The Reurns o Scale Effec in abour Produciviy Growh Hideyuki Mizobuchi June 4 2 Absrac abour produciviy is defined as oupu per uni of labour inpu. Economiss acknowledge ha echnical progress as well as growh in capial inpus increases labour produciviy. However lile aenion has been paid o he fac ha changes in labour inpu alone could also impac labour produciviy. Since his effec disappears for he consan reurns o scale shor-run producion fronier we call i he reurns o scale effec. We decompose he growh in labour produciviy ino wo componens: ) he join effec of echnical progress and capial inpu growh and 2) he reurns o scale effec. We propose heoreical measures for hese wo componens and show ha hey coincide wih he inde number formulae consising of prices and uaniies of inpus and oupus. We hen apply he resuls of our decomposiion o U.S. indusry daa for 987 27. I is acknowledged ha labour produciviy in he services indusries grows much more slowly han in he goods indusries. We conclude ha he reurns o scale effec can eplain a large par of he gap in labour produciviy growh beween he wo indusry groups. ey Words: abour produciviy inde numbers Malmuis inde Törnvis inde oupu disance funcion inpu disance funcion JE classificaion: C4 D24 O47 O5 Faculy of Economics Ryukoku Universiy 67 Fukakusa Tsukamoo-cho Fushimi-ku yoo 62-8577 Japan; mizobuchi@econ.ryukoku.ac.jp
. Inroducion Economiss broadly hink of produciviy as measuring he curren sae of he echnology used in producing he firm s goods and services. The producion fronier consising of inpus and he maimum oupu aainable from hem characerizes he prevailing sae of echnology. Produciviy growh is ofen idenified by he shif in he producion fronier reflecing changes in producion echnology. 2 However produciviy growh can also be driven by movemen along he producion fronier. Even in he absence of changes in he producion fronier changes in he inpus used for producion can lead o produciviy growh moving along he producion fronier and making use of is curvaure. Produciviy growh ha is induced by he movemen 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 properly evaluae improvemens in he underlying producion echnology reflecing he shif in he producion fronier we mus disenangle he reurns o scale effec from labour produciviy. Produciviy measures can be classified ino wo ypes: oal facor produciviy (TFP) and parial facor produciviy. The former 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 produciviy (P) among several measures of parial facor produciviy. P is defined as oupu per labour inpu in he simple one-oupu one-labour-inpu case. Economy-wide P is he criical deerminan of a counry s sandard of living in he long-run. For eample U.S. hisory reveals ha increases in P have ranslaed o nearly one-for-one increases in per capia income over a long period of ime. 3 The imporance of P as a source for he progress of economic wellbeing promps many researchers o invesigae wha deermines P growh. 4 Technical progress and capial inpu growh have been emphasized as he main deerminans of a counry s enormous 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 more eplanaory facor o P growh. 5 See Griliches (987). The same inerpreaion is also found in Chambers (988). 2 In principle produciviy improvemen can occur hrough echnological progress and gains in echnical efficiency. Technical efficiency is he disance beween he producion plan and he producion fronier. The presen paper assumes a firm s profi-maimizing behaviour and in our model he curren producion plan is always on he curren producion fronier. The assumpion of profi maimizaion is common in economic approaches o inde numbers. See Caves Chrisensen and Diewer (982) and Diewer and Morrison (986). 3 See he 2 Economic Repor of he Presiden. 4 The P growh and he capial inpu growh are he abbreviaions for he growh raes of P and capial inpu. In his paper for eample 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. 5 If he number of workers or he number of hours worked are adoped as he measure of labour inpu changes in characerisics of labour inpu also affecs P. These auhors also found an imporan role of labour ualiy growh (in oher words human capial accumulaion) for eplaining changes in heir measure of P ha is defined by using he number of workers or he number of hours worked. Since we allow wages o vary across differen ypes of labour inpu he ualiy of each labour inpu is differeniaed in our measure of labour inpu. Thus we ignore he role of he labour ualiy growh for eplaining P growh hroughou his paper. See Foonoe 6 for he unmeasured improvemen in labour ualiy.
P relaes labour inpus o oupus holding echnology and capial inpus fied. The shor-run producion fronier which consiss of labour inpus and he maimum oupu aainable from hem represens he capaciy of curren echnology 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 by movemen along he shor-run producion fronier has never been eposed. We decompose P growh ino wo componens: ) he join effec of echnical progress and capial inpu growh 2) he reurns o scale effec. 6 Firs we propose heoreical measures represening he wo effecs by using disance funcions. Second we derive he inde number formulae consising of prices and uaniies and show ha hey coincide wih heoreical measures assuming he ranslog funcional form for he disance funcions and he firm s profi-maimizing behaviour. Our approach o implemening heoreical measures is drawn from Caves Chrisensen and Diewer (982) (hereafer CCD). Using he disance funcion CCD formulae he (heoreical) Malmuis produciviy inde which measures he shif in he producion fronier and show ha he Malmuis produciviy inde and he Törnvis produciviy inde coincide assuming he ranslog funcional form for he disance funcions and he firm s profi-maimizing behaviour. The Törnvis produciviy inde is a measure for he TFP growh calculaed by he Törnvis uaniy indees. I is an inde number formula consising of prices and uaniies of inpus and oupus. Euivalence beween he wo indees breaks down if he underlying echnology does no ehibi consan reurns o scale. CCD shows ha is difference depends on he degree of reurns o scale in he underlying echnology which capures he curvaure of he producion fronier. Thus following Diewer and Nakamura (27) and Diewer and Fo (2) we can inerpre ha CCD decompose he TFP growh ha is calculaed by he Törnvis uaniy indees ino Malmuis produciviy inde and he reurns o scale effec. 7 The former componen capures TFP growh induced by he shif in he producion fronier. The laer componen which is he difference beween he Malmuis produciviy inde and he Törnvis produciviy inde capures TFP growh induced by he movemen along he producion fronier eploiing is curvaure. Many researchers have been concerned wih he growh in TFP induced by movemen along he underlying producion fronier. For eample ovell (23) calls i he scale effec. In he lieraure of Daa Envelopmen Analysis (Balk 2 Coelli e al. 23) he produc of scale efficiency change and inpu mi effec or ha of scale efficiency change and oupu mi effec summarizes he TFP growh induced by movemen along he producion fronier and i can be inerpreed as he reurns o scale effec. 8 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 more 6 In case when our measure of labour inpus fails o capure he improvemen in labour ualiy TFP growh induced by ha unmeasured improvemen in labour ualiy is inerpreed as ha induced by echnical progress. Thus i is capured by he join effec of echnical progress and capial inpu growh. 7 CCD use he word of scale facor for he reurns o scale. 8 For he decomposiion of Nemoo and Goo (25) we inerpre he produc of scale change and inpu and oupu mi effecs as he reurns o scale effec. Their resul idenifies he combined effec of changes in he composiion of inpus and ha of oupus. 3
imporan in eplaining P growh han in eplaining TFP growh. When he underlying echnology ehibis consan reurns o scale he reurns o scale effec disappears from TFP growh. However i sill plays a role for P growh. This is because even if he underlying echnology ehibis consan reurns o scale he shorrun producion fronier is likely no o ehibi consan reurns o scale. Triple and Bosworh (24 26) and Bosworh and Triple (27) observed ha P growh in he service indusries was much less han in he goods indusries in U.S. economy since he early 97s. As we discussed above here are wo underlying facors o P growh; herefore possible eplanaions for he low P growh in he services indusries are as follows: ) he join effec of echnical progress and increases in capial inpus is modes; 2) an increase in labour inpus induces negaive reurns o scale effecs; 3) boh ) and 2). We apply our decomposiion resul o U.S. indusry daa o compare he relaive conribuions of he wo effecs. Secion 2 illusraes he wo effecs underlying P growh graphically. Secion 3 discusses he measure of he join effec of echnical progress and capial inpu growh in he muliple-inpus muliple-oupus case. Secion 4 discusses he measure of he reurns o scale effec in he muliple-inpus muliple-oupus case. We show ha he produc of he join effec of echnical progress and capial inpu growh and he reurns o scale effec coincides wih P growh. Secion 5 includes he applicaion o he U.S. indusry daa. Secion 6 presens he conclusions. 2. Two Sources of abour Produciviy Growh We display graphically wha derives P growh using a simple model of one oupu y and wo inpus: labour inpu and capial inpu. Suppose ha a firm produces oupus y and y using inpus ( ) and ( ). Period producion echnology is described by he period producion funcion y = f ( ) for = and. e us begin by 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 capial inpu growh posiively affec he producive capaciy of labour. The lower curve represens he period shor-run producion fronier. I indicaes how much oupu can be produced by using a specified uaniy of labour given he capial and echnology available in period. Similarly he higher curve represens he period shor-run producion fronier. I indicaes how much oupu can be produced by using a specified uaniy of labour given he capial and echnology available in period. Since he shor-run producion fronier shifs upward he oupu aainable from 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 measure of he disance beween he shor-run producion froniers of periods and in he direcion of he y 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 capial inpu growh can be capured hroughou by measuring he shif in he shor-run producion fronier. [Place Figure appropriaely here] 4
Any uaniy of labour inpu can produce more oupu in period han in period reflecing he posiive join effec of echnical progress and capial inpu growh. The firm increases is demand for labour inpu from o eploiing he increased producive capaciy of labour inpu. Suppose ha producion akes place a A for period and a B for period. The slope of he ray from he origin o A and B indicaes he P of each period. Since y / is smaller han y / 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. 9 The pah from A o B can be divided ino wo pars: he verical jump from A o A and he movemen along he period shor-run producion fronier from A o B. Along he verical jump from A o A he P changes from y / o f ( )/. Is raio (y / )/(f ( )/ ) is considered o be he growh in P induced by he shif in he shor-run producion fronier which is he join effec of echnical progress and capial inpu growh. P growh is offse by he change in labour inpu from o. The movemen along he period shor-run producion fronier from A o B reduces P from f ( )/ o y /. We call he P growh induced by movemen along he shor-run producion fronier (y / )/(f ( )/ ) he reurns o scale effec. In his secion we illusrae wo sources of P growh using he simple one-oupu wo-inpus model. However he division of he pah from A o B ino wo seps from A o A and from A o B is merely an eample. I is also possible o decompose he change from A o B ino he movemen along he period shor-run producion funcion from A o B and he verical jump from B o B. In his case he former movemen reflecs he reurns o scale effec and he laer jump reflecs he join effec of echnical progress and capial inpu growh. For measuring he join effec of echnical progress and capial inpu growh he imporan consideraion is he uaniy of labour inpu a which he disance beween wo shor-run producion froniers is evaluaed. For measuring he reurns o scale effec wheher we consider he movemen along he period or shor-run producion fronier maers. Hereafer we generalize our discussion o he more general muliple-inpus muliple-oupus case and propose measures for he wo effecs ha are immune from choosing he arbirary benchmark. 3. Join Effec of Technical Progress and Capial Inpu Growh A firm is considered as a producive eniy ransforming inpus ino oupus. We assume here are M (ne) oupus y = [y y M ] T and P + Q inpus consising of P ypes of capial inpus = [ P ] T and Q ypes of labour inpus = [ Q ] T. The period producion possibiliy se S consiss of all feasible combinaions of inpus and oupus and i is defined as () S {( y ) :( )can produce y}. 9 This is jus an eample of he fac ha he shif in he shor-run producion fronier is no he only one conribuion facor o P growh. We do no eclude he case ha P increases under he ouward shif in he shor-run producion fronier. Oupus include inermediae inpus. If oupu m is an inermediae inpu hen y m <. Hence he nominal value of oal (ne) oupus p y is he value-added ha a firm generaes. 5
We assume S saisfies Färe and Primon s (995) aioms ha guaranee he eisence of oupu and inpu disance funcions. The period producion fronier which is he boundary of S is represened by he period inpu reuiremen funcion G. I is defined as follows: (2) F ( y ) min { :( y ) S }. I represens he minimum amoun of he firs capial inpu ha a firm can use a period producing oupu uaniies y holding oher capial inpus = [ 2 P ] T and labour inpus fied. This funcion which is originally formulaed for characerizing he period producion fronier also can be used for characerizing he period shor-run producion fronier. Given period capial inpu he se of labour inpus and oupus y saisfying = F (y ) forms he period shor-run producion fronier. CCD measure he shif in he producion fronier by using he oupu disance funcion. Adjusing heir approach we also use he oupu disance funcion o measure he shif in he shor-run producion fronier. Using he inpu reuiremen funcion he period oupu disance funcion for = and is defined as follows: y ( ) min :. (3) DO y F Given capial inpus and labour inpus D O (y ) is he minimum conracion of oupus y so ha he conraced oupus y/d O (y ) capial inpus and labour inpus are on he period producion fronier. If (y ) is on he period producion fronier D O (y ) euals. Noe ha D (y ) is linearly homogeneous in y. We also can relae i o he shor-run producion fronier. Given labour inpus D O (y ) is he minimum conracion of oupus so ha he conraced oupus y/d O (y ) and labour inpu are on he period shor-run producion fronier. Thus D O (y ) provides a radial measure of he disance of y o he period shorrun producion fronier. We measure he shif in he shor-run producion fronier by comparing he radial disances of y o he shor-run producion froniers of he periods and. I is defined as follows: DO ( y ) (4) SHIFT ( y ). D ( y ) O If echnical progress and capial inpu growh have a posiive effec on he producive capaciy of labour beween periods and he shor-run producion fronier shifs ouward. Given labour inpus more oupus can be produced. Thus he minimum conracion facor for given oupus y declines such ha D O (y ) D O (y ) leading o SHIFT (y ). Similarly he negaive join effec of echnical progress and capial inpu growh leads o SHIFT (y ). Each choice of reference vecors (y ) migh generae a differen measure of he shif in he shor-run producion fronier from periods o. We calculae wo measures using differen reference vecors (y ) and (y ). Since hese reference oupus CCD and Färe e al (994) inroduce a measure of he shif in he producion fronier by using he raio of he oupu disance funcion. Given (y ) Färe e al (994) measure he shif in he producion fronier by D O (y )/D O (y ). 6
and labour inpus are in fac chosen in each period hey are eually reasonable. Following Fisher (922) and CCD we use he geomeric mean of hese measures as a heoreical measure of he join effec of echnical progress and capial inpu growh SHIFT as follows: 2 (5) SHIFT SHIFT ( y ) SHIFT ( y ). The case of one oupu and one labour inpu offers a graphical inerpreaion of SHIFT. In Figure i is reduced o he following formula: (6) SHIFT ( f ( ) / y )( y / f ( )). Given a uaniy of labour inpu he raio of he oupu aainable from such a labour inpu a period o he oupu aainable a period represens he een o which he shor-run producion funcion epands. SHIFT is he geomeric mean of hose raios condiional on and. SHIFT is a heoreical measure defined by he unknown disance funcions and here are alernaive ways of implemening i. We show ha he heoreical measure coincides wih a formula of price and uaniy observaions under he assumpion of a firm s profi-maimizing behaviour and a ranslog funcional form for he oupu disance funcion. 3 Our approach is drawn from CCD which implemens he Malmuis produciviy inde a heoreical measure of he shif in he producion fronier. They show ha he Malmuis produciviy inde coincides wih a differen inde number formula of price and uaniy observaions called he Törnvis produciviy inde under similar assumpions. 4 CCD also show ha he firs-order derivaives of he disance funcion D wih respec o uaniies a he period acual producion plan (y ) are compuable from price and uaniy observaions. Their euivalence resul beween he Malmuis and Törnvis produciviy indees relies on hese relaionships. Uilizing he same relaionships we also show ha SHIFT coincides wih an inde number formula of price and uaniy observaions. Euaions (7) (6) already have been derived by CCD bu for compleeness of discussion we ouline below how o compue he firsorder derivaives of he disance funcions from price and uaniy observaions. The implici funcion heorem is applied o he inpu reuiremen funcion F (y/δ ) = o solve for δ = D O (y ) around (y ). 5 Is derivaives are 2 Since he firm s profi maimizaion is assumed i is possible o adop a differen formulaion for he measure of he shif in he shor-run producion fronier: SHIFT = (D O (y )/D O (y )) /2 (D O (y )/D O (y )) /2. This formulaion is closer o he Malmuis produciviy inde inroduced by CCD. 3 Alernaive approaches involve esimaing he underlying disance funcion by economeric or linear programming approaches. Eiher approach reuires sufficien empirical observaions. Our approach originaed by CCD is applicable so long as price and uaniy observaions are available for he curren and he reference periods. See Nishimizu and Page (982) for he applicaion of he economeric echniue and see Färe e al. (994) for he applicaion of he linear programming echniue. 4 CCD jusify he use of he Törvis produciviy inde which is he Törvis oupu uaniy inde divided by he Törvis inpu uaniy inde. In addiion o he ranslog funcional form for he oupu disance funcion CCD assume he firm s cos-minimizing and revenue-maimizing funcions. Their approach is known as he eac inde number approach which consrucs a formula of price and uaniy observaions ha approimae heoreical measures. 5 We assume he following hree condiions are saisfied for = and : F is differeniable a he poin (y ) y >> M and y y F(y ) >. 7
represened by he derivaives of F (y ). We have he following euaions for = and : (7) ydo ( y ) F ( ) y y y F ( y ) y (8) D ( ) O y F ( ( ) F ) y y y y (9) DO ( y ) F ( y ). y F ( y ) y We assume he firm s profi-maimizing behaviour. Thus (y ) >> N+P+Q is a soluion o he following period profi maimizaion problem for = and 6 : () p y rf y r w ma{ ( ) }. 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 Q ] T >>. Noe ha r = [r r P ] T. The period profi maimizaion problem yields he following firs-order condiions for = and : p y ( () r F y ) s (2) r F y ) r ( w ( s (3) r F y ). By subsiuing () (3) ino (7) (9) we obain he following euaions for = and : (4) D ( ) / y O y p p y (5) ] r DO ( y ) [ r / p y ] [/ p y F ( y ) r (6) D O ( y ) w / p y. Euaions (4) (6) allow us o compue derivaives of he disance funcion wihou knowing he oupu disance funcion iself. Informaion concerning he derivaives is useful for calculaing 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 (y ) in euaions (4) (6) for = and. The disance funcions evaluaed a he hypoheical producion plan such as (y ) and (y ) also consiue SHIFT. Hence he above euaions are insufficien for implemening SHIFT. In addiion o a firm s profi maimizaion we furher assume a following ranslog funcional form for he period oupu disance funcion for = and. I is defined as 6 We assume ha here always eiss a soluion o he firm s profi maimizaion problem. Thus we implicily eclude he case ha he underlying echnology ehibis increasing reurns o scale. However i is possible ha he underlying echnology ehibis consan or decreasing reurns o scale. 8
(7) M M M O y m m m i j i j i j D ( ) y (/ 2) y y P P P p p (/ 2) i j i p i j j Q Q Q (/ 2) i j i i j j M P M Q y y m p m p m p m m m P Q p p p where he parameers saisfy he following resricions: (8) i j j i for all i and j such ha i < j M; (9) i j j i for all i and j such ha i < j P; (2) i j j i for i and j such ha i < j Q; N n (2) ; n (22) M i m for m = M; i M (23) m p for p = P; m (24) M m for = Q. i Resricions (2) (24) guaranee he linear homogeneiy in y. The ranslog funcional form characerized in (7) (24) is a fleible funcional form so ha i can approimae an arbirary oupu disance funcion o he second order a an arbirary poin. Thus he assumpion of his funcional form does no harm any generaliy of he oupu disance funcion. Noe ha he coefficiens for he linear erms and he consan erm are allowed o vary across periods. Thus echnical progress under he ranslog disance funcion is by no means limied o Hicks neural and is able o represen a variey of echnical progress. Under he assumpions of he profi-maimizing behaviour and he ranslog funcional form a heoreical measure SHIFT is compued from price and uaniy observaions. Proposiion : Assume ha he oupu disance funcions D O and D O have he ranslog funcional form defined by (7) (24) and ha a firm follows compeiive profi-maimizing behaviour in periods = and. Then he join effec of echnical progress and capial inpu growh SHIFT can be compued from observed prices and uaniies as follows: M y (25) Q m SHIFT s m m s ym where s m and s are he average value-added shares of oupu m and labour inpu beween periods and such ha: p m ym pm ym s m and s 2 p y p y w w 2 p y p y. 9
The inde number formula in (25) can be inerpreed as he raio of a uaniy inde of oupu o a uaniy inde of labour inpu. Noe ha no daa on price and uaniy of capial inpus appear in his formula. Alhough he shif in he shor-run producion fronier reflecs echnical progress as well as he change in capial inpu we can measure is shif wihou resor o capial inpu daa eplicily. 4. Reurns o Scale Effec As shown in Figure he shif in he shor-run producion fronier is no he only facor conribuing o he growh in P. Even when here is no change in he shor-run producion fronier he movemen along he fronier could raise P eploiing he curvaure of he shor-run producion fronier. We refer o P growh induced by he movemen along he shor-run producion fronier as he reurns o scale effec. In he simple model 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 from he previous period o he curren period coincides wih he growh rae of oupu divided by he growh rae of labour inpu. Since he reurns o scale effec is he P growh induced by he movemen along he shor-run producion fronier i is compued by he growh raes of oupu and labour inpu beween he wo end poins of he movemen. Figure 2 shows how he movemen along he period shor-run producion fronier from poin C o D affecs P. Comparing poins C and D he growh rae of oupu is f ( )/f ( ) and he growh raio of labour inpu is /. The growh rae of P beween he wo poins coincides wih he growh rae of oupu divided by ha of labour inpu so ha (f ( )/f ( ))/( / ) = (f ( )/ )/(f ( )/ ). [Place Figure 2 appropriaely here] We generalize he growh raes of labour inpu and oupu beween wo poins on he period shor-run producion fronier in order o measure he reurns o scale effec in he muliple-inpus muliple-oupus case. Firs we invesigae he counerpar of he growh rae of labour inpus in he muliple-inpus muliple-oupus case. CCD define he inpu uaniy inde which is he counerpar of he growh rae of oal inpus by comparing he radial disance beween he 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 by CCD we inroduce he labour inpu disance funcion ha measures he radial disance of labour inpus o he period producion fronier. The period labour inpu disance funcion for = and is defined as follows: D ( y ) ma : F y. (26) Given oupus y and capial inpus D (y ) is he maimum conracion of labour inpus so ha he conraced labour inpus /D (y ) and capial inpus wih oupus y are on he period producion fronier. If (y ) is on he period producion fronier D (y ) euals. Noe ha D (y ) is linearly homogeneous in. We can also relae i o he shor-run producion fronier. Given oupus y D (y ) is he maimum conracion of labour inpus so ha he conraced labour inpus
/D (y ) and oupus y are on he period shor-run producion fronier. Thus D (y ) provides a radial measure of he disance of o he period shor-run producion fronier condiional on y. We consruc he counerpar of he growh rae of labour inpu by comparing wo labour inpus and o he period shor-run producion fronier condiional on y. I is defined as follows: (27) ABOUR ( y) D ( y ) / D ( y ) If labour inpus increase beween wo periods moves furher away from he origin han meaning ha he labour inpu vecor is larger han he labour inpu vecor. The maimum conracion facor for producing oupus y wih he period capial inpus using he period echnology increases such ha D (y ) D (y ) leading o ABOUR( y). Similarly if labour inpu shrinks beween wo periods moves closer o he origin han does leading o ABOUR ( y). Second we generalize he growh rae of oupus beween wo poins on he period shor-run producion fronier. In he muliple-inpus muliple-oupus case oupus aainable from given capial inpus and labour inpus are no uniuely deermined by he shor-run producion fronier. e P ( ) be he porion of he period producion fronier ha is condiional on capial inpus and labour inpus consising of he se of maimum oupus ha are aainable from and using echnology available a period. I is defined as follows: P ( ) { y : F ( y ) }. (28) We can also relae i o he shor-run producion fronier. The porion of he period shor-run producion fronier ha is condiional on is represened by P ( ). Since D O (y ) provides a radial measure of he disance of y o he period shorrun producion fronier condiional on i can also be inerpreed as a radial measure of he disance of y o P ( ). We consruc he counerpar of he growh rae of oupus beween wo poins on he period shor-run producion fronier by measuring he disance beween P ( ) and P ( ). We sar wih he reference oupus vecor y. We measure he disance beween P ( ) and P ( ) comparing he radial disances from oupus y o P ( ) and P ( ). I is defined as follows: (29) OUTPUT y D y D y. ( ) O( ) / O( ) If he labour inpu growh makes i possible o produce more oupus while holding capial inpu fied and using he same echnology he se of oupus aainable from P ( ) shifs ouward o ha of oupus aainable from P ( ). Thus he minimum conracion facor for given oupus y declines such ha D O (y ) D O (y ) leading o OUTPUT ( y). Similarly if he change in labour inpus reduces oupus aainable from given capial and labour inpus leads o OUTPUT ( y). Using he counerpars of he growh rae of oupus and labour inpus beween wo poins on he period shor-run producion fronier we can propose a measure for he P growh beween hese wo poins. When we consider he movemen along he
period shor-run producion and use oupus y as reference he reurns o scale effec is defined as follows 7 : SCAE( y) OUTPUT ( y) / ABOUR ( y) (3) DO ( y ) D ( y ). DO ( y ) D ( y ) Each choice of reference shor-run producion fronier and reference oupu vecor y migh generae a differen measure of he reurns o scale effec going from period o period. We calculae wo measures by using shor-run producion froniers and oupu vecors ha are available a he same period: period shor-run producion fronier and period oupu vecor y ; period shor-run producion fronier and period oupu vecor y. Since hese ses of shor-run producion froniers and oupu vecors are eually reasonable we use he geomeric mean of hese measures as a heoreical inde of he reurns o scale effec SCAE as follows: (3) SCAE SCAE( y ) SCAE( y ). The case of one oupu and one labour inpu offers us a graphical inerpreaion of SCAE. In Figure (3) can be reduced o he following formula: f ( ) y (32) SCAE. y 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 by he change in labour inpu. SCAE is he geomeric mean of hose raios condiional on he period and shor-run producion froniers. CCD apply he implici funcion heorem o he inpu reuiremen funcion wih he oupu disance funcion such as F (y /δ ) = where δ = D O (y ). Accordingly we apply he implici funcion heorem o he inpu reuiremen funcion wih he labour inpu disance funcion such as F (y /δ) = where δ = D (y ). In his case D (y ) is differeniable around he poin (y ). 8 Is derivaives are represened by he derivaives of F (y ). We have he following euaions for = and : (33) yd ( y ) F ( y ) y F ( y ) (34) D ( ) y ( ( ) F ) F y y (35) D ( y ) F ( y ). F ( y ) 7 This formulaion is a counerpar of he scale effec on TFP growh ha is proposed by ovell (23; 45). ovell s definiion is based on he inpu disance funcion insead of he labour inpu disance funcion. 8 We also assume he following hree condiions are saisfied for = and : F is differeniable a he poin (y ) >> M and F(y ) >. 2
We assume ha (y ) >> N+P+Q is a soluion o he period profi maimizaion problem () for = and. By subsiuing () (3) obained from he profi maimizaion ino (33) (35) we obain he following euaions for = and : (36) D ( ) / y y p w r (37) D ( y ) [/ w ] r (38) D ( y ) w / w. Euaions (36) (38) allow us o compue he derivaives of he labour inpu disance funcion wihou knowing he labour inpu disance funcion iself. Informaion concerning he derivaives is useful for calculaing he values of SCAE which is defined by 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 (y ) in euaions (36) (38) for = and. The disance funcions evaluaed a he hypoheical producion plan such as (y ) and (y ) also consiue SCAE. Hence he above euaions are insufficien for obaining SCAE. In addiion o he firm s profi maimizaion we furher assume he ranslog funcional form for he period labour inpu disance funcion for = and. I is defined as follows: (39) M M M y m m m i j i j i j D ( ) y (/ 2) y y P P P p p (/ 2) i j i p i j j Q Q Q (/ 2) i j i i j j M P M Q y y m p m p m p m m m P Q p p p where he parameers saisfy he following resricions: (4) i j j i for all i and j such as i < j M; (4) i j j i for all i and j such as i < j P; (42) i j j i for i and j such as i < j Q; Q (43) ; Q (44) i for = Q; i Q (45) for m = M; m Q (46) p for p = P. Euaion (39) is he same funcional form defined by (7) ha we assumed for he oupu disance funcion in he discussion of SHIFT. However he resricions on parameers on he labour inpu disance funcion differ from hose on he oupu 3
disance funcion. We replace resricions (2) (24) wih (43) (46). While resricions (2) (24) guaranee he linear homogeneiy in oupus y for he oupu disance funcion resricions (43) (46) guaranee he linear homogeneiy in labour inpus for he labour inpu disance funcion. The ranslog funcional form characerized by (39) (46) is a fleible funcional form so ha i can approimae an arbirary labour inpu disance funcion o he second order a an arbirary poin. Thus he assumpion of his funcional form does no harm any generaliy of he labour inpu disance funcion. Noe ha he coefficiens for he linear erms and he consan erm are allowed o vary across periods. Thus echnical progress under he ranslog disance funcion is by no means limied o Hicks neural and a variey of echnical progress is possible. Under he assumpions of profimaimizing behaviour and he ranslog funcional form a heoreical inde of he reurns o scale SCAE is compuable from price and uaniy observaions. Proposiion 2: Assume he following: oupu disance funcions D O and D O have he ranslog funcional form defined by (7) (24); labour inpu disance funcions D and D have he ranslog funcional form defined by (39) (46) and a firm follows compeiive profi-maimizing behaviour in periods = and. Then he reurns o scale effec SCAE can be compued from observable prices and uaniies as follows: Q (47). Q. SCAE s s. where s is he average value-added shares of labour inpu and s is he average labour-compensaion share of labour inpu beween periods and such ha w w s and s 2 p y p y w w 2 w w The inde number formula on he righ-hand side of (47) can be inerpreed as he raio of he uaniy indees of labour inpus. Boh erms in (47) are he weighed geomeric average of he growh raes for labour inpus. The firs erm uses he raio of labour compensaion for a paricular ype of labour inpu o he value-added as weigh and he second erm uses he raio of labour compensaion for a paricular ype of labour inpu o he oal labour compensaion as weigh. Thus if labour income share which is he raio of he oal labour compensaion o he value-added is large he difference beween wo erms (47) becomes small making he magniude of SCAE smaller. Conversely if labour income share is small he magniude of SCAE becomes larger. Saring from he undersanding ha he wo conribuion facors eis for he P growh we reached he inde number formula for hese facors independenly. Our resul however does no deny he possibiliy ha oher unknown facors eplain P growh. Forunaely wo facors of SHIFT and SCAE can fully eplain he P growh. The produc of SHIFT and SCAE coincides wih he inde of he P growh as follows Corollary : Assume he following: oupu disance funcions D O and D O have he ranslog funcional form defined by (7) (24); labour inpu disance funcions D and D have he ranslog funcional form defined by (39) (46) and a firm follows. 4
compeiive profi-maimizing behaviour periods = and. Then he produc of SHIFT and SCAE can be compued from observed prices and uaniies as follows: M y (48) Q m SHIFT SCAE s m m s ym where s m is he average value-added shares of oupu m and s is he average labourcompensaion share of labour inpu beween periods and such ha p m ym pm ym s m and s 2 p y p y w w 2 w w The righ side of euaion (48) represens he logarihm of P growh. The firs erm of he righ-hand side coincides wih he Törnvis uaniy inde of oupus and he second erm is he Törnvis uaniy inde of labour inpus. Euaion (48) allows us o decompose P growh fully ino wo componens SHIFT and SCAE when muliple inpus and oupus are employed. This decomposiion is jusifiable as a generalizaion of he one-inpu one-oupu case in which P growh is induced by he shif in he producion fronier and he movemen along he producion fronier.. 5. An Applicaion o U.S. Indusry Daa Having discussed he heory of he decomposiion we now eplore is empirical significance wih indusry daa. The indusry daa covering he period 987 27 is aken from he Bureau of abour Saisics (BS) mulifacor produciviy daa. We use gross oupu inermediae inpu and labour inpu a curren and consan prices by 58 indusries which consiue he non-farm privae business secor. abour inpu a consan prices measures he number of hours worked. 9 These indusries are caegorized eiher as goods-producing indusries (goods indusries hereafer) or services-providing indusries (services indusries hereafer). [Place Table appropriaely here] Table compares P growh and is componens across he non-farm privae business secor he goods indusries and he services indusries. For he enire sample period 987 27 he reurns o scale effec had a negaive impac on P growh of 2.7 percen per year in he non-farm privae business secor. While he average rae of he join effec of echnical progress and capial inpu growh was 2.53 percen i was largely offse by he reurns o scale effec of.36 percen. Triple and Bosworh (24 26) and Bosworh and Triple (27) found in he U.S. indusry daa ha P growh in he services indusries was sagnan and lower han P growh in he goods indusries. 2 Tha difference in P growh beween he goods and he services indusries is also documened in our daase. During 987 27 he average growh rae of he goods indusries P of 2.76 percen was almos 9 Thus his measure of labour inpu does no appropriaely capure changes in labour ualiy. The join effec of echnical progress and capial inpu growh includes he P growh ha is induced by changes in he characerisics of labour inpu. 2 Triple and Bosworh (26) call he siuaion ha P growh in he services indusries is likely o sagnae Baumol s disease. They argue ha his disease has been cured in he middle 99s. 5
wice ha of he services indusries P of.95 percen. Alhough he reurns o scale effec was suble and negligible in he goods indusries for he period a significan negaive reurns o scale effec appeared in he services indusries and P growh in he services indusries rose significanly when he reurns o scale effec was ecluded. More han half he difference in P growh beween he goods and he services indusries can be eplained by he difference in he reurns o scale effec. The join effecs of echnical progress and capial inpu growh in boh indusry groups were on average very close over he period 987 27: 2.7 percen for he goods indusries and 2.47 percen for he services indusries. This reflecs ha he increase in labour inpu was occurring primarily in he services indusries. [Place Table 2 appropriaely here] Table 2 summarizes he growh in labour inpu for he non-farm privae business secor he goods indusries and he services indusries. Boh he weighed and he unweighed average of he deailed indusries show labour inpu in he services indusries increases more rapidly han ha in he goods indusries by a leas.9 percen per year. I is useful o divide he enire sample period 987 27 ino wo periods: he produciviy slowdown period 987 995 and he produciviy resurgence period 995 27. A produciviy slowdown in U.S. economy sared in he early 97s wih an average annual growh rae of.39 percen for he non-farm privae business secor during he period 987 995. Produciviy growh surged afer 995 wih an average annual growh rae of 2.69 percen during he period 995 25. As Triple and Bosworh (24 26) and Bosworh and Triple (27) poined ou he services indusries P grew slowly especially during he produciviy slowdown period wih an average growh rae of.22 percen during he period 97 995. During he same period he goods indusries grew a an average annual rae of.82 percen. However once we conrol for he reurns o scale effec and consider only he join effec of echnical progress and capial inpu growh he services indusries wih he average annual rae of.83 percen come very close o he goods indusries wih an average annual rae of.88 percen. Thus alhough P growh was lesser in he services indusries han i was in he goods indusries he producive capaciy of labour which is he oupu aainable from given labour inpus increased in he services indusries as much as i did in he goods indusries. I reflecs ha he large increase in labour inpu in he services indusries resrained P from increasing significanly. While labour inpu in he goods indusries slighly increased during he period 987 995 i even fell during he period 995 27 leading o he posiive reurns o scale effec. On he oher hand labour inpu in he services indusries increased hroughou he sample period leading o he negaive reurns o scale effec. Thus he par of he gap in P growh eplained by he gap in he reurns o scale effec beween he goods and he services indusries became larger in he period 995 27. [Place Tables 3 and 4 appropriaely here] Tables 3 and 4 show P growh and is componens and growh in labour inpu and labour income share by indusry during he periods 987 995 and 995 27. The paern found in he aggregae sudy based on he secor daa in Table is also documened in he deailed indusries. Mos indusries in he services indusries show he negaive reurns o scale effec. I is especially significan during he period 987 6
995. On he oher hand many indusries in he goods indusries show a modes reurns o scale effec during he period 987 995 and a posiive reurns o scale effec during he period 995 27. There are ecepional indusries in boh he goods and he services indusries. abour inpu grew largely wih he average growh rae of more han 2 percen in hree indusries wihin he goods indusries leading a significanly negaive reurns o scale effec: plasics and rubber producs indusry during he period 987 995 suppor aciviies for mining indusry and consrucion indusry during he period 995 27. On he oher hand labour inpu fell for few indusries in he services indusries. Especially he average growh raes of labour inpu in boh he periods 987 995 and 995 27 are negaive for hree indusries: uiliies indusry rail ransporaion indusry and pipeline ransporaion indusry. In hese indusries here is a rend of decrease in labour inpu for he enire sample period. The accumulaed posiive reurns o scale effecs was significan. The reurns o scale effec depends on labour income share as well as growh in labour inpu. The deailed indusry sudy reveals cases when he reurns o scale effec induced by labour inpu growh is miigaed by he small labour income share. The larges average growh rae of labour inpu is 6.36 percen per year in oher ransporaion and suppor aciviies indusry during he period 987 995. However is reurns o scale effec is.47 which is no he larges among all he indusries. The larges and he second larges reurns o scale effec during he period 987 995 are found in informaion and daa processing services indusry and renal and leasing services and lessors of inangible asses indusry. The average annual rae is 2.43 percen for he former indusry and 2.29 percen for he laer indusry. The average growh rae of labour inpu of oher ransporaion and suppor aciviies indusry eceeds ha in he above wo indusries by a leas 2 percen. This is because he impac of he rapid increase in labour inpu of oher ransporaion and suppor aciviies indusry is miigaed by is large labour income share of 77.8 percen. 2 6. Conclusion In his paper we disinguished wo effecs on P growh by eamining he shor-run producion fronier. The join effec of echnical progress and capial inpu growh appears as he growh in P ha is induced by he shif in he shor-run producion fronier. The reurns o scale effec appears as he P growh induced by he movemen along he shor-run producion fronier. The P growh calculaed by Törnvis uaniy indees is fully decomposed ino he produc of hese wo effecs. We applied his decomposiion resul o U.S. indusry daa for he period 987-27. I is shown ha a large par of he difference in P growh beween he goods indusries and he services indusries can be eplained by he reurns o scale effec. In his paper we assumed he firm s profi-maimizing behaviour and ruled ou inefficien producion processes. If we rela he firm s profi-maimizing behaviour anoher facor echnical efficiency change appears in he decomposiion of P growh. Even wih no change in he shor-run producion fronier and no change in labour inpu a firm can approach closer o he shor-run producion fronier by 2 The impac of he rapid increase in labour inpu is also miigaed by he large labour income share in Compuer sysems design and relaed services indusry. 7
improving echnical efficiency. For eample a firm improves echnical efficiency by increasing oupu up o he maimum level aainable from given labour inpus under curren echnology. For he implemenaion of he decomposiion of he P growh wihou assuming a firm s profi-maimizing behaviour we can esimae he disance funcion by using economeric echniues or Daa Envelopmen Analysis s linear programming echniue. However we leave his eercise o he fuure research. 8
Appendi A Proof of Proposiion DO ( y ) DO ( y ) SHIFT 2 DO ( y ) 2 DO ( y ) DO ( y ) DO ( y ) 2 DO ( y ) 2 DO ( y ) Since he firm s profi maimizaion is assumed he period producion plan is on he period producion fronier for = and. M DO ( y ) DO ( y ) m 4 ym ym M DO ( y ) DO ( y ) m 4 DO ( y ) DO ( y ) y m ym ym ym DO ( M DO ( y ) DO ( y ) y m m 2 ym ym ym Q DO ( y ) DO ( y ) 2 M DO ( y ) DO ( y ) m 4 ym ym M DO ( y ) DO ( y ) m 4 y ) D ( ) O y using he ranslog ideniy in CCD DO ( y ) DO ( y ) y m ym ym ym DO ( y ) DO ( y ) M DO ( y ) DO ( y ) y m m 2 ym ym ym Q DO ( y ) DO ( y ) 2 2 y M m m m m m m p y p y ym p y p y 2 w Q p y p y w from he euaion (7). 9
Proof of Proposiion 2 DO ( y ) D ( y ) SCAE 2 DO ( ) D ( ) y y D ( ) ( ) O y DO y 2 DO ( ) DO ( ) y y subsiuing euaions (4) and (6). DO ( y ) D ( y ) 2 DO ( ) D ( ) y y D ( ) ( ) y D y 2 D ( ) D ( ) y y Since he firm s profi maimizaion is assumed he period producion plan is on he period producion fronier for = and. M DO ( y ) DO ( y ) m 4 M 4 m DO ( y ) DO ( y ) D ( y ) D ( y ) D ( y ) D ( y ) M DO ( y ) DO ( y ) m 2 Q D ( y ) D ( y ) 2 M DO ( y ) DO ( y ) m 4 M D ( y ) D ( y ) m 4 using he ranslog ideniy in CCD DO ( y ) DO ( y ) D ( y ) D ( y ) 2
2 ( ) ( ) 2 ( ) ( ) 2 M O O m Q D D D D y y y y from he euaion (7) and (39). Q Q w w w w 2 2 w w y p y p subsiuing euaions (6) and euaions (38).
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Figure : Sources of Secoral abour Produciviy Growh 987-27 987-995 995-27 Non-farm privae business secor abour produciviy growh 2.7.39 2.69 Technical progress and capial inpu growh 2.53.82 3. Reurns o scale effec -.36 -.43 -.3 Goods indusries abour produciviy growh 2.76.82 3.38 Technical progress and capial inpu growh 2.7.88 3.26 Reurns o scale effec.5 -.6.2 Services indusries abour produciviy growh.95.22 2.43 Technical progress and capial inpu growh 2.47.83 2.89 Reurns o scale effec -.52 -.6 -.46 Noe : All figures are average annual percenages. Figure 2: abour Inpu Growh and abour Income Share 987-27 987-995 995-27 abour inpu growh (un-weighed average) Privae non-farm business.62.22.22 Goods indusries -.6 -.26 -.59 Services indusries.63 2.9.33 abour inpu growh (weighed average) Privae non-farm business.3.4.95 Goods indusries -.2.2 -.48 Services indusries.7.99.5 abour income share Privae non-farm business 69.8 69.57 68.92 Goods indusries 67.44 69.29 66.2 Services indusries 69.88 69.69 7. Noe : All figures are average annual percenages. The un-weighed average growh rae of labour produciviy is he arihmeic mean of he growh rae of indusry labour produciviy. The weighed average growh rae of labour produciviy is calculaed using labour income in each indusry divided by he sum of indusry labor 24