu quaniy indee can be defined a: Q P Q L wi i w j j, (7) i= j= wi i w j j, (8) i= j= Q F (Q P Q L )(2). (9) Oupu inpu quaniy indee are all ha are needed o define a meaure of he econd concep of TFPG ince an oupu quaniy inde i a meaure of growh for oal oupu an inpu quaniy inde i a meaure of growh for oal inpu. For he Paache, Lapeyre Fiher indee, we how in Secion 2.3 ha a meaure of he fir concep of TFPG can be defined by rearranging he aggregae defining he meaure of he econd concep. However, in order o pecify meaure of he hird fourh concep for he muliple inpu, muliple oupu cae, price indee are needed oo. Price indee can be conruced wih any of he funcional form ued for quaniy indee by revering he role of he price quaniie in he quaniy inde. Thu he Paache Lapeyre oupu inpu price indee can be defined a: P P P P P L i= p i y i M p j y j, (20) j= wi i w j j, (2) i= j= pi y i i= j= M p j y j, (22)

6 32 DIEWERT AND NAKAMURA P L wi i w j j. (23) i= j= The Fiher oupu inpu price indee are given by P F (P P P L ) (2) (24) P F (P P P L )(2). (25) A price inde i he implici counerpar of a quaniy inde if he produc rule i aified. Thi rule require ha he produc of he quaniy price indee mu equal he oal co raio for inpu ide indee or he oal revenue raio for oupu ide indee. 7 Uually he implici price inde will no have he ame funcional form a he quaniy inde i i aociaed wih. For eample, he Paache price inde i he implici counerpar of a Lapeyre quaniy inde, he Lapeyre price inde i he implici counerpar of a Paache quaniy inde. The Fiher indee are unuual in ha he Fiher price inde aifie he produc e rule when paired wih a Fiher quaniy inde. In defining proving equaliie for he meaure of he four concep of TFPG for a general muliple inpu, muliple oupu producion iuaion, we ue he following implicaion of he produc rule: for he Paache, Lapeyre Fiher indee, on he inpu ide we have Q P P L = Q L P P = Q F P F = C C, (26a) on he oupu ide we have Q P P L = Q L P P = Q F P F = R R. (26b) 2.3. TFPG Meaure for he N-M Cae The radiional definiion of a oal facor produciviy growh inde in he inde number lieraure i a a raio of oupu inpu quaniy indee: TFPG QQ. (27) Thu he Paache, Lapeyre Fiher TFPG indee can be defined uing he Paache, Lapeyre Fiher quaniy indee. Given a choice of any one of hee hree funcional form, we will prove ha he correponding muliple inpu, muliple oupu cae meaure are all equal for he four concep of TFPG inroduced in Secion. To eablih hee equaliie, we ue he produc rule reul o define Paache, Lapeyre Fiher TFPG(3) meaure. Then we ue he definiion of he componen of he TFPG(3) meaure o define eablih he equaliy of TFPG(2) TFPG() meaure.

7 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 33 The definiion equaliie for hee meaure are a follow: TFPG P = Q P Q P uing (27) (26) = = (R R )P L (C C )P L M M p m y m p m y m M p m ym M p m ym N w n n N w n n N w n n N w n n TFPG(3) P = M M p m y m p m y m N w n n N w n n TFPG(2) P uing (0), () alo (22) (23) M N = m y m w n n M N p m y m w n n TFPG() P (28) TFPG L = Q L Q L uing (27) (26) = = (R R )P P (C C )P P M M p m y m p m y m M p m ym M p m ym N w n n N w n n N w n n N w n n TFPG(3) L = M M p m y m p m y m N w n n N w n n TFPG(2) L uing (0), () alo (20) (2) M N = m y m w n n M N p m y m w n n TFPG() L (29) TFPG F = Q F Q F uing (27) (26) = = (R R )P F (C C )P F TFPG(3) F [( R ) ] 2 [( R ) ] 2 PL PP R R [( C ) ] C P 2 [( C ) ] L C P 2 = P uing (6), (26), (0), (), (20) (23) [ M ] 2 [ M = [ M p m y m N w n n p m ym N w n n ] 2 [ M ] 2 p m y m N w n n p m ym N w n n [ M p m y m M p m ym [ N w n n N w n n ] 2 [ M ] 2 [ N ] 2 p m y m M p m ym w n n N w n n ] 2 TFPG(2) F ] 2 TFPG() F (30)

8 34 DIEWERT AND NAKAMURA TFPG(4) i he rae of growh in he margin afer conrolling for price change. In he general N-M cae, ju a in he - one, he margin m i given for = 0,,...,T by + m R C. (3) Depending on wheher Lapeyre, Paache or Fiher price indee are ued o deflae he co revenue componen of he margin, he epreion for TFPG(3) given in (28), (29) (30) can be rewrien a: TFPG(4) P [( + m )( + m )][PL P L], (32) TFPG(4) L [( + m )( + m )][PP P P], (33) TFPG(4) F [( + m )( + m )][P F P F]. (34) Noice ha if he margin m are zero, regardle of he reaon, hen each of hee epreion for TFPG(4) reduce o he raio of he inpu price inde o he oupu price inde Oher Inde Number Formula Many oher inde number formula have been propoed beide he Paache, Lapeyre Fiher. 9 Here we will ue Q G P G Q G P G o denoe any given oupu inpu quaniy price indee ha aify he produc rule o ha Q G P G = (R R ) Q G P G = (C C ). From hee produc rule reul (3), i i eaily een ha he following meaure of concep 2, 3 4 of TFPG are equal: (R R )P G (C C )P G TFPG(3) G = Q G Q G TFPG(2) G = [( + m )( + m )][P P] TPFG(4). (35) Bu wha abou TFPG() G? A meaure of he growh in he rae of ranformaion of oal inpu ino oal oupu ideally hould be defined uing meaure of oal oupu oal inpu ha are comparable for period in he ene ha he micro level quaniie for boh period are aggregaed uing he ame price weigh. 0 The quaniy aggregae ha are he componen of he Paache, Lapeyre Fiher TFPG() meaure defined in he fir line of (28), (29) (30) aify hi comparabiliy over ime ideal. However, here are many oher inde number formula for which i i no poible o define hi or of an ideal TFPG() meaure ha alo equal he correponding meaure for he oher hree concep of TFPG. For any pair of quaniy price indee aifying he produc e, from (35) he produc rule implicaion we ee ha he following epreion equal hoe defined in (35)

9 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 35 for TFPG(2) G, TFPG(3) G TFPG(4) G : Q G Q G = (R R )P (C C )P = M N ( ) p m PG y M m p m y m ( ) w n P G N n w n. (36) n In he la of hee epreion, he price vecor (p P G ) (w PG ) appearing in he period oupu inpu quaniy aggregae are he period price epreed in period dollar. If we chooe hi epreion a he meaure of TFPG() G, hen for any inde number formula oher han he Paache, Lapeyre or Fiher, hi meaure will no be ideal in he ene of uing he ame price weigh o compare he period period quaniie. However, here i an approimae oluion o hi problem for indee ha aify he produc rule are alo wha i ermed uperlaive. Thi approimae oluion make ue of he Fiher funcional form: a funcional form for which we have an ideal TFPG() meaure, defined in (30). Diewer coined he erm uperlaive for an inde number funcional form ha i eac in ha i can be derived algebraically from a producer or conumer behavioral equaion ha aifie he Diewer fleibiliy crierion. According o hi crierion, an equaion i fleible if i can provide a econd order approimaion o an arbirary wice coninuouly differeniable linearly homogeneou funcion. Diewer (976, 978) Hill (2000) eablihed ha all of he commonly ued uperlaive inde number formula (including he Fiher, alo he Törnqvi implici Törnqvi funcional form inroduced below) approimae each oher o he econd order when evaluaed a an equal price quaniy poin. Thi i a numerical analyi approimaion reul ha doe no rely on any aumpion of economic heory. Becaue he Fiher quaniy price indee alo aify he produc rule, we have Q G P G = (R R ) = Q F P F Q G P G = (C C ) = Q F P F, dividing hrough by P G PG, repecively, yield [ ][ ] Q G Q F PF P G Q = G Q F PF. (37) P G From (37), (35) (30) we ee ha if we define he meaure for he fir concep of TFPG a [ ] PF P G TPFG() G TPFG() F PF, (38) P G hi meaure will equal TFPG(2) G, TFPG(3) G TFPG(4) G a defined in (35). However, hi meaure doe no have he ideal comparabiliy over ime propery. In hi TFPG() G meaure, he period price vecor, p w, of he TFPG() F componen are replaced by (p (P F P G )) (w (PF P G )). A a conequence, unle he given price indee are Lapeyre or Paache or Fiher one, he period period quaniie compared by he meaure will no be aggregaed uing he ame price weigh when here have been change in relaive price. Neverhele, from (38) he Diewer (976, 978)Hill (2000) approimaion reul for uperlaive inde number, i follow ha when he choen quaniy price indee are any of he commonly ued uperlaive indee uch a he Törnqvi or implici Törnqvi, hen we can ue he reul ha TFPG() G = TFPG()F.

10 36 DIEWERT AND NAKAMURA 2.5. The Törnqvi (or Tranlog) Indee 2 Törnqvi (936) indee are weighed geomeric average of growh rae for he microeconomic daa (he quaniy or price relaive). Thee indee appear more complicaed han he oher defined o far, bu are alo widely ued. I i he formula for he naural logarihm of a Törnqvi inde ha i uually hown. For he oupu quaniy inde, hi i [( M ) ( M )] ln Q T = (2) + ln ( ym ) y m. (39) p m y m i= p i y i p m y m The Törnqvi inpu quaniy inde Q T i defined analogouly, wih inpu quaniie price ubiued for he oupu quaniie price in (39). Revering he role of he price quaniie in he formula for he Törnqvi oupu quaniy inde yield he Törnqvi oupu price inde, P T,defined by [( M ) ( M )] ln P T = (2) + ln ( pm ) p m. (40) p m y m i= p i y i p m y m The inpu price inde PT i defined in a imilar manner. The implici Törnqvi oupu quaniy inde, denoed by Q T,idefined implicily by 3 (R R )P T Q T, he implici Törnqvi inpu quaniy inde, Q,idefined analogouly uing he co raio PT. The implici Törnqvi oupu price inde, P T,i T given by (R R )Q T P T, he implici Törnqvi inpu price inde, P,idefined T analogouly. Uing he Törnqvi quaniy he implici Törnqvi price indee, or he implici Törnqvi quaniy he Törnqvi price indee, meauremen formula for concep 2 4 of TFPG can be pecified a in (35) above. A already noed, when Törnqvi or implici Törnqvi indee are ued, i i no poible o define a TFPG() meaure ha i ideal in he ene dicued in Secion 2.4. However, hee are uperlaive indee for which he Secion 2.4 approimaion reul applie; ha i, we have TFPG() T = TFPG()F TFPG() T = TFPG() F. The eence of hi reul i ha he Fiher TFPG() meaure, which i ideal in he ene dicued in Secion 2.4, hould be a good approimae meaure for he fir concep of TFPG when he Törnqvi or he implici Törnqvi indee are eleced. j= j= p j y j p j y j 3. An Inpu-Oupu Coefficien Decompoiion of TFPG [I] i our hypohei ha non-financial financial conrol yem erve very differen role in organizaion. The non-financial yem are ued for day-o-day operaion conrol. (Armiage Akinon, 990, The Choice of Produciviy Meaure in Organizaion: A Field Sudy of Pracice, p. 4) A Armiage Akinon repor, buine urvey reveal ha ingle facor produciviy efficiency meaure are widely ued for operaion monioring. Thu, i i imporan o

11 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 37 relae he ingle facor meaure o overall TFPG. Forunaely, he TFPG(3) form of a oal facor produciviy growh inde can be convenienly decompoed in erm of inpu-oupu coefficien or heir reciprocal. A eablihed in he previou ecion, any TFPG inde can be epreed in he form TFPG = [R P][C P ] (4) R C where P P are he implici counerpar of Q Q. For any given ime period, he raio of oal nominal revenue o oal nominal co can be rewrien a [ ( )] R C = pm y m wn n = = [ ( )] wn n p m ym [ ( )( w n p m n ym ) ]. (42) I i clear from (42) ha, afer conrolling for price change, improvemen in any ingle facor efficiency meaure, ay (ym n ), conribue o an improved revenue co raio, a migh be epeced. Subiuion of (42) ino (4) yield: M { N [( )( )][ ]} w TFPG = n P pm P n y m M { N [ ][ ]} w n p m n y. (43) m Thi epreion can be ued o conider he relaive effec on TFPG of alernaive ingle facor produciviy improvemen, or of anicipaed change in any of he real price raio for individual inpuoupu combinaion. 4. The Aiomaic (or Te) Approach o Chooing Among Alernaive Inde Number Formula Which funcional form hould be ued for he price indee in (4) or he quaniy indee in (27)? Hiorically, inde number heori have relied on wha i called he aiomaic or e approach o addre hi choice problem. An overview of hi approach i provided here, concenraing on he oupu ide quaniy price indee ince he approach i he ame on he inpu ide. A before, P denoe an oupu price inde Q denoe an oupu quaniy inde. The aiomaic approach o he deerminaion of he funcional form for P Q work a follow. The aring poin i a li of mahemaical properie ha he price inde hould aify baed on a priori reaoning. Thee are he inde number heory e or aiom. Mahemaical reaoning i applied o deermine wheher he a priori e are muually conien wheher hey uniquely deermine, or uefully narrow he range of choice

12 38 DIEWERT AND NAKAMURA for, he funcional form for he price inde. 4 The produc e rule i hen applied o olve for he funcional form of he quaniy inde. A already noed in Secion 2.2, on he oupu ide he Produc Te ae ha he produc of he price quaniy indee, P Q, hould equal he nominal revenue raio for period : 5 PQ = R R. (44) If he funcional form for he oupu price inde P i given, hen impoing he produc rule mean ha he funcional form for he oupu quaniy inde mu be given by he epreion Q = (R R )P. (45) If he produc e i impoed a a rule on boh he oupu inpu ide, hen once he funcional form have been pecified for he price indee, he funcional form have alo been deermined for he quaniy indee for he TFPG inde. 6 There are many oher aiomaic e ha can be applied for chooing among he quaniy price inde pair ha aify he produc e. Since he indee ha make up hee pair are no deermined independenly, we only need conider aiom ha apply o he price indee, wih he proce of elecion being he ame for inpu indee a for he oupu ide one conidered here. We conclude hi overview of he aiomaic approach by liing four of hee e. The Ideniy or Conan Price Te i 7 P(p, p, y, y ) =. (46) Wha hi mean i ha if p = p = p = (p,...,p M ) o ha all price are equal in he wo period, hen he price inde hould be one regardle of he quaniy value for period. The Conan Bake Te, alo called he Conan Quaniie Te, i 8 P(p, p, y, y) = pi y i p j y j. (47) i= j= Thi e ae ha if he quaniie are conan over he wo period o ha y = y = y (y,...,y M ), hen he level of price in period compared o period hould equal he value of he conan bake of quaniie evaluaed a he period price divided by he value of hi ame bake evaluaed a he period price. The Proporionaliy in Period Price Te i 9 P(p,λp, y, y ) = λp(p, p, y, y ) for λ>0. (48) According o hi e, if each of he elemen of p i muliplied by he poiive conan λ, hen he level of price in period relaive o period will differ by he ame muliplicaive facor λ. Our final eample of a price inde e i he Time Reveral Te: 20 P(p, p, y, y ) = P(p, p, y, y ). (49)

13 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 39 If hi e i aified, hen when he price quaniie for are inerchanged, he reuling price inde will be he reciprocal of he original price inde. The Fiher price inde P F aifie all four of hee addiional e. The Paache Lapeyre indee, P P P L, fail he ime reveral e (49). The Törnqvi inde, P T, fail he conan bake e (47), he implici Törnqvi inde, P T, fail he conan price e (48). When a more eenive li of e i compiled, he Fiher price inde coninue o aify more e han oher leading cidae. 2 However, he Paache, Lapeyre, Törnqvi, implici Törnqvi indee all how up quie well according o he aiomaic approach. 5. The Eac Inde Number Approach An alernaive approach o he deerminaion of he funcional form for a meaure of oal facor produciviy growh i o derive he inde from a producer behavioral model. Diewer (976) eac inde number approach yemaize he procedure for developing equivalencie beween differen propoed definiion for inde number opimizing model of economic behavior. Uing hee equivalencie, a choice can be made among alernaive produciviy inde number formula baed on he preferred properie for a pecified behavioral equaion. A firm echnology can be ummarized by i period producion funcion f. Focuing on oupu, he period producion funcion can be repreened for = 0,,...,T a y = f (y 2, y 3,...,y M,, 2,..., N ), (50) which give he amoun of oupu he firm can produce uing he echnology available in any given period if i alo produce y 2 uni of oupu 2, y 3 uni of oupu 3,..., y M uni of oupu M uing uni of inpu, 2 uni of inpu 2,..., N uni of inpu N. The producion funcion f can be ued o define he period co funcion, c, a follow: c (y, y 2,...,y M, w, w 2,...,w N ) { } min w n n : y = f (y 2,...,y M,,..., N ). (5) Thi co funcion give he minimum co of producing he oupu y,...,y M uing he period echnology inpu price. Under he aumpion of co minimizing behavior, he oberved period co of producion, C, i equal o he minimum co. Hence for = 0,,...,T,wehave C wn n = c( y,...,y M, w,...,w ) N. (52) We need ome way of relaing he co funcion for period = 0,,...,T o each oher. A common way of doing hi i o aume ha c ( y,...,y M, w,...,w N ) = (a )c ( y,...,y M, w,...,w N ), (53) where a > 0 denoe a period relaive efficiency parameer c denoe an aemporal

14 40 DIEWERT AND NAKAMURA co funcion. The normalizaion a 0 i uually impoed. Given (53), a naural meaure of produciviy change for a producive uni in going from period o i he following raio: a a. (54) If hi raio i greaer han, hen efficiency i aid o have improved. Taking he naural logarihm of boh ide of (53), we have ln c ( y,...,y M, w,...,w N ) = ln a + ln c ( y,...,y M, w,...,w N ). (55) If a ranlog funcional form i aumed for ln c, where c i he aemporal co funcion on he righ h ide of (55), hen we have ln c ( y,...,y M, w,...,w ) N M = b 0 + b m ln ym + c n ln wn + (2) d ij ln yi ln y j + (2) i= j= i= j= M f ij ln wi ln w j + g mn ln ym ln w n. (56) An advanage of he ranlog funcional form i ha i doe no impoe a priori rericion on he admiible paern of ubiuion beween inpu oupu. 22 However, hi pecificaion alo ha a large number of unknown parameer. There are M + of he b m parameer; N of he c n parameer; M(M +)2 independen d ij parameer afer impoing he requiremen ha d ij = d ji for i < j M; N(N + )2 independen f ij parameer even afer alo impoing he requiremen ha f ij = f ji for i < j N; MN of he g mn parameer in he aemporal co funcion (56), in addiion o anoher T independen a parameer in (55). If homogeneiy of degree one in he inpu price i impoed a well for he co funcion, hen we have c n = ; f ij = 0 for i =,...,N, g mn = 0 for m =,...,M. j= Wih hee addiional rericion, he number of independen parameer i reduced o T + M(M + )2 + N(N + )2 + MN. Thi i ill a larger number han he number of obervaion which i T +. Boh economeric inde number mehod require he impoiion of furher rericion for evaluaion of a meaure baed on (55) (56). 23 If he producer i minimizing co, hen he following dem relaionhip will hold: 24 n = c( y,...,y M, w,...,w N ) wn (58) for n =,...,N = 0,,...,T. Since ln c i a quadraic funcion in he variable ln y,ln y 2,..., ln y M, ln w,ln w 2,...,ln w N, (57)

15 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 4 hen Diewer (976, p. 9) logarihmic quadraic ideniy can be applied yielding: 25 ln c ln c = (2) + (2) [ y m ln c [ w m y m (y, w ) + ym lnc ] (y, w ) ln ( y ) m y y m m lnc (y, w ) + w lnc m (y, w ) w m w m [ lnc ] + (2) (y, w lnc ) + (y, w ) a a [ = (2) ym + (2) ln c y m ] ln ( wm ) w m ln(a a ) (59) (y, w ) + ym lnc ] (y, w ) ln ( y ) m y y m m [( w n n C ) + ( wn n C )] ln ( wm ) w m + (2)[ + ( )] ln(a a ). (60) We can furher implify (60) by impoing he addiional aumpion of compeiive profi maimizing behavior. More pecifically, uppoe we can aume ha he oberved period oupu y,...,y M olve he following profi maimizaion problem for = 0,,...,T : { M maimize y,...,y M pm y m c ( y,...,y M, w,...,w ) } N. (6) Thi lead o he uual price equal marginal co relaionhip ha reul when compeiive price aking behavior i aumed; i.e., we now have p m = c( y,...,y M, w,...,w N ) ym, m =,...,M. (62) Making ue of he definiion of oal co in (52) a well a (62), epreion (60) can now be rewrien a: ln(c C ) = (2) + (2) [( p m ym C ) + ( pm y m C )] ln ( ym ) y m [( w n n C ) + ( wn n C )] ln ( wm ) w m ln(a a ). (63)

16 42 DIEWERT AND NAKAMURA Co can be oberved in each period, a can oupu inpu price quaniie. Thu he only unknown in (63) i he produciviy change meaure. Solving (63) for hi yield { M } a a ( ) = y m y (2)[(p m ym C )+(pm y m C )] m Q T, (64) where Q T i he implici Törnqvi inpu quaniy inde ha i defined analogouly o he implici Törnqvi oupu quaniy inde inroduced in Secion 2.5. Formula (64) can be implified furher uing he aumpion ha he echnology ehibi conan reurn o cale. If co grow proporionally wih oupu, hen he co funcion mu be linearly homogeneou in he oupu quaniie (Diewer 974, pp ). In ha cae, wih compeiive profi maimizing behavior, revenue mu equal co in each period o ha c (y, w ) = C = R. (65) Uing (65), we can replace C C in (64) by R R, repecively, (64) become a a = Q T Q T (66) where Q T i he Törnqvi oupu quaniy inde. The hypohei of conan reurn o cale invoked in moving from epreion (64) o (66) i very rericive. However, if he underlying echnology i ubjec o diminihing reurn o cale (or equivalenly, o increaing co), we can conver he echnology ino an arificial one ill ubjec o conan reurn o cale by inroducing an era fied inpu, N+ ay, eing hi era fied inpu equal o one (ha i, N= for each period ). The correponding period price for hi inpu, w N+, i e equal o he firm period profi, R C. Wih hi era facor, he period co i redefined o be he adjued co given for = 0,,...,T by N+ C A = C + w N+ N+ = wn n = R. (67) The derivaion can now be repeaed uing he adjued co C A raher han he acual co C. Wha reul i he ame produciviy change formula ecep ha Q T i now he implici ranlog quaniy inde for N + inead of N inpu. The inde number mehod for meauring he produciviy change erm a a ha i illuraed by he formula on he righ-h ide of (64) or by (66) can be ued even wih hou of oupu inpu. In deriving hee epreion uing he eac inde number approach, we did aume compeiive (i.e., price aking) profi maimizing behavior. When he aumpion of compeiive profi maimizing behavior break down, he relaion (62) are no valid. However, all of he formula defined in Secion 2 ( many oher no dicued) can ill be direcly evaluaed uing he oberved price quaniy daa he aiomaic approach can ill be ued o chooe among alernaive inde number formula. I i only he economic heory inerpreaion ha i lo.

17 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS Producion Funcion Baed Meaure The eac inde number approach, like he aiomaic approach o inde number heory, can be helpful in chooing a paricular inde formula from he many ha have been propoed. In addiion, when a TFPG inde can be relaed o a pecific producer behavioral relaionhip, hi may enable a deeper undering of he meaning of he inde may alo be helpful o manager concerned wih finding operaional way o enhance oal facor produciviy. 6.. Technical Progre Reurn o Scale in he Simple - Cae When we know he producion funcioning correponding o a TFPG inde, one of he hing hi knowledge enable i a decompoiion of TFPG ino a echnical progre erm, TP, a reurn o cale erm, RS. One reaon for inere in hi decompoiion i ha he policy opion for enhancing TFPG gain from echnical progre veru reurn o cale are ofen differen. 26 Thi decompoiion of a TFPG inde i imple o under in he one inpu, one oupu cae. We aume knowledge of he period period inpu oupu quaniie he rue period period producion funcion. Tha i, we aume we have: y = f ( ) (68) y = f ( ). (69) If echnical progre i characerized a a hif in he producion funcion, i i ill neceary o pecify he ype of hif. Four of he variou poible hif meaure we could define make ue of he acual oupu inpu quaniy daa for period (i.e.,,, y, y ), can be eaily relaed o ard managerial accouning concep. We focu on hee four TP meaure here. Hypoheical quaniie are needed o define hee meaure: wo on he oupu ide wo on he inpu ide. The oupu ide hypoheical quaniie are y f ( ) (70) y f ( ). (7) The quaniy y i he oupu ha could be produced wih he period inpu quaniy uing he period producion echnology embodied in f, he quaniy y i he oupu ha could be produced wih he period inpu quaniy uing he period echnology. The inpu ide hypoheical quaniie,, are defined implicily by y = f ( ) (72) y = f ( ). (73)

18 44 DIEWERT AND NAKAMURA The quaniy i he hypoheical amoun of he inpu facor ha would be required o produce he acual period oupu, y, uing he more recen period echnology. Hence will uually be le han. The quaniy i he inpu quaniy ha would be required o produce he period oupu y uing he older period echnology, o i epeced o be larger han. The fir wo of he four echnical progre indee defined are oupu baed: 27 TP() y y = f ( ) f ( ), (74) TP(2) y y = f ( ) f ( ). (75) Each of hee decribe he percenage increae in oupu reuling olely from wiching from he period o he period producion echnology. For TP(), echnical progre i meaured wih he inpu level fied a, wherea for TP(2) he inpu level i fied a. The oher wo indee of echnical progre defined here are inpu baed: 28 TP(3) (76) TP(4). (77) Each of hee give he reciprocal of he percenage decreae in inpu uage reuling olely from wiching from he period o he period producion echnology. For TP(3), echnical progre i meaured wih he oupu level fied a y wherea for TP(4) he oupu level i fied a y. Each of he echnical progre meaure defined above i relaed o TFPG a follow: TFPG = TP(i) RS(i) (78) where for i =, 2, 3, 4 depending on he eleced TP meaure, he RS erm i or RS() [ y ][ ] y, (79) RS(2) [ y ][ ] y, (80) RS(3) [ y ][ ] y, (8) RS(4) [ y ][ ] y. (82) In he TFPG decompoiion in (78), he echnical progre erm, TP(i), correpond o a hif. 29 The reurn o cale erm, RS(i), correpond o a movemen along a fied producion funcion due olely o change in inpu quaniy uilizaion. Each reurn o cale meaure will be greaer han one if oupu divided by inpu increae a we move along he producion urface. For wo period, ay = 0 =, wih ju one inpu facor one oupu good, he four meaure of TP defined in (74) (77) he four meaure of reurn o cale

19 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 45 y y y f () B y f 0 () y * y 0* y 0 A 0 0 * 0 * Figure. Producion funcion baed meaure of echnical progre. defined in (79) (82) can be illuraed graphically, a in Figure. (Here he ubcrip i dropped for boh he ingle inpu he ingle oupu.) The lower curved line i he graph of he period 0 producion funcion; i.e., i i he e of poin (, y) uch ha 0 y = f 0 (). The higher curved line i he graph of he period producion funcion; i.e., i i he e of poin (, y) uch ha 0 y = f (). The oberved daa poin are A for period 0 wih coordinae ( 0, y 0 ), B for period wih coordinae (, y ). 30 Applying formula (2) from Secion, for hi eample we have TFPG = [y ][y 0 0 ] which i he lope of he raigh line OB divided by he lope of he raigh line OA in Figure. The reader can ue Figure he definiion provided above o verify ha each of he four decompoiion of TFPG given by (78) correpond o a differen combinaion of hif of movemen along a producion funcion ha ake u from oberved poin A o oberved poin B. 3 Geomerically, each of he reurn o cale meaure i he raio of wo oupu-inpu coefficien, ay [y j j ] divided by [y k k ] where he poin (y j, j ) ( k, y k ) are on he ame fied producion funcion wih j > k. Thu, if RS(i) = [y j j ][y k k ] i greaer han, he producion funcion ehibi increaing reurn o cale, while if RS(i) = we have conan reurn o cale if RS(i) < we have decreaing reurn. If he reurn o cale are conan, hen RS(i) = TP = TFPG. 32 Figure illurae ha conan reurn o cale aumpion need no be aumed for evaluaing he TFPG meaure in Secion 2. 33

20 46 DIEWERT AND NAKAMURA 6.2. Malmqui Indee The fundamenal inigh of he eac inde number approach i ha, if he echnology of a producion uni can be repreened in each ime period by ome known muliple inpu, muliple oupu producion funcion, hen quaniy TFPG indee can be derived wihin an opimizing model of producer behavior. We do hi here. A in (50), we le y denoe he amoun of oupu produced in period for = 0,,...,T. Here we alo le ỹ [y2, y 3,...,y M ] denoe he vecor of oher (han good ) oupu joinly produced in each period uing he vecor of inpu quaniie [, 2,..., N ]. Thu, he producion funcion for oupu in period period can be repreened compacly a: y = f (ỹ, ) y = f (ỹ, ). (83) In defining Malmqui oupu indee, we begin by defining hree alernaive formulaion on he oupu ide. 34 The fir Malmqui oupu inde, α,idefined a he number which aifie y α = f (ỹ α, ). (84) Thi inde α meaure oupu growh uing he comparion period echnology inpu vecor. I value i he number which ju deflae he period vecor of oupu, y [y, y 2,...,y M ], ino an oupu vecor y α ha can be produced wih he period vecor of inpu,, uing he period echnology. Due o ubiuion, when he number of oupu good, M, i greaer han, hen y α will no uually be equal o he acual period oupu vecor, y. However, if here i only one oupu, equaion (84) become y α = f ( ) = y, we have α = y y. A econd Malmqui oupu inde, α,idefined a he number which aifie α y = f (α ỹ, ). (85) The inde α meaure oupu growh uing he curren period echnology inpu vecor. I i he number ha inflae he period vecor y ino α y, an oupu vecor ha can be produced wih he period inpu vecor uing he period echnology. The vecor α y will no uually be equal o y when here are muliple oupu. However, when M =, hen (85) become α y = f ( ) = y α = y y reduce o he (ingle) oupu growh rae. When here i no reaon o prefer α or α no need o diinguih beween hem, we recommend defining he geomeric mean of α α a he Malmqui oupu quaniy inde: α [α α ] 2. (86) When here are only wo oupu good, he Malmqui oupu indee α α can be illuraed a in Figure 2 for ime period = = 0. The lower curved line repreen he e of oupu {(y, y 2 ) : y = f 0 (y 2, 0 )} ha can be produced wih period 0 echnology inpu. The higher curved line repreen he e of oupu {(y, y 2 ) : y = f (y 2, )} ha can be produced wih period echnology inpu. The period oupu poibiliie e

21 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 47 y 2 y 2 0 y 0 y 2 0 y 0 (y 0, y 2 0 ) y 2 y (y, y 2 ) y 2 0 y 0 (0, 0) y 0 y 0 y 0 y y Figure 2. Alernaive economic oupu indee illuraed. will generally be higher han he period 0 one for wo reaon: (i) echnical progre (ii) inpu growh. 35 In Figure 2, he poin α y 0 = [α y,α y2 ] i he raigh line projecion of he period 0 oupu vecor y 0 = [y 0, y0 2 ] ono he period oupu poibiliie e, y α 0 = [y α0, y2 α0 ] i he raigh line conracion of he oupu vecor y = [y, y 2 ] ono he period 0 oupu poibiliie e. We now urn o he inpu ide. A fir Malmqui inpu inde, β,idefined a follow: y = f (ỹ, β ) f ( y 2,...,y M, β,..., N β ). (87) Thi inde meaure inpu growh uing he period echnology oupu vecor. A econd Malmqui inpu inde, denoed by β, i he oluion o he following equaion y = f (ỹ,β ) f ( y 2,...,y M,β,...,β N ). (88) I meaure inpu growh uing he period echnology oupu vecor. When here i no paricular reaon o prefer eiher he inde β o β no need o diinguih beween hem, we recommend defining heir geomeric average a he Malmqui inpu quaniy inde: β [β β ] 2. (89) Figure 3 illurae he Malmqui indee β β for he cae where here are ju wo inpu good for he ime period = = 0.

22 48 DIEWERT AND NAKAMURA ( 0, 2 0 ) (, 2 ) 2 0 (0, 0) Figure 3. Alernaive Malmqui inpu indee illuraed. The lower curved line in Figure 3 repreen he e of inpu needed o produce he vecor of oupu y 0 uing period 0 echnology. Thi i he e {(, 2 ) : y 0 = f 0 (ỹ 0,, 2 )}. The higher curved line repreen he e of inpu ha are needed o produce he period vecor of oupu y uing period echnology. Thi i he e {(, 2 ) : y = f (ỹ,, 2 )}. 36 The poin β 0 = [β 0,β 2 0] i he raigh line projecion of he inpu vecor 0 [ 0, 2 0] ono he period inpu requiremen e. The poin β 0 i he raigh line conracion of he inpu vecor [, 2 ] ono he period 0 inpu requiremen e. Once heoreical Malmqui quaniy indee meauring he growh of oal oupu inpu have been defined, hen a Malmqui TFPG inde for he general N-M cae can be defined oo. Uing he preferred Malmqui oupu inpu quaniy indee given in (96) (99), he definiion we recommend for he Malmqui TFPG inde i TFPG M αβ. (90) In he - cae, epreion (90) i readily een o reduce o he meaure of TFPG in Secion Direc Evaluaion of Malmqui Indee for he N-M Cae Recall ha α [α α ] 2 β [β β ] 2 are he Malmqui oupu inpu quaniy indee defined in (86) (89). Uing he eac inde number approach, Cave, Chrienen

23 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 49 Diewer (982, pp ) give condiion under which he following equaliie hold: α = Q T (9) β = Q T, (92) where Q T i he Törnqvi oupu quaniy inde defined by (39) Q T i he Törnqvi inpu quaniy inde defined analogouly. If he aumpion ha juify (9) (92) are aified, hen we can evaluae he Malmqui meaure TFPG M defined in (90) by aking he raio of he Törnqvi oupu inpu indee: TFGP M = αβ = Q T Q T TFPG T. (93) The aumpion required o derive (9) (92) are, roughly peaking: (i) price aking, revenue maimizing behavior, (ii) price aking, co minimizing behavior, (iii) a ranlog echnology. 37 The pracical imporance of (93) i ha he epreion can be evaluaed from obervable price quaniie wihou knowledge of he parameer of f f. However, change in he value of hi inde canno by decompoed ino echnical progre reurn o cale componen wihou knowing or eimaing he producion funcion. 7. Diewer-Morrion Decompoiion of an Alernaive Produciviy Meaure In Secion 5, we ued he period producion funcion f o define he period co funcion, c. Here we ue he period producion funcion o define he period (ne) revenue funcion: r (p, ) ma{p y : y (y, y 2,...,y M ); y = f (y 2,...,y M ; )}, (94) y where p (p,...,p M ) i an oupu price vecor ha he producer face (,..., N ) i he inpu vecor. 38 Following Diewer Morrion (986), revenue funcion for period he period are ued o define anoher family of heoreical produciviy change indee: RG(p, ) r (p, )r (p, ). (95) From (94), i can be een ha RG(p, ) i he raio of he oupu ha can be produced uing he period veru he period echnology wih he reference inpu vecor wih boh of he reuling oupu vecor evaluaed uing he reference price vecor, p. Thi i a differen approach o he problem of conrolling for oal facor inpu uilizaion in judging he ucce of he period veru he period producion oucome. Two pecial cae of (95) are of inere: RG RG(p, ) = r (p, )r (p, ) RG RG(p, ) = r (p, )r (p, ). (96)

24 50 DIEWERT AND NAKAMURA RG i he heoreical produciviy inde obained by chooing he reference vecor p o be he period oberved oupu price vecor p inpu quaniy vecor.rg i he heoreical produciviy inde obained by chooing he reference vecor o be he period oberved oupu price vecor p inpu quaniy vecor. 39 Under he aumpion of revenue maimizing behavior in boh period, we can idenify he denominaor of RG he numeraor of RG. Tha i, we have: p y = r (p, ) p y = r (p, ). (97) Our problem in evaluaing he heoreical produciviy indee RG RG i ha we canno direcly oberve he hypoheical revenue, r (p, ) r (p, ). The fir of hee i he hypoheical revenue ha would reul from uing he period echnology wih he period inpu quaniie oupu price. The econd i he hypoheical revenue ha would reul from uing he period echnology wih he period inpu quaniie oupu price. However, hee hypoheical revenue figure can be evaluaed from obervable daa if we know ha he period revenue funcion ha he following ranlog funcional form: M ln r (p, ) α0 + αm ln p m + βn ln n + (2) α ij ln p i ln p j j= + (2) β ij ln i ln j + γ mn ln p m ln n, (98) where α ij = α ji β ij = β ji he parameer aify variou oher rericion o enure ha r (p, ) i linearly homogeneou in he componen of he price vecor p. 40 Noe ha he coefficien vecor α0,α m β n can be differen in each period bu he quadraic coefficien are aumed o be conan over he wo period. Diewer Morrion (986; p. 663) how ha under he above aumpion, he geomeric mean of he wo heoreical produciviy indee defined in (96) can be idenified uing he obervable price quaniy daa ha perain o he wo period; i.e., we have [RG RG ] 2 = abc (99) where a, b c are given by a p y p y, (00) ln b (2) [( pm m y p y ) + ( pm m y p y )] ln ( pm ) p m, (0) ln c = (2) [( wn n p y ) + ( wn n p y )] ln ( n ) n. (02) If we have conan reurn o cale producion funcion f f, hen he value of oupu will equal he value of inpu in each period we have p y = w. (03) i= j=

25 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS 5 The ame reul can be derived wihou he conan reurn o cale aumpion if we have a fied facor ha aborb any pure profi or loe, wih hi fied facor defined a in (67). Subiuing (03) ino (02), we ee ha epreion c become he Törnqvi inpu inde Q T. By comparing (0) (40), we ee alo ha b i he Törnqvi oupu price inde P T. Thu ab i an implici Törnqvi oupu quaniy inde. If (03) hold, hen we have he following decompoiion for he geomeric mean of he produc of he heoreical produciviy change indee defined in (03): [RG RG ] 2 = [p y p y ][P T Q T ] (04) where P T i he Törnqvi oupu price inde Q T i he Törnqvi inpu quaniy inde, boh of which are funcion of he price quaniie in boh period. Diewer Morrion (986) ue he period revenue funcion o define wo heoreical oupu price effec for m =,...,M which how how revenue change a a ingle oupu price change: P m r ( p,...,p m, p m, p m+,...,p M, ) r (p, ), (05) P m r (p, ) r ( p,...,p m, p m, p m+,...,p M, ), m =,...,M. (06) Thee heoreical indee give he proporional change in he value of oupu ha would reul if we changed he price of he mh oupu from i period level pm o i period level pm holding conan all oher oupu price he inpu quaniie a reference level uing he ame echnology in boh iuaion. For he heoreical inde defined in (05), he reference oupu price inpu quaniie echnology are he period one, wherea for he inde defined in (06) hey are he period one. Now define he heoreical oupu price effec b m a he geomeric mean of he wo effec defined by (05) (06): b m [ P m P m] 2, m =,...,M. (07) Diewer Morrion (986) Kohli (990) how ha he b m given by (07) can be evaluaed by he following obervable epreion if condiion (97), (98) (03) hold: ln b m = (2) [( p m y m p y ) + ( p m y m p y )] ln ( p m p m ), m =,...,M. (08) Comparing (0) wih (08), i can be een ha we have he following decompoiion for b: b = M b m = P T. (09) Thu he overall Törnqvi oupu price inde can be decompoed ino a produc of he individual oupu price effec, b m. Diewer Morrion (986) alo ue he period revenue funcion in order o define for n =,...,N wo heoreical inpu quaniy effec: Q n r ( p,,..., n, n, n+,..., N ) r (p, ), (0)

26 52 DIEWERT AND NAKAMURA Q n r (p, ) r ( p,,..., n, n, n+,..., N ). () Thee heoreical indee give he proporional change in he value of oupu ha would reul from changing inpu n from i period level n o i period level n, holding conan all oupu price oher inpu quaniie a reference level uing he ame echnology in boh iuaion. For he heoreical inde defined by (0), he reference oupu price, inpu quaniie echnology are he period one, wherea for he inde in () hey are he period one. Now define he heoreical inpu quaniy effec c n a he geomeric mean of he wo effec defined by (00) (0): c n [ Q n Q n] 2, n =,...,N. (2) Diewer Morrion (986) how ha he c n defined by (2) can be evaluaed by he following empirically obervable epreion provided ha aumpion (97) (98) hold: ln c n = (2) [( wn n p y ) + ( wn n p y )] ln ( n ) n (3) ln c n = (2) [( wn n w ) + ( wn n w )] ln ( n ) n. (4) The epreion (4) follow from (3) provided ha he aumpion (03) alo hold. Comparing (3) wih (2), i can be een ha we have he following decompoiion for c: c = N c n (5) c = Q T, (6) where (6) follow from (5) provided ha he aumpion (03) alo hold. Thu if aumpion (97), (98) (93) hold, he overall Törnqvi inpu quaniy inde can be decompoed ino a produc of he individual inpu quaniy effec, he c n for n =,...,N. Having derived (09) (5), we can ubiue hee decompoiion ino (99) rearrange he erm o obain he following very ueful decompoiion: p y p y = [RG RG ] 2 M N b m c n (7) Thi i a decompoiion of he growh in he nominal value of oupu ino he produciviy growh erm [RG RG ] 2 ime he produc of he oupu price growh effec b m ime he produc of he inpu quaniy growh effec (he c n ). The effec on he righ-h ide of epreion (7) can be calculaed uing only he obervable price quaniy daa for he wo period. 4 An inereing pecial cae of (7) reul when here i only one inpu in he vecor i i fied. Then he inpu growh effec c i uniy variable inpu appear in he y vecor wih negaive componen. In hi pecial cae, he lef-h ide of (7) become he pure profi raio ha i decompoed ino a produciviy effec ime variou price effec (he b m ).

27 INDEX NUMBER CONCEPTS, MEASURES AND DECOMPOSITIONS Concluion Before umming up our finding, we mu menion eplicily ome limiaion of our analyi. Direc evaluaion of he meaure preened in hi paper depend on four problemaic precondiion: (i) he li of inpu ued oupu produced mu remain conan over he curren period he comparion period ; (ii) quaniy eiher uni price or oal value informaion mu be available for each of he wo period for all inpu purchaed oupu old; (iii) i mu be poible o calculae uer co or renal price in an unambiguou way for all capial inpu (i.e., for all durable inpu whoe iniial co hould be pread over he muli-period life of he good);, (iv) for ome or of udie i i imporan for he difference beween he e ane epeced price he e po realized price o be negligible. We briefly dicu why each of hee condiion i problemaic below. The fir econd limiaion are problemaic becaue of daa collecion horcoming becaue of he ongoing appearance of new good. 42 There are alo iue having o do wih he choice of oupu inpu ha hould be included in an analyi ince firm produciviy i affeced by facor eernal o he firm uninended oupu uch a polluan. 43 The hird limiaion of our analyi i problemaic becaue of unreolved concepual iue concerning he meauremen of uer co for durable inpu. For inance, when a durable inpu i purchaed, he purchae price hould be pread over i ueful lifeime. Co accouning depreciaion allowance aemp o do hi, bu he radiional accouning reamen of depreciaion in an inflaionary environmen i unaifacory here i diagreemen on how hi radiional pracice hould be alered. For inance, wha inere rae hould be ued in deermining he value of financial capial ied up in he ownerhip of durable good? Should impued equiy inere co be included oo? There are alo more baic unreolved concepual problem aociaed wih he meauremen of capial inpu. For eample, hould he quaniy of he capial ervice provided by a machine in each accouning period be reaed a conan (ha i, hould i be meaured a an average per uni ime period) over he lifeime of he machine, or hould he quaniy be reduced each period by a deerioraion facor o reflec he decline in efficiency of he machine? The fir view lead o a gro capial ervice concep he econd o a ne capial ervice concep. Thee wo view can lead o ignificanly differen meaure of capial ervice inpu, hence, o ignificanly differen meaure of produciviy. 44 The fourh limiaion i problemaic becaue during inflaionary ime period ubanial difference can develop beween e ane e po price. Many capial inpu canno be adjued inananeouly (i.e., hey canno be bough or old inananeouly); herefore, a co minimizing producer form a priori epecaion abou he purchae dipoal price a well a fuure inere rae, depreciaion rae, a rae in order o calculae he e ane uer co of capial inpu. However, a reearcher, we can only oberve e po price, inere rae, depreciaion rae, a rae; hu we can only calculae e po uer co. If he epecaion abou fuure price rae are no realized, hen he e ane uer co he price which hould appear in our co funcion in he eac inde number formula may differ ignificanly from he e po uer co.