On the Tang and Wang Decomposition of Labour Productivity Growth into Sectoral Effects

Transcription

1 O the Tag ad Wag Decomoitio of Labour Productivity Growth ito Sectoral Effect W. Erwi Diewert, Aril 5, 8 Dicuio Paer 8-6, Deartmet of Ecoomic, Uiverity of Britih Columbia, Vacouver, B.C., Caada, V6T Z. diewert@eco.ubc.ca Abtract Tag ad Wag rovided a decomoitio of ecoomy wide labour roductivity ito ectoral cotributio effect. The reet ote rework their methodology to rovide a more traaret ad imle decomoitio. Thi ew decomoitio i the related to aother decomoitio due to Gii ad aalyzed by Balk. Overall growth i labour roductivity i due to three factor: (i) growth i the labour roductivity of idividual ector; (ii) chage i real outut rice of the ector ad (iii) chage i the allocatio of labour acro ector. Joural of Ecoomic Literature Claificatio umber C43, D4. Keyword Idex umber, labour roductivity, decomoitio of aggregate labour roductivity ito ectoral effect.. Itroductio Jiami Tag ad Weimi Wag (4; 46) rovided a iteretig decomoitio for ecoomy wide labour roductivity ito ectoral cotributio effect. However, the iterretatio of the idividual term i their decomoitio i ot comletely clear ad o i ectio, we rework their methodology i order to rovide a more traaret ad imle decomoitio. I ectio 3, we urue a omewhat differet aroach which i due to Gii (937) ad i a geeralizatio of the Fiher (9) ideal idex umber methodology to aggregate that are roduct of three factor rather tha two.. The Tag ad Wag Methodology Reworked Let there be ector or idutrie i the ecoomy. Suoe that for eriod t,, the outut (or real value added) of ector i Y t with correodig eriod t rice P t ad The author thak Bert Balk ad Jiami Tag for helful commet o earlier draft of thi ote.

2 labour iut L t for,...,. We aume that thee labour iut ca be added acro ector ad that the ecoomy wide labour iut i eriod t i L t defied a () L t L t ; t,. Idutry labour roductivity i eriod t, t, i defied a idutry outut divided by idutry labour iut: () t Y t /L t ; t, ;,...,. It i ot etirely clear how aggregate labour roductivity hould be defied ice the outut roduced by the variou idutrie are meaured i heterogeeou, ocomarable uit. Thu we eed to weight thee heterogeeou outut by their rice, um the reultig eriod t value ad the divide by a geeral outut rice idex, ay P t for eriod t, i order to make the ecoomy wide omial value of aggregate outut comarable i real term acro eriod. Thu with a aroriate choice for the aggregate outut rice idex P t, the eriod t ecoomy wide labour roductivity, t, i defied a follow: (3) t P t Y t / P t L t ; t,. We ca imlify the exreio for aggregate labour roductivity i eriod,, by a judiciou choice of uit of meauremet for each idutry outut. We will chooe to meaure each idutry outut i term of the umber of uit of the idutry outut that ca be urchaed by oe dollar i eriod. The effect of thee choice for the uit of meauremet i to et the rice of each idutry outut equal to uity i eriod ; i.e., we have: 3 (4) P ;,...,. We will alo ormalize the ecoomy wide rice idex to equal uity i eriod ; i.e., we have: 4 (5) P. Uig defiitio (3) for t alog with the ormalizatio (4) ad (5), it ca be ee that the eriod ecoomy wide labour roductivity i equal to the followig exreio: (6) Y / L L / L uig defiitio () Thi follow the methodological aroach take by Tag ad Wag (4; 45). 3 I reality, each idutry will be roducig may roduct ad o P will be ay the Fiher (9) rice idex for all of the idutry roduct goig from eriod to. 4 Tyically, P will be the Fiher rice idex goig from eriod to where the eriod ad rice ad quatity vector are the eriod t idutry rice ad quatity vector, [P t,...,p t ] ad [Y t,...,y t ] reectively for t,.

3 3 L where the hare of labour ued by idutry i eriod t, L t, i defied i the obviou way a follow: (7) L t L t / L t ;,..., ; t,. Thu aggregate labour roductivity for the ecoomy i eriod i a (labour) hare weighted average of the ectoral labour roductivitie, a quite eible reult. Uig defiitio (3) for t ad the defiitio (7) for t lead to the followig exreio for aggregate labour roductivity i eriod : 5 (8) P Y / P L [P /P ] [Y /L ] [L /L ] [P /P ] L L uig defiitio () ad (7) for t where the eriod t idutry real outut rice, t, i defied a the idutry t outut rice P t, divided by the aggregate outut rice idex for eriod t, P t ; i.e., we have the followig defiitio: 6 (9) t P t /P t ;,..., ; t,. Thu ecoomy wide labour roductivity i eriod,, i ot equal to the (labour) hare weighted average of the ectoral labour roductivitie, L. Itead, i equal to L, o that the labour roductivity of ay ector which ha exerieced a real outut rice icreae (o that i greater tha oe) get a weight that i greater tha it traightforward labour hare weighted cotributio, L ; i.e., ector get the weight L. U to thi oit, our aalyi follow that of Tag ad Wag (4; 45-46) excet that Tag ad Wag did ot bother with the ormalizatio (4) ad (5). However, i what follow, we hoefully rovide ome additioal value added to their aalyi. Firt, we defie the value added or outut hare of idutry i eriod, Y, a follow: () Y P Y / i P i Y i ;,..., Y / i Y i uig the ormalizatio (4). ote that the roduct of the ector labour hare i eriod, L, with the ector labour roductivity i eriod,, equal the followig exreio: 5 Equatio (8) correod to equatio () i Tag ad Wag (4; 46). 6 Thee defiitio follow thoe of Tag ad Wag (4; 45).

4 4 () L [L /L ][Y /L ] ;,..., Y /L. Uig (), we ca etablih the followig equalitie: () L / i Li i [Y /L ] / i [Y i /L ] ;,..., Y / i Y i Y uig (). ow we are ready to develo a exreio for the rate of growth of ecoomy wide labour roductivity. Uig exreio (6) ad (8), we have: (3) / L / L [ L / L ] [ / ] L / L [ L / L ] [ / ] Y [ / ] [ L / L ] [ / ] Y uig () uig (4) ad (5). Thu overall ecoomy wide labour roductivity growth, /, i a outut hare (ee the term Y i (3) above) weighted average of three growth factor aociated with idutry. The three growth factor are: /, (oe lu) the rate of growth i the labour roductivity of idutry ; L / L, (oe lu) the rate of growth i the hare of labour beig utilized by idutry ad / [P /P ]/[P /P ] which i (oe lu) the rate of growth i the real outut rice of idutry. Thu i lookig at the cotributio of idutry to overall (oe lu) labour roductivity growth, we tart with a traightforward hare weighted cotributio factor, Y [ / ], which i the eriod outut or value added hare of idutry i eriod, Y, time the idutry (oe lu) rate of labour roductivity growth, /. Thi traightforward cotributio factor will be augmeted if real outut rice growth i oitive (if / i greater tha oe) ad if the hare of labour ued by idutry i growig (if L / L i greater tha oe). The decomoitio of overall labour roductivity growth give by the lat lie of (3) eem to be ituitively reaoable ad fairly imle a ooed to the rather comlex decomoitio obtaied by Tag ad Wag (4; 46). 3. A Alterative Decomoitio due to Gii Rewrite (3), makig ue of (4), (5) ad (9) a follow: (4) / L / L. Suoe that we wat to decomoe /, the overall chage i aggregate roductivity, ito the roduct of three effect:

5 5 Oe effect that hold cotat the ectoral labour hare L t ad the ectoral roductivitie t ad jut give u the effect of the chage i the real rice ; Aother effect that hold cotat the real rice ad the ectoral roductivitie t ad give u the effect of the chage i the ectoral labour hare L t ad A fial effect that hold cotat the idividual labour hare L t ad real rice ad give u the effect of the chage i the ectoral roductivitie t. Thi i a well kow roblem that ha bee tudied exteively by Balk (/3) ad Balk ad Hoogeboom-Sijker (3) ad by may other. I articular, the geeralizatio of the Fiher (9) ideal idex to a aggregate that i the roduct of 3 differet factor made by Gii (937; 7) eem to be aroriate for the reet ituatio. A relatively imle way to derive Gii formula i a follow. / i equal to the ratio L / L. Let u write thi ratio a a roduct of three imilar ratio, where i each of thee three ratio, oe of the factor i the umerator i et equal to either or L or ad the ame factor i the deomiator i et equal to either or L or. The remaiig factor i the umerator ad deomiator are cotat. There are oly 6 way thi ca be doe ad the reultig decomoitio are a follow: (5) L L L L L L P()S()() ; (6) L L L L L L P()S()() ; (7) L L L L L L P(3)S(3)(3) ; (8) L L L L L L P(4)S(4)(4) ; (9) L L L L L L P(5)S(5)(5) ; () L L L L L L P(6)S(6)(6) where P() i defied a the rice idex which i the firt term i bracket o the right had ide of (5), S() i defied a the hare idex which i the ecod term i bracket

6 6 o the right had ide of (5) ad () i the roductivity idex which i the third term i bracket o the right had ide of (5) ad o o for the defiitio i (6)-(). All of the decomoitio of the ratio / are equally valid o it eem eible to defie a overall idex of rice chage, ay P, a a ymmetric average of the idividual rice idexe P()-P(6) which aeared i (5)-(). It i alo atural to follow the examle of Fiher (9) ad Gii (937; 7) ad take geometric mea o that the idexe will atify the time reveral tet ad alo reerve the exact decomoitio of / ito the roduct of three exlaatory factor. Hece lettig t, t ad t be the dimeioal vector of the real rice i eriod t, t, the labour hare i eriod t, L t, ad the ectoral roductivitie i eriod t, t, reectively, we have the followig exreio for the Gii rice chage cotributio factor to overall labour roductivity growth: () P(,,,,, ) [P()P()...P(6)] /6 { L L L L L L L L } /6. I a imilar maer, we ca derive the followig exreio for the Gii labour hare chage cotributio factor to overall labour roductivity growth: () S(,,,,, ) [S()S()...S(6)] /6 { L L L L L L L L } /6. Fially, we ca derive the followig exreio for the Gii ure roductivity chage cotributio factor to overall labour roductivity growth (which hold cotat the effect of chagig real outut rice ad chagig ector labour hare): (3) (,,,,, ) [()()...(6)] /6 { L L L L L L L L } /6. Balk (/3; ) ugget ome axiom that idex umber formulae of the tye defied by ()-(3) hould atify. 7 It ca be verified that the above Gii idexe atify all of Balk uggeted tet. Aother iteretig aect of the Gii formulae i that if the labour hare are cotat acro the two eriod, o that, the the labour hare cotributio factor S(,,,,, ) defied by () i uity, the real rice chage cotributio factor P(,,,,, ) defied by () reduce to the ordiary Fiher rice idex, P F, ad the 7 Balk (/3; ) alo ote with aroval the Gii formulae defied by ()-(3) ad give additioal hitorical referece to the literature.

7 7 ure roductivity chage cotributio factor (,,,,, ) defied by (3) reduce to the ordiary Fiher quatity idex Q F, where P F ad Q F are defied a follow: (4) P F (,,, ) [ / ] / ; (5) Q F (,,, ) [ / ] /. Similarly, if the real rice are cotat acro the two eriod, the the real rice chage cotributio factor P(,,,,, ) i uity, the labour hare cotributio factor S(,,,,, ) collae to the Fiher idex [ / ] / ad the ure roductivity chage cotributio factor (,,,,, ) reduce to the Fiher quatity idex [ / ] /. Each of the cotributio factor defied by ()-(3) ha a iterretatio a a idex of chage of rice, labour hare ad ectoral labour roductivitie, holdig cotat the other two factor. However the iterretatio of () ad () i ot comletely traightforward (a it i the cae of ormal idex umber theory) ice hare by defiitio caot all grow from oe eriod to the ext ad o the iterretatio of () a a weighted average of the idividual hare growth rate, /, while valid doe ot eem to be very ituitive. Similarly, the iterretatio of () a a weighted average of the growth rate of the ectoral real outut rice, /, alo eem to lack ituitive aeal ice the average of the real rice t for each eriod t will ecearily be cloe to oe, ad hece, it will ot be oible for all of the relative rice, /, to exceed uity uder ormal coditio. Fortuately, it i oible to reiterret each of the cotributio factor defied by () ad () a idicator of tructural chage a we will ow how. I order to derive thee alterative iterretatio of () ad (), it i firt eceary to develo a idetity that wa ued by Bortkiewicz (93; ) i a idex umber cotext. Suoe that we have two dimeioal vector x [x,...,x ] ad y [y,...,y ] ad a dimeioal vector of oitive hare weight [,..., ]. 8 We ue thee hare i order to defie the hare weighted average of x ad y, x * ad y * reectively ad the hare weighted covariace betwee x ad y, Cov (x,y;): (6) x * x ; y * y ; Cov(x,y;) (x x * )(y y * ). It i traightforward to ue the above defiitio i order to derive the followig idetity: (7) x y Cov(x,y;) + x * y *. ow coider a geeric hare idex of the tye defied by S() to S(6) i (5)-(). We have the followig decomoitio of uch a idex, which we label a S: 9 (8) S / 8 We aume that the hare um to uity; i.e.,. 9 The geeric ector real outut rice will be equal to or ad the geeric ector labour roductivity level will be equal to or.

8 8 ( / ) / x (z /z * ) defiig x / ; z ad z * z x y defiig y z /z * for,...,. ote that the hare weighted mea of the x ad y are both equal to oe; i.e., we have: (9) x * x ( / ) ; (3) y * y (z /z * ) z * /z *. ote that each z i equal to the roduct of the geeric real outut rice i ector,, which will tyically be cloe to oe, time the geeric roductivity level of ector,. Thu z * i the weighted average of thee ector real rice weighted roductivity level,. Hece y / j j j j i the real rice weighted geeric roductivity level of ector relative to a weighted average of thee ame rice weighted roductivity level. ow aly the idetity (7) to the lat lie i (8) ad we obtai the followig decomoitio for the geeric S: (3) S (x x * )(y y * ) + x * y * (x )(y ) + uig (9) ad (3) Cov (x,y; ) +. Thu the geeric labour hare chage cotributio factor to overall labour roductivity growth S defied by the firt equatio i (8) will be greater tha oe if ad oly if the Cov(x,y; ) i oitive. Thu if the hare weighted correlatio betwee the x / (oe lu the rate of chage of the ectoral labour hare) ad the ectoral real rice weighted roductivity level relative to their hare weighted average level y / j j j j i oitive, the S will be greater tha oe. Put aother way, if the labour hare goig from eriod to chage i uch a way that higher hare go to higher roductivity ector, the the cotributio factor S to overall labour roductivity growth will be oitive. Thu the Gii labour hare cotributio factor S(,,,,, ) defied by () will be greater tha oe if all 6 of the covariace Cov (x,y, ) of the tye defied i (3) are oitive for the ecific idexe defied by S() to S(6). Thu the Gii labour hare cotributio factor ca be iterreted a a meaure of tructural hift of labour acro idutrie of varyig roductivity level. ow coider a geeric real outut rice idex of the tye defied by P() to P(6) i (5)-(). We have the followig decomoitio of uch a idex, which we label a P: (3) P / ( / ) / x (z /z * ) defiig x / ; z ad z * z x y defiig y z /z * for,...,. The geeric ector labour hare will be equal to or ad the geeric ector labour roductivity level will be equal to or.

9 9 ote that the hare weighted mea of the y i equal to oe but we caot etablih the ame equality for the mea of the x ; i.e., we have: (33) x * x ( / ) ; (34) y * y (z /z * ) z * /z *. ote that each z i equal to the roduct of the eriod real outut rice i ector,, which will tyically be cloe to oe, time the geeric roductivity level of ector,. Thu z * i the geeric weighted average of thee ector real rice weighted roductivity level,. Hece y / j j j j i the real rice weighted geeric roductivity level of ector relative to a weighted average of thee ame rice weighted roductivity level. ow aly the idetity (7) to the lat lie i (3) ad we obtai the followig decomoitio for the geeric P: (35) P (x x * )(y y * ) + x * y * (x x * )(y ) + x * uig (34) Cov (x,y;) + x *. The iterretatio of (35) i ot a traightforward a wa the iterretatio of (3). The rice cotributio factor P defied by (3) will be greater tha oe if the um of the covariace term Cov (x,y;), equal to (x x * )(y ), ad the mea real rice chage x *, equal to ( / ), i greater tha oe. The iterretatio of the x * term i traightforward. P i equal to thi traightforward effect (which will geerally be cloe to oe) lu the covariace term, (x x * )(y ). Recallig that x i equal to (oe lu) the rate of growth of the ector real outut rice, /, ad that y i the roductivity level of ector relative to a average roductivity level, it ca be ee that thi covariace will be oitive if the ector which have high rate of growth of real outut rice are aociated with ector that have high relative roductivity level. 4. Cocluio Our cocluio at thi oit i that the Gii (937) decomoitio of aggregate labour roductivity ito ectoral cotributio factor ad the aociated tructural hift eem romiig. I term of imlicity, the decomoitio give by (3) alo eem attractive. But it aear that there are may very reaoable decomoitio ad at thi tage it i difficult to ay which i bet. It aear that there i room for additioal reearch i thi area, articularly i develoig the axiomatic aroach to the toic, a aroach which wa iitiated by Balk (/3). A ecoomic aroach may alo be ueful i idicatig what a bet decomoitio might be. Referece Fiher, I. (9), The Makig of Idex umber, Boto: Houghto-Miffli.

10 Balk, B.M. (/3), Ideal Idice ad Idicator for Two or More Factor, Joural of Ecoomic ad Social Meauremet 8, 3-7. Balk, B.M. ad E. Hoogeboom-Sijker (3), The Meauremet ad Decomoitio of Productivity Chage: Exercie o the etherlad Maufacturig Idutrie, Voorburg: Statitic etherlad. Bortkiewicz, L. vo (93), Zweck ud Struktur eier Preiidexzahl, ordik Statitik Tidkrift, Gii, C. (937), Method of Elimiatig the Ifluece of Several Grou of Factor, Ecoometrica 5, Tag, J. ad W. Wag (4), Source of Aggregate Labour Productivity Growth i Caada ad the Uited State, The Caadia Joural of Ecoomic 37,