December 2012 SECTORAL CONTRIBUTIONS TO LABOUR PRODUCTIVITY GROWTH IN CANADA: DOES THE CHOICE OF DECOMPOSITION FORMULA MATTER?

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1 December Slater Street, Sute 710 Ottawa, Ontaro K1P 5H , Fax csls@csls.ca CENTRE FOR THE STUDY OF LIVING STANDARDS SECTORAL CONTRIBUTIONS TO LABOUR PRODUCTIVITY GROWTH IN CANADA: DOES THE CHOICE OF DECOMPOSITION FORMULA MATTER? CSLS Research Report December 2012

2 Sectoral Contrbutons to Labour Productvty Growth n Canada: Does the Choce of Decomposton Formula Matter? Rcardo de Avllez December 28, 2012 Abstract Usng three decomposton formulas (TRAD, CSLS, and GEAD), ths artcle estmates sectoral contrbutons to busness sector labour productvty growth n Canada durng the perod. Although at the aggregate economy level there was substantal agreement among the three formulas wth most of busness sector labour productvty growth beng explaned by wthn-sector productvty mprovements, contrbuton estmates vared wdely at the sectoral level. In partcular, there were sgnfcant dfferences n the estmated contrbutons of constructon, manufacturng, and mnng and ol and gas extracton. Ultmately, these dfferences reflect the fact that tradtonal decomposton formulas (TRAD and CSLS) and the GEAD formula measure dstnct economc phenomena. Instead of seeng estmates constructed by the GEAD and tradtonal formulas as competng narratves, the artcle concludes t s more useful to see them as provdng complementng stores about the role of dfferent sectors n drvng aggregate labour productvty growth. 1 Introducton An mportant part of productvty analyss s the estmaton of sectoral contrbutons to aggregate labour productvty growth. Several decomposton formulas have been developed for ths purpose. Unfortunately, dfferent decomposton formulas frequently yeld sgnfcantly dfferent results. The objectve of ths artcle s to compare estmates produced by three such formulas n the case of labour productvty growth n the Canadan busness sector durng the perod. In the past 20 years, statstcal agences around the world have ncreasngly favoured chaned ndexes n place of fxed-base ndexes when calculatng real output. The advantages of chaned ndexes over ther fxed-base counterparts are well establshed n the economcs lterature (see, for Rcardo de Avllez s an economst at the Centre for the Studes of Lvng Standards. The author would lke to thank Erwn Dewert, Marshall Rensdorf, and Janmn Tang for comments, correctons, and crtcsms. Emal: rcardo.avllez@csls.ca 1

3 nstance, Whelan (2002)), but a few ponts are worth hghlghtng here. Generally speakng, real output estmates n constant prces.e. calculated usng fxed-base ndexes use relatve prces from an arbtrarly chosen base perod as the bass for comparson wth all the other perods; on the other hand, real output estmates n chaned prces.e. calculated usng chaned ndexes take nto account the fact that relatve prces are constantly changng. The growth rate of real output n constant prces depends on the choce of the base year, becomng ncreasngly unbalanced as one moves further and further away from t. Ths does not happen wth real output n chaned prces, snce relatve prces are updated every perod. Thus, n general, chaned ndexes produce better qualty estmates of real output. A dsadvantage of usng chaned ndexes, however, s that real output estmates cease to be addtve. When real output s computed usng fxed-base Laspeyres quantty and Paasche prce ndexes, 1 aggregate real output s exactly equal to the sum of ts ndvdual components. Ths s not true when chaned ndexes are used. In ths case, aggregate real output s equal to the sum of ts ndvdual components only for the chosen reference year (when real output s also equal to nomnal output). The dfference between the two ncreases as one moves away from the reference year. Even though statstcal agences have updated ther methods, productvty analysts stll often use technques that assume real output s addtve. Ths s partcularly true when t comes to decomposton formulas used to fnd sectoral or regonal contrbutons to aggregate labour productvty growth. Labour productvty estmates are constructed usng real output and labour nput estmates. Snce offcal real output estmates are now calculated manly wth chaned ndexes, labour productvty growth decomposton formulas that assume real output n constant prces generate sectoral or regonal contrbutons that do not sum up exactly to aggregate labour productvty growth. Tang and Wang (2004) addressed ths ssue wth ther Generalzed Exactly Addtve Decomposton (GEAD) formula. By takng nto account changes n relatve prces, the GEAD formula s able to generate sectoral contrbuton estmates that are perfectly addtve rrespectve of how real output s calculated. Despte ths clear advantage over other decomposton formulas, the GEAD formula s stll not wdely used by productvty analysts, possbly due to some of ts results beng perceved as counterntutve. Dumagan (2012) compared the GEAD formula to a tradtonal decomposton formula (TRAD) that assumed real output n constant prces. He concluded that the GEAD formula produced superor estmates from both an emprcal and an analytcal pont of vew. The purpose of ths 1 A Laspeyres formula uses base perod prces (or quanttes, n the case of a prce ndex) as weghts, whereas a Paasche formula uses current perod prces (or quanttes, n the case of a prce ndex) as weghts. The Fsher formula s a geometrc average of the Laspeyres and Paasche formulas. For a dscusson on these and other ndex number formulas, as well as on many topcs central to ndex number theory, see Dewert (1993). 2

4 artcle s to expand Dumagan s nvestgaton on how estmates produced by the GEAD formula compare to those produced by tradtonal labour productvty growth decomposton formulas. In order to do so, a thrd decomposton formula s added to the mx and sectoral contrbuton estmates for all three formulas are analysed n the case of labour productvty growth n the Canadan busness sector durng the perod. Ths artcle s organzed as follows. Secton two presents the TRAD decomposton formula, a varaton of that formula developed by the Centre for the Study of Lvng Standards (CSLS), and the GEAD formula. Secton three compares the sectoral contrbutons calculated by the three formulas for the Canadan busness sector durng the perod. Secton four concludes. 2 Decomposng Aggregate Labour Productvty Growth Ths secton descrbes three formulas commonly used to calculate sectoral contrbutons to aggregate labour productvty growth. The frst decomposton formula s the TRAD formula, whch, accordng to Dumagan (2012), can be traced back to Denson (1962). It assumes that real output s measured n constant prces more specfcally, usng fxed-base Laspeyres quantty and Paasche prce ndexes, so that aggregate real output corresponds to the sum of sectoral real output. A second decomposton formula, referred to here as the CSLS decomposton, was developed by the Centre for the Study of Lvng Standards (CSLS) and s used n several of ts artcles and reports, ncludng Sharpe (2008), Sharpe (2010) and Sharpe and Thomson (2010). The CSLS formula s essentally a varaton of the TRAD formula. It also assumes real output n constant prces, but dffers sgnfcantly from the TRAD formula n the way t accounts for the contrbuton of each sector to aggregate productvty growth. Fnally, the last formula was developed by Tang and Wang (2004) (see also Dewert (2008) for an alternatve formulaton) and s perfectly addtve regardless of how real output s measured. Followng Dumagan (2012), the Tang and Wang decomposton s referred to here as the Generalzed Exactly Addtve Decomposton (GEAD) formula. 2 Ths secton s dvded nto four parts. The frst three parts present the TRAD, CSLS, and GEAD formulas, respectvely. The fourth part compares the three formulas, notng how dfferent specfcatons can lead to very dfferent sectoral contrbutons to aggregate labour productvty growth. Ths secton draws from Dumagan (2012), especally regardng the choce of notaton and the dervatons of the TRAD and GEAD formulas. 2 The reader should bear n mnd that there are many other labour productvty decompostons that are not dscussed or used n ths artcle. See, for nstance, Nordhaus (2002), Rensdorf et al. (2002), Dewert (2008), and Rensdorf and Yuskavage (2010). 3

5 2.1 TRAD Decomposton The TRAD decomposton formula s stll wdely used to measure the contrbuton of dfferent sectors to aggregate productvty growth (see, for example, Dekle and Vandenbroucke (2006), IMF (2006), and Usu (2011)). The underlyng assumpton of ths formula s that real output s calculated n constant prces usng fxed-base Laspeyres quantty and Paasche prce ndexes at both the aggregate and sectoral levels. When ths happens, the sum of sectoral real output (net of ntermedate nputs,.e. value added) X t s equal to the economy s real output X t,.e. X t = X t, where the superscrpt = 1, 2,..., N denotes the sector and the subscrpt t = 1, 2,..., T denotes the tme perod. Defnng labour productvty as output per unt of labour nput, aggregate labour productvty Zt = X t and sectoral productvty Zt = X t L t L where L t and L t represent labour t nput used n the aggregate economy and n sector (respectvely) such that L t = L t. In ths case, snce real output s addtve: Zt = X t = X t = Z t L t L t L t L t = Z t l t (1) where lt = L t. Thus, aggregate labour productvty Zt s equal to the weghted sum of sectoral L t labour productvty Zt across all s, where the weghts lt are each sector s labour nput shares. Sectoral contrbutons to aggregate labour productvty growth can be computed by lookng at productvty changes between two perods of tme: G t = Z t Z t 1 Z t 1 = (Z t l t Z t 1 l t 1 ) Z t 1 Addng and subtractng ltz t 1 to the numerator of equaton (2) and collectng terms: (2) Defnng G t = Z t Z t 1 Zt 1 G t = [l t(z t and subtracted to equaton (3) so that: and notng that Z t Z t 1 ) Z t 1 (l t l t 1 )] Z t 1 Z t lt = Z t Zt L t = X t L t X t, Z t 1 Zt 1 l t 1G t (3) can be added Zt 1 G t = Zt 1 = [ X t 1 Xt 1 [l t 1G t G t + (l t l t 1) + (l t l t 1)G t ] (4) + Z t 1 Zt 1 t 1 Zt 1 l t + Z ] ltg t Equaton (5) s the TRAD decomposton formula. Accordng to ths formula, sectoral contrbutons to aggregate productvty growth can be broken down nto three effects. The frst term of (5) 4

6 equaton (5) represents the wthn-sector effect (WSE). 3 As the name mples, t measures the contrbuton to aggregate productvty growth due solely to productvty ncreases experenced by ndvdual sectors. If sectoral labour shares reman unchanged over tme ( l t = 0), the second and thrd terms of equaton (5) equal zero and the contrbuton of each sector collapses to the frst term, whch s the sectoral labour productvty growth weghted by the sector s real share n aggregate real output ( X t X t ). The other two terms of equaton (5) represent two dfferent sectoral reallocaton effects. The second term of equaton (5) captures the reallocaton level effect (RLE). 4 As Denson (1962) realzed, aggregate labour productvty can ncrease even when sectoral labour productvty remans constant, as long as labour moves from sectors wth below average labour productvty levels towards sectors wth above average labour productvty levels. In the TRAD decomposton, ths effect s postve when l t > 0. The rato between the sector s labour productvty level and the aggregate labour productvty level scales the magntude of the effect, ether ncreasng t (when Z t Zt decreasng t (when Z t Z t < 1). > 1) or The thrd term s the reallocaton growth effect (RGE). 5 It captures a phenomenon smlar to Baumol s cost dsease (see Baumol (1967) and Baumol et al. (1985)) that s, the propensty of labour to move towards sectors where labour productvty s stagnant or declnng (G t 0). In the TRAD decomposton, ths effect wll be postve ether when labour has moved towards a sector wth postve labour productvty growth ( lt > 0 and G t > 0) or when labour has moved away from a sector wth negatve labour productvty growth ( lt < 0 and G t < 0). The magntude of the reallocaton growth effect depends not only on the lt and G t but also on the rato between the sector s labour productvty level and the aggregate labour productvty level ( Z t Zt ). 2.2 CSLS Decomposton Lke the TRAD formula, the CSLS decomposton formula also assumes real output n constant prces calculated usng fxed-base Laspeyres quantty and Paasche prce ndexes, so that sectoral real output sums up to aggregate real output. Startng from equaton (1), the absolute change n labour productvty between two perods of tme s: 3 Tang and Wang (2004) call ths effect the pure productvty growth effect, whle Dumagan (2012) labels t the wthn-sector productvty growth effect 4 Tang and Wang (2004) name ths effect the relatve sze change effect, notng that t was an analog of the Denson effect n Nordhaus (2002) (p.427) Dumagan (2012), n turn, calls ths effect the statc structural reallocaton effect. 5 Tang and Wang (2004) smply label ths effect as the nteracton term, whle recognzng that t s smlar to the Baumol effect n Nordhaus (2002). Dumagan (2012) refers to ths effect as the dynamc structural reallocaton effect. 5

7 Z t = Z t Z t 1 = (Z t l t Z t 1l t 1) (6) Addng and subtractng Zt 1 l t 1, Z t 1 l t, and Zt lt 1 to equaton (6) and collectng terms: Z t = [(Z t Zt 1)l t 1 + Zt 1(l t lt 1) + (Zt Zt 1)(l t lt 1)] (7) = Z t lt 1 + Z t 1 lt + Z t lt (8) Subtractng Z t 1 l t and Z t l t from equaton (8), t becomes becomes: 6 Z t = Z t l t 1 + (Z t 1 Z t 1) l t + ( Z t Z t ) l t (9) Equaton (9) s the CSLS decomposton formula. Note that, whle the TRAD decomposton stated the contrbuton of ndvdual sectors n terms of percentage ponts (relatve to the per cent growth n aggregate labour productvty), the CSLS formula looks at absolute changes,.e. ncreases n constant dollars per unt of labour. The CSLS formula can be easly modfed so that sectoral contrbutons are stated n percentage ponts; one needs only dvde both sdes of equaton (9) by Z t 1 : G t = Z t Z t 1 Z t 1 = Z t lt 1 Zt 1 + (Z t 1 Z t 1 ) l t Zt 1 + ( Z t Zt ) lt Zt 1 Analogous to the TRAD decomposton, the frst term of the CSLS decomposton formula accounts for the WSE, whle the two other terms represent the RLE and the RGE, respectvely. Although the CSLS formula appears to specfy the WSE dfferently wth sectoral labour shares (lt) nstead of real output shares ( X t WSE. Recallng that G t Z t l t 1 Z t 1 X t Z = Z t t 1 Zt 1 = Z t 1 l t 1 Z t 1 whch corresponds to the TRAD s WSE. (10) ) used as weghts, the two formulas actually have the same, the frst term of equaton (10) can be rewrtten as Z t Z t 1 = Xt 1 L t 1 L L t 1 t 1 Xt 1 G t = ( X L t 1 t 1 Xt 1 ) G t (11) The crucal dfference between the two formulas les n how the RLE and RGE are specfed and nterpreted. Much lke n the TRAD decomposton, the CSLS s RLE captures aggregate labour productvty changes caused by labour nput shfts to sectors wth above- or below-average labour productvty levels. In the CSLS decomposton, ths effect s postve ether when a sector wth above-average labour productvty level (Z t > Z t ) experences an ncrease n ts labour nput share 6 As Sharpe (2010) notes, because l t = 0, the terms Z t 1 l t and Z t l t both sum to zero when aggregated across all sectors. 6

8 (lt > lt 1 ) or when a sector wth below-average labour productvty level (Z t < Zt ) experences a reducton n ts labour nput share (lt < lt 1 ). In both the TRAD and CSLS formulas, the magntude of the RLE s a functon of l t. In the case of the TRAD formula, however, the effect depends on the rato between the sectoral labour productvty level and the aggregate labour productvty level, whle n the CSLS formula t depends on the absolute dfference between the two. The thrd term of the CSLS decomposton formula represents the RGE, whch measures the mpact of shfts of labour nput to sectors wth above- or below-average labour productvty growth. Ths effect s postve ether when l t > 0 and Z t Z t > 0 or when l t < 0 and Z t Z t < 0. Both the TRAD s and the CSLS s RGE are a functon of l t. The dfference between the two s reflected, once agan, by the fact that the TRAD s RGE depends on the rato between a sector s labour productvty level and the aggregate labour productvty level, whereas the CSLS s RGE s a functon of the dfference between sectoral labour productvty growth and aggregate labour productvty growth. The dfferences between the TRAD s and the CSLS s reallocaton effects are explored n more detal n secton GEAD The TRAD and the CSLS decompostons share a common assumpton: that output s measured n constant prces usng fxed-base Laspeyres quantty and Paasche prce ndexes. Ths guarantees that real aggregate output s equal to the sum of real sectoral output, whch leads to equaton (1). The Generalzed Exactly Addtve Decomposton (GEAD) formula has a dfferent startng pont. Notng that: 1. Aggregate nomnal output (Y t ) s always addtve across sectors (Y t = Y t, where Yt represents output of sector ); 2. By defnton, Y t = P t X t and Yt = Pt Xt, where X t and Xt are aggregate and sectoral real output (respectvely) and P t and Pt are ther correspondng prce ndexes; then Y t = Y t = P t X t. Defnng relatve prces (p t) as the rato between prces n sector and economy-wde prces so that p t = P t P t, t can be noted that the relatonshp between aggregate labour productvty (Z t = X t ) and sectoral labour productvty (Zt = X t ) s: L t Z t = X t L t = Y t P t = Y t = L t P t L t P t X t P t L t = P t P t L t L t X t L t L t = p tl tz t (12) Whle equaton (1) holds only when output s calculated usng fxed-base Laspeyres quantty and Paasche prce ndexes, equaton (13) s true regardless of the ndex number formula used to 7

9 calculate real output, because Y t = P t X t and Y t = P t X t always hold. In partcular equaton (13) s true when chaned ndex number formulas are used to measure real output, n whch case the addtvty of real output does not hold (X t X t). The dervaton of the GEAD formula follows smlar steps to that of the TRAD formula: G t = Z t Z t 1 = (p tltz t p t 1 l t 1 Z t 1 ) (13) Z t 1 Z t 1 Addng and subtractng p tltz t 1 to the numerator of equaton (13): G t = [p tl t(z t Z t 1 ) Z t 1 (p tl t p t 1 l t 1 )] Z t 1 (14) Defnng G t = Z t Zt 1 and notng that Z t p Z t 1 Z tlt = Z t Pt L t t Z t P t L t added and subtracted to equaton (14) so that = Y t, Y t Z t 1 Z t 1 p t 1l t 1G t can be Z t 1 G t = [p Z t 1lt 1G t + (p tlt p t 1lt 1) + (p tlt p t 1lt 1)G t] (15) t 1 = [ Y ] t 1 G t + Z t 1 (p Y t 1 Z tlt p t 1lt 1) + Z t 1 (p t 1 Z tlt p t 1lt 1)G t (16) t 1 Equaton (16) s the GEAD formula. The smlartes as well as the dfferences between equaton (16) and equaton (5) are mmedately notceable. Much lke the TRAD and CSLS decompostons, the GEAD breaks down sectoral contrbuton nto three effects: WSE, RLE, and RGE. The frst term of the GEAD represents the WSE. Notce that, whle n the TRAD formula the WSE was defned as sectoral labour productvty growth weghted by the sector s real output share ( X t ), n the GEAD sectoral growth s weghted by the sector s nomnal output share ( Y t ). X t Y t Regardng the reallocaton effects, the GEAD s RLE and RGE do not depend only on how sectoral labour nput shares changed over tme (as was the case n the TRAD decomposton), but also on relatve prce movements. Thus, a declne n a sector s labour nput share can be offset by an ncrease n the sector s relatve prces. In the GEAD formula, a sector wll have a postve RLE when p tl t > p t 1 l t 1. The RGE wll be postve ether when p tl t > p t 1 l t 1 and G t > 0 or when p tl t < p t 1 l t 1 and G t < 0. If there are no movements n relatve prces (p t = 1.0, for all t), then the GEAD formula becomes equal to the TRAD formula. 7 7 Dewert (2008) notes the dffculty n nterpretng the GEAD s RLE and RGE as actual reallocaton effects, snce the effects of changes n labour shares are mxed wth the effects of changes n relatve prces. Thus, he proposes an alternatve formulaton of the GEAD formula, where the two reallocaton effects are replaced by a labour nput effect and a prce effect. Although Dewert s formulaton of the GEAD breaks down sectoral contrbutons nto terms that have a more straghtforward nterpretaton, the overall sectoral contrbutons estmated by hs formula wll be exactly the same as those estmated by the orgnal GEAD formula. 8

10 2.4 Comparson between Dfferent Decomposton Formulas Table 1 compares the dfferent specfcatons of the WSE, RLE, and RGE found n the TRAD, CSLS, and GEAD decomposton formulas. As mentoned prevously, despte the apparent dssmlarty between the TRAD and CSLS wthn-sector effect, the two effects are actually the same. The dfferences between the TRAD and CSLS decompostons arse n ther treatment of the RLE and RGE. These dfferences can be better understood wth a smple example. Table 1: Comparson between the TRAD, CSLS, and GEAD Decomposton Formulas TRAD CSLS GEAD Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect Xt 1 Xt 1 G t Z t l t 1 Z t 1 Y t 1 Y t 1 G t Zt 1 Zt 1 lt (Z t 1 Z t 1) l t Z t 1 Z t 1 Z t 1 (p tl t p t 1l t 1) Zt 1 Zt 1 ( Z t l tg t Z t ) l t Z t 1 Z t 1 Z t 1 (p tl t p t 1l t 1)G t Table 2: Two-Sector Economy Example: Nomnal Output, Real Output n Constant Prces, Prces, Labour Input, and Labour Productvty Perod 0 Y 0 Y0 Y 0 X 0 X0 X0 P 0 p 0 L 0 l 0 Z 0 Total Economy Sector A Sector B Perod 1 Total Economy Sector A Sector B Y1 Y1 X X1 1 Y 1 X1 P1 p 1 L 1 l1 Z [+77%].. [+15%].. [+54%].. [+0%].. [+15%] [+82%].. [+0%].. [+82%].. [-33%].. [+50%] [+60%].. [+60%].. [+0%].. [+50%].. [+7%] Notes: 1) All varables as defned prevously; 2) Numbers n square brackets ndcate per cent growth experenced between perod 0 and perod 1. Assume a two sector economy, where sector A s a hgh productvty sector responsble for three-fourths of the economy s real output (net of ntermedate goods) n perod 0 and sector B s a low productvty sector that accounts for the remander of that economy s producton. In perod 9

11 1, sector A experences substantal productvty gans, promptng frms to reduce the total amount of labour nput used n the sector whle mantanng the same level of producton. Sector B also experences an ncrease n labour productvty and, at the same tme, soaks up the excess labour from sector A. Table 2 summarzes these developments. Note that t s ntally assumed that real output s calculated usng fxed-base Laspeyres quantty and Paasche prce ndexes so that t s addtve,.e. the economy s real output s the sum of real output n sectors A and B. Snce labour productvty n both sectors ncreased, t can be expected that both sectors wll have postve WSE. On the other hand, the fact that labour moved from a sector that had aboveaverage labour productvty level and growth (sector A) to a sector that had below-average labour productvty level and growth (sector B) mples that, on aggregate, the contrbutons of the RLE and RGE to aggregate labour productvty growth wll be negatve. Lastly, t s also known that the overall magntude of the WSE wll be greater than that of the jont reallocaton effects because the economy experenced postve labour productvty growth durng the perod. Table 3: Two-Sector Economy Example: TRAD, CSLS, and GEAD Decompostons of Aggregate Labour Productvty Growth Usng Real Output n Constant Prces A) TRAD decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B B) CSLS decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B C) GEAD decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B Table 3 confrms our ntuton by presentng the contrbutons to aggregate labour productvty 10

12 growth for sectors A and B accordng to the TRAD and CSLS formulas. As expected, sector A and sector B had postve WSE, wth dentcal contrbutons to aggregate labour productvty growth n both decompostons. The aggregate contrbutons of the RLE and RGE were also dentcal n both formulas. However, sectoral RLEs and RGEs n the CSLS and TRAD formulas were very dfferent. Sector B had postve RLE and RGE n the TRAD decomposton, but negatve effects n the CSLS decomposton. Sector A had negatve contrbutons n both decomposton formulas, but the magntude of both the RLE and RGE were substantally greater n the TRAD decomposton. To understand the dfferences between the reallocaton effects n the two formulas, ther dfferent specfcatons must be analysed. As noted earler, the reallocaton effects n the TRAD formula are a functon of the rato between each sector s labour productvty level and aggregate labour productvty level. If lt > 0 and G t > 0, then the sector s reallocaton effects are never negatve, even f the sector had below-average labour productvty level and growth. In case the sector had below-average labour productvty level and growth however, ths postve contrbuton wll always be offset by the negatve contrbuton of the sector wth above-average labour productvty level and growth where l t < 0. Ths s exactly what happens n ths example, where the postve reallocaton effects of sector B were completely offset by the negatve reallocaton effects n sector A. The reallocaton effects n the CSLS formula work n a dfferent way. What matters here s not the rato, but the dfference between a sector s labour productvty level (growth) and aggregate labour productvty level (growth). Thus, f labour moves from a sector wth above-average labour productvty level (growth) to a sector wth below-average labour productvty level (growth), the reallocaton effects of both sectors are negatve. In ths example, the TRAD formula penalzes only the sector wth above-average labour productvty level (growth) that suffered a reducton n ts labour share, whereas the CSLS formula penalzes both the sector that had above-average labour productvty level (growth) and the sector that had below-average labour productvty level (growth). Thus, whle the TRAD and CSLS formulas yeld the exact same results at the aggregate economy level.e., the sum of the WSE, RLE, and RGE across all sectors s always the same, the total estmated contrbuton of each sector to aggregate labour productvty growth.e. the sum of each sector s WSE, RLE, and RGE can be very dfferent. In ths example, for nstance, the TRAD decomposton shows that sector B was responsble for the entrety of labour productvty growth experenced by the economy durng the perod, because the postve WSE of sector A was completely offset by the sector s negatve reallocaton effects. The story told by the CSLS formula s almost the exact opposte, wth sector A explanng all of aggregate labour productvty growth. In fact, the CSLS formula says somethng more: aggregate labour productvty growth could have been even hgher, f not for the ncrease n sector B s labour share. Table 2 also presents nomnal output and prce deflator estmates for sectors A and B, whch 11

13 allows sectoral contrbutons to aggregate labour productvty growth to be calculated usng the GEAD formula. To fully exemplfy the dfferences between the GEAD formula and the TRAD and CSLS formulas, t s assumed that prces were set to 1.0 n a perod pror to perod 0, so that some relatve prce movements can already be observed n perod 0. 8 Furthermore, from perod 0 to perod 1, sector A prces almost double whle sector B prces reman constant. How do these movements n relatve prces affect the estmated contrbutons to aggregate labour productvty growth? Frst, note that the GEAD s WSE s determned by nomnal output shares n perod t 1 (n ths case perod 0) nstead of real output shares. Sector A s nomnal output share s slghtly larger than ts real output share because the prce ncrease experenced n perod 0 was more than enough to offset the lack of real output growth n the sector. Snce sector A s also the sector whch experenced above-average labour productvty growth, the larger weght of ts nomnal output share mples a stronger WSE. Not only that, ths stronger contrbuton more than makes up for the smaller contrbuton of sector B s WSE, leadng to an overall larger contrbuton of the WSE to aggregate labour productvty than the one observed n the TRAD and CSLS decompostons. Second, a sgnfcant ncrease n sector A s relatve prce n perod 1 dampens the magntude of the sector s RLE and RGE although (n ths example) t s not enough to change the negatve sgn of both reallocaton effects. Recall that n the TRAD decomposton, sector A s WSE was completely offset by ts negatve reallocaton effects. Ths does not happen n the GEAD decomposton, where the WSE completely domnates the two reallocaton effects. The GEAD decomposton also changes sector B s reallocaton effects. In the TRAD decomposton, both reallocaton effects were postve for sector B. In the GEAD, snce sector B s prces remaned constant n the two perods whle sector A s prces almost doubled, sector B s relatve prce fell substantally, causng the sector s RLE to be slghtly negatve and ts RGE to be zero. In the end, the overall reallocaton effects n the GEAD formula had the same negatve sgn as n the TRAD and CSLS formulas, but wth a larger magntude, whch was caused by a stronger reallocaton level effect. In all three cases dscussed above, t s clear that the sum of sectoral contrbutons was exactly equal to aggregate labour productvty growth. In the cases of the TRAD and CSLS formulas, ths also happens, but only because the real output measure used to calculate labour productvty was n constant prces. In the case of the GEAD formula, the contrbutons wll always sum up to aggregate labour productvty growth, regardless of how real output was calculated. Table 4 shows the same economy descrbed n Table 2, but ths tme real output s n chaned prces, calculated usng chaned Fsher quantty and prce ndexes. Note that real output remans the same at the sectoral level (t s assumed that ths s the lowest possble level of aggregaton), but not at the total economy level. In partcular, the addtvty of 8 If prces were set to 1.0 n perod 0, then nomnal output shares would be equal to real output shares n that perod, causng the contrbuton of the wthn-sector effect to be the same n all three formulas. 12

14 real output no longer holds, so that total economy real output can be above or below the sum of sectoral real output. As a consequence, real output shares no longer sum up to 1.0. Obvously, ths change n total economy real output also affects mplct prce deflators and aggregate labour productvty. Table 4: Two-Sector Economy Example: Nomnal Output, Real Output n Chaned Prces, Prces, Labour Input, and Labour Productvty Perod 0 Y0 X X0 0 P0 p 0 L 0 l0 Z0 Y 0 X 0 Total Economy Y 0 Sector A Sector B Perod 1 Total Economy Sector A Sector B Y 1 Y 1 X 1 Y 1 X1 X 1 P1 p 1 L 1 l1 Z [+77%].. [+17%].. [+51%].. [+0%].. [+17%] [+82%].. [+0%].. [+82%].. [-33%].. [+50%] [+60%].. [+60%].. [+0%].. [+50%].. [+7%] Notes: 1) All varables as defned prevously; 2) Numbers n square brackets ndcate per cent growth experenced between perod 0 and perod 1. Table 5 provdes estmates for the sectoral contrbutons to aggregate labour productvty growth accordng to the TRAD, CSLS, and GEAD formulas when real output s calculated n chaned prces. Although the estmates change slghtly from those n Table 3, the man stores are the same. In all three formulas, the postve WSE domnates the negatve (jont) reallocaton effects. In the TRAD formula, sector B contnues beng the sole contrbutor to aggregate labour productvty growth, whle n the CSLS formula t actually hnders productvty growth. In the GEAD formula, sector A stll accounted for most of the aggregate labour productvty growth, although sector B also had a postve (albet small) contrbuton. The man dfference between these estmates and the ones presented n Table 3 s that the sectoral contrbutons generated by the TRAD and CSLS formulas no longer sum up to aggregate labour productvty growth. Because real output was calculated usng chaned ndexes, sectoral contrbutons sum up to aggregate labour productvty growth only n the case of the GEAD formula. 13

15 Table 5: Two-Sector Economy Example: TRAD, CSLS, and GEAD Decompostons of Aggregate Labour Productvty Growth Usng Real Output n Chaned Prces A) TRAD decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B B) CSLS decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B C) GEAD decomposton Total Contrbuton Wthn-Sector Effect Reallocaton Level Effect Reallocaton Growth Effect (percentage pont contrbuton to aggregate labour productvty growth) Total Economy Sector A Sector B Sectoral Contrbutons to Busness Sector Labour Productvty Growth n Canada, Ths secton looks at how dfferent sectors contrbuted to aggregate labour productvty growth n Canada durng the perod accordng to the TRAD, CSLS, and GEAD decomposton formulas. The secton s dvded nto four parts. The frst part provdes an overvew of the data used n ths artcle, descrbng adjustments made and possble data lmtatons. The second part descrbes nomnal output, real output, prces, labour nput, and labour productvty trends observed n Canada durng the perod for both the busness sector as a whole and two-dgt NAICS sectors. The thrd part analyses sectoral contrbuton estmates to aggregate labour productvty growth n the Canadan busness sector durng the perod. Sx sets of estmates are presented, two for each of the three decomposton formulas. The fourth part dscusses whch of the three formulas produces better estmates. 14

16 3.1 Data Statstcs Canada constructs productvty estmates for the Canadan busness sector, two-dgt NAICS sectors, and three-dgt NAICS subsectors. These estmates span a varable perod of tme, dependng on whch program they belong to estmates from the multfactor productvty program currently go all the way back to 1961 and up to 2011 (CANSIM Tables /22), whle estmates from the labour productvty program span the perod (CANSIM Table ). Both sets of productvty estmates use real GDP n chaned 2002 dollars. Snce Statstcs Canada constructs real GDP estmates n both constant 2002 dollars and chaned 2002 dollars, an opton would have been to use Statstcs Canada s offcal productvty numbers and calculate a new set of productvty estmates usng real GDP n constant 2002 dollars. Dong so, however, could create addtonal problems. In partcular, there s no guarantee that all the data adjustments made by Statstcs Canada would have been replcable, causng our labour productvty estmates n constant prces to not be perfectly comparable to the offcal Statstcs Canada estmates n chaned prces. In order to crcumvent ths problem, two sets of labour productvty estmates one n chaned 2002 dollars and the other n constant 2002 dollars were constructed by the author usng Statstcs Canada data on nomnal GDP, real GDP, and hours worked. These estmates span the perod and refer to the Canadan busness sector as a whole and two-dgt NAICS sectors. Below, detals on how these estmates were constructed are provded: 1. Statstcs Canada provdes nomnal GDP estmates for the aggregate busness sector and two-dgt NAICS sectors (excludng non-busness sector actvtes) n Canada up to Two man adjustments were made to Statstcs Canada numbers. Frst, nomnal GDP from mputed rent of owner-occuped dwellngs was, per usual practce, excluded from total GDP of the fnance, nsurance, real estate, rental and leasng (FIRE) sector. Second, the busness sector components of educatonal servces, health care and socal assstance, and other servces (except publc admnstraton) were combned under a sngle aggregate called other prvate servces. After these adjustments were made, our nomnal GDP estmates matched Statstcs Canada s estmates almost perfectly. 2. Statstcs Canada calculates real GDP n both constant 2002 dollars and chaned 2002 dollars (CANSIM Table ). At the two-dgt NAICS level, however, these estmates refer to total economc actvty,.e. they nclude both busness sector and non-busness sector actvtes. Thus, they are not consstent wth our nomnal GDP estmates, whch nclude 9 Nomnal GDP seres were expanded to 2010 by usng provncal sectoral shares (CANSIM Table ) and provncal nomnal GDP at basc prces (CANSIM Table ), both of whch go all the way up to 2010, to calculate natonal estmates. Combnng these two data sets provdes us wth nomnal GDP estmates for two-dgt NAICS sectors (total economc actvty) and total economy. To obtan busness sector estmates, we assumed that busness sector shares of two-dgt NAICS sectors remaned at ther 2008 levels. 15

17 only busness sector actvtes. To construct real GDP estmates consstent wth our nomnal GDP estmates, t was assumed that busness sector mplct prce deflators were equal to total economy prce deflators. Ths allowed real GDP estmates n both constant 2002 dollars and chaned 2002 dollars to be constructed. 10 For most sectors, ths assumpton s not partcularly problematc almost the entrety of economc actvty n agrculture, forestry, fshng and huntng, for nstance, s consdered a busness sector actvty. The only sectors where ths can be consdered a strong assumpton are educatonal servces; health care and socal assstance; arts, entertanment and recreaton; and other servces (except publc admnstraton). 3. The labour nput measure used to construct our estmates was actual hours worked, taken drectly from Statstcs Canada s labour productvty program. Labour productvty was calculated as real GDP per hour worked. Overall, our labour productvty estmates were reasonably close to the offcal numbers. Whle Statstcs Canada s offcal estmates show busness sector labour productvty growng at an average annual rate of 0.76 per cent (chaned 2002 dollars) durng the perod, our estmates show an average annual growth of 0.91 per cent (chaned 2002 dollars) and 0.86 per cent (constant 2002 dollars). The dfference between our estmates and the offcal ones s caused, as mentoned above, by the use of dfferent mplct prce deflators. 3.2 Output, Prces, Labour Input, and Labour Productvty Trends n Canada, Nomnal GDP n the Canadan busness sector ncreased at an average annual rate of 4.10 per cent durng the perod (Table 6). 11 Nomnal GDP growth was partcularly strong n mnng and ol and gas extracton (8.88 per cent per year) and constructon (8.22 per cent per year). Due to ths above-average growth, the nomnal GDP of the two sectors as a share of busness sector GDP ncreased substantally n the case of mnng and ol and gas extracton from 7.9 per cent 10 Two specfc adjustments were made to our real GDP estmates: 1) Imputed rent for owner-occuped dwellngs was subtracted from total real GDP n the FIRE sector, for both the constant 2002 dollar and chaned 2002 dollar estmates; 2) Real GDP for other prvate servces (the aggregaton of educatonal servces, health care and socal assstance, and other servces (except publc admnstraton)) was calculated usng a chaned Fsher quantty ndex for the estmates n chaned 2002 dollars and a fxed-base Laspeyres quantty ndex for the estmates n constant 2002 dollars. 11 Instead of usng only startng and end ponts of a partcular seres, growth rates presented n ths artcle were calculated as the perod average of annual growth rates, e.g. for the perod, reported growth rates are the average of the annual growth experenced n 2001, 2002,..., Ths ensures that growth rates are consstent wth our estmates of sectoral contrbutons to aggregate labour productvty growth, whch were calculated on a year-to-year bass. Although sectoral contrbuton estmates could have been calculated for the entre perod, snce contrbutons are crucally dependent of output and labour shares n the ntal perod, large swngs n these shares n subsequent perods could bas estmated contrbutons sgnfcantly. 16

18 n 2000 to 9.7 per cent n 2010, whle n constructon from 6.5 per cent to 9.4 per cent. At the same tme, nomnal GDP n the manufacturng sector fell 1.22 per cent per year, causng a marked declne n the sector s share of busness sector nomnal GDP, from 24.4 per cent n 2000 to 14.4 per cent n Table 6: Nomnal GDP n Canada, Busness Sector and Two-Dgt NAICS Sectors, Y 2000 Y 2010 Ave. Annual Growth Y 2000 Y 2000 Y 2010 Y 2010 ($ mllons) ($ mllons) (%) (%) (%) Busness sector ndustres 769,682 1,141, Agrculture, forestry, fshng and huntng 22,137 24, Mnng and ol and gas extracton 60, , Utltes 26,242 33, Constructon 49, , Manufacturng 187, , Wholesale trade 50,931 80, Retal trade 51,311 86, Transportaton and warehousng 43,896 67, Informaton and cultural ndustres 32,150 53, FIRE 108, , Professonal, scentfc and techncal servces 43,566 76, ASWMRS 20,367 37, Arts, entertanment and recreaton 7,009 10, Accommodaton and food servces 23,263 33, Other Prvate Servces 42,522 73, Notes: 1) All varables as defned prevously; 2) FIRE - Fnance, nsurance, real estate, rental and leasng; ASWMRS - Admnstraton and support, waste management and remedaton servces; 3) Growth rates calculated as the arthmetc average of annual growth rates. Table 7 presents two sets of prce deflators, the frst one calculated usng a fxed-base Paasche prce ndex and the second one calculated usng a chaned Fsher prce ndex. It s clear from the numbers that, for the tme perod n queston, the dfferences between the two sets of deflators are mnor. The only two sectors where there s a (potentally) sgnfcant dfference between prce deflators are manufacturng (whch saw prces rsng 0.70 per cent per year accordng to the constant prce deflator and 0.44 per cent per year accordng to the chaned prce deflator) and wholesale trade (1.52 per cent per year vs per cent per year). Almost 60 per cent of the total ncrease n busness sector nomnal GDP was caused by rsng prces. At the busness sector level, prces rose at an average annual rate of 2.38 per cent accordng to the constant prce deflator and 2.33 per cent accordng to the chaned prce deflator. The sectors that saw the most sgnfcant prce ncreases were mnng and ol and gas extracton ( per cent per year) and constructon ( per cent per year). Although prces n manufacturng ncreased at a much slower rate than overall busness sector prces, the fact that they dd ncrease 17

19 mples that real GDP growth n the sector was even lower than nomnal GDP growth. In fact, ths s exactly what Table 8 shows, wth real GDP n the manufacturng sector declnng 1.85 per cent per year accordng to the constant prce deflator or 1.64 per cent per year accordng to the chaned prce deflator. It s nterestng to note, also, that most of the nomnal GDP growth experenced by the mnng and ol and gas extracton sector was due to prce ncreases. Real GDP growth n the sector was only per cent per year durng the perod. The constructon sector, on the other hand, saw not only rapd prce ncreases but also fast real GDP growth. The sector had the second hghest real growth rate among all two-dgt NAICS sectors ( per cent per year), only behnd retal trade ( per cent per year). Other sectors that experenced robust real GDP growth were FIRE ( per cent per year) and admnstraton and support, waste management and remedaton servces (ASWMRS) ( per cent per year). At the busness sector level, real GDP grew at an average annual rate of per cent durng the perod. Whle nomnal GDP n mnng and ol and gas extracton as a share of busness sector GDP ncreased substantally from 2000 to 2010, the sector s real GDP share actually fell durng the perod, from 6.6 per cent n 2000 to per cent (dependng on whch deflator s used). Manufacturng also saw a marked declne n ts real GDP share, from per cent to per cent, although the magntude of ths declne was not as marked as t was n nomnal terms. The constructon sector s real GDP share, on the other hand, ncreased n the perod, from per cent to 7.9 per cent. Another nterestng development was the ncrease n the FIRE sector s real GDP share, from 14.1 per cent to per cent. Hours worked n the Canadan busness sector ncreased at an average annual rate of 0.78 per cent durng the perod (Table 9). Hours worked saw partcularly fast growth n mnng and ol and gas extracton (3.91 per cent per year), constructon (3.66 per cent per year), and ASWMRS (3.37 per cent per year). Conversely, agrculture, forestry, fshng and huntng (-2.91 per cent per year) and manufacturng (-2.50 per cent per year) experenced a declne n total hours worked. The changes n sectoral hours worked as a share of total hours worked n the busness sector durng the 2000s were not as drastc as those seen n terms of nomnal GDP or real GDP shares. The two most sgnfcant changes were n constructon, where the hours worked share ncreased from 8.3 per cent n 2000 to 11.0 per cent n 2010, and manufacturng, where the share decreased from 18.3 per cent to 13.1 per cent. Usng our two sets of real GDP estmates and hours worked, labour productvty estmates were constructed for the busness sector and two-dgt NAICS sectors n Canada from 2000 to 2010 (Table 10). Durng the perod, labour productvty n the Canadan busness sector grew at an average annual rate of 0.86 per cent accordng to the constant 2002 dollar estmates from $35.57 per hour n 2000 to $ per hour n 2010 and 0.91 per cent accordng to the chaned

20 Table 7: Implct Prce Deflators for Canada, Busness Sector and Two-Dgt NAICS Sectors, A) Constant 2002 Dollars P 2000 P 2010 Ave. Annual Growth p 2000 p 2010 (2002 = 100) (2002 = 100) (%) (%) (%) Busness sector ndustres Agrculture, forestry, fshng and huntng Mnng and ol and gas extracton Utltes Constructon Manufacturng Wholesale trade Retal trade Transportaton and warehousng Informaton and cultural ndustres FIRE Professonal, scentfc and techncal servces ASWMRS Arts, entertanment and recreaton Accommodaton and food servces Other Prvate Servces B) Chaned 2002 Dollars P 2000 P 2010 Ave. Annual Growth p 2000 p 2010 (2002 = 100) (2002 = 100) (%) (%) (%) Busness sector ndustres Agrculture, forestry, fshng and huntng Mnng and ol and gas extracton Utltes Constructon Manufacturng Wholesale trade Retal trade Transportaton and warehousng Informaton and cultural ndustres FIRE Professonal, scentfc and techncal servces ASWMRS Arts, entertanment and recreaton Accommodaton and food servces Other Prvate Servces Notes: 1) All varables as defned prevously; 2) FIRE - Fnance, nsurance, real estate, rental and leasng; ASWMRS - Admnstraton and support, waste management and remedaton servces; 3) Growth rates calculated as the arthmetc average of annual growth rates. 19

21 Table 8: Real GDP n Canada, Busness Sector and Two-Dgt NAICS Sectors, A) Constant 2002 Dollars X 2000 X 2010 Ave. Annual Growth X 2000 X 2000 X 2010 X 2010 ($ mllons) ($ mllons) (%) (%) (%) Busness sector ndustres 785, , Agrculture, forestry, fshng and huntng 26,312 28, Mnng and ol and gas extracton 52,183 54, Utltes 26,914 30, Constructon 51,655 72, Manufacturng 190, , Wholesale trade 51,256 69, Retal trade 51,766 75, Transportaton and warehousng 46,670 54, Informaton and cultural ndustres 32,126 43, FIRE 110, , Professonal, scentfc and techncal servces 45,731 59, ASWMRS 21,697 30, Arts, entertanment and recreaton 7,634 8, Accommodaton and food servces 24,406 26, Other Prvate Servces 45,633 57, B) Chaned 2002 Dollars X 2000 X 2010 Ave. Annual Growth X 2000 X 2000 X 2010 X 2010 ($ mllons) ($ mllons) (%) (%) (%) Busness sector ndustres 785, , Agrculture, forestry, fshng and huntng 25,549 27, Mnng and ol and gas extracton 51,519 54, Utltes 27,560 30, Constructon 51,757 73, Manufacturng 188, , Wholesale trade 52,511 68, Retal trade 52,536 75, Transportaton and warehousng 46,472 54, Informaton and cultural ndustres 32,242 43, FIRE 110, , Professonal, scentfc and techncal servces 46,068 59, ASWMRS 21,750 30, Arts, entertanment and recreaton 7,665 8, Accommodaton and food servces 24,362 26, Other Prvate Servces 45,622 56, Notes: 1) All varables as defned prevously; 2) FIRE - Fnance, nsurance, real estate, rental and leasng; ASWMRS - Admnstraton and support, waste management and remedaton servces; 3) Growth rates calculated as the arthmetc average of annual growth rates. 20