THE RETURNS TO SCALE EFFECT IN LABOUR PRODUCTIVITY GROWTH. Hideyuki Mizobuchi. January 23, 2011

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1 THE RETURNS TO SCAE EFFECT IN ABOUR PRODUCTIVITY GROWTH Hideuki Mizobuchi Januar 23 2 Absrac abour producivi is defined as oupu per uni of labour inpu. Econoiss acknowledge ha echnical progress as well as growh in capial inpus increases labour producivi. However lile aenion has been paid o he fac ha changes in labour inpu alone could also ipac labour producivi. Since his effec disappears for he consan reurns o scale shor-run producion fronier we call i he reurns o scale effec. We decopose he growh in labour producivi ino wo coponens: ) he join effec of echnical progress and capial inpu growh and 2) he reurns o scale effec. We propose heoreical easures for hese wo coponens and show ha he coincide wih he inde nuber forulae consising of prices and uaniies of inpus and oupus. We hen appl he resuls of our decoposiion o U.S. indusr daa for I is acknowledged ha labour producivi in he service secor grows uch ore slowl han in he goods-producing secor. We conclude ha he reurns o scale effec can eplain a large par of he gap in labour producivi growh beween he wo secors. e Words abour producivi inde nubers Maluis inde Törnvis inde oupu disance funcion inpu disance funcion Journal of Econoic ieraure Nubers C4 D24 O47 O53 Research Fellow of he Japan Socie for he Prooion of Science eio Econoic Observaor eio Universi Mia Minao-ku Toko Japan ; Eail: hideuki.izobuchi@gail.co

2 . Inroducion Econoiss broadl hink of producivi as easuring he curren sae of he echnolog used in producing he fir s goods and services. The producion fronier consising of inpus and he aiu oupu aainable fro he characerizes he prevailing sae of echnolog. Producivi growh is ofen idenified b he shif in he producion fronier reflecing changes in producion echnolog. 2 However producivi growh can also be driven b oveen along he producion fronier. Even in he absence of changes in he producion fronier changes in he inpus used for producion can lead o producivi growh oving along he producion fronier and aking use of is curvaure. Producivi growh ha is induced b he oveen along he producion fronier is called he reurns o scale effec. This effec does no reflec changes in he producion fronier. Thus in order o properl evaluae iproveens in he underling producion echnolog reflecing he shif in he producion fronier we us disenangle he reurns o scale effec fro labour producivi. Producivi easures can be classified ino wo pes: oal facor producivi (TFP) and parial facor producivi. The forer inde relaes a bundle of oal inpus o oupus whereas he laer inde relaes a porion of oal inpus o oupus. The presen paper deals wih labour producivi (P) aong several easures of parial facor producivi. P is defined as oupu per labour inpu in he siple one-oupu one-labour-inpu case. Econo-wide P is he criical deerinan of a counr s sandard of living in he long-run. For eaple U.S. hisor reveals ha increases in P have ranslaed o nearl one-for-one increases in per capia incoe as well as real wages over a long period of ie. The iporance of P as a source for he progress of econoic well-being props an researchers o invesigae wha deerines P growh. 3 Technical progress and capial inpu growh have been idenified as he ain deerinans of a counr s enorous P growh over long periods (Jorgenson and Siroh 2 Jones 22) as well as he wide differences in P across counries (Hall and Jones 999). The presen paper adds one ore eplanaor facor o P growh. P relaes labour inpus o oupus holding echnolog and capial services fied. The shor-run producion fronier which consiss of labour inpus and he aiu oupu aainable fro he represens he capaci of curren echnolog o ranslae labour inpus o oupus. Boh echnical progress and capial inpu growh which have been idenified as he sources of P growh induce P growh hroughou he shif in he shor-run producion fronier. However he reurns o scale effec which is he een of P growh induced b oveen along he shor-run producion fronier has never been eposed. See Griliches (987). The sae inerpreaion is also found in Chabers (988). 2 In principle producivi iproveen can occur hrough echnological progress and gains in echnical efficienc. Technical efficienc is he disance beween he producion plan and he producion fronier. The presen paper assues a fir s profi aiizing behaviour and in our odel he curren producion plan is alwas on he curren producion fronier. The assupion of profi aiizaion is coon in econoic approaches o inde nubers. See Caves Chrisensen and Diewer (982) and Diewer and Morrison (986). 3 The P growh and he capial inpu growh are he abbreviaions for he growh raes of P and capial inpu. In his paper he growh rae of P beween he curren and previous periods is he raio of P in he curren period o P in he previous period.

3 We decopose P growh ino wo coponens: ) he join effec of echnical progress and capial inpu growh 2) he reurns o scale effec. Firs we propose heoreical easures represening he wo effecs b using disance funcions. Second we derive he inde nuber forulae consising of prices and uaniies and show ha he coincide wih heoreical easures assuing a ranslog funcional for for he underling echnolog and he fir s profi aiizing behaviour. Our approach o ipleening heoreical easures is drawn fro Caves Chrisensen and Diewer (982) (CCD). Using he disance funcion CCD forulae he (heoreical) Maluis producivi inde which easures he shif in he producion fronier and show ha he Maluis producivi inde and he Törnvis producivi inde coincide assuing he ranslog funcional for for he underling echnolog and he fir s profi aiizing behaviour. The Törnvis producivi inde is a easure for he TFP growh calculaed b he Törnvis uani indees. I is an inde nuber forula consising of price and uani observaions. Euivalence beween he wo indees breaks down if he underling echnolog does no ehibi consan reurns o scale. Is difference represens he reurns o scale effec which is he growh in TFP induced b oveen along he producion fronier. Thus like Diewer and Nakaura (27) and Diewer and Fo (2) we can inerpre ha CCD decopose he TFP growh ha is calculaed b he Törnvis uani indees ino Maluis producivi inde and he reurns o scale effec. The forer coponen capures TFP growh induced b he shif in he shor-run producion fronier. Man researchers have been concerned wih he growh in TFP induced b oveen along he underling producion fronier. CCD (982) call i he reurns o scale facor Diewer and Nakaura (27) call i he reurns o scale coponen and ovell (23) calls i he scale effec. In he lieraure of Daa Envelopen Analsis (Balk 2 and Coelli e al. 23) he produc of scale efficienc change inpu i effec and oupu i effec suarizes he TFP growh induced b oveen along he producion fronier and i is inerpreed as he reurns o scale effec. Alhough scholars have recognized he significance of he reurns o scale effec for TFP growh is effec on P growh has never been addressed even hough i is ore iporan in eplaining P growh han in eplaining TFP growh. When he underling echnolog ehibis consan reurns o scale he scale econoies effec disappears fro TFP growh. However i sill plas a role for P growh. This is because even if he underling echnolog ehibis consan reurns o scale he shorrun producion fronier is likel no o ehibi consan reurns o scale. Triple and Bosworh (24 26) and Bosworh and Triple (27) observed ha P growh in he service secor was uch less han in he service secor in U.S. econo since he earl 97s. As we discussed above here are wo underling facors o P growh; herefore wo eplanaions are possible for he low P growh in he services secor: ) he join effec of echnical progress and increases in capial inpus is odes or 2) an increase in labour inpus induces negaive reurns o scale effecs. We appl our decoposiion resul o U.S. indusr daa o copare he relaive conribuions of he wo effecs. Secion 2 illusraes he wo effecs underling P growh graphicall. Secion 3 discusses he easure of he join effec of echnical progress and capial inpu growh in he uliple-inpus uliple-oupus case. Secion 4 discusses he reurns o scale 3

4 effec in he uliple-inpus uliple-oupus case. We show ha he produc of he join effec of echnical progress and capial inpu growh coincides wih P growh. Secion 6 includes he applicaion o he U.S. indusr daa. Secion 5 presens he conclusions. 2. Two Sources of abour Producivi Growh We displa graphicall wha derives P growh using a siple odel of one oupu and wo inpus: labour inpu and capial inpu. Suppose ha a fir produces oupus and using inpus ( ) and ( ). Period producion echnolog is described b he period producion funcion = f ( ) for = and. e us begin b considering how his join effec of echnical progress and capial inpu growh raises P. Figure illusraes he case when he join effecs of echnical progress and increases in capial inpus posiivel affec he eploen of labour in producion. The lower curve represens he period shor-run producion fronier. I indicaes how uch oupu can be produced in period b using a specified uani of labour given he capial and echnolog available in period. Siilarl he higher curve represens he period shor-run producion fronier. I indicaes how uch oupu can be produced fro a given labour inpu in period given he capial and echnolog available in period. Since he shor-run producion fronier shifs upward he oupu aainable fro a given labour inpu increases beween he wo periods such ha f ( ) > f ( ). The corresponding P also grows such ha f ( )/ > f ( )/. Thus he raio f ( )/f ( ) = (f ( )/ )/(f ( )/ ) capures he join effec on P growh of echnical progress and capial inpu growh. Noe ha he raio is also a easure of he disance beween he shor-run producion froniers of periods and in he direcion of he ais evaluaed a. The raio increases as he disance beween he period and he period shor-run producion froniers increases. Therefore he join effec of echnical progress and increase in capial inpus can be capured hroughou b easuring he shif in he shor-run producion fronier. [Place Figure appropriael here.] An uani of labour inpu can produce ore oupu in period han in period reflecing he posiive join effec of echnical progress and increase in capial inpus. The fir increases is deand for labour inpu fro o eploiing he increased producive capaci of labour inpu. Producion akes place a A for period and a B for period. The slope of he ra fro he origin o A and B indicaes he P of each period. Since / is saller han / P declines beween he wo periods. The fac ha P can decline despie he ouward shif in he shor-run producion fronier suggess ha anoher facor conribues o P growh. The pah fro A o B can be divided ino wo pars: he verical jup fro A o A and he oveen along he period shor-run producion fronier fro A o B. Along he verical jup fro A o A he P changes fro / o f ( )/. Is raio ( / )/(f ( )/ ) is considered o be he growh in P induced b he shif in he shor-run producion fronier which is he join effec of echnical progress and an increase in capial inpus. P growh is offse b he change in labour inpu fro o. The oveen along he period shor-run producion fronier fro A o B reduces P fro f ( )/ o ( / ). We call he P growh induced b 4

5 oveen along he shor-run producion fronier ( / )/(f ( )/ ) he reurns o scale effec. In his secion we illusrae wo sources of P growh using he siple one-oupu wo-inpus odel. However he division of he pah fro A o B ino wo seps fro A o A and fro A o B is erel an eaple. I is also possible o decopose he change fro A o B ino he oveen along he period shor-run producion funcion fro A o B and he verical jup fro B o B. In his case he forer oveen reflecs he reurns o scale effec and he laer oveen reflecs he join effec of echnical progress and capial inpu growh. For easuring he join effec of echnical progress and capial inpu growh he iporan consideraion is he uani of labour inpu a which he disance beween wo shor-run producion froniers is evaluaed. For easuring he reurns o scale effec wheher we consider he oveen along he period or shor-run producion fronier aers. Hereafer we generalize our discussion o he ore general uliple-inpus uliple-oupus case and propose easures for he wo effecs ha are iune fro choosing he arbirar benchark. 3. Join Effec of Technical Progress and Capial Inpu Growh A fir is considered as a producive eni ransforing inpus ino oupus. We assue ha here are M (ne) oupus = [ M ] T and P + inpus consising of P pes of capial inpus = [ P ] T and pes of labour inpus = [ ] T. 4 The period producion possibili se S consiss of all he feasible cobinaions of inpus and oupus and i is defined as follows: () S {( ):( ) can produce } We assue S is a closed se ha ehibis a free disposabili proper and includes he origin. 5 The period producion fronier which is he boundar of S is represened b he period inpu reuireen funcion G. I is defined as follows: (2) F ) in { :( ) S } ( I represens he iniu aoun of he firs capial inpu ha a fir can use a period producing oupu uaniies holding oher capial inpus = [ 2 P ] T and labour inpus fied. This funcion which is originall forulaed for he period producion fronier can be used for characerizing he period shor-run producion fronier. Given period capial inpu he se of oupus and labour inpus saisfing = F ( ) fors he period shor-run producion fronier. CCD (982) easures he shif in he producion fronier b using he oupu disance funcion. Adjusing heir approach we also use he oupu disance funcion o easure he shif in he shor-run producion fronier. Using he inpu reuireen funcion he period oupu disance funcion for = and is defined as follows: 4 Oupus include inerediae inpus. If oupu is an inerediae inpu hen <. Hence he noinal value of oal (ne) oupus p is he value-added ha a fir generaes. 5 Free disposabili proper eans ha given ( ) S if (* * *) ( ) (* * *) S. 5

6 (3) D : F O( ) in. Given capial inpus and labour inpus D O ( ) is he iniu conracion of oupus so ha he conraced oupus /D O ( ) capial inpus and labour inpus are on he period producion fronier. If ( ) is on he period producion fronier D O ( ) euals one. Noe ha D ( ) is linearl hoogeneous in. We can also relae i o he shor-run producion fronier. Given labour inpus D O ( ) is he iniu conracion of oupus so ha he conraced oupus /D O ( ) and labour inpu are on he period shor-run producion fronier. Thus D O ( ) provides a radial easure of he disance of o he period shorrun producion fronier. We easure he shif in he shor-run producion fronier b coparing he disance of o he shor-run producion froniers of he period and. I is defined as follows: 6 (4) SHIFT DO ( ) ). D ( ) ( O If echnical progress and capial inpu growh have a posiive effec on he producive capaci of labour beween period and he shor-run producion fronier shifs ouward. Given labour inpus ore oupus can be produced. Thus he iniu conracion facor for given oupus declines such ha D O ( ) D O ( ) leading o SHIFT( ). Siilarl he negaive join effec of echnical progress and capial inpu growh leads o SHIFT( ). Each choice of reference vecors ( ) igh generae a differen easure of he shif in he shor-run producion fronier going fro period o period. We calculae wo easures b using differen reference vecors ( ) and ( ). Since hese reference oupus and labour inpus are acuall chosen in each period he are euall reasonable. Following Fisher (922) and CCD (982) we use he geoeric ean of hese easures as a heoreical easure of he join effec of echnical progress and capial inpu growh SHIFT as follows: 7 (5) SHIFT SHIFT ( ) SHIFT ( ). The case of one oupu and one labour inpu offers us a graphical inerpreaion of SHIFT. In Figure i reduces o he following forula. (6) SHIFT ( f ( ) / )( / f ( )). Given a uani of labour inpu he raio of he oupu aainable fro such a labour inpu a period o he oupu aainable a period represens he een o which he 6 CCD (982) and Färe e al (994) inroduce a easure of he shif in he producion fronier b using he raio of he oupu disance funcion. Given ( ) Färe e al (994) easure he shif in he producion fronier b D O ( )/D O ( ). 7 Since he fir s profi aiizaion is assued i is possible o adop a differen forulaion for he easure of he shif in he shor-run producion fronier: SHIFT = (D O ( )/D O ( )) /2 (D O ( )/D O ( )) /2. This forulaion is closer o he original Maluis producivi (TFP growh) inde which was inroduced b CCD (982). 6

7 shor-run producion funcion epands. SHIFT is he geoeric ean of hose raios condiional on and. SHIFT is a heoreical easure defined b he unknown disance funcions and here are alernaive was of ipleening i. We show ha he heoreical easure coincides wih a forula of price and uani observaions under he assupion of a fir s profi aiizing behaviour and a ranslog funcional for for he oupu disance funcion. 8 Our approach is drawn fro CCD (982) ha ipleen he Maluis producivi inde which is he heoreical easure of he shif in he producion fronier. The show ha he Maluis producivi inde coincides wih a differen inde nuber forula of price and uani observaions which is called Törnvis producivi inde under soe assupion. 9 CCD (982) also show ha he firs-order derivaives of he disance funcion D wih respec o uaniies a he period acual producion plan ( ) are copuable fro price and uani observaions. Their euivalence resul beween he Maluis and Törnvis producivi indees relies on hese relaionships. Uilizing he sae relaionships we also show ha SHIFT coincides wih an inde nuber forula of price and uani observaions. Alhough all he following euaions (7)- (6) have been alread derived b CCD (982) we ouline how o copue he firsorder derivaives of he disance funcions fro price and uani observaions below for copleeness of discussion. The iplici funcion heore is applied o he inpu reuireen funcion F (/δ ) = o solve for δ = D O ( ) around ( ). Is derivaives are represened b he derivaives of F ( ). We have he following euaions for = and : (7) DO ( ) F ( ). F ( ) (8) D ( ) O F ( ( ) F ). (9) DO ( ) F ( ). F ( ) We assue he fir s profi aiizing behaviour. Thus ( ) >> N+P+ is a soluion o he following period profi aiizaion proble for = and : () a{ p r F ( ) r w } 8 Alernaive approaches are o esiae he underling disance funcion b econoeric and linear prograing approaches. Eiher of he reuires sufficien epirical observaions. Our approach originaed b CCD (982) is applicable as long as price and uani observaions are available a he curren and he reference periods. See Nishiizu and Page (982) for he applicaion of he econoeric echniue and see Färe e al. (994) for he applicaion of he linear prograing echniue. 9 CCD (982) jusif he forula ha is Törvis oupu uani inde divided b Törvis uani inde. In addiion o he ranslog funcional for for he oupu disance funcion he assue he fir s cos iniizing and revenue aiizing. Their approach is called he inde nuber approach finding a forula of price and uani observaions ha approiae a heoreical easure. We assue he following hree condiions are saisfied for = and : ) F is differeniable a he poin ( ): 2) >> M and 3) F( ) >. 7

8 Oupus are sold a he posiive producer prices p = [p p M ] T >> capial inpus are purchased a he posiive renal prices r = [r r P ] T >> and labour inpus are purchased a he posiive wages w = [w w ] T >>. Noe ha r = [r r P ] T. The period profi aiizaion proble ields he following firs order condiions for = and : p ( () r F ). s (2) r F ). r ( w ( s (3) r F ). B subsiuing ()-(3) ino (7)-(9) we obain he following euaions for = and : (4) D ( ) / O p p. (5) ] r DO ( ) [ r / p ] [/ p. F ( ) r (6) D O ( ) w / p. The above euaions (4)-(6) allow us o copue he derivaives of he disance funcion wihou knowing he oupu disance funcion iself. The inforaion of he derivaives is useful for calculaing he values of he oupu disance funcions. However one disadvanage is ha he derivaives of he period oupu disance funcion need o be evaluaed a he period acual producion plan ( ) in euaions (4)-(6) for = and. The disance funcions evaluaed a he hpoheical producion plan such as ( ) and ( ) also consiue SHIFT. Hence he above euaions are no enough for ipleening SHIFT. In addiion o a fir s profi aiizaion we furher assue a following ranslog funcional for for he period oupu disance funcion for = and. I is defined as follows; (7) D O ( ) M P p p M P p P (/ 2) (/ 2) (/ 2) p p p p p M M i j P P i j i j p where he paraeers saisf he following resricions: (8) i j j i for all i and j such as i < j M; (9) i j j i for all i and j such as i < j P; (2) i j j i for i and j such as i < j ; N n (2) ; n i j i j i j i j i j M i j 8

9 (22) M i for =...M ; i M (23) p for p =...P ; (24) M for =... i Resricions (2)-(24) guaranee he linear hoogenei in. The ranslog funcional for characerized (7)-(24) is a fleible funcional for so ha i can approiae an arbirar oupu disance funcion o he second order a an arbirar poin. Thus assuing his funcional for does no har an generali of he oupu disance funcion. Noe ha he coefficiens for he linear ers and he consan er are allowed o var across periods. Thus echnical progress under he ranslog disance funcion is b no eans liied o Hicks neural and is able o represen a vas varie of echnical progress is possible. Under he assupion of he profi aiizing behaviour and he ranslog funcional for a heoreical easure SHIFT is copued fro price and uani observaions. Proposiion : Assue ha he oupu disance funcions D O and D O have he ranslog funcional for defined b (7)-(24) and a fir follows copeiive profi aiizing behaviour in period = and. Then he join effec of echnical progress and capial inpu growh SHIFT can be copued fro observed prices and uaniies as follows: M (25) SHIFT s s where s and s are he average value-added shares of oupu and labour inpu beween periods and such ha; p p s and s 2 p p w w 2 p p The inde nuber forula in (25) can be inerpreed as he raio of a uani inde of oupus o a uani inde of labour inpu. Noe ha no daa of price and uani of capial inpus appear in his forula. Even hough he shif in he shor-run producion fronier reflecs echnical progress as well as he change in capial inpu we can easure is shif wihou resor o capial inpu daa eplicil. 4. Reurns o Scale Effec As is shown in Figure he shif in he shor-run producion fronier is no he onl facor conribuing o he growh in P. Even when here is no change in he shor-run producion fronier he oveen along he fronier could raise P eploiing he concavi of he shor-run producion fronier. We call he P growh induced b he oveen along he shor-run producion fronier he reurns o scale effec. In he siple odel consising of one oupu and one labour inpu P is defined as oupu per one uni of labour inpu. Therefore P growh which is he growh rae of P beween he curren period and he previous period coincides wih he growh rae of oupu divided b he growh rae of labour inpu. Since he reurns o scale effec is he P growh induced b he oveen along he shor-run producion fronier i is 9

10 copued b he growh raes of oupu and labour inpu beween he wo end poins of he oveen. Figure 2 shows how he oveen along he period shor-run producion fronier fro poin C o D affecs P. Coparing wo poins C and D he growh rae of oupu is f ( )/f ( ) and he growh raio of labour inpu /. The growh rae of P beween wo poins coincides wih he growh rae of oupu divided b ha of labour inpu so ha (f ( )/f ( ))/( / ) = (f ( )/ )/( f ( )/ ). [Place Figure 2 appropriael here.] We generalize he growh raes of labour inpu and oupu beween wo poins on he period shor-run producion fronier in order o easure he reurns o scale effec in he uliple-inpus uliple-oupus case. Firs we invesigae how o easure he counerpar of he growh rae of labour inpus in he uliple-inpus uliple-oupus case. CCD (982) define he inpu uani inde which is he counerpar of he growh rae of oal inpus b coparing he radial disance beween wo inpu vecors and he period producion fronier. The inpu disance funcion is used for he radial scaling of oal inpus. Adaping he inpu disance funcion used b CCD (982) we inroduce he labour inpu disance funcion ha easure he radial disance of labour inpus o he period producion fronier. The period labour inpu disance funcion for = and is defined as follows: (26) D ) a : F. ( Given oupus and capial inpus D ( ) is he aiu conracion of labour inpus so ha he conraced labour inpus /D ( ) and capial inpus wih oupus are on he period producion fronier. If ( ) is on he period producion fronier D ( ) euals one. Noe ha D ( ) is linearl hoogeneous in. We can also relae i o he shor-run producion fronier. Given oupus D ( ) is he aiu conracion of labour inpus so ha he conraced labour inpus /D ( ) and oupus are on he period shor-run producion fronier. Thus D ( ) provides a radial easure of he disance of o he period shor-run producion fronier. We consruc he counerpar of he growh rae of labour inpu b coparing wo labour inpus and o he period shor-run producion fronier. I is defined as follows; (27) ABOUR ( ) D ( ) / D ( ) If labour inpus grow beween wo periods oves furher awa fro he origin han eaning ha he labour inpu vecor is larger han he labour inpu vecor. The aiu conracion facor for producing oupus wih he period capial inpus using he period echnolog increases such ha D ( ) D ( ) leading o ABOUR( ). Siilarl if labour inpus shrinks beween wo periods oves closer o he origin han leadings o ABOUR( ). Second we invesigae how o easure he growh rae of oupus on he period shor-run producion fronier. In he uliple-inpus uliple-oupus case here is he se of oupus aainable fro given labour inpus wih period echnolog and capial inpus. I is a par of he period shor-run producion fronier. Since D O ( ) provides a radial easure of he disance of o he period shor-run

11 producion fronier D O ( ) provides a radial easure of he disance of o he se of oupus aainable fro labour inpus wih period echnolog and capial inpu. We consruc he counerpar of he growh rae of oupus on he period shor-run producion fronier b coparing oupus o he ses of oupus aainable fro and wih period capial inpu. I is defined as follows: (28) OUTPUT D D ( ) O( ) / O( ) If he labour inpu growh akes i possible o produce ore oupus wih given capial inpu using echnolog he se of oupus aainable fro shifs ouward o ha of oupus aainable fro. Thus he iniu conracion facor for given oupus declines such ha D O ( ) D O ( ) leading o OUTPUT( ). Siilarl if he change in labour inpus reduces oupus aainable fro given capial and labour inpus leads o OUTPUT( ). Using he counerpars of he growh rae of oupus and labour inpus beween wo poins we can propose he easure of P growh beween he wo poins on he period shor-run producion fronier. When we use he period shor-run producion and oupus as reference echnolog and oupus he reurns o scale effec is defined as follows: SCAE( ) OUTPUT ( ) / ABOUR ( ) (29) DO ( ) D ( ). DO ( ) D ( ) Each choice of he reference echnolog igh generae a differen easure of he reurns o scale effec going fro period o period. We calculae wo easures b using wo differen reference echnologies: he period and shor-run producion froniers. Since hese reference echnolog are euall reasonable we use he geoeric ean of hese easures as a heoreical inde of he reurns o scale effec SCAE as follows; (3) SCAE SCAE( ) SCAE ( ). The case of one oupu and one labour inpu offers us a graphical inerpreaion of SHIFT. In Figure i reduces o he following forula. f ( ) (3) SCAE. f ( ) Given he period shor-run producion fronier he raio of he P associaed wih o he P associaed wih represens he P growh induced b he change in labour inpu. SCAE is he geoeric ean of hose raios condiional on he period and shor-run producion froniers. As CCD (982) appl he iplici funcion heore o he inpu reuireen funcion wih he oupu disance funcion such as F ( /δ ) = where δ = D O ( ) we appl he iplici funcion heore o he inpu reuireen funcion wih he labour inpu disance funcion such as F ( /δ) = where δ = D (

12 ). In his case D ( ) is differeniable around he poin ( ). Is derivaives are represened b he derivaives of F ( ). We have he following euaions for = and : (32) D ( ) F ( ). F ( ) (33) D ( ) ( ( ) F ). F (34) D ( ) F ( ). F ( ) We assue ha ( ) >> N+P+ is a soluion o he period profi aiizaion proble () for = and. B subsiuing ()-(3) obained fro he profi aiizaion ino (32)-(34) we obain he following euaions for = and : (35) D ( ) / p w. r (36) D ( ) [/ w ]. r (37) D ( ) w / w. The above euaions (35)-(37) allow us o copue he derivaives of he labour inpu disance funcion wihou knowing he labour inpu disance funcion iself. The inforaion of he derivaives is useful for calculaing he values of SCAE which is defined b he disance funcions. However one disadvanage is ha he derivaives of he period disance funcion need o be evaluaed a he period acual producion plan ( ) in euaions (35)-(37) for = and. The disance funcions evaluaed a he hpoheical producion plan such as ( ) and ( ) also consiue SCAE. Hence he above euaions are no enough for obaining SCAE. In addiion o he fir s profi aiizaion we furher assue he ranslog funcional for for he period labour inpu disance funcion for = and. I is defined as follows; (38) D ( ) P p M P M p p P p p (/ 2) (/ 2) (/ 2) M M i j P P i j i j p p p p where he paraeers saisf he following resricions: (39) i j j i for all i and j such as i < j M; i j i j i j i j i j M i j We also assue he following hree condiions are saisfied for = and : ) F is differeniable a he poin ( ): 2) >> M and 3) F( ) >. 2

13 (4) i j j i for all i and j such as i < j P; (4) i j j i for i and j such as i < j ; (42) ; (43) i for =... ; i (44) for =... M ; (45) p for p =... P. The euaion (38) is he sae funcional for defined b (7) ha we assue for he oupu disance funcion in he discussion of SHIFT. However he resricions on paraeers on he labour inpu disance funcion differ fro hose on he oupu disance funcion. We replace he resricions (2)-(24) b (42)-(45). While he resricions (2)-(24) guaranee he linear hoogenei in oupus for he oupu disance funcion he resricions (42)-(45) guaranees he linear hoogenei in labour inpus for he labour inpu disance funcion. The ranslog funcional for characerized b (38)-(45) is a fleible funcional for so ha i can approiae an arbirar labour inpu disance funcion o he second order a an arbirar poin. Thus assuing his funcional for does no har an generali of he labour inpu disance funcion. Noe ha he coefficiens for he linear ers and he consan er are allowed o var across periods. Thus echnical progress under he ranslog disance funcion is b no eans liied o Hicks neural and a vas varie of echnical progress is possible. Under he assupion of he profi aiizing behaviour and he ranslog funcional for a heoreical inde of he reurns o scale SCAE is copuable fro price and uani observaions. Proposiion 2: Assue ha he oupu disance funcions D O and D O have he ranslog funcional for defined b (7)-(24) he labour inpu disance funcions D and D have he ranslog funcional for defined b (38)-(45) and a fir follows copeiive profi aiizing behaviour in period = and. Then he reurns o scale effec SCAE can be copued fro observable prices and uaniies as follows: (46). SCAE s s. where s is he average value-added shares of labour inpu and s is he average labour-copensaion share of labour inpu beween periods and such ha; w w s and s 2 p p w w 2 w w The inde nuber forula in he righ-hand side of (46) can be inerpreed as he raio of he uani indees of labour inpus. Boh ers in (46) are he weighed geoeric average of he growh raes for labour inpus. The firs er uses he value added share and he second er uses he labour copensaion share. Thus if capial share which is he noinal share of capial inpus o he value added is large he. 3

14 difference beween wo ers (46) also becoes large aking he value of SCAE larger. Conversel in he case when no capial inpus are uilized for producion he reurns o scale effec disappears. Saring fro undersanding ha he wo conribuion facors eiss for he P growh we reached he inde nuber forula for hese facors independenl. Our resul however does no den he possibili of he eisence of oher eplanaor facors o he P growh. Forunael wo facors of SHIFT and SCAE can full eplain he P growh. The produc of SHIFT and SCAE coincides wih he inde of he P growh as follows: Corollar : Assue ha he oupu disance funcions D O and D O have he ranslog funcional for defined b (7)-(24) he labour inpu disance funcions D and D have he ranslog funcional for defined b (38)-(45) and a fir follows copeiive profi aiizing behaviour in period = and. Then he produc of SHIFT and SCAE can be copued fro observed prices and uaniies as follows: M (47) SHIFT SCAE s s where s is he average value-added shares of oupu and s is he average labourcopensaion share of labour inpu beween periods and such ha; p p s and s 2 p p w w 2 w w The righ side of he euaion (47) represens he logarih of P growh. The firs er of he righ hand side coincides wih he Törnvis uani inde of oupus and he second er is he Törnvis uani inde of labour inpus. The euaion (47) allows us o full decopose P growh ino wo coponens such as SHIFT and SCAE in he case of uliple inpus and oupus. I is a generalizaion of he siple eplanaion in he case of one inpu and one oupu ha he P grows is induced b he shif in he producion fronier and he oveen along he producion fronier.. 5. An Applicaion o U.S. Indusr Daa We appl he decoposiion resul o U.S. indusr daa. Daase covering he period is aken fro he coprehensive indusr daase called he EU EMS Growh and Producivi Accouns. 2 Indusr daa drawn fro he daabase consis of gross oupus and inerediae inpus a curren and consan prices and hours worked of eploen b 3 indusries. These indusr daa are organized according o he Sse of Indusr Classificaion (SIC) adoped b he U.S. official saisics. 3 2 Daa are downloaded fro he EU EMS websie (hp:// The deailed eplanaion abou his coprehensive inernaional daabase is found in O Mahon and Tier (29). The U.S. indusr daa of EU EMS is consruced b Dale Jorgenson and his research group. See Jorgenson Ho and Siroh (28). 3 For each indusr here eis one pe of gross oupu and one pe of inerediae inpu. Their deflaor varies across indusries. abour inpu is hours worked b oal eploen. Toal eploen in each indusr includes eploees and he self-eploed engaged in he producion of he indusr. 4

15 The are caegorized eiher as he goods producing secor (goods secor hereafer) or he services providing secor (services secor hereafer). [Place Table appropriael here.] Table copares P growh and is coponens across he whole econo he goods secor and he services secor. For he enire saple period he reurns o scale effec has a negaive ipac on P growh reflecing he average growh rae of hours worked of.43 percen. While he average rae of he join effec of echnical progress and capial inpu growh is 2.4 percen i is largel offse b he reurns o scale effec of -.52 percen. There is a hpohesis so called Bauol s disease saing ha P in he services secor is likel o be sagnaed and lower han ha of he goods secor which has been widel advocaed b Triple and Bosworh (24 26) and Bosworh and Triple (27). Such difference in P growh beween wo secors is also docuened in our daase. The average growh rae of he goods secor P of 2.33 percen is alos wice as large as ha of he goods secor P of.27 percen. While he reurns o scale effec is suble and negligible in he goods secor he significan negaive reurns o scale effec is found in he services secor. Reoving he reurns o scale effec largel raises he services secor P growh. The ore han half of he difference in P growh beween wo secors can be eplained b he difference in he reurns o scale effec. The join effecs of echnical progress and capial inpu growh in boh secors are ver close on average over he period 97-25: 2.33 percen for he goods secor and 2. percen for he services secor. I reflecs ha he increase in hours worked is aking place osl in he services secor. While hours worked for he goods secor grows a he average growh rae of.5 percen onl hours worked for he services secor grows a he average growh rae of.97 percen. I is convenien o divide he enire saple period ino wo periods: he producivi slowdown period and he producivi resurgence period Producivi slowdown in U.S. econo sared since he earl 97s wih he average growh rae of.2 percen during he period Producivi growh surged afer 995 wih he average growh rae of 2.3 percen during he period As Triple and Bosworh (24 26) and Bosworh and Triple (27) poined ou he services secor P grows slowl especiall during he producivi slowdown period wih he average growh rae of percen during he period A he sae period he goods secor grows wih he average growh rae of.57 percen. However once we conrol he reurns o scale effec and consider he join effec of echnical progress and capial inpu growh onl he services secor wih he average rae of.82 percen forereaches he goods secor wih he average rae of.62 percen. Thus even hough P growh is saller in he services secor han in he goods secor he producive capaci of labour which is he oupu aainable fro given labour inpus increases ore in he services secor han in he goods secor. I reflecs ha he large increase in hours worked in he services secor resrains P fro growing largel. While he hours worked in he goods secor grow slighl during he period i even declines during he period leading o he posiive reurns o scale effec. On he oher hand hours worked in he services secor increases hroughou he saple period leading o 4 I is known ha he sagnaion of he U.S. econo sared since 973. Since he daase is available since 97 and he period is oo enough o idenif he underling rend we deal wih he period

16 he negaive reurns o scale effec. Thus he par of he gap in P growh eplained b he gap in he reurns o scale effec beween wo secors becoe even larger in he period [Place Table 2 appropriael here.] The reurns o scale effec growh in hours worked and capial share b indusr are shown in Table 2. 5 The paer found in he aggregae sud based on he secor daa in Table is also docuened in he deailed indusries. Mos of he indusries in services secor show he negaive reurns o scale effec. I is especiall significan fro On he oher hand an indusries in he goods secor show odes reurns o scale effec fro and he posiive reurns o scale effec fro Teiles eile producs leaher and foowear (SIC7-9) and coke refined peroleu producs and nuclear fuel (SIC 23) are ecepional indusries in he goods secor. Reflecing largel decreasing hours worked he posiive and large reurns o scale effecs are observed in boh indusries wihin he goods secor in he period The reurns o scale effec depends on capial share as well as increases in hours worked. The deailed indusr sud reveals cases when he reurns o scale effec ges larger for differen reasons. On average over he whole saple period he os significan reurns o scale effec is found in Real esae aciviies (SIC 7) averaging percen. Rening of euipen and oher business aciviies (SIC 7-74) shows he second larges reurns o scale effec in agniude. However is average growh rae is -.97 percen which is uch saller han he value of he real esae aciviies. The reason wh he reurns o scale effec becoes large differs beween wo indusries. Rening of euipen and oher business aciviies has he highes growh of hours worked averaging 4.68 percen leading o he second large reurns o scale effec. However hours worked in real esae aciviies does no grow significanl wih he average growh rae of 2.6 percen. For eaple alhough hours worked in healh and social work (SIC N) grows larger wih he average growh rae of 3.3 percen is rae of reurns effec is -.56 on average which is saller han even one uarer of real esae aciviies. The reason of he larges reurns o scale effec is because he real esae aciviies has he large capial share 9.52 percen. The ipac of he growh in hours worked on he reurns o scale effec is largel aplified b he large capial share. 6. Conclusion In his paper we disinguished wo effecs on P growh b eaining he shor-run producion fronier. The join effec of echnical progress and capial inpu growh appears as he growh in P ha is induced b he shif in he shor-run producion fronier. The reurns o scale effec appears as he P growh induced b he oveen along he shor-run producion fronier. The P growh calculaed b Törnvis uani indees is full decoposed ino he produc of hese wo effecs. We applied his decoposiion resul o U.S. indusr daa for he period I is shown ha a large par of he difference in P growh beween he goods secor and he services secor can be eplained b he reurns o scale effec. 5 Since privae households wih eploed persons is no endowed wih capial inpus and all he producion has been conduced b using labour inpus onl here is no reurns o scale effec for his indusr. Thus his indusr is ecluded fro Table 2. 6

17 In his paper we assued he fir s profi aiizing behaviour and ruled ou inefficien producion processes. If we rela he fir s profi aiizing behaviour anoher facor echnical efficienc change appears in he decoposiion of P growh. Even wih no change in he shor-run producion fronier and no change in labour inpu a fir can approach closer o he shor-run producion fronier b iproving echnical efficienc. For eaple a fir iproves echnical efficienc b increasing oupu up o he aiu level aainable fro given labour inpus under curren echnolog. For he ipleenaion of he decoposiion of he P growh wihou assuing a fir s profi aiizing behaviour we can esiae he disance funcion b using econoeric echniues or Daa Envelopen Analsis s linear prograing echniue. However we leave his eercise o he fuure research. 7

18 Appendi A Proof of Proposiion DO ( ) DO ( ) SHIFT 2 DO ( ) 2 DO ( ) DO ( ) DO ( ) 2 DO ( ) 2 DO ( ) Since he fir s profi aiizaion is assued he period producion plan is on he period producion fronier for = and. M DO ( ) DO ( ) 4 M DO ( ) DO ( ) 4 DO ( ) DO ( ) DO ( M DO ( ) DO ( ) 2 DO ( ) DO ( ) 2 M DO ( ) DO ( ) 4 M DO ( ) DO ( ) 4 ) DO ( ) using he ranslog ideni in CCD (982) DO ( ) DO ( ) DO ( ) DO ( ) M DO ( ) DO ( ) 2 DO ( ) DO ( ) 2 2 M p p p p 2 w p p w fro he euaion (7). 8

19 Proof of Proposiion 2 DO ( ) D ( ) SCAE 2 DO ( ) D ( ) D ( ) ( ) O DO 2 DO ( ) DO ( ) subsiuing euaions (4)-(6). DO ( ) D ( ) 2 DO ( ) D ( ) D ( ) ( ) D 2 D ( ) D ( ) Since he fir s profi aiizaion is assued he period producion plan is on he period producion fronier for = and. M DO ( ) DO ( ) 4 M D ( ) D ( ) 4 DO ( ) DO ( ) D ( ) D ( ) M DO ( ) DO ( ) 2 D ( ) D ( ) 2 M DO ( ) DO ( ) 4 M D ( ) D ( ) 4 using he ranslog ideni in CCD (982) DO ( ) DO ( ) D ( ) D ( ) 9

20 2 ( ) ( ) 2 ( ) ( ) 2 M O O D D D D fro he euaion (7) and (38). w w w w 2 2 w w p p subsiuing euaions (4)-(6) and euaions (35)-(37).

21 References Balk B.M. (2) Divisia Price and uani Indees: 75 Years Afer Deparen of Saisical Mehods Saisics Neherlands Bosworh B.P. and J.E. Triple (27) The Earl 2s Cenur U.S. Producivi Epansion is Sill in Services Inernaional Producivi Monior Caves D.W.. Chrisensen and W.E. Diewer (982) The Econoic Theor of Inde Nubers and he Measureen of Inpu Oupu and Producivi Econoerica Chabers R.G. (988) Applied Producion Analsis: A Dual Approach New York: Cabridge Universi Press. Diewer W.E. (974) Applicaions of Duali Theor in M.D. Inriligaor and D.A. endrick (ed.) Froniers of uaniaive Econoics Vol. II Aserda: Norh-Holland pp Diewer W.E. (976) Eac and Superlaive Inde Nubers Journal of Econoerics Diewer W.E. (978) Superlaive Inde Nubers and Consisenc in Aggregaion Econoerica Diewer W.E. and C.J. Morrison (986) Adjusing Oupu and Producivi Indees for Changes in he Ters of Trade Econoic Journal Diewer W.E. and A.O. Nakaura (27) The Measureen of Producivi for Naions in J.J. Heckan and E.E. eaer (ed.) Handbook of Econoerics Vol. 6 Chaper 66 Aserda: Elsevier pp Fare R. S. Grosskopf M. Norris and Z. Zhang (994) Producivi Growh Technical Progress and Efficienc Change in Indusrialized Counries Aerican Econoic Review Griliches Z. (987) Producivi: Measureen Probles pp. -3 in J. Eawell M. Milgae and P. Newan (ed.) The New Palgrave: A Dicionar of Econoics. New York: McMillan. Hall R.E. and C.I. Jones (999). Wh do soe counries produce so uch ore oupu per worker han ohers? uarerl Journal of Econoics Hoelling H. (932) Edgeworh s Taaion Parado and he Naure of Deand and Suppl Funcions Journal of Poliical Econo Hulen C.R. (978) Growh Accouning wih Inerediae Inpus Review of Econoic Sudies Jones C.I. (22) Sources of U.S. Econoic Growh in a World of Idea Aerican Econoic Review Jorgenson D.W. and.j. Siroh (2) Raising he Speed ii: U.S. Econoic Growh in he Inforaion Age Brookings Papers on Econoic Acivi Jorgenson D.W. M.S. Ho and.j. Siroh (28) A Rerospecive ook a he U.S. Producivi Growh Resurgence Journal of Econoic Perspecive

22 ohli U. (24b) Real GDP Real Doesic Incoe and Ters of Trade Changes Journal of Inernaional Econoics Nishiizu M. and J.M. Page (982) Toal Facor Producivi Growh Technical Progress and Technical Efficienc Change: Diensions of Producivi Change in Yugoslavia Econoic Journal O Mahon M. and M.P. Tier (29) Oupu Inpu and Producivi Measures a he Indusr evel: he EU EMS Daabase Econoic Journal 9 F374- F43. Sauelson P.A. (953) Prices of Facors and Goods in General Euilibriu Review of Econoic Sudies 2-2. Triple J.E. and B.P. Bosworh (24) Services Producivi in he Unied Saes: New Sources of Econoic Growh Washingon D.C.: Brookings Insiuion Press. Triple J.E. and B.P. Bosworh (26) Bauol s Disease Has Been Cured: IT and Muli-facor Producivi in U.S. Services Indusries The New Econo and Beond: Pas Presen and Fuure. Dennis W. Jansen (eds.) Chelenha: Edgar Elgar. 22

23 Table : abour Producivi Growh and Is Coponens (% ) Whole Econo abour Producivi Growh Technical Progress and Capial Inpu Growh Reurns o Scale Effec Hours Worked Capial Share Goods Secor abour Producivi Growh Technical Progress and Capial Inpu Growh Reurns o Scale Effec Hours Worked Capial Share Services Secor abour Producivi Growh Technical Progress and Capial Inpu Growh Reurns o Scale Effec Hours Worked Capial Share

24 Table 2: Indusr abour Producivi and Reurns o Scale Effec (% ) abour Producivi Reurns o Scale Effec Hours Worked Capial Share Goods Secor Agriculure huning foresr and fishing Mining and uarring Food producs beverages and obacco Teiles eile producs leaher and foowear Oher anufacuring Wood and producs of wood and cork Pulp paper paper producs prining and publishing Coke refined peroleu producs and nuclear fuel Cheicals and cheical producs Rubber and plasics producs Oher non-eallic ineral producs Basic eals and fabricaed eal producs Machiner nec Elecrical and opical euipen Transpor euipen Elecrici gas and waer suppl Consrucion Services Secor Sale and repair of oor vehicles and oorccles Wholesale rade and coission rade Reail rade repair of household goods Transpor and sorage Pos and elecounicaions Financial inerediaion Rening of euipen and oher business aciviies Hoels and resaurans Oher couni social and personal services Public adin and defence; copulsor social securi Educaion Healh and social work Real esae aciviies

25 f ( ) B B f f ( ( ) ) f ( ) A A Figure : abour Producivi Growh and Shif in he Shor-run Producion Fronier f ( ) D f ( ) f ( ) C Figure 2: Reurns o Scale Effec and Moveen along he Shor-run Producion Fronier 25