Estimation of R&D Depreciation Rates: A Suggested Methodology and Preliminary Application

Transcription

1 Esimaion of R&D Depreciaion Raes: A Suggesed Mehodology and Preliminary Applicaion Ning Huang and Erwin Diewer Ocober 9, 009 Discussion Paper 09- Deparmen of Economics Universiy of Briish Columbia Vancouver, Canada V6T Z. Addresses: Ning.Huang@sacan.gc.ca and diewer@econ.ubc.ca Absrac The 008 version of he SNA will recommend capializaion of R&D expendiures. To implemen his recommendaion, measures for he sock of R&D capial mus be consruced, and his implies a need o deermine he depreciaion rae of R&D capial. In his paper, we develop a simple model, based on a producion funcion mehod ha allows for monopolisic compeiion, o esimae he annual depreciaion rae of R&D capial. We rea R&D capial as a echnology shifer insead of as an explici inpu facor. Boh he R&D sock and he ime variable are used o capure echnological progress. Modeling R&D capial in his manner can beer represen he role R&D plays in economic growh. Esimaed R&D depreciaion raes and markup facors are presened for he U.S. manufacuring secor and four U.S. knowledge inensive indusries, namely chemical producs (SIC 8), non-elecrical machinery (SIC 35), elecrical producs (SIC 36) and ransporaion equipmen (SIC 37). Keywords Research and developmen, R&D depreciaion raes, R&D capial socks, producion heory, obsolescence, flexible funcional forms, echnology shifs, monopolisic compeiion. Journal of Economic Lieraure Classificaion Codes C3, C43, C8, D4, D43 D9. Saisics Canada and he Universiy of Briish Columbia respecively. The views expressed in his paper are hose of he auhors and do no reflec hose of Saisics Canada. The auhors hank he SSHRC for financial suppor and acknowledge very helpful discussions wih Alice Nakamura.

2 . Inroducion In a recen paper presening an innovaive new indicaor of echnical change, Michelle Alexopoulos and Jon Cohen (008) wrie ha: Alhough echnical change is cenral in much of modern economics, our abiliy o idenify empirically he facors ha shape is pace, naure, and impac are consrained by daa limiaions. The relevance of his saemen is underlined by boh he curren imperaive for naional economies o achieve beer produciviy growh and new developmens on he inernaional scene in accouning for R&D in he Sysem of Naional Accouns. Alhough research and developmen (R&D) expendiures accoun for only a small porion of GDP, heir imporance in creaing new echnologies and promoing produciviy growh is widely recognized. This necessiaes he re-examinaion of measuremen issues relaed o hese expendiures ha have broad implicaions for economiss conducing analyses of he conribuion of knowledge capial o endogenous growh and o he naion s wealh, as well as for policy makers who may wan o subsidize R&D invesmens in order o maximize economic growh. A widely used definiion of R&D is given in he 993 Frascai Manual (OECD, 994). In his documen, R&D is defined as creaive work underaken on a sysemaic basis in order o increase he sock of knowledge, including knowledge of man, culure and sociey, and he use of his sock of knowledge o devise new applicaions. Three aciviies are covered: basic research, applied research, and experimenal developmen. The key crieria ha disinguish R&D from he oher relaed aciviies are he presence in R&D of an appreciable elemen of novely and he resoluion of scienific and/or echnological uncerainy. The SNA 993, which is sill in effec as he inernaional sandard, does no rea R&D expendiures as invesmen. Thus, in he Naional Income and Produc Accouns (NIPA s) of Unied Saes, R&D expendiures are reaed as an inermediae inpu for business and curren consumpion for he non-profi insiuions and governmen. However, he SNA 008 ha will soon supplan he SNA 993 does recommend he capializaion of R&D expendiures. Successful R&D invesmens add o he sock of knowledge, and his sock in urn provides a flow of services over ime, raher han in one period. Thus, R&D resembles invesmen more closely han inermediae inpu or curren consumpion. However, hough he idea of capializing R&D expendiures has been acceped in principle for he SNA 008, implemenaion requires he deerminaion of depreciaion raes for R&D capial. Due o he measuremen difficulies involved, Nadiri and Prucha (996) observe ha, researchers doing applied work ypically assume an arbirary depreciaion rae of 0 o 5 percen o consruc he sock of R&D capial using he perpeual invenory mehod. However, i would be preferable o have a scienific mehod for deermining his depreciaion rae. This version of he Frascai Manual formed he mehodological basis for he influenial 00 OECD STI Scoreboard. See hp://library.cerh.gr/libfiles/mobility-portal/mon-5-oecd-scoreboard-00- A9.pdf. The version of he Frascai Manual currenly in use is 00 (OECD 00).

3 3 There have been some aemps o empirically measure R&D depreciaion raes. Pakes and Shankerman (98) esimaed he rae of depreciaion in he privae value of paens using European daa on paen renewal fees and raes of renewal; Nadiri and Prucha (996) measured he depreciaion rae of R&D sock using a facor requiremens funcion and resriced cos funcion, and reaing he R&D capial as a normal reproducible capial inpu, as is also he case for Bernsein and Mamuneas (005) who esimaed R&D depreciaion raes in an ineremporal cos minimizaion framework. However, he approaches adoped in previous sudies generally do no allow for non-r&d effecs ha improve he echnology, such as knowledge diffusion hrough educaion and learning by doing. Thus all of he echnological improvemen is inappropriaely aribued o R&D invesmen. 3 Also, he esimaion is ypically conduced in a framework of compeiive pricing behaviour. However, privae R&D invesmens are ofen underaken wih he explici goal of achieving shor-run monopolisic advanages over compeiors. Thus he assumpion of compeiive behaviour on oupu markes is unsuiable. Anoher weakness wih mos sudies is ha R&D invesmens are reaed similarly o invesmens in ordinary physical capial, bu R&D invesmens are quie differen in heir effecs, as is explained subsequenly. In his paper, we propose o rea he sock of R&D capial no as an explici inpu facor; insead we define he sock of R&D capial o be he echnology index ha locaes he economy s producion fronier. An increase in he sock of R&D shifs he producion fronier ouwards. In our model, he R&D capial depreciaion rae is esimaed wihin a monopolisic compeiion framework using gross R&D invesmen daa. We esimae R&D depreciaion raes for he U.S. oal manufacuring secor and four U.S. knowledge inensive indusries: chemical producs (SIC 8), non-elecrical machinery (SIC 35), elecrical producs (SIC 36) and ransporaion equipmen (SIC 37). The res of he paper is organized as follows. Secion explains he consrucion of he R&D sock and he reasons for R&D depreciaion. Secion 3 develops our basic model for esimaing he rae of depreciaion for R&D capial. Secion 4 presens he esimaion mehodology and resuls. Secion 5 concludes. Appendix A shows he derivaion of he esimaing equaions. Differen ses of break poins used in he linear spline model and quadraic spline model are given in appendix B.. Consrucion of he R&D Sock According o he 993 Frascai Manual definiion, he objecive of conducing R&D is o increase he sock of knowledge. Thus we define R&D capial as he knowledge asse creaed by R&D invesmen. Hence he sock of R&D capial can be regarded as a proxy for sociey s echnological level. R&D invesmens ac as a mechanism for shifing ouward sociey s producion possibiliy fronier. This reamen is he major disinguishing feaure of our approach. The sudies of ohers rea R&D capial as an explici facor inpu in a manner ha is similar o he reamen of ordinary physical capial.. R&D Capial and Ordinary Physical Capial 3 The work of Bernsein and Mamuneas (005) is no subjec o his criicism.

4 4 Physical capial like machinery, equipmen and buildings wears ou hrough use and he efficiency ends o decline over ime. Successful R&D venures creae new knowledge for he firms conducing he R&D. This new knowledge can be eiher a new cos-saving process, or a new echnology for an innovaive or qualiy-improved produc. R&D capial does no wear ou because of is uilizaion and is absolue efficiency level does no change wih he passage of ime. However, R&D capial is subjec o a ype of depreciaion because processes or producs can become obsolee over ime. Alhough physical capial and R&D capial are boh called capial asses, here are some fundamenal differences. Firs, physical capial, such as machines and equipmen, can be reproduced over muliple periods. Wih physical capial, reproducibiliy makes i possible for us o observe a he same poin in ime renal prices of differen vinages, and also used asse prices of differen vinages, of a capial asse, and his in urn allows us o esimae depreciaion raes. In conras, for R&D capial, once, say, a new blueprin has been produced, i can be made available o many economic unis wihou furher producive aciviy. Typically, we canno collec price informaion for differen vinages of R&D capial a he same poin in ime 4. Secondly, physical capial and R&D capial suppor producion in differen ways. As Pizer (004) poined ou, R&D capial acs as if i were producing recipes while physical capial is regarded as one of he producive inpus 5 ha are consumed during a producion process. Thirdly, he reasons for depreciaion are differen for hese wo ypes of capial asses. We reurn o his issue subsequenly. Because of hese differences, we believe ha he reamen of R&D asses should necessarily be differen from he reamen of reproducible capial asses.. Consrucing he Sock of R&D Capial Lacking a good measure of R&D oupu, we use inpu informaion in he form of gross (real) R&D expendiures as a proxy measure. Because all he echnologies, new or old, are creaed by R&D invesmens, he R&D sock can be wrien as a funcion of a series of pas R&D invesmens. Thus, he period R&D sock is defined as follows: () R θ I + θ I + θ 3 I 3 + θ 4 I 4 + θ 5 I where θ n is he period efficiency index represening how much he R&D invesmen ha was made n periods before period conribues o he echnology or knowledge sock in period. The R&D lags and weighs are all incorporaed in hese efficiency indexes. 4 However, we can someimes observe he marke price for he righs o some new echnology. If he same echnology is again sold in a fuure period, hen we could collec price informaion for differen periods and infer a depreciaion rae for he R&D invesmen 5 Pizer (004) defines producive inpu as: Fundamenal o producion is he noion ha inpus are proporional in some sense o oupus. Oupus are creaed by combining a paricular collecion of inpus in a paricular manner. If more oupus are desired, hen more inpus are necessary, and usually, more of all inpus. I may no be necessary o double he inpus o double he oupus, bu more inpus are necessary o produce more oupus.

5 5 I n is he R&D invesmen made in year n. We expec ha he furher back he R&D invesmen was made, he less i conribues o he prevailing knowledge. Thus he following relaionships should hold among all he efficiency indexes 6 : () θ θ θ 3 θ 4 θ We add he superscrip because hese efficiency indexes may vary wih ime. The speed of he echnological upgrading is one facor which helps o deermine he size of he efficiency indexes. If he new echnology is creaed quickly, he pas invesmen may become irrelevan a a fas pace wih he newly updaed-knowledge. For example, if here were more new knowledge creaed in year compared o year, we would expec he following inequaliies o hold: (3) θ > θ and θ < θ ; θ 3 < θ 3 ; θ 4 < θ 4 ;.... These inequaliies show us ha he previous invesmens become less imporan a a faser speed as he pace of echnology improvemen speeds up. The series of R&D socks can be wrien as: (4) R = θ I + θ I + θ 3 I θ I + θ I 5 R 0 ; R - = θ I + θ I 3 + θ 3 I θ I + θ R 0 ; R - = θ I 3 + θ I 4 + θ 3 I θ I + θ R 0 ;... where R 0 is he iniial knowledge sock. The efficiency index varies wih R&D invesmens and he echnology updaing frequency. To simplify our analysis, we assume ha he efficiency indexes decline a a consan geomeric rae; ha is, we assume ha he following relaionships hold among he efficiency indexes: (5) θ n = ( δ) n and 0 δ. Based on he above simplificaions, he sock of R&D capial can be consruced as follows: (6) R = I + ( δ)r - where δ can be regarded as he R&D depreciaion rae, which is assumed o be consan over ime. From (6), we see ha he sock of R&D capial in period is consruced from he previous R&D invesmen, I, and he depreciaed R&D sock of period. This is a widely used mehod o consruc capial socks. However, i can be seen ha we need a resricive assumpion abou he efficiency index o end up wih his simple equaion. According o his equaion, R&D capial accumulaion depends on wo opposie forces: he addiion of he new knowledge sock, 6 Anoher way o jusify inequaliy () is follows. Suppose ha he indusry or firm oupu is no homogeneous bu consiss of a mix of new and old producs. Over ime, indusry oupu shifs away from he older more obsolee producs and owards he newer improved producs. Hence he use of he old echnology ha produces he older obsolee producs diminishes over ime and he iniial good effecs of invesmens in echnological improvemens ha were made many periods ago gradually wiher away. Thus we have a jusificaion for why he iniial good efficiency effecs of R&D invesmens diminish or depreciae over ime.

6 6 which is creaed by he curren period R&D invesmens, 7 and he depreciaion of he old knowledge sock. We use he sock of R&D capial as a echnology index, indicaing he posiion of he producion fronier. If he newly creaed knowledge sock is a leas as large as he depreciaion of he old knowledge sock, we will no have a backward shifing of he producion fronier. I is someimes argued ha he R&D depreciaion rae is zero, since old knowledge is preserved. However, new innovaions end o make he innovaions from pervious periods obsolee. Also, consumer preferences can shif over ime, causing a drop in he demand for old producs, and leading o obsolescence of pas R&D invesmens. Hence he older R&D invesmens do suffer depreciaion. Indeed, i is possible ha he depreciaion of old echnologies ouweighs he incremenal effecs of he new echnologies and hence resuls in a ne decrease of he R&D capial sock. R&D invesmen in he Chemical Producs (SIC 8) and he Non-elecrical Machinery (SIC 35) caegories has increased smoohly over he pas 40 years, while R&D invesmen in Transporaion Equipmen (SIC 37) has flucuaed more. These differen rends in he R&D invesmen imply differen frequencies for creaing new echnology and resul in differen depreciaion raes for he four indusries. In he res of his secion, we discuss he reasons for R&D capial depreciaion in more deail..3 Reasons for R&D Capial Depreciaion The depreciaion of a angible asse can be esimaed by using he informaion on he price of he used asse (provided ha i is no a uniquely consruced angible asse). If we rea R&D expendiures as invesmen, here exiss a similar problem: he decline in he uilizaion and efficiency of he knowledge capial. Alhough here is no apparen wear or ear of he inangible asse, we canno assume ha he knowledge capial has an infinie service life. Boh price and quaniy changes are responsible for changes in he value of R&D. R&D capial depreciaion, under our invesigaion, means he real quaniy change due o he change in efficiency and uilizaion of he knowledge asse. The relaive efficiency of he knowledge capial will decline over ime due o he following reasons: The obsolescence of he echnology. When newer echnologies are creaed by R&D invesmen, he old echnology may be parly or enirely replaced by he newly creaed echnologies and consequenly, he relaive efficiency and he uilizaion of he old knowledge would decline. The changing preferences of consumers. Consumer s ases may change for differen reasons, such as he implemenaion of new healh resricions, he emergence of new producs, and changes in he consumer s capabiliies. Changing ases may shif away he demand for some producs ha heavily rely on older echnologies, and cause relaed 7 Equaion (6) implies ha one uni of R&D invesmen can creae one uni knowledge. This can be regarded as he simples funcional form for a knowledge producion funcion.

7 7 marke shrinkage. Responding o he shifing demand, firms would reduce he uilizaion level of he older echnology ha produces he oumoded producs. Because here are very few observed marke prices for old echnologies, we canno observe he depreciaion on R&D capial as we can for a angible asse which rades on second hand markes. In he following secion, we will se up a framework for esimaing he R&D capial depreciaion rae. 3. The Esimaion Framework In our esimaion framework, each indusry is reaed as facing a monopolisic compeiion environmen. In he producion funcion, he R&D sock is reaed as a echnology index for he posiion of he producion fronier. We use an exension of a model due o Diewer and Lawrence (005). 3. The Basic Framework We allow for he possibiliy of increasing reurns o scale in he indusry. Because he assumpion of compeiive profi maximizing behaviour is no suiable for he modelling of he indusry s behaviour, we rea he indusry as engaging in monopolisic profi maximizaion. We assume ha each indusry has an aggregae producion funcion f, wih he form y = f(x,r,) so ha f is a funcion ha depends on he usual inpu vecor x, he R&D sock R, and he ime variable which represens non R&D sources of echnical change for he producion funcion. 8 Thus boh R and shif he producion funcion over ime. Defining he producion funcion in his way, we can avoid he overesimaion of he effecs of R&D capial on echnological improvemen, compared o producion funcions ha have only an R&D variable R as a shif variable. The aggregae demand funcion for he oupu of an indusry in year is represened by an inverse demand funcion of he form p = P(y,). Under his siuaion, each indusry solves he following monopolisic profi maximizaion problem a each period by choosing inpus and he nex period s echnology level: Max x,r β{p (y, )y w x Pr, Ir,} = 0 subjec o : y f (x, R, ) and R I ( )R, 0,,... (8) = = r, + δ = where β is he period discoun facor, w is an inpu price vecor, I r, is R&D invesmen in period, and P r, is he corresponding price index. In his model, we assume ha each indusry maximizes he discouned fuure monopolisic profis wih full informaion abou fuure prices. Ignoring he uncerainy of fuure prices is no realisic, bu i dramaically simplifies he problem. 8 These non R&D sources of echnical progress could include learning by doing effecs, freely available research, informaion on new echnologies made available a rade fairs and so on.

8 8 The firs order necessary condiions for solving he above maximizaion problem are: (9) (0) p xf (x, R, ) + [ P(y, ) / y]y xf (x, R, ) = w, = 0,,...T, P( ) f ( ) f ( ) β + y + + p + + Pr, + ( δ) y R + R + = β Pr,, where p is he oupu price and x f(x,r,) is he vecor of firs order parial derivaives of he producion funcion wih respec o he componens of he inpu vecor x. Facoring p and x f(x,r,) ou on he lef hand side of (9), we obain he following simplificaion of equaion (9): P(y, ) / y p ( + ) xf (x,r, ) = w () p / y. Applying similar algebraic rearrangemens o equaion (0), we have: () P ( ) / y f (x,r, ) p β + + = Pr, ( δ) Pr, + p + / y + R + β +, where β /β = +r and where r is he nominal ineres rae prevailing a ime. The erm [ P(y,)/ y]/[p/y] is he inverse of he price elasiciy of demand, and reflecs how oupu (demand) changes wih respec o he price change. If we use ε o denoe his price elasiciy, we can define is inverse as he period nonnegaive markup, denoed by m, as follows: P(y, ) / y m = 0 (3) p / y ε and he markup facor M can be defined as follows: P(y,) / y M = m = + = + (4) ε p / y. If we assume ha he markup facor is consan over ime, hen we can rewrie () and () as: (5) w,n = p M [ f(x,r,)/ x n ], n =,,...,N ; (6) (+r )P r, ( δ)p r, = p M [ f(x,r +,)/ R ] where n denoes he n-h facor in he inpu vecor x. Deails of hese equaions are given in appendix A.

9 9 The lef hand side of equaion (6) is he user cos of one uni of R&D invesmen purchased in period. 9 Equaions (5) and (6) form our sysem of esimaing equaions. Including equaion (6) as an exra esimaing equaion is helpful for disinguishing R and. However we may also inroduce some esimaion problems by using anicipaed variables in his equaion, where he anicipaions are formed in period when we purchase I r, a he price P r,. To simplify our analysis, we use he acual daa a period o approximae he prediced variables. The lef hand side user cos in (6) depends on he depreciaion rae δ. In order o compare log likelihoods across alernaive depreciaion models, we need he lef hand side variable o be consan across models. Thus we rewrie equaion (6) as: (6a) (+r )P r, = ( δ)p r, + p M [ f(x,r +,)/ R ]. Equaion (6a) is sill no ideal for economeric esimaion because i involves a lagged dependen variable. However, if we divide boh sides of equaion (6a) by P r,, his leads o he following esimaing equaion: (6b) (+r ) = ( δ)(p r, /P r, ) + (p /P r, ) M [ f(x,r +,)/ R ]. We rea R&D capial similarly o he ime variable in ha he R&D sock shifs he echnology like oher sources of produciviy improvemen (approximaed by he ime variable). 0 Hence, in our model, we have labour, inermediae, and non-r&d capial service inpus. In order o help idenify some parameers in he model, we add he producion funcion o he esimaing sysem. Thus our final esimaing sysem includes he following five equaions: (7) w, /p = M f(x,r,)/ x ; (8) w, /p = M f(x,r,)/ x ; (9) w,3 /p = M f(x,r,)/ x 3 ; (0) (+r ) = ( δ)(p r, /P r, ) + (p /P r, ) M [ f(x,r +,)/ R ] ; () y = f(x,r,). 3. The Choice of he Funcional Form for he Producion Funcion To specify esimaing equaions, we need o choose a funcional form for he producion funcion. As a saring poin, we use he following varian of a normalized quadraic funcional form: 9 I may be a be surprising iniially ha he ne effec of purchasing an R&D invesmen in period can be expressed in such a simple manner as is given in equaion (6). However, under our perfec foresigh assumpions, he producer needs o purchase unis of R&D in period in order o adjus he sock of R&D o precisely he righ level in period ; he R&D sock for period can be adjused o he righ level by purchasing addiional unis of R&D in period and so on. 0 However, he R&D variable is differen from he ime variable because we regard he ime effec as being enirely exogenous in our model whereas he R&D sock is endogenously deermined by producers.

10 0 () f(x,r,) b + c x + c x + c 3 x 3 + g x + g x + g 3 x 3 + h x R + h x R + h 3 x 3 R + e + e R {(/)x T Sx/(φ x + φ x + φ 3 x 3 )} where x is he labour inpu, x is he inermediae inpu, x 3 is he non-r&d capial inpu, and R is he sock of R&D capial. In addiion, S [s ij ] is a 3 by 3 symmeric posiive semi-definie subsiuion marix of unknown parameers and he φ i are predeermined posiive parameers. In our empirical work, we calculae he sample mean of he x i, say x i *, and hen se he φ i equal o x i * /(x * + x * + x 3 * ). The unknown parameer b deermines he degree of reurns o scale: if b = 0, we have consan reurns o scale in producion; if b is less han 0, hen here are increasing reurns o scale; and if b is greaer han 0, here are decreasing reurns o scale. The wo parameers e and e are echnical progress parameers. In order o idenify all of he parameers and o reduce muli-collineariy, i is necessary o impose some linear resricions on he marix S. Our linear resricions are as follows: (3) j= 3 s nj = 0 ; n =,,3. The normalized quadraic producion funcion defined by () and (3), wih he parameers b, e and e se equal o 0, is flexible in he class of consan reurn o scale producion funcions. The addiional parameer b allows us o es he degree of local reurns o scale. The main advanage of choosing his flexible funcional form is ha he flexibiliy properies would no be desroyed by imposing curvaure condiions; see Diewer and Wales (988). In our example, imposing posiive semi-definieness condiions on he symmeric marix S means ha we can wrie S in erm of he following marix produc: (4) S = UU T where U [u ij ] is a 3 by 3 lower riangular marix and U T is he ranspose of U. The linear resricions (3) on S can be imposed on U oo; ha is, we impose he following resricions: (5) u + u + u 3 = 0 ; u + u 3 = 0 ; u 33 = 0. We find ha here are only hree independen parameers in he U marix: u, u 3 and u 3. The main diagonal parameers u ij can be represened in erms of he off diagonal parameers u ij. Parially differeniaing he producion funcion defined by equaion () wih respec o he inpus, x i, and wih respec o nex period s R&D sock, R, and subsiuing he resuling derivaives ino he esimaing equaions (7) o (0), we can rewrie he esimaing equaions of our basic model as follows: 3 (6) w,n / p = s T j njx, j x = φ Sx M cn gn h nr T n T φ x ( φ x ) for n =,, 3, and

11 (7) + r p = M h Pr, { x + h x + h x + e } 3 3 Pr, + + ( δ) Pr, T φ x where j = φ j x 3, j. 3.3 The Problem of Trending Elasiciies Diewer and Lawrence (00) poin ou ha, for he normalized quadraic funcional form, he esimaed elasiciies ofen have srong rends when here are srong rends in he price and quaniy daa. They also sugges one way o solve his problem. We adop heir echnique here. In he iniial funcional form, he subsiuion marix S is consan over ime. To handle he rending elasiciy problem, we le he producion funcion be flexible a wo sample poins, which means he marix S is allowed o change over ime. We use he following weighed average subsiuion marix: (8) S = ( (/T))A + (/T)B ; = 0,,,...,T where T+ is he oal number of periods covered by he esimaion daa sample. We have daa for he years , so T = 47. Using his weighed average subsiuion marix, he echnological progress capured by he ime variable no only affecs he consan erms in he esimaing sysem, bu also he subsiuion possibiliies. As in he basic case, we can impose he curvaure condiions by seing A and B equal o UU T and VV T respecively, where U and V are lower riangular marices; ha is, we se: (9) A = UU T and B = VV T ; U and V are lower riangular. Similarly, we can impose he following normalizaions on hese wo marices U and V: (30) U T 3 = 0 3 and V T 3 = 0 3 where 3 and 0 3 are 3-dimensional vecors of s and 0 s, respecively. Wih hese consrains, we only add hree addiional independen parameers o he iniial model; hese new parameers are v, v 3 and v 3. Making hese changes, our producion funcion can be wrien as follows for =0,,...,47: (3) f(x,r,) b + c T x + g T x + h T x R + e + e R {(/)x T [( (/T))UU T + (/T)VV T ]x/(φ x + φ x + φ 3 x 3 )}. (3) Here we wrie he producion funcion in marix form in order o simplify our noaion. In equaion (3), c T [c,c,c 3 ], g T [g,g,g 3 ] and h T [h,h,h 3 ]. In he esimaion, if he rending elasiciy problem exiss, we will see significan increases in he value of he log-likelihood funcion as we add he parameers in he V marix o hose in he U marix. We do see his for our resuls.

12 3.4 Problems due o Non-Smooh Technical Progress Anoher problem relaed o he iniial producion funcion model is ha i does no allow for nonsmooh change in echnical progress. We now add more feaures o he model in order o capure changes in he direcion of echnical progress over ime. Technological progress ypically does no proceed smoohly. Thus we add linear splines or quadraic splines in he ime variable o allow for he differen change paerns of echnological progress a differen periods. The modified producion funcion in period can be wrien as follows: (3) f(x,r,) b + c T x + j= 3 g j () x j, + h T x R + e () + e R {(/)x T [( (/T))UU T + (/T)VV T ]x /φ T x }. where e () and g j () are linear spline funcions of ime. The number of spline segmens depends on he break poins chosen by invesigaing he plos of preliminary esimaions. A break poin is a posiive ineger less han he maximum number of he ime variable; so here, i is less han 47. We will illusrae how o define e () if we choose hree break poins, 0 < < < 3 <47: (33) e () e for = 0,,,..., ; e + e ( ) for = +, +,..., ; e + e ( ) + e 3 ( ) for = +, +,..., 3 ; e + e ( ) + e 3 ( 3 ) + e 3 ( 3 ) for = 3 +, 3 +,...,47. In equaions (33), he e j are he unknown parameers o be esimaed. From he above example, we know ha wih n break poins, here are n+ parameers o be esimaed. Similarly, we can generae linear splines for he funcions g j (). The subscrip j means we allow differen splines for differen inpus j. Adding linear spline induces, perhaps arificially, kinks in he direcion of echnical change. For smooh change, he linear splines can be replaced by quadraic splines, in which case e () and he g j () are quadraic spline funcions. Wih hree break poins, he quadraic spline funcions can be defined as follows: On seing up quadraic splines in a normalized quadraic model, see Diewer and Wales (99). Normalized quadraic cos funcions are modeled here whereas we are modeling normalized quadraic producion funcions

13 3 (34) e () e + (/)e ; 0 ; e + (/)e + ( )(e + e ) + (/)e 3 ( ) ; < ; e + (/)e + ( )(e + e ) + (/)e 3 ( ) + ( )(e + e + e 3 ) + (/) e 4 ( ) ; < 3 ; e + (/)e + ( )(e + e ) + (/)e 3 ( ) + ( 3 )(e + e + e 3 ) + (/)e 4 ( 3 ) + ( 3 )( e + e + e 3 + e 4 3 ) + (/)e 5 ( 3 ) ; 3 < 47. As in he linear spline case, he e j are he unknown parameers o be esimaed. If we choose n break poins, hen here are n+ addiional parameers ha need o be esimaed for each equaion. Adding splines increases he flexibiliy of he funcional form bu a he cos of esimaing more echnical change parameers. As was he case for linear splines, choosing differen break poins will generally resul in differen esimaes. The following feaures disinguish our model from he previous lieraure: Insead of reaing R&D capial as one explici facor inpu like ordinary physical capial, we rea R&D capial as a echnological index indicaing he posiion of he producion fronier. R&D capial, which is he knowledge asse creaed by he R&D invesmen, is no consumed like he physical capial in he producion; i jus indicaes he echnology level. Holding all he usual flow inpus consan, an increase of he sock of R&D capial would shif he producion fronier ouwards. Thus he R&D sock variable is reaed in a manner similar o he ime variable. Boh he ime variable and R&D capial sock variable are included in he producion funcion model. The R&D sock frequenly grows in a roughly linear fashion, so he inclusion of he variables R and in he regression equaions can lead o a mulicollineariy problem. However, if he ime variable,, is dropped from he model, hen he R variable becomes he only echnical change shif variable, and frequenly, he resuling rae of reurn o R&D invesmens is unrealisically large. Therefore, including boh he ime variable and he R&D variable in he model, we will generally no aribue all of he echnological progress o R&D invesmens, and avoid oversaemen of he effecs of R&D invesmens on boh echnological progress and on produciviy growh o some exen. The model allows for he possibiliy of monopolisically compeiive behaviour ha is consisen wih increasing reurn o scale. Thus our model esimaes he markup facor and he degree of reurns o scale along wih he R&D depreciaion rae. Because of hese differen feaures of our model, our resuls may be differen from he resuls obained from radiional models ha rea R&D as jus anoher capial sock. 4. Empirical Esimaion and Resuls If he esimaed markup facor M urns ou o equal, hen we have compeiive behavior.

14 4 Here we describe our daa and empirical resuls. We esimae R&D depreciaion raes for he period of for U.S. manufacuring and four U.S. echnology inensive indusries: chemical and allied producs (SIC 8), non-elecrical machinery (SIC 35), elecrical producs (SIC 36) and ransporaion equipmen (SIC 37). In 998, he R&D expendiures of hese four indusries accouned for 54.35% of he R&D expendiures of all indusries and 76.37% of he manufacuring R&D expendiures. 4. Esimaion Mehodology The basic esimaion sysem wih he curvaure condiions (4) and linear resricions (5) imposed is given by equaions (), (6) and (7). 3 To specify hese esimaing equaions, we mus define he normalized quaniies and he differences for he normalized quaniies. The nh normalized quaniy, q n, is: (35) q n x n /φ T x ; n =,,3. The differences beween he normalized quaniies can be defined in he following way: (36) q q q ; q 3 q 3 q ; q 3 q 3 q. Using he above definiions and subsiuing resricions (4) and (5) ino equaion (6), we can express he firs order necessary condiions for profi maximizaion wih respec o he choice of he differen inpus in he following way: (37) (38) (39) w w w,,,3 / p / p / p c + g + hr + (u + u3)(u q + u3q3) = M + 0.5φ[(u q + u3q3) + (u3q3 ) ], c + g + h R u (u q + u3q3) + (u3 ) q3 = M + 0.5φ[(u q + u3q3) + (u3q3 ) ], and c3 + g3 + h3r u3(u q + u3q3) (u3 ) q3 = M φ3[(u q + u3q3) + (u3q3 ) ] The producion funcion is: (40) y = b + c x + e + e + c x + c R 0.5( φ x 3 x 3 + φ + g x + g x x + φ 3 x 3 )[(u + g q 3 x 3 + h x R + h x + u 3 q 3 ) + (u R + h x 3 3 3q3 ) ]. R The above four equaions plus (7) form our basic esimaing sysem wih 6 parameers o be esimaed. 3 Thus for his basic model, here are no splines and no rending subsiuion marices.

15 5 Due o he non-lineariy of he equaions, we use non-linear maximum likelihood esimaion opion in he SHAZAM. To find he esimaes for he R&D depreciaion raes, we consruc a grid of depreciaion raes: δ = 0.0, δ = 0.0, δ = 0.03, and δ =. Based on hese depreciaion raes, we build he iniial sock of R&D capial using he following formula: (4) R(0) = I r,0 /(δ+γ r ) ; δ = 0, 0.0, 0.0,..., 0.99, where I r,0 denoes he R&D invesmen a he firs period, and γ r denoes he geomeric growh rae of R&D invesmen over he sample period and can be calculaed as: (4) γ r (I r,47 /I r,0 ) /47. For he remaining periods, he R&D sock is calculaed using equaion (6) in he second secion. All ogeher, we have 0 ses of R&D socks, corresponding o he 0 possible choices for an R&D depreciaion rae. Using hese alernaive R&D sock series, we can esimae he five equaions. For each depreciaion rae, we obain he value of he log-likelihood funcion. Comparing hese values of he log-likelihood funcion, we can locae he depreciaion rae corresponding o he maximum value of he likelihood funcion. According o our esimaing procedure, we believe ha he depreciaion rae ha maximizes he value of log-likelihood is he bes esimaor for he R&D depreciaion rae. 4. Daa Consrucion To conduc he esimaion, we need quaniy series and price series for indusrial inpu and oupu, and price and quaniy series for R&D invesmens. Indusrial inpu and oupu daa oher han R&D relaed daa are obained from he Mulifacor Produciviy daa ses provided by he Bureau of Labour Saisics (BLS). R&D relaed daa are derived from he websie of he Naional Science Foundaion (NSF). From he BLS, we obained value series in curren dollar and price index series in 996 consan dollar. 4 For each indusry, we have daa on secoral oupu, labour inpu (L), capial service inpu (K), energy inpu (E), non-energy maerials (M) inpu, and purchased business services (S) inpu. The BLS uses he Törnqvis index number formula o consruc he aggregae daa. Labour is measured as he hours worked by all persons engaged in a secor. Capial inpu is defined as he flow of services from physical asses, which include equipmen, srucures, invenories and land. Service flows are assumed o be proporional o socks. The descripion of he measures and he mehodology for consrucing all of hese daa ses are given in Chaper 0 and of BLS Hand Book of Mehods, and in Gullickson and Harper (987). Our measure of inermediae inpu is a Törnqvis aggregae of energy, maerial and purchased services. Wih value series and price series, we can consruc he implici quaniy series. Indusrial inpu and oupu daa ses are relaively well consruced over he R&D daa ses, bu we face a double couning problem when we ry o explicily model he role of R&D capial because R&D expendiures have already been included in he iniial (BLS) inpu daa. 4 We hank Mike Harper for providing us wih manufacuring daa no available on he BLS websie.

16 6 The indusrial R&D expendiure daa are obained from he websie of NSF. For he years , he daa are aken from he Indusrial R&D Informaion Sysem (IRIS) ha uses he Sandard Indusrial Classificaion (SIC) o classify he indusry. For oher years, he daa are obained from Research and Developmen in Indusry, for he year 000 and 00. These wo publicaions use he Norh American Indusrial Classificaion Sysem (NAICS). We use he 998 daa as he bridge o link he series based on he differen indusrial classificaion and reclassify he daa o conform o he SIC. Using R(Syear) o denoe he daa based on SIC and R(Nyear) o denoe he daa based on NAICS, he consruced daa we need for esimaion can be obained by using he following formula: (43) R(S999) = R(N999) [R(S998)/R(N998)]. Equaion (43) describes how he wo daa ses are linked for 999. Similar adjusmens can be made for 000 and 00. The daa on he cos componens of R&D expendiures come from various issues of Research and Developmen in Indusry. According o he NSF s classificaion, he ype of R&D expendiure includes: wage and salaries, maerials, R&D depreciaion, and oher coss. There are wo imporan problems associaed wih his daa se: one is ha he NSF does no use he above classificaion consisenly over years; anoher problem is ha daa are no available for quie a few years. To deal wih hese problems, we mus make assumpions and creae approximae daa. For example, for he years 953 o 96, we do no have daa relaed o he ype of cos. Thus, we assume ha he cos srucure for hese years was he same as ha in 96. Similarly, for he period 977 o 997, we have daa every wo years. To obain daa for he missing years, we use moving average mehods o deermine he missing daa. Finally, we group he R&D expendiures ino hree caegories: Wage and Salaries (labour), Maerials (inermediaes) and Capial expendiure. Unforunaely, he cos caegory, overhead or oher coss, accouns for a big porion of he expendiure. We allocae hese expendiures o wage and salaries, maerials and capial expendiure according o he BLS indusry cos shares. From he above descripions, we should be aware ha assumpions made o fill in gaps in he R&D daa can have a direc effec on he qualiy of he resuling daa ses and on he resuls of analyses based on hese daa ses. Afer consrucing R&D cos componen informaion, we can make adjusmens o he iniial BLS inpu daa ses. R&D labour cos, wage and salaries, is subraced from he oal labour cos; he maerial componen of R&D expendiure is subraced from he oal inermediae inpu cos; and he capial expendiure par of R&D is subraced from he oal capial service cos. Dividing hese adjused value series by an implici price index, yields quaniy series for labour, inermediae and capial service inpu. The quaniy series of R&D invesmen is consruced by using he Törnqvis index formula. Price indexes for he hree componens of R&D expendiure are assumed o be same as indusrial inpu price indexes. Finally, we need nominal ineres raes for he year 953 o 000 o consruc he discoun facors. Nominal ineres raes are obained from he on-line daa of Federal Reserve Sysem. We choose he long-erm nominal ineres rae: marke yield on U.S. Treasury securiies a 0-year consan mauriies, quoed on an invesmen basis, o consruc he series of discoun facors.

17 7 4.3 Esimaion Resuls Maximum likelihood esimaion, which is he non-linear opion in SHAZAM, is sensiive o he choice of saring poin, which also affecs he number of ieraions. We sar from a simple regression wih M, b, e and e in equaion (7) and equaions (36) o (40) se equal o zero iniially. The esimaed parameers from his regression are used as he saring poin for he nex regression, which adds addiional parameers. As we proceed, we also check for big jumps in he log likelihood of he model (if he jumps are small, hen he inclusion of he exra parameers is no warraned bu in general, we obain significan increases in he log likelihood as we add he exra parameers). We esimae he following four models: Model I: This is he basic model defined by equaion (7) and equaions (36) o (40). Wihou splines and wihou a weighed subsiuion marix, his model may no properly reflec real world complexiy. We expeced relaively low values of he log-likelihood for his class of models. Model II: This model adds a weighed subsiuion marix o equaions (36) o (40). I can deal wih he possible rending elasiciy problem. If our model does have his rending elasiciy problem, we would expec o see a big increase in he value of he loglikelihood funcion. Model III: This model adds linear splines in he ime variable based on he las model. Non-smooh change in echnical progress can be capured by adding hese splines. Model IV: This model adds quadraic splines for he ime variable o Model II (raher han linear splines as in Model III). The following able liss he esimaes of he depreciaion rae, he value of maximum loglikelihood, and he value of markup facor for he above 4 models for U.S. manufacuring and he four seleced knowledge inensive indusries. 5 Table. Depreciaion Raes Maximizing he Log-likelihood Funcion Model Model Model 3(a) Model 3(b) SIC 8 SIC 35 SIC 36 SIC 37 Manuf Dep Rae Markup Facor Log-likelihood Dep Rae Markup Facor Log-likelihood Dep Rae ** Markup Facor Log-likelihood Dep Rae ** 0.08 Markup Facor The break poins for he spline models are repored in Appendix B.

18 8 Model 3(c) Model 4(a) Model 4(b) Model 4(c) Log-likelihood Dep Rae 0.0** ** Markup Facor Log-likelihood Dep Rae ** Markup Facor Log-likelihood Dep Rae Markup Facor Log-likelihood Dep Rae Markup Facor Log-likelihood From he above able, we can see ha he value of he maximum log-likelihood generally improves considerably from Model I o Model II. This suggess we have rending elasiciy problems. Wih linear splines or quadraic splines added o he model, he value of maximum log-likelihood funcion increases dramaically. In comparison wih he resuls obained from he linear spline model (Model III) and he quadraic spline model (Model IV), he value of he loglikelihood funcion increases in Model IV for Elecrical Producs, and decreases for Chemical Producs and Transporaion Equipmen. For Non-elecrical Machinery and he manufacuring secor, he changes of he log-likelihood value from Model III o Model IV are mixed. We also ry differen break poins for Model III and Model IV 6. Unforunaely, i urns ou ha he esimaes are sensiive o he choice of he break poins. Consequenly, in order o choose an appropriae value for he depreciaion rae of R&D capial, we have o use our subjecive judgemen. This is one drawback of our modeling sraegy. In our example, we choose he depreciaion rae for R&D based on values of boh he markup facor and he log-likelihood funcion. According o he definiion of markup facor given in he second secion, a reasonable value should be less han. Our final choices of he depreciaion raes are given by able. Table. Depreciaion Raes and Markup Facors Depreciaion Rae Markup Facor SIC 8 SIC 35 SIC 36 SIC 37 Manufacuring * 0.7*** 0.9*** (0.866) (.588) (3.0986) (5.3064) (0.534) (0.06) (0.0494) (0.0554) (0.04) (0.09) Noes: : The values in he brackes are log-likelihood saisics, which is defined as: χ = [ lg( L 0 ) lg( L )]. *: Saisically significan wih 0% confidence level; ***: Saisically significan wih a leas % confidence level. 6 Differen break poins corresponding o he esimaions lised in Table are given in he appendix B.

19 9 : The values in he brackes are he sandard errors. From he esimaed markup facors, we can derive he markup for he oal manufacuring secor and he four indusries. The markup is 5.3 percen for he oal manufacuring secor, 3. percen for chemical producs, 5. percen for non-elecrical machinery, 8. percen for elecrical producs, and 3.7 percen for ransporaion equipmen. As shown in able, he esimaed depreciaion raes for SIC 8 and SIC 35 are no significanly differen from zero. We may inerpre his as he verificaion of some economis s belief ha he knowledge asse should no depreciae over ime. Anoher possible explanaion for he small depreciaion raes is ha our model does no fi he daa well in hese indusries. In order o esimae accuraely he R&D depreciaion rae, we need some large flucuaions in R&D invesmens. As we have poined our in secion, R&D invesmens in SIC 8 and SIC 35 increase relaively smoohly in our sample period. The lack of flucuaions in R&D invesmens may cause our model fail o find he appropriae depreciaion rae. Our oher esimaes fall in he range of R&D depreciaion raes ha are found in he lieraure. Depreciaion raes for SIC 37 and Manufacuring are very close o each oher, which may due o he similar paern in he changes of R&D invesmens in hese wo secors. Using our esimaes of he depreciaion rae for R&D capial, we can consruc he series of R&D capial socks for manufacuring and he four knowledge inensive indusries. The following figure shows how R&D socks change over he period of Figure. R&D Socks ( ) R&D S ocks ( ) SI C 8 SI C 35 SI C 36 SI C 37 Manuf ac ur i ng The geomeric growh rae of R&D socks for manufacuring is 3.4%, and he geomeric growh raes of he R&D socks for he four knowledge inensive indusries, namely SIC 8, SIC 35, SIC 36 and SIC 37, are 4.5%, 5.%, 3.9% and 3.%, respecively. 5. Conclusion In his paper, we have developed a simple model based on a producion funcion o esimae he depreciaion raes of R&D capial for he U.S. oal manufacuring secor and he four knowledge

20 0 inensive indusries, including chemical producs (SIC 8), non-elecrical machinery (SIC 35), elecrical producs (SIC 36) and ransporaion equipmen (SIC 37). We rea R&D capial as a echnology shifer insead of as an ordinary inpu in he model. Using boh he R&D sock variable and ime variable as echnology shifers can avoid he overesimaion of R&D capial s conribuion o produciviy growh. Along wih he esimaion of he depreciaion rae, we have esimaed he markup facor for he U.S. manufacuring and he four seleced indusries. The esimaed R&D depreciaion rae is 9 percen for U.S. oal manufacuring secor, percen for chemical producs, 3 percen for non-elecrical machinery, 4 percen for elecrical producs and 7 percen for ransporaion equipmen. The corresponding markup is 5.3 percen for he oal manufacuring secor, 3. percen for chemical producs, 5. percen for non-elecrical machinery, 8. percen for elecrical producs, and 3.7 percen for ransporaion equipmen. Based on he esimaed depreciaion rae, he geomerical growh rae of he R&D sock is 3.4 percen for he manufacuring secor, 4.5 percen for chemical producs, 5. percen for nonelecrical machinery, 3.9 percen for elecrical producs, and 3. percen for ransporaion equipmen. The resuls repored here are preliminary. We have no incorporaed some imporan feaures associaed wih R&D invesmen, such as he uncerainy of R&D invesmen and he exernaliy of he creaed knowledge. Also, we have imposed some resricive assumpions o simplify he problem, such as consan depreciaion raes over years, consan markup facors, and full informaion abou he fuure prices. In addiion, he robusness of he resuls should be checked agains alernaive funcional forms for he producion funcion and agains alernaive ways of consrucing he sock of R&D capial. 7 Appendix A: The Indusry s Profi Maximizaion Problem Each indusry s profi maximizaion problem can be wrien as follows: (A) Max β x R + = 0 's, 's y = f (x,r,) { P (y,)y w x P I } subjec o R and = Ir, + ( δ) R. r, Subsiuing he consrains ino he objecive funcion, we have he following equivalen problem: (A) +... Max x 's,r 'sβ {P (f (x, R, ), )f (x, R, ) w x Pr, (R ( δ)r )} + + +β {P (f (x,r, + ), + )f (x,r, + ) w x Pr, (R ( δ)r )} +β {P (f (x,r 3, + ), + )f (x,r, + ) w x Pr, (R 3 ( δ)r )} Therefore he firs order necessary condiions wih respec o vecor x can be wrien as: (A3) p xf (x,r,) + [ P(y,) / y]y xf (x,r, ) = w, = 0,,...T, 7 Our resuls may also be subjec o some aggregaion bias since we have used indusry daa insead of firm daa.

21 where p is he oupu price and x f(x,r,) is he vecor of firs order parial derivaives of he period producion funcion wih respec o he componens of he inpu vecor x. Facoring ou he oupu price p and x f(x,r,) on he lef hand side of equaion (A3), we have he following simplified form for equaion (A3): (A4) P(y,) / y p (+ ) xf (x,r,) = w p / y. The firs order necessary condiions wih respec o R&D sock variable R are as follows: (A5) β P( ) f ( ) f ( ) y + p + Pr, ( δ) y R R = β Pr,. Afer some rearrangemen, we can rewrie he above equaion as follows: (A6) P f (x,r ) ( ) / y + +, β p + = Pr, ( δ) Pr, P / y R β +. Assuming ha β /β = +r where r is he nominal ineres rae, he above equaion can be rewrien as: (A7) P f (x,r, ) ( ) / y + + p + = (+ r )Pr, ( δ) Pr, P / y R +. Define he period non-negaive markup as follows: (A8) P(y, ) / y m = 0 p / y ε. The corresponding period markup facor M can be defined as: (A9) M P(y, ) / y = m = + = + ε p / y. If we assume ha markup facors are consan over ime, we can rewrie our sysem of firs order condiions as: (A0) w = p M xf (x,r,), and f (x,r, + ) (+ r )Pr, ( δ)pr, + = p M R (A).

22 Moving he nd erm o he righ hand side of equaion (A) and dividing hrough by P r,, we obain: (A) p f (x,r, + ) Pr, + r = M + ( δ) Pr, R Pr,. Equaions (A0) and (A) form our final sysem of esimaing equaions. Appendix B: Break Poins Table B. Break Poins for Model III and Model IV Eq. SIC 8 SIC 35 SIC 36 SIC 37 MANF 9,, 9, 34, 44 4, 3, 39 0, 38, 46 3, 7, 3, 4 4,, 6, 9, 34, 38 Model 5,, 3, 9, 40 9, 9, 7, 39, 7, 30, 38 9,, 3, 36, 40 3(a) and Model 4(a) , 8, 35, 4 8, 3,, 3, 4,, 38, 8, 3, 38 3, 8, 36, 4, 47 4, 8, 35, 4 30, 39 6, 40, 8, 35, 40,, 9, 36, 40 5,, 9, 3, 39 0, 3, 38 9,, 30, 43 4,, 9, 34, 34, Model 5,, 43 0, 9, 30, 9,, 36, 4 6,, 30,39 9,, 3, 36, 40 3(b) 38 and 3 3, 8, 35, 8, 3,,,, 30, 3, 7, 3, 38 3,, 36, 4, Model 4 38, 4 40, (b) 4, 8, 35, 9, 30, 39 4, 37 6,, 4, 8,,, 9, 36, 4 33, ,, 9, 34, 44 3, 39 0, 38 9,, 30, 43 5,, 9, 34, 39, 44 Model 5,, 30, 0, 9, 30, 8,, 4 7,, 30,37 9,, 3, 8, 3(c) and , 46 Model 4(c) 3 3, 8, 35, 45 8, 4,, 37, 4,, 30, 45, 8, 3, 38 3, 8, 36, 4, 45 4, 8, 35, 4 9, 30, 39 4, 37 3, 8, 35, 39 9,, 9, 36, 40 Noe: Equaions ()-(3) are esimaing equaions for he hree inpus; equaion (4) is he producion funcion.