Decompositions of Productivity Growth into Sectoral Effects

Transcription

1 Decomposiions of Produciviy Growh ino Secoral Effecs W. Erwin Diewer (Universiy of Briish Columbia, Canada, and UNSW, Ausralia) Paper Prepared for he IARIW-UNSW Conference on Produciviy: Measuremen, Drivers and Trends Sydney, Ausralia, November 26-27, 213 Session 7B: ICT and Produciviy Time: Wednesday, November 27, 3:3-4:15

2 1 Decomposiions of Produciviy Growh ino Secoral Effecs W. Erwin Diewer, 1 Revised November 23, 213 Discussion Paper 13-5, School of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1Z1. erwin.diewer@ubc.ca Absrac The paper provides some new decomposiions of labour produciviy growh and Toal Facor Produciviy (TFP) growh ino secoral effecs. These new decomposiions draw on he earlier work of Tang and Wang (24). The economy wide labour produciviy growh rae urns ou o depend on he secoral produciviy growh raes, real oupu price changes and changes in secoral labour inpu shares. The economy wide TFP growh decomposiion ino explanaory facors is similar bu some exra erms due o real inpu price change make heir appearance in he decomposiion. Journal of Economic Lieraure Classificaion Numbers C43, C82, D24. Key Words Toal Facor Produciviy, labour produciviy, index numbers, secoral conribuions o growh. 1 W. Erwin Diewer: School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, Universiy of New Souh Wales, Sydney, Ausralia ( erwin.diewer@ubc.ca). The auhor hanks Ber Balk, Derek Burnell, Ricardo de Avillez, Alice Nakamura, Hiu Wei, Marshall Reinsdorf and wo referees for helpful commens and graefully acknowledges he financial suppor of he SSHRC of Canada.

3 2 1. Inroducion Denison (1962) poined ou ha measures of economy wide produciviy change canno be obained as a simple weighed sum of he corresponding indusry measures; he showed ha changes in he allocaion of resources across he indusries also played an imporan role in conribuing o aggregae produciviy change. However, he Denison decomposiion of aggregae labour produciviy change ino explanaory effecs ignored he role of changes in indusry oupu prices. In an imporan paper, Tang and Wang (24; 426) exended he Denison decomposiion o ake ino accoun changes in real oupu prices in heir decomposiion of economy wide labour produciviy ino explanaory conribuion effecs. However, Tang and Wang combined he effecs of changes in real oupu prices wih he effecs of changes in inpu shares and so in secion 2, we rework heir mehodology in order o provide a decomposiion of aggregae labour produciviy growh ino separae conribuion erms due o secoral produciviy growh, changes in inpu shares and changes in real oupu prices. In secion 3, we presen some alernaive inerpreaions of he basic decomposiion. In secion 4, we generalize he resuls in secions 2 and 3 in order o provide a decomposiion of economy wide Toal Facor Produciviy (or Mulifacor Produciviy) growh ino indusry explanaory facors. Secion 5 concludes. This exension o measuring aggregae MFP growh adds a fourh se of explanaory erms: he effecs of changes in real inpu prices. An Appendix illusraes he decomposiions using Ausralian daa for 16 indusries over he period The Tang and Wang Mehodology Reworked Le here be N secors or indusries in he economy. 2 Suppose ha for period =,1, he oupu (or real value added or volume) of secor n is Y n wih corresponding period price P n 3 and labour inpu L n for n = 1,...,N. We assume ha hese labour inpus can be added across secors and ha he economy wide labour inpu in period is L defined as follows: (1) L n=1 N L n ; =,1. Indusry n labour produciviy in period, X n, is defined as indusry n real oupu divided by indusry n labour inpu: (2) X n Y n /L n ; =,1 ; n = 1,...,N. 2 The maerial in his secion follows Diewer (24). 3 These indusry real oupu aggregaes Y n and he corresponding prices P n are indexes of he underlying micro ne oupus produced by indusry n. The exac funcional form for hese indexes does no maer for our analysis bu we assume he indexes saisfy he propery ha for each and n, P n Y n equals he indusry n nominal value added for period.

4 3 I is no enirely clear how aggregae labour produciviy should be defined since he oupus produced by he various indusries are measured in heerogeneous unis, which are in general, no comparable. Thus we need o weigh hese heerogeneous oupus by heir prices, sum he resuling period values and hen divide by an appropriae oupu price index, say P for period, in order o make he economy wide nominal value of aggregae oupu comparable in real erms across periods. 4 Thus wih an appropriae choice for he aggregae oupu price index P, he period economy wide labour produciviy, X, is defined as follows: 5 (3) X n=1 N P n Y n /P L = n=1 N (P n /P )Y n /L = n=1 N p n Y n /L ; =,1 where he period indusry n real oupu price, p n, is defined as he indusry oupu price P n, divided by he aggregae oupu price index for period, P ; i.e., we have he following definiions: 6 (4) p n P n /P ; n = 1,...,N ; =,1. Using definiions (2) and (3), i is possible o relae he period aggregae produciviy level X o he indusry produciviy levels X n as follows: 7 (5) X n=1 N P n Y n /P L = n=1 N p n [Y n /L n ][L n /L ] = n=1 N p n s Ln X n using definiions (2) where he share of labour used by indusry n in period, s Ln, is defined in he obvious way as follows: 4 The main problem is how exacly should he aggregae period oupu level, Y, be defined? In pracice, a sequence of value added oupu levels Y (and he corresponding value added oupu price indexes P ) will be defined by aggregaing individual indusry oupus (and inermediae inpus enered wih negaive signs) ino economy wide oupu levels, using bilaeral Laspeyres, Paasche, Fisher (1922) or oher indexes as building blocks. In he Appendix, we use chained Fisher oupu indexes over he marke indusry volume levels Y n wih he corresponding Ausralian Bureau of Saisics (ABS) official indusry price indexes P n used as price weighs for he Y n. Thus for each year, our Fisher index produc P Y will equal he ABS sum of indusry price imes volumes for year, n=1 N P n Y n, which in urn is equal o marke secor nominal value added for year. Thus our Y = n=1 N P n Y n /P and so equaion (3) can be rewrien as X = Y /L. We noe ha he ABS uses chained Laspeyres volume indexes o aggregae value added over indusries o form aggregae marke secor real value added bu he resuling differences in he Y are very small. 5 This follows he mehodological approach aken by Tang and Wang (24; 425). As noed in he previous foonoe, he aggregae oupu price index P can be formed by applying an index number formula o he indusry oupu prices (or value added deflaors) for period, (P 1,...,P N ), and he corresponding real oupu quaniies (or indusry real value added esimaes) for period, (Y 1,...,Y N ). The applicaion of chained superlaive indexes would be appropriae in his conex bu again, he exac form of index does no maer for our analysis as long as P Y equals period aggregae nominal value added. 6 These definiions follow hose of Tang and Wang (24; 425). 7 Equaion (5) corresponds o equaion (2) in Tang and Wang (24; 426).

5 4 (6) s Ln L n /L ; n = 1,...,N ; =,1. Thus aggregae labour produciviy for he economy in period is a weighed sum of he secoral labour produciviies where he weigh for indusry n is p n, he real oupu price for indusry n in period, imes s Ln, he share of labour used by indusry n in period. Up o his poin, our analysis follows ha of Tang and Wang (24; ) bu now our analysis will diverge from heirs. 8 Firs, we define he value added or oupu share of indusry n in oal value added for period, s Yn, as follows: (7) s Yn P n Y n / N i=1 P i Y i = p n Y n / N i=1 p i Y i =,1 ; n = 1,...,N using definiions (4). Noe ha he produc of he secor n real oupu price imes is labour share in period, p n s Ln, wih he secor n labour produciviy in period, X n, equals he following expression: (8) p n s Ln X n = p n [L n /L ][Y n /L n ] ; =,1 ; n = 1,...,N = p n Y n /L. Now we are ready o develop an expression for he rae of growh of economy wide labour produciviy. Using definiion (3) and equaion (5), aggregae labour produciviy growh (plus 1) going from period o 1, X 1 /X, is equal o: (9) X 1 /X = N n=1 p 1 n s 1 Ln X 1 n / N n=1 p n s Ln X n = N n=1 [p 1 n /p n ][s 1 Ln /s Ln ][X 1 n /X n ][p n Y n /L ]/ N i=1 [p i Y i /L ] using (8) = N n=1 [p 1 n /p n ][s 1 Ln /s Ln ][X 1 n /X n ] s Yn using definiions (7). Thus overall economy wide labour produciviy growh, X 1 /X, is an oupu share weighed average of hree growh facors associaed wih indusry n. The hree growh facors are: X 1 n /X n, (one plus) he rae of growh in he labour produciviy of indusry n; s 1 Ln /s Ln, (one plus) he rae of growh in he share of labour being uilized by indusry n and p 1 n /p n = [P 1 n /P n ]/[P 1 /P ] which is (one plus) he rae of growh in he real oupu price of indusry n. 8 Tang and Wang (24; ) combined he effecs of he real price for indusry n for period, p n, wih he indusry n labour share s Ln for period by defining he relaive size of indusry n in period, s n, as he produc of p n and s Ln ; i.e., hey defined he indusry n weigh in period as s n p n s Ln. They hen rewroe equaion (5), X = N n=1 p n s Ln X n, as X = N n=1 s n X n. Thus heir analysis of he effecs of he changes in he weighs s n did no isolae he separae effecs of changes in indusry real oupu prices and indusry labour inpu shares.

6 5 Thus in looking a he conribuion of indusry n o overall (one plus) labour produciviy growh, we sar wih a sraighforward share weighed conribuion facor, s Yn [X n 1 /X n ], which is he period oupu or value added share of indusry n in period, s Yn, imes he indusry n rae of labour produciviy growh (plus one), X n 1 /X n. This sraighforward conribuion facor will be augmened if real oupu price growh is posiive (if p n 1 /p n is greaer han one) and if he share of labour used by indusry n is growing (if s Ln 1 /s Ln is greaer han one). The decomposiion of overall labour produciviy growh given by he las line of (9) seems o be inuiively reasonable and fairly simple as opposed o he decomposiion obained by Tang and Wang (24; 426) which does no separaely disinguish he effecs of real oupu price change from changes in he indusry s labour share. 3. Alernaive Expressions and Discussion The lieraure on aggregae labour produciviy decomposiions 9 has focused on decomposiions ha decompose aggregae labour produciviy growh ino explanaory facors ha are funcions of growh raes (percenage changes in variables) raher han growh facors (one plus he growh raes). Thus in his secion, we will rewrie (9) in growh rae form as opposed o is presen conribuion facor form. Define he aggregae labour produciviy growh rae, he secoral labour produciviy growh raes n, he secoral real oupu price growh raes n and he secoral labour inpu share growh raes n beween periods and 1 as follows: (1) (X 1 /X ) 1 ; (11) n (X 1 n /X n ) 1 ; n = 1,...,N; (12) n (p 1 n /p n ) 1 ; n = 1,...,N; (13) n (s 1 Ln /s Ln ) 1 ; n = 1,...,N. Now subsiue definiions (1)-(13) ino (9) and we obain he following decomposiion for he aggregae labour produciviy growh rae : 1 (14) = n=1 N s Yn {[1+ n ][1+ n ][1+ n ] 1} = n=1 N s Yn { n + n + n + n n + n n + n n + n n n } = n=1 N s Yn n + n=1 N s Yn n + n=1 N s Yn n + n=1 N s Yn n n + n=1 N s Yn n n + n=1 N s Yn n n + n=1 N s Yn n n n. The above exac expressions for aggregae labour produciviy growh ell us ha is a quadraic funcion in he indusry growh raes for labour produciviy n, real oupu price growh raes n and indusry labour inpu share growh raes n. The oal conribuion o he overall growh rae from indusry n is he nh erm in he firs equaion in (14), s Yn {[1+ n ][1+ n ][1+ n ] 1}. This expression is relaively easy o 9 See Tang and Wang (24) and de Avillez (212) for references o his lieraure. 1 We use he fac ha he indusry period value added oupu shares s Yn sum o one.

7 6 inerpre. If indusry n s real oupu price and labour inpu share remained consan, hen he growh raes n and n would be and he conribuion of indusry n o economy wide labour produciviy growh would be is oupu share in period, s Yn, imes [1+ n ][1+][1+] 1 which is equal o n. Thus under hese condiions, he conribuion of indusry n o economy wide labour produciviy growh is equal o he indusry n labour produciviy growh rae n imes is period value added share in economy wide value added, s Yn. This is an enirely sensible resul. In he case where n and n are posiive, one plus he indusry n labour produciviy growh, 1+ n, is augmened by he indusry n real oupu price growh facor 1+ n and furher augmened by he indusry n labour inpu share growh facor 1+ n and hen 1 is subraced from he produc of hese facors o give us he indusry n augmened labour produciviy growh facor, [1+ n ][1+ n ][1+ n ] 1. Augmening 1+ n by 1+ n is reasonable since he increased labour share for indusry n in period 1 will indicae a relaive increase in labour inpu ino he indusry and increase he imporance of indusry n in overall labour produciviy. Augmening 1+ n by 1+ n is more difficul o explain bu he increase in he real price of indusry n s oupu will increase he imporance of he oupu of indusry n in he economy wide aggregae oupu index and evidenly, muliplying 1+ n by 1+ n will reflec his increased imporance. The las equaion in (14) ells us ha is equal o an oupu share weighed average of he indusry produciviy growh raes, n=1 N s Yn n, plus a share weighed average of he indusry real oupu price growh raes, n=1 N s Yn n, plus a share weighed average of he indusry labour inpu share growh raes, n=1 N s Yn n, plus he quadraic erms in he indusry growh raes, n=1 N s Yn n n + n=1 N s Yn n n + n=1 N s Yn n n, plus he cubic erms in he indusry growh raes, n=1 N s Yn n n n. Since growh raes are generally small, he firs hree ses of erms on he righ hand side of (15) will generally be he dominan ones. The las four ses of erms represen second and hird order ineracion erms. I is possible o give inerpreaions for he firs hree ses of erms on he righ hand side of he las equaion in (14). The firs se of erms, n=1 N s Yn n, can be inerpreed as economy wide labour produciviy growh provided ha all real oupu price growh raes n are equal o zero and all labour inpu share growh raes n are equal o zero. This se of erms is jus he sraighforward aggregaion of indusry produciviy growh raes and could be called he direc effec. The second se of erms, n=1 N s Yn n, can be inerpreed as economy wide labour produciviy growh provided ha all indusry labour produciviy growh raes n are equal o zero and all labour inpu share growh raes n are equal o zero. Thus even if all indusry labour produciviy levels remain consan and all labour inpu shares remain consan, economy wide labour produciviy growh can change due o changes in indusry real oupu prices. As menioned above, his effec is due o he changes in oupu prices leading o changes in he price weighs for he indusry oupu growh raes, which in urn affecs aggregae labour produciviy growh. This effec could be called he oupu price weighing effec. The hird se of erms, n=1 N s Yn n, can be inerpreed as economy wide labour produciviy growh provided ha all indusry labour produciviy growh raes n are equal o zero and all real oupu growh

8 7 raes n are equal o zero. Thus even if all indusry labour produciviy levels remain consan and all indusry real oupu prices remain consan, economy wide labour produciviy growh can change due o changes in indusry labour inpu shares. This effec could be called he labour inpu reallocaion effec. 11 Noe ha i is no possible o see his reallocaion effec in he indusry n conribuion erm, s Yn n. This erm correcly gives he conribuion o economy wide labour produciviy growh of an increase in indusry n s labour share (so ha n is greaer han in his case) bu he overall effec of his increase in n s labour share is offse by a decrease in oher indusry labour shares and i is he ne effec of he change in n s labour share ha gives rise o he reallocaion effec. However, if N is greaer han wo, i is no possible o deermine precisely how he increase in labour share for indusry n is offse by decreases in shares for he oher indusries. Neverheless, i is possible o deermine he overall labour inpu reallocaion effec. Similarly, alhough he overall oupu price weighing effec can be deermined as he weighed sum N n=1 s Yn n of he indusry oupu price changes n, one canno inerpre he indusry n conribuion erm s Yn n as he independen effec of a change in indusry n s real oupu price because an increase in indusry n s nominal price P 1 n will affec he economy wide price index P and hus he indusry real prices p n = P n /P canno vary independenly, jus as he indusry labour inpu shares s Ln canno vary independenly. 12 The general decomposiion formula (14) can be specialized o give Denison s (1962) decomposiion formula. Suppose ha he real oupu price growh raes n are all equal o. Then (14) reduces o he following decomposiion of aggregae labour produciviy growh: (15) = n=1 N s Yn {[1+ n ][1+ n ] 1} = n=1 N s Yn { n + n + n n } = n=1 N s Yn n + n=1 N s Yn n + n=1 N s Yn n n. De Avillez (212) calls he decomposiion given by (15) he radiional labour produciviy decomposiion. However, i only provides an exac under he resricive assumpion ha all n =. 13 The reader may have noiced ha here seems o be a sligh asymmery in he labour produciviy growh formula (9) in ha he oupu value added shares for he base period, s Yn, are used as weighs for he symmeric growh facors p n 1 /p n, s Ln 1 /s Ln and X n 1 /X n. However, i is possible o develop a decomposiion for he reciprocal of aggregae produciviy growh where he period 1 value added shares, s Yn 1, are used as weighs in his alernaive decomposiion. Thus using he same noaion and seps ha were used o esablish (9), we can esablish he following decomposiion: This ype of effec was firs noiced by Denison (1962). I is called he reallocaion level effec by de Avillez (212). 12 On he oher hand, he indusry produciviy growh raes n can vary independenly. 13 De Avillez (212) called he firs erm in he las line of (15) he wihin effec, he second erm he reallocaion level effec and he hird erm he reallocaion growh effec. 14 This derivaion of (16) is due o Balk (28) who noiced he asymmery in (9).

9 8 (16) X /X 1 = n=1 N p n s Ln X n / n=1 N p n 1 s Ln 1 X n 1 = n=1 N [p n /p n 1 ][s Ln /s Ln 1 ][X n /X n 1 ][p n 1 Y n 1 /L 1 ]/ i=1 N [p i 1 Y i 1 /L 1 ] using (8) = n=1 N [p n /p n 1 ][s Ln /s Ln 1 ][X n /X n 1 ] s Yn 1. Now ake reciprocals of boh sides of (16) and using definiions (12)-(15), we obain he following alernaive decomposiion for aggregae produciviy growh : (17) = [ n=1 N s Yn 1 {(1+ n )(1+ n )(1+ n )} 1 ] 1 1 Boh (14) and (17) provide exac decomposiions of aggregae labour produciviy growh using he indusry growh raes n, n and n and he indusry value added shares s Yn as explanaory variables. Bu i can be seen ha he decomposiion given by (14) is much easier o inerpre so we will no discuss (17) furher. We reurn o he produciviy decomposiion given by he las line of (14). There are 7 ses of explanaory erms on he righ hand side of his las equaion. Each of hese ses of erms could be repored for each indusry as conribuions o overall aggregae labour produciviy growh bu pracical economic analyss will find i a bi burdensome o consider so many explanaory facors. Moreover, in mos siuaions, only he firs order erms, n=1 N s Yn n + n=1 N s Yn n + n=1 N s Yn n, will be imporan numerically wih he second and hird order erms being very small in magniude. Thus we propose o consolidae hese seven ses of erms ino he following hree ses of erms. For n = 1,...,N, define he indusry n conribuions (o overall labour produciviy growh) due o he change in indusry n labour produciviy, X n ; due o he change in indusry n real oupu prices, p n ; and due o changes in indusry n labour inpu shares, s Ln, as follows, for n = 1,...,N: (18) X n s Yn n {1 + (1/2) n + (1/2) n + (1/3) n n }; (19) p n s Yn n {1 + (1/2) n + (1/2) n + (1/3) n n }; (2) s Ln s Yn n {1 + (1/2) n + (1/2) n + (1/3) n n }. I can be shown ha he above conribuions sum up o he aggregae labour produciviy growh rae defined by (14); i.e., we have (21) = n=1 N X n + n=1 N p n + n=1 N s Ln. Essenially, we have simplified (14) by assigning he second and hird order erms in (14) o corresponding firs order erms in a symmeric, even handed manner A similar allocaion has been applied o he Benne (192) decomposiion of a value difference, say p n 1 q n 1 p n q n, ino he wo erms, (1/2)(p n 1 + p n ) q n and (1/2)(q n 1 + q n ) p n, where q n q n 1 q n and p n p n 1 p n. The quaniy change erm has he decomposiion (1/2)(p n 1 + p n ) q n = p n q n + (1/2) p n q n while he price change erm has he decomposiion (1/2)(q n 1 +q n ) p n = q n p n + (1/2) p n q n. Noe ha he value change can also be wrien as he sum of he following 3 erms: p n q n + q n p n + p n q n. Thus ½ of he second order ineracion erm p n q n is assigned o overall quaniy change and he oher ½ is assigned

10 9 In he following secion, we will show how he analysis presened in secions 2 and 3 can be generalized o provide a decomposiion of economy wide Toal Facor Produciviy growh. 4. An Exension o a Decomposiion of Aggregae Mulifacor Produciviy Growh Again, le here be N secors or indusries in he economy and again suppose ha for period =,1, he oupu (or real value added or volume) of secor n is Y n wih corresponding period price P n. However, we now assume ha each secor uses many inpus and index number echniques are used o form indusry inpu aggregaes Z n wih corresponding aggregae indusry inpu prices W n for n = 1,...,N and =,1. 16 Indusry n Toal Facor Produciviy (TFP) in period, X n, is defined as indusry n real oupu Y n divided by indusry n real inpu Z n : (22) X n Y n /Z n ; =,1 ; n = 1,...,N. As in secion 2, economy wide real oupu in period, Y, is defined as oal value added divided by he economy wide oupu price index P. Thus we have 17 (23) Y = n=1 N P n Y n /P = n=1 N p n Y n ; =,1 where he period indusry n real oupu price is defined as p n P n /P for n = 1,...,N and =,1. We define economy wide real inpu in an analogous manner. Thus we form an economy wide inpu price index for period, W, in one of wo ways: 18 By aggregaing over all indusry micro economic inpu prices using microeconomic inpu quaniies as weighs o form W (single sage aggregaion of inpus) or o overall price change in he Benne decomposiion of value change. We are following a similar symmeric assignmen scheme in allocaing higher order ineracion erms o he firs order erms. For more on he properies of he Benne decomposiion, see Harberger (1971), Diewer (25) and Diewer and Mizobuchi (29). 16 These indusry inpu aggregaes Z n and he corresponding price indexes W n are indexes of he underlying micro inpus uilized by indusry n. The exac funcional form for hese indexes does no maer for our analysis bu we assume he indexes saisfy he propery ha for each and n, W n Z n equals he indusry n inpu cos for period. 17 As in secions 2 and 3, P n Y n is nominal indusry n value added in period so ha Y n is deflaed (by he indusry n value added price index P n ) indusry n value added. I need no be he case ha P n Y n is equal o W n Z n ; i.e., i is no necessary ha value added equal inpu cos for each indusry. 18 In eiher case, we assume ha he produc of he oal economy inpu price index for period, W, imes he corresponding aggregae inpu quaniy or volume index, Z, is equal o oal economy nominal inpu cos. If Laspeyres or Paasche price indexes are used hroughou, hen he wo sage and single sage inpu aggregaes will coincide. If superlaive indexes are used hroughou, hen he wo sage and single sage inpu aggregaes will approximae each oher closely using annual daa; see Diewer (1978).

11 1 By aggregaing he indusry aggregae inpu prices W n (wih corresponding inpu quaniies or volumes Z n ) ino he aggregae period inpu price index W using an appropriae index number formula (wo sage aggregaion of inpus). Economy wide real inpu in period, Z, is defined as economy wide inpu cos divided by he economy wide inpu price index W : 19 (24) Z = n=1 N W n Z n /W = n=1 N w n Z n ; =,1 where he period indusry n real inpu price is defined as: (25) w n W n /W ; n = 1,...,N and =,1. The economy wide level of TFP (or MFP) in period, X, is defined as aggregae real oupu divided by aggregae real inpu: (26) X Y /Z ; =,1. We denoe he oupu share of indusry n in period, s Yn, by (7) again and we define he inpu share of indusry n in economy wide cos in period, s Zn, as follows: (27) s Zn W n Z n / N i=1 W i Z i = w n Z n / N i=1 w i Z i =,1 ; n = 1,...,N where he second equaion in (27) follows from he definiions w n W n /W. Subsiue (23) and (24) ino definiion (26) and we obain he following expression for he economy wide level of TFP in period : (28) X = N n=1 p n Y n / N i=1 w i Z i = N n=1 p n (Y n /Z n )Z n / N i=1 w i Z i = N n=1 (p n /w n )X n w n Z n / N i=1 w i Z i = N n=1 (p n /w n )X n s Zn =,1 using (22) using (27). Now we are ready o develop an expression for he rae of growh of economy wide Toal Facor Produciviy. Using (28), aggregae TFP growh (plus 1) going from period o 1, X 1 /X, is equal o: (29) X 1 /X = n=1 N (p n 1 /w n 1 )X n 1 s Zn 1 / i=1 N (p i /w i )X i s Zi = n=1 N (p n 1 /p n )(w n /w n 1 )(X n 1 /X n )(s Zn 1 /s Zn )(p n /w n )X n s Zn / i=1 N (p i /w i )X i s Zi = n=1 N s Yn (p n 1 /p n )(w n /w n 1 )(s Zn 1 /s Zn )(X n 1 /X n ). The las equaion in (29) follows from he following equaions for n = 1,...,N: 19 Noe ha W Z equals period oal economy inpu cos for each.

12 11 (3) (p n /w n )X n s Zn = (p n /w n )(Y n /Z n )(w n Z n / i=1 N w i Z i ) using (22) and (27) = p n Y n / i=1 N w i Z i. Thus one plus economy wide TFP growh, X 1 /X, is equal o an oupu share weighed average (wih he base period weighs s Yn ) of one plus he indusry TFP growh raes, imes an augmenaion facor, which is he produc (p n 1 /p n )(w n /w n 1 )(s Zn 1 /s Zn ). Thus formula (29) is very similar o our previous labour produciviy growh formula (9) excep we have an addiional muliplicaive conribuion facor, which is w n /w n 1, he reciprocal of one plus he rae of growh of real inpu prices for secor n. We can use definiions (11) and (12) in secion 3 o rewrie he decomposiion (29), which is in conribuion facor form, ino a growh rae form. We need o add he following definiions: (31) n (s 1 Zn /s Zn ) 1 ; n = 1,...,N. (32) n (w n /w 1 n ) 1 ; n = 1,...,N. Noe ha 1 + n equals w n /w n 1 so ha n is a reciprocal growh rae of real inpu prices for secor n. Now subsiue (11), (12), (31) and (32) ino (29) and we obain he following decomposiion for economy wide TFP growh (X 1 /X ) 1 ino explanaory indusry conribuion erms: (33) = n=1 N s Yn {[1+ n ][1+ n ][1+ n ][1+ n ] 1}. Thus he conribuion erm for indusry n is he nh erm in he above summaion, which depends only on indusry n TFP growh n, he oupu share s Yn, he real oupu price growh n, he real reciprocal inpu price growh n and indusry n inpu cos share growh n. The indusry n conribuion erm s Yn [1+ n ][1+ n ][1+ n ][1+ n ] 1} can be wrien ou as he sum of 4 firs order erms, 6 second order ineracion erms, 4 hird order ineracion erms and 1 fourh order ineracion erm or 15 separae conribuion effecs in all. As was he case wih he labour produciviy conribuion formula (14), his will be oo many erms for analyss o handle and so we sugges he following counerpar o he labour produciviy decomposiion defined earlier by (21): for n = 1,...,N, define he indusry n conribuions (o overall MFP or TFP growh) due o he changes in indusry n TFP, X n ; due o he changes in indusry n real oupu prices, p n ; due o he changes in indusry n real reciprocal inpu prices, w n ; and due o he changes in indusry n inpu cos shares, s Zn, as follows, for n = 1,...,N: (34) X n s Yn n {1+(½) n +(½) n +(½) n +(⅓) n n +(⅓) n n +(⅓) n n +(¼) n n n }; (35) p n s Yn n {1+(½) n +(½) n +(½) n +(⅓) n n +(⅓) n n +(⅓) n n +(¼) n n n }; (36) w n s Yn n {1+(½) n +(½) n +(½) n +(⅓) n n +(⅓) n n +(⅓) n n +(¼) n n n }; (37) s Zn s Yn n {1+(½) n +(½) n +(½) n +(⅓) n n +(⅓) n n +(⅓) n n +(¼) n n n };

13 12 I can be shown ha he above conribuions sum up exacly o he aggregae MFP growh rae defined by (29); i.e., we have (38) = n=1 N X n + n=1 N p n + n=1 N w n + n=1 N s Zn. Our empirical resuls for Ausralia indicae ha while individual indusry erms for any of he above conribuion erms on he righ hand side of (38) can be quie significan, when we sum he las hree ses of erms in (38), we find ha he sum over indusries is close o zero; i.e., for each ime period, n=1 N p n n=1 N w n n=1 N s Zn. Thus while he real price conribuions and he indusry share conribuions, p n, w n and s Zn can be individually quie subsanial for many indusries n, when we sum hese effecs over all indusries, he overall sum of hese conribuions is close o zero for each ime period. Thus he individual indusry MFP conribuion erms, X n, are generally more imporan in conribuing o aggregae produciviy growh han he price and inpu reallocaion erms. 5. Conclusion If one wishes o decompose aggregae labour produciviy growh ino explanaory facors ha depend on he indusries in he aggregae, hen he decomposiions given by (9) and he firs line in (14) appear o be he simples ones in he lieraure o dae. These decomposiions have he advanage ha only four indusry variables need o be repored in order o explain each indusry conribuion erm: he indusry shares of aggregae value added in he base period s Yn, he indusry labour produciviy growh raes n, he indusry real oupu price growh raes n and he indusry labour inpu share growh raes n beween he wo periods under consideraion. The overall indusry n conribuion erm is s Yn {[1+ n ][1+ n ][1+ n ] 1}. If a decomposiion of aggregae TFP growh is desired, hen he decomposiions given by (29) and (33) are also very simple. In his framework, he overall indusry n conribuion erm is s Yn {[1+ n ][1+ n ][1+ n ][1+ n ] 1} where 1+ n equals w n /w n 1, which in urn is he reciprocal of one plus he real inpu price growh for indusry n. However, if a decomposiion of aggregae TFP growh ino explanaory facors is desired ha depend on a sum of erms involving n, n, n and n separaely, hen we recommend he decomposiion of aggregae labour produciviy growh defined by (18)-(21) and he decomposiion of TFP growh defined by (34)-(38). In our empirical example using Ausralian daa, we found ha he main driver for he TFP decomposiion was he firs se of erms in he decomposiion (38); i.e., in he aggregae, he sum of he effecs involving he n, n and n was close o zero. However, using he Ausralian daa, his unimporance resul did no hold for he labour produciviy decomposiion defined by (21): he effecs on Marke Secor labour produciviy growh of changes in indusry shares of labour inpu proved o be significan. Appendix: Empirical Applicaion o Ausralian Marke Secor Daa;

14 13 We apply he produciviy decomposiions given by (21) (for aggregae labour produciviy growh) and (38) (for TFP or MFP growh) using official indusry daa from he Ausralian Bureau of Saisics (ABS) for he June years From Table 9 of he ABS (212), we can obain indexes of indusry real value for he following 16 Marke Secor indusries: 1. Agriculure, Foresry and Fishing; 2. Mining; 3. Manufacuring; 4. Elecriciy, Gas, Waer and Wase Services; 5. Consrucion; 6. Wholesale Trade; 7. Reail Trade; 8. Accommodaion and Food Services; 9. Transpor, Posal and Warehousing; 1. Informaion, Media and Telecommunicaions; 11. Financial and Insurance Services; 12. Renal, Hiring and Real Esae Services; 13. Professional, Scienific and Technical Services; 14. Adminisraive and Suppor Services; 15. Ars and Recreaion Services and 16. Oher Services. From Tables 9 and 1 of he ABS (212), we can obain qualiy adjused indexes of labour inpu and capial services for he same 16 indusries for he June years From Table 14 of he same publicaion, we obained he inpu cos share of labour and capial by he 16 indusries. Finally, from he ABS (213), esimaes of value added by indusry in curren dollars for he 16 indusries were obained. Dividing hese nominal value added esimaes by indusry by he corresponding indexes of indusry real value added gives us implici price indexes for each indusry oupu. These price indexes were normalized o equal 1 in 1995 and hese normalized indusry price indexes, P n, are lised in Table 1 below. The indusry n year nominal value added was divided by he corresponding normalized price index P n in order o obain an esimae of real value added for indusry n for year, Y n. The Y n are lised in Table 2 below. The unis of measuremen are in billions of consan 1995 dollars. Table 1: Indusry Value Added Oupu Prices P n P 1 P 2 P 3 P 4 P 5 P 6 P 7 P P 9 P 1 P 11 P 12 P 13 P 14 P 15 P 16 2 A June year is an aggregaion of a rolling year of four quarers of daa ending in June of he indicaed year.

15 Table 2: Indusry Value Added Oupu Volumes Y n Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y Y 9 Y 1 Y 11 Y 12 Y 13 Y 14 Y 15 Y 16 We assume ha value added is equal o inpu cos by indusry so ha wih our esimaes of nominal value added by indusry and he ABS cos shares, we can obain esimaes for he value of labour inpu and capial services inpu by indusry. These nominal cos values can be divided by he ABS index esimaes of real labour and capial services inpu o give us implici price indexes for labour and capial services by indusry. These price indexes were normalized o equal 1 in 1995 and are lised in Tables 3 and 5 below as W Ln and W Kn for indusry n in year. Finally, nominal indusry n labour cos in year was divided by W Ln o give he indusry n quaniy of labour inpu in year, Q Ln, and nominal indusry n capial services cos in year was divided by W Kn o give he indusry n quaniy of capial services inpu in year, Q Kn. The Q Ln and Q Kn are lised in Tables 4 and 6 below. Table 3: Indusry Qualiy Adjused Wages W Ln W L1 W L2 W L3 W L4 W L5 W L6 W L7 W L W L9 W L1 W L11 W L12 W L13 W L14 W L15 W L16

16 Table 4: Indusry Qualiy Adjused Labour Inpu Q Ln L 1 L 2 L 3 L 4 L 5 L 6 L 7 L L 9 L 1 L 11 L 12 L 13 L 14 L 15 L 16 Table 5: Indusry Prices of Capial Services W Kn W K1 W K2 W K3 W K4 W K5 W K6 W K7 W K W K9 W K1 W K11 W K12 W K13 W K14 W K15 W K16 Table 6: Indusry Capial Services Volumes Q Kn

17 In order o calculae indusry measures of Mulifacor produciviy, we will need esimaes of aggregae inpu prices and quaniies (or volumes) by indusry. Thus we compued chained Fisher (1922) price and quaniy indexes, W n and Z n for indusry n in year and hese indexes are lised in Tables 7 and 8 respecively. 21 Table 7: Indusry Inpu Price Indexes W n W 1 W 2 W 3 W 4 W 5 W 6 W 7 W W 9 W 1 W 11 W 12 W 13 W 14 W 15 W 16 Table 8: Indusry Inpu Volume (Quaniy) Indexes Z n Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z Z 9 Z 1 Z 11 Z 12 Z 13 Z 14 Z 15 Z 16 The above Tables provide he daa ha are necessary o calculae Indusry and Marke Secor MFP esimaes. However, in order o calculae Indusry and Marke Secor Labour Produciviy esimaes, we will require an addiional Table. From he ABS (212), Table 21 The price and quaniy (volume) series lised in Tables 3-6 were he inpus ino he Fisher index number formula for each indusry.

18 17 9, we can obain indexes of hours worked in he 16 indusries as well as for he Marke Secor. Denoe hese index series for year as H 1 -H 16 and H. We ran an ordinary leas squares regression of H on H 1 -H 16 (wih no consan erm) in order o recover he acual indusry hours worked by indusry (up o a facor of proporionaliy). 22 Denoe he esimaed indusry regression coefficiens by 1-16 and define Marke Secor hours worked in year by L H and he corresponding year indusry n measures of hours worked by L n n H n for n = 1,...,16. The resuling measures of Labour inpu are lised in Table 9 below. Noe ha for each year, L = n=1 16 L n. Table 9: Marke Secor Hours Worked L and Indusry Hours Worked L n L L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 L 9 L 1 L 11 L 12 L 13 L 14 L 15 L Wih he above indusry daa lised, we can now use equaions (2) in he main ex o calculae he labour produciviy level of indusry n in year, X n as Y n /L n, where he Y n are lised in Table 2 and he L n are lised in Table 9 above. We normalize hese indusry labour produciviy levels by dividing each X n by X 1995 n for = 1995,...,212 and n = 1,...,16. Denoe he normalized indusry labour produciviies by X * n X n /X 1995 n. These normalized labour produciviy esimaes are lised in Table 1 below. The nex sep is o consruc a measure of aggregae marke secor real value added, Y for each year. We will use Fisher chained indexes of he indusry oupus Y n (wih price weighs P n ) in order o consruc he Y. The P n are lised in Table 1 and he corresponding Y n are lised in Table 2. The chained Fisher Marke Secor oupu price index ha corresponds o Y is denoed by P. The P and Y are lised in Table 15 below. Noe ha hese indexes saisfy he ideniy Y = 16 n=1 P n Y n /P for each. Wih Y and L defined, he year Marke Secor labour produciviy level is defined as X Y /L for = 1995,...,212. We normalize hese Marke Secor labour produciviy levels by dividing 22 The R 2 for he regression urned ou o be 1. so we are confiden ha we recovered he acual hours worked by indusry from our procedure (up o a facor of proporionaliy).