THE TYRANNY OF THE IDENTITY: GROWTH ACCOUNTING REVISITED

Transcription

1 THE TYRANNY OF THE IDENTITY: GROWTH ACCOUNTING REVISITED Jesus Felpe Economcs and Research Deparmen Asan Developmen Bank Manla (Phlppnes) e-mal: JSL McCombe Cenre for Economc and Publc Polcy Unversy of Cambrdge Cambrdge (UK) e-mal: Sepember 8, 2005 Absrac: Ths paper shows, frs, ha he recen proposal by Barro (1999) and Hseh (1999, 2002) o perform growh accounng by drecly dfferenang he NIPA deny s smply an exercse n he manpulaon of an accounng deny whou any foundaon. Second, smulaons show ha growh accounng performed wh aggregae daa s no equvalen o he rue rae of echnologcal progress mpled by he mcro daa. Thrd, we conclude ha he rae of oal facor producvy growh usually esmaed s smply a measure of dsrbuonal ncome. JEL Classfcaon: O11, O16, O47, O53 We are graeful o Rana Hasan, Frankln M. Fsher and he parcpans n he Conference on Undersandng Economc Growh: New Drecons n Theory and Polcy, Cambrdge, U.K., Sepember 1-3, 2005, for her commens and suggesons. The usual dsclamer apples. Ths paper represens he vews of he auhors and does no represen hose of he Asan Developmen Bank, s Execuve Drecors, or he counres ha hey represen.

2 THE TYRANNY OF THE IDENTITY: GROWTH ACCOUNTING REVISITED The concluson o be drawn from hs bref dscusson s ha aggregaon s a serous problem affecng he magnude, he sably, and he dynamc changes of oal facor producvy. We need o be cauous n nerpreng he resuls ha depend on he exsence and specfcaon of he aggregae producon funcon Tha he use of he aggregae producon funcon gves reasonably good esmaes of facor producvy s due manly o he narrow range of movemen of aggregae daa raher han he sold foundaon of he funcon. In fac, he aggregae producon funcon does no have a concepual realy of s own. Nadr (1970, p ). 1. INTRODUCTION A seres of papers (Barro 1999; Hseh 1999, 2002) have recenly argued ha growh accounng exercses, specfcally he dervaon of he dual measure of oal facor producvy growh (TFPG), can be performed by smply dfferenang he Naonal Income and Produc Accouns (NIPA) deny. Accordng o hs, value added equals he wage bll plus oal profs, ha s, V Y M W + Π w L + r J (1) where V s value added, Y s gross oupu, M s he value of nermedae maerals, W s he oal wage bll, Π denoes oal profs (operang surplus n he NIPA ermnology), w s he average real wage rae, L s employmen, r s he average expos prof rae (hs s dscussed below), and J s he deflaed value of he sock of capal, wh all varables expressed n real erms. Expressng equaon (1) n growh raes yelds: Vˆ a ŵ + (1 a )rˆ + a Lˆ + ( 1 a ) Ĵ 1

3 TFPG + a Lˆ + (1 a ) Ĵ (2) where ^ denoes a proporonal growh rae, 1 a r J / V a w L / V s he share of labor n oupu, s he share of capal. Equaon (2) can be rearranged o gve: TFPG Vˆ (1 a ) Ĵ a Lˆ a ŵ + (1 a ) rˆ (3) whch s nerpreed as he rae of oal facor producvy growh. Ths approach o growh accounng sands n marked conras o he argumens of Felpe and McCombe (2003). There was argued ha precsely because of he exsence of he accounng deny, growh accounng exercses amoun o no more han manpulaons of he ex-pos naonal ncome accounng deny, and, as such, hey are auologes whou necessarly any behavoral conen. Ths accounng deny, as shown above, s he one ha relaes he value of oupu o he sum of he wage bll plus profs. The reason for he problem ha Felpe and McCombe (2003) hghlgh s ha here s an mporan dfference beween usng physcal daa, only avalable a a dsaggregaed level, and usng daa expressed n value erms, he ype daa used a any level of aggregaon. For purposes of growh accounng exercses and for he compuaon of he rae of TFPG n parcular, daa n value erms on oupu, capal and labor (a physcal quany) are relaed hrough he above accounng deny. As a consequence, hs paper shows ha he rae of TFPG esmaed wh aggregae daa has lle relaon wh he rae of echncal progress esmaed usng physcal daa. I should be noed, however, ha esmaes of TFPG have recenly been calculaed usng physcal oupu a he seven-dg SIC produc classfcaon (Foser e al. 2005). We brefly dscuss hs work and argue ha does no avod he problems ha hs paper dscusses. 2

4 Ths paper has a dual purpose. The frs one s o provde an evaluaon of wha superfcally seems o be a useful mehodologcal; ha s, o derve he rae of TFPG from he NIPA, as suggesed by Barro and Hseh. The second purpose s o sudy usng smulaons wheher he rae of oal facor producvy growh (TFPG) usng aggregae daa (n value erms) s equvalen o he rue rae of echncal progress (ϕ ) esmaed usng dsaggregaed daa (n physcal erms). The dfference beween daa n physcal erms and n value erms s prces, ypcally unobserved. The wo objecves of he paper are, n fac, relaed. Indeed, as we shall see, he rae of TFPG s deermned by he accounng deny. In dong hs, we rase some fundamenal mehodologcal problems of he growh accounng approach, and exend he argumens n Felpe and McCombe (2003) n he conex of he well-known and seemngly nexhausble debae of he sources of growh n Eas Asa ha began wh he growh accounng sudes of Young (1992, 1995) and economerc analyss of Km and Lau (1994), and whch Hseh (1999, 2002) has revsed. 1 Neverheless, he argumen goes beyond hs parcular debae and for hs reason we also ake Barro s (1999) survey on growh accounng, who also adops he same approach as Hseh, as our sarng pon. Fernald and Neman (2003) also dscuss he debae abou he sources of growh n Eas Asa and carry ou her growh accounng analyss from he NIPA accounng deny. Har (1996) also shares he same vew. We dscuss hs below. Hseh s (1999, 2002) approach s based on he assumpon ha growh accounng can be performed by smply dfferenang he NIPA deny, as shown above. 2 The use of he dual s no new and, ndeed, forms he raonale behnd Jorgenson and Grlches (1967) semnal growh accounng sudy. As Hseh pus : 1 Hseh s (1999, 2002) cenral ene abou he Eas Asan economes s he observaon ha f he capal-oupu rao n he NIEs ndeed ncreased as much as he daa ndcaes, he reurn o capal should have fallen dramacally as capal accumulaon encouners dmnshng reurns. Ths s a pon also made by Nelson and Pack (1999) and elaboraed upon by Felpe and McCombe (2001). Hseh s calculaons and daa sources have been subjec o crcsm by Young (1998). 2 Hseh (2002, p.502) argues: I s useful o hnk abou hs as an accounng deny. 3

5 wh only he condon ha oupu equals facor ncomes, we have he resul ha he prmal and dual measures of he Solow resdual are equal. No oher assumpons are needed for hs resul: we do no need any assumpon abou he form of he producon funcon, bas of echnologcal change, or relaonshp beween facor prces and her socal margnal producs. We do no even need o assume ha he daa s correc. For example, f he capal sock daa s wrong, he prmal esmae of he Solow resdual wll clearly be a based esmae of aggregae echnologcal change. However, as long as he oupu and facor prce daa are conssenly wrong, he dual measure of he Solow resdual wll be exacly equal o he prmal measure, and consequenly, equally based. The wo measures of he Solow resdual can dffer when naonal oupu exceeds he paymens o capal and labor (Hseh 2002, p.505; alcs added). Barro (1999) also concurs: he dual approach can be derved readly from he equaly beween oupu and facor ncome (Barro 1999, p.123). To show hs, he wres he ncome accounng deny, dfferenaes, and expresses n erms of growh raes (Barro 1999, equaons (7) and (8)). Barro and Hseh agree ha: s mporan o recognze ha he dervaon of equaon (8) [he growh accounng equaon n hs paper] uses only he condon V = r J + w L. No assumpons were made abou he relaons of facor prces o socal margnal producs or abou he form of he producon funcon (Barro 1999, pp.123). 3 Barro connues: If he condon V = r J + w L holds, hen he prmal and dual esmaes of TFP growh nevably concde [ ] If he condon V = w L + r J holds, hen he dscrepancy beween he prmal and dual esmaes of TFP has o reflec he use of dfferen daa n he wo calculaons (Barro 1999, p ). In he lgh of Barro s (1999) and Hseh s (1999, 2002) papers, s worh elaborang her argumens and conrasng hem wh hose of Felpe and McCombe (2003). Whle Hseh and Barro acknowledge ha hey manpulae an deny, no whsandng he quoaon above s clear ha here are neoclasscal assumpons 3 The noaon n Barro s quoaons has been changed o make conssen wh ha n hs paper. 4

6 mplc n her analyss. 4 Our argumen s ha because of he underlyng accounng deny n value erms, s no possble o es hese assumpons (and poenally refue hem) and hence o dchoomze he growh of oupu no ha caused by he (weghed) growh of he facors of producon and a resdual, whch s nerpreed as he rae of echncal change or, more generally, as he rae of ncrease n effcency of he economy. We conclude ha he procedure Hseh and Barro propose s whou any foundaon. The res of he paper s srucured as follows. Secon 2 summarzes he Barro- Hseh approach o growh accounng. The secon also remnds he reader of he nsurmounable problems posed by he Cambrdge Capal Theory debaes and he aggregaon problem for he noon of aggregae producon funcon. Secon 3 explans why mos ofen esmaon of aggregae producon funcons wh me seres leads o poor resuls; and shows why esmang economercally aggregae producons, f done correcly, mus always lead o he same paradoxcal fndng. Secon 4 uses a seres of smple smulaons o shed lgh on wha growh accounng s acually measurng, and consders wheher or no he rae of TFPG usng aggregae daa s equvalen o he rue rae of echncal progress esmaed usng dsaggregaed daa and hen added up. The answer s no. Gven hese resuls, we propose n secon 5 a renerpreaon of growh accounng exercses and of he rae of TFPG conssen wh he non-exsence of he aggregae producon funcon. We also brefly consder Har s (1996) and Foser e al. s (2005) analyses. Secon 6 concludes. 2. THE NIPA, GROWTH ACCOUNTING AND THE BARRO-HSIEH APPROACH The Barro-Hseh hess s ha growh accounng exercses can be drecly performed usng daa from he NIPA, accordng o whch value added equals he paymens o he facors of producon, as expressed n equaon (1). From our pon of vew, s mporan o emphasze ha he symbol ndcaes ha expresson (1) s an 4 Ths s very clear n Hseh (2002, p.502), where he refers o he renal prce of capal and he margnal produc of capal. These conceps are heory dependen. 5

7 accounng deny, whch has, per se, no behavoral mplcaons and wll hold rrespecve of he sae of compeon and wheher or no an aggregae producon funcon acually exss. I mus be added ha he NIPA does no provde he decomposon on he rgh-hand sde of (1), bu only he aggregae sum of he paymens o he facors of producon, ha s, V wage bll W + Π. The decomposon of he ( W ) and he operang surplus ( Π ) no he producs of he facor prces mes he quanes s defnonal, bu arfcal. Moreover, wll always be rue ha he wage bll can be wren as he produc of he average wage (w) rae mes employmen (L). On he oher hand, wheher he wage rae equals he margnal produc of labor or no, s que anoher maer. Neverheless, daa on labor and he consan prce value of he capal sock are avalable from he sascal offces such as he US Bureau of Labor Sascs and he OECD. Why do Hseh and Barro argue ha growh accounng can be performed by smply hnkng n erms of he NIPA? Equaon (3), under he usual neoclasscal assumpons, s formally equvalen o he resdual measure of oal TFPG. The frs par, TFPG = Vˆ, s dencal o he growh accounng equaon derved from a neoclasscal aggregae producon funcon, mposng he condons for producer equlbrum, and s referred o as he prmal measure of TFPG. The second par of equaon (3), TFPG (1 a ) Ĵ a Lˆ a ŵ + ( 1 a ) rˆ, s he dual measure of TFPG, derved n neoclasscal economcs from he cos funcon. The laer s he expresson Hseh used n hs emprcal analyses. Wha are he neoclasscal assumpons referred o above ha lead Barro and Hseh o argue ha equaon (1) allows a growh accounng exercse? Implc n he neoclasscal approach s he exsence of an aggregae producon funcon wh Hcks neural echncal change,.e., ). Facors are pad her margnal producs ( f, =L, J) and Euler s heorem gves he resul ha V = f L f J where f L and V = A( )F( L,J L + f J are equal o he facor prces, whch equal he margnal socal producs. Moreover, he oupu elasces are equal o he relevan facor shares. In our vew, J 6

8 he mos conroversal assumpon underlyng he growh accounng approach s he exsence of such an aggregae producon funcon. Ths s a queson no even consdered n orhodox analyses. We reurn o hs ssue a he end of he secon. If he aggregae producon funcon does exs, may be wren generally n growh rae form as Vˆ = TFPG + α Lˆ + β Ĵ (4) where α and β are he oupu elasces of labor and capal, respecvely; and TFPG s nerpreed as he growh rae of echncal progress, or he Solow resdual. As here are usually no relable esmaes of α or β, by nvokng he frs order condons (from prof maxmzaon and compeve markes), he facor elasces are aken o equal he approprae facor shares,.e., α = a and β = 1 a. Hence: Vˆ = TFPG + a Lˆ + (1 a ) Ĵ (5) These are no nnocuous assumpons. The nerpreaon of TFPG n equaons (2) and (3) as he rae of echnologcal progress follows drecly from a comparson wh (5); oherwse, on wha grounds can TFPG n equaons (2) and (3) be referred o as he rae of echncal progress? As we have noed above, he aggregae producon funcon, ogeher wh he condons for producer equlbrum (he frsorder condons), provde he underlyng heory for he sandard nerpreaon of he growh accounng exercse. In fac, s argued ha he aggregae producon funcon provdes a heory of he ncome sde of he NIPA (Presco 1998, p.532). Ths follows from he alleged lnk ha s beleved o exs beween he accounng deny, he aggregae producon funcon (assumed o be homogeneous of frs degree, and o have posve frs order paral dervaves and negave second order paral dervaves) and, as menoned above, Euler s heorem (Hulen 2000, p.11). The heory mples ha V = f L f J. From he frs-order condons f L = ( F / L ) = w and L + J 7

9 f J = ( F / J ) =ν, follows ha V = wl + νj, (where ν s he renal prce of capal) whch s hough o be he deny (1). Somehow, he neoclasscal framework consders ha he producon funcon, hrough Euler s heorem, mples he deny. Bu he mporan hng o noe s ha under hs argumen, hs deny has become a vrual deny, one ha holds f and only f he heory under whch was bul holds. On he oher hand, our argumen s ha he wage bll and oal profs can be spl no he respecve producs (.e., equaon (1)) whou recourse o any heory and do no requre ha wage and prof raes equal her respecve margnal producs. Our argumen (Felpe and McCombe 2003) s ha he deny V W + Π wl + rj holds rrespecve of wheher any of hese condons are fulflled. Hseh s quoaon ced above ha we do no need any assumpon abou he form of he producon funcon, bas of echnologcal change, or relaonshp beween facor prces and her socal margnal producs s msleadng o he exen ha could be nerpreed as mplyng ha none of hese condons are requred for growh accounng. Wha smply means s ha f we have an deny X Y + Z, hen hs may be expressed as θŷ Xˆ ( 1 θ ) Ẑ, where θ Y/X. If we erm he rgh-hand sde he prmal, hen by defnon mus equal he lef-hand sde, wheher s called he dual or anyhng else. Consequenly, any measuremen error on eher he rgh-hand sde or he lefhand sde of he equaon n growh rae form mus lead o an equal error on he oppose sde. Bu hs should no lead o overlook he necessary assumpons ha underle he growh accounng approach, f eher he dual or he prmal s o be nerpreed as a measure of echncal progress, or of he ncrease n effcency. If here are ncreasng reurns o scale, hen he Solow resdual could sll be calculaed as he prmal as (α+β >1), bu clearly hs resdual would no be equal o he weghed average of he growh raes of he wage and prof raes (.e., TFPG = Vˆ αlˆ βjˆ a ŵ + (1 a ) rˆ ). Facors of producon do need o be pad her margnal socal values f he Solow resdual s o be gven an economc nerpreaon. Moreover, 8

10 whle s rue ha we do no need o know he precse funconal form of he aggregae producon funcon, we do need o assume ha exss. Consequenly, bearng hs qualfcaon n mnd, from he accounng deny, f a conssen daa se (.e., one ha makes he deny (1) hold) s used for calculang he dual and he prmal measures of TFPG, hen, by defnon, hey mus be equal. The deny consss of fve varables, namely, V, w, L, r, and J. If values of each are obaned ndependenly hen s possble ha he deny, equaon (1), wll no hold because of measuremen error. For hs reason, o ensure conssency, one of he varables mus be obaned resdually. The NIPA provdes daa on V, W and Π; daa on w and L can be obaned from wage and labor force sascs; and an esmae of J may be obaned by he perpeual nvenory mehod (some sascal offces also compue ). Consequenly, s ofen r ha s obaned resdually. Bu f here were ndependen esmaes of r publshed or calculaed, one could equally oban J (or any of he oher varables for ha maer) resdually. 5 However, he wo approaches could dffer f he deny s no conssen. For example, f r and J are calculaed ndependenly, may well be ha r J V - w L. Ths may be useful n erms of he growh accounng approach o hghlgh possble measuremen errors. However, he accounng deny mus hold and so eher r or J mus be adjused o ensure ha he deny s conssen. I s mporan o menon ha, n hs calculaons, Hseh dd no use he accounng deny exacly as descrbed above,.e., by calculang resdually one of he fve varables. Hseh dd no calculae he fve seres ndependenly; nsead of calculang r resdually, he compued he renal prce of capal ( v ) as v = q [ ρ + δ q& ] (Jorgenson 1963) where q s he prce of capal, ρ s a measure of he cos of capal, δ s he deprecaon rae and q& s he capal gan or loss. If hs 5 Consder he followng. From he NIPA we can oban he labor share as a (W / V ). Now suppose we oban ndependen daa on he average wage rae (w) and employmen (L), and fnd ha a L / V ). However hs deny s calculaed, he four (w seres have o preserve. The same apples o he varables ha make up he capal share. 9

11 s used n he deny wh he oher varables ndependenly measured, hen here may be a sascal or measuremen problem along he lnes oulned above. Neverheless, n Felpe and McCombe (2005a, 2005b) we show ha hs ssue does no pose a concepual problem for our argumen. 6 Before concludng hs analyss of he Barro-Hseh approach, s worh nong ha, as ndcaed above, a crucal assumpon underlyng hs whole approach s ha a well-behaved aggregae producon funcon exss. Ths s aken o be self-evden and s rarely, f ever, explcly dscussed. The producon funcon s a relaonshp beween he amoun of oupu produced and he quanes of npus used (n a conex of opmal allocaon), ha s, Q = A( )F( L,K ), where Q, L and K are oupu, labor and capal, respecvely, all measured n physcal erms. However, hs s no an nnocuous assumpon and ndeed here are wo heorecal reasons for belevng ha he aggregae producon funcon canno exs. The frs sems from he resuls of he Cambrdge Capal Theory Conroverses, whch seem o be long forgoen and rarely menoned n he leraure. 7 Neverheless, however nconvenen for he aggregae neoclasscal approach, hese resuls sll sand. See, for example, he recen rerospecve vew of Cohen and Harcour (2003). Secondly, here are he aggregaon problems, recenly surveyed by Felpe and Fsher (2003). The mos mporan resuls of hs leraure can be summarzed as follows: 6 Hseh dd no use, and hence dd no calculae, all he seres n he deny as he concenraed on he dual measure of TFPG. We hank Hseh for ponng hs ou n correspondence wh hm. However, when he calculaed he reurn o capal derved from he naonal accouns, mplcly he was usng all he seres n he deny and calculaed one of hem (he reurn o capal) resdually. 7 Bernanke (1987), n passng, whle dscussng an emprcal paper by Romer where he laer esed a seres of endogenous growh models, made he followng remark: I would be useful, for example, o hnk a b abou he meanng of hose arfcal consrucs oupu, capal, and labor, when hey are measured over such long me perods (he Cambrdge- Cambrdge debae and all ha) (Bernanke 1987, p.203; alcs added). Sylos-Labn (1995) recenly wroe: I s worh recallng hese crcsms, snce an ncreasng number of young and alened economss do no know hem, or do no ake hem serously, and connue o work ou varans of he aggregae producon funcon and nclude, n addon o echncal progress, oher phenomena, for example, human capal (Sylos-Labn 1995, p.487). 10

12 Excep under consan reurns, aggregae producon funcons are unlkely o exs a all. Even under consan reurns, he condons for aggregaon are so very srngen as o make he exsence of aggregae producon funcons n real economes a non-even. Ths s rue no only for he exsence of an aggregae capal sock bu also for he exsence of such consrucs as aggregae labor or even aggregae oupu. One canno escape he force of hese resuls by argung ha aggregae producon funcons are only approxmaons. Whle, over some resrced range of he daa, approxmaons may appear o f, good approxmaons o he rue underlyng echncal relaons requre close approxmaon o he srngen aggregaon condons, and hs s no a sensble hng o suppose. Suffce o say here ha boh hese crques addressed, from very dfferen pons of vew, he queson of wheher or no he aggregae producon funcon, wh he usual neoclasscal properes, s a legmae concep, even when vewed as an approxmaon. Boh crcsms come o he same concluson on hs pon: he exsence and hence he use- of aggregae producon funcons s very problemac. Ths mples ha he connecon made n neoclasscal macroeconomcs beween he deny, he aggregae producon funcon and Euler s heorem s very enuous, o say he leas. Indeed, hs lne of reasonng s unenable f he aggregae producon funcon does no exs; 8 and as Fsher poned ou, If aggregae capal does no exs, hen of course one canno beleve n he margnal producvy of aggregae capal (Fsher 1971a, p.405; alcs n he orgnal). Why s hen he aggregae producon funcon sll so wdely used? The answer, alhough s rarely explcly saed, mus be ha ever snce he early esmaons of Douglas and hs colleagues, he sascal esmaon of aggregae producon funcons gves good fs wh he esmaes of he oupu elasces close 8 As a consequence, parameers such as he elascy of subsuon are meanngless. In he words of Fsher e al. (1977): he elascy of subsuon n hese producon funcons s an esmae of nohng; here s no rue aggregae parameer o whch corresponds (Fsher e al. 1977, p.312). 11

13 o he observed facor shares, alhough me-seres resuls are known for her fragly (see Sylos-Labn 1995; see Felpe and Fsher 2003 for a dealed dscusson). Ths argumen reles on Fredman s (1953) nsrumenal mehodologcal jusfcaon ha wha maers are he successful predcons of a heory, regardless of he unrealsm of he underlyng assumpons. No only s hs vew no longer wdely held by phlosophers of scence, bu our argumen shows ha hs jusfcaon s flawed, even on s own erms (McCombe 1998a). 3. ESTIMATING THE AGGREGATE PRODUCTION FUNCTION Barro (1999) brefly compared growh accounng o he economerc esmaon of he producon funcon and favored he former mehod. He argued ha growh accounng s preferable because he laer has some mporan dsadvanages (Barro 1999, pp ): () he growh raes of capal and labor are no exogenous varables wh respec o he growh of oupu; () he growh of capal s usually measured wh error. Ths ofen leads o low esmaes of he conrbuon of capal accumulaon; and () he regresson framework mus be exended o allow for varaons n facor shares and he TFP growh rae. I s useful o address he las pon frs snce he argumens concernng render he frs wo largely rrelevan. For hs purpose, le us reurn o equaon (2) and assume ha n hs economy facor shares are consan, for example, because frms pursue a consan mark-up prcng polcy. 9 Suppose also ha n hs economy wage and prof raes grow a consan raes. 10 Then, hs expresson becomes: Vˆ aŵ + ( 1 a )rˆ + alˆ + ( 1 a ) Ĵ 9 The consancy of he facor shares does no mply a Cobb-Douglas producon funcon, as Fsher s (1971b) smulaon exercses showed. 10 The sylzed facs are ha he rend growh of he real wage rae s consan, whle he rae of prof shows no sysemac rend. The rae of prof, and o a lesser exen he real wage rae, exhb procyclcal varaon. The consancy of facor shares s a furher sylzed fac. 12

14 λ + alˆ + (1 a ) Ĵ (6) If hese assumpons hold, equaon (6) wll connue beng he accounng deny n growh raes. I mples nohng abou he echnologcal condons of producon. Inegrang equaon (6) and akng an-logarhms gves: where 0 a 1-a V A exp( λ ) L J (7) λ aŵ + (1 a ) rˆ and s he consan of negraon. I should be noed ha A 0 hs s merely anoher way of wrng he deny, alhough n hs case wh he wo assumpons. Ths resul ndcaes ha f one ges daa for he economy n queson and regresses oupu (V) on labor (L), capal (J), and an exponenal me rend (),.e., b b 2 3 V = B0 exp(b1 ) L J exp( ε ), where ε s he random dsurbance, and he wo assumpons made happen o be correc, s obvous, by comparson wh (7), ha he f wll be perfec, b 1 aŵ + ( 1 a ) rˆ λ, b 2 a, and 1 a. Under a neoclasscal nerpreaon, he equaly of he elasces o he facor shares would be nerpreed as valdaon of he neoclasscal heory of facor prcng and, consequenly, ha markes are compeve. Moreover, he resul ndcaes consan reurns o scale. Bu hs economy could well be a counry wh a command economy where facors are no pad her margnal producs. All we have used n dervng equaon (7), o paraphrase Hseh and Barro, s he condon ha oupu equals he paymen o he facors of producon, ogeher wh he wo sylzed facs. The fac ha he esmaed oupu elasces closely approxmae he facor shares does no mply ha markes are compeve, and ha here are consan reurns o scale. Ths correspondence merely follows from he accounng deny. Wha f he assumpons abou he facor shares and he wage and prof raes are ncorrec (and hs addresses Barro s hrd pon)? The esmaon of eher equaons (6) or (7) wll, n hese crcumsances, gve a poor sascal f wh he esmaed coeffcens wh large sandard errors and dvergng sgnfcanly from he b 3 13

15 facor shares. Bu hs problem s no nsurmounable. For mos economes facor shares are suffcenly consan so ha he esmaon of (6) or (7) wll sll yeld remarkably good resuls. Bu even f hs were no he case, all ha would be needed s o specfy a funcon ha racks he shares correcly, hen subsue no (2) and proceed as above. Fndng hs funcon mgh be more or less complcaed emprcally, bu concepually he ssue s sraghforward. 11 The second assumpon o derve equaon (7) s he one abou he consancy of he growh raes of he wage and prof raes. Emprcally, s hs one ha causes mos problems n he esmaon of producon funcons usng me-seres daa. From he deny, we know ha TFPG a ŵ + (1 a ) rˆ. Plos of he Solow resdual ypcally show a procyclcal flucuaon around s mean growh rae (see, for example, Solow, 1957). Consequenly, n spe of he srong rend underlyng he growh rae, usng a lnear me rend (n he log-levels specfcaon) or a consan (n he growh rae specfcaon) does no accuraely explan he varaon n hs varable. In fac, he problem n he regresson s akn o one of omed varable bas. For all praccal purposes, TFPG s omed from he regresson, hus basng he esmaes of capal and labor. Ths s he problem o whch Barro refers. Wha s he soluon? Gven he ypcal pah of TFPG, a complex rgonomerc funcon, raher han a smple me rend, wll do a much beer job a rackng he pah of TFPG a ŵ + (1 a ) rˆ. Bu once hs s done, we wll ge back o he deny, elasces wll equal he facor shares, and we wll fnd puave consan reurns o scale. Summng up: f he producon funcon s esmaed correcly (.e., f he funconal form chosen correcly approxmaes he accounng deny), no daa se can refue he null hypoheses ha he elasces equal he facor shares and consan reurns o scale. I s no possble o es he exsence of an aggregae producon funcon One could, for example, ry a Box-Cox ransformaon, whch s analogous o a CES producon funcon. 12 The problem s a lle more complcaed han hs n ha by judcous choce of a complex me-rend one could ge a perfec sascal f wh any esmaed value of he oupu elasces ha one lkes. To see hs, consder Dˆ = Vˆ φ( Ĵ Lˆ ) where 0 φ 1. If a 14

16 There s a dfferen soluon o he conundrum ha Barro hns a n hs second pon, when he menons ha he capal sock s unlkely o correspond o he sock currenly ulzed n producon (Barro 1999, p.123). He refers o he use of a capal sock correced for ulzaon. Ths does no maer for he accounng deny, bu could poenally solve he problem of esmang he producon funcon. The ssue s as follows: as we have noed, f one looks a he componens of TFPG a ŵ + (1 a ) rˆ, he one ha manly vares s rˆ, and does so procyclcally. As we are argung ha hs varable s beng omed from he regresson, he ncluson of any oher varable ha moves procyclcally, namely he adjused for capacy ulzaon capal sock, wll serve as a proxy and wll do a good job. If capal s share s roughly consan, mparng a procyclcal flucuaon n he capal sock wll reduce ha exhbed by he rae of prof. Barro s rgh, bu for he wrong reason. The answers o Barro s frs and second pons on why growh accounng s preferable o he esmaon of he producon funcon follow from he above dscusson. I does no maer wheher he growh of capal and labor are exogenous or endogenous, f all ha s beng esmaed s an deny. Moreover, a he heorecal level, hese consrucs do no sasfy Fsher s aggregaon condons, so s dubous wha emprcal relevance hey have. Economercally, he ssue s no one of nsrumenal varables or un roos snce, once agan, we are dealng wh an deny. Fnally, he possbly of measuremen error s poenally a serous ssue, bu wll affec boh economerc esmaon and growh accounng. 4. SOME SIMULATION EXPERIMENTS To llusrae he above argumens, we underook some smple smulaons wh a vew o sudyng wha growh accounng s acually measurng. The key queson behnd he complex me rend can be found ha gves a near perfec f o D() n he equaon lnv = lnd() + b 4 ln(j L), hen he esmaon of hs Cobb-Douglas equaon wll gve b 4 = φ. We have no way of decdng beween any of he specfcaons, as, by defnon, hey all gve perfec fs. 15

17 exercse s he followng: f an aggregae producon funcon does no exs, wha s beng measured as producvy, n parcular as oal facor producvy growh, n exercses ha use aggregae daa? Ths has he followng mplcaons: () f he answer s ha some average of ndvdual frms producves hen, does he aggregae mehod sll yeld such an average? If no, why no?; and () more generally, how and o wha exen are we msled by he resuls of aggregae growh accounng? I should be emphaszed ha we are no performng Mone Carlo smulaons. The purpose of hs exercse s merely o generae a daa se whose underlyng srucure s known, n boh physcal and value erms n order o hghlgh he dfferen resuls ha he wo ypes of daa lead o n growh accounng exercses. Mone Carlo smulaons sudyng relaed ssues, bu answerng dfferen quesons, were performed by Felpe and Holz (2001). See also Fsher (1971b) and Shakh (1980). Table 1 summarzes he characerscs of he smulaons. We assume ha here are well-defned mcro-producon funcons, whch are specfed n physcal erms, as deally hey should be. The consan prce value of oupu was calculaed hrough a mark-up and he capal socks were generaed resdually hrough he NIPA deny, equaon (1). I s assumed ha each frm produces a homogenous oupu, whch may or may no be he same for all frms. The analyss does no depend on hs assumpon so long as, n he former case, s no possble o recover he physcal quanes from he value daa. In he laer case,.e., oupu s no homogeneous across frms, we canno, of course, esmae a cross-secon producon funcon usng physcal daa. I s mporan o emphasze ha we generae wo ypes of daa, n physcal erms and n value erms. The former are assumed and generaed so as o gve a good f o he Cobb-Douglas funcon. The nvesgaor does no know hs. Value daa, as ndcaed above, are generaed hrough he accounng deny. The nvesgaor knows all he value daa bu canno recover he physcal daa from hem. I should also be noed ha n he case where a cross-frm producon funcon s esmaed, no aggregaon problem à la Fsher s nvolved. Ths s an mporan pon. If we need o esmae a producon funcon usng oupus and npus summed over dfferen frms, we encouner all he well-known aggregaon problems. As value daa 16

18 has o be used n esmang hs aggregae producon funcon, we can explan why regressons usng hese daa gve a good f o he aggregae daa when heorecally hey should no (Fsher, 1971b, Felpe and McCombe, 2005c). Bu he problem s even more fundamenal han hs. As wll be demonsraed, he accounng deny presens nsurmounable problems of nerpreaon, even when here are no aggregaon problems of any knd regardng funconal forms, or affecng oupu, labor or capal (see Felpe and Fsher 2003) or problems of he ype dscussed n he conex of he Cambrdge debaes regardng he naure and consrucon of capal socks (Cohen and Harcour 2003). The mporan aspec of our smulaons s ha hey show how he use of value daa can gve resuls a varance wh he rue magnudes of he underlyng producon funcons and, herefore, msleadng numercal esmaes of boh he parameers of he producon funcon and of he rae of echncal progress. For clary purposes, we wll confne he erm echncal progress o ha calculaed usng physcal daa; and he erm oal facor producvy o ha calculaed usng value daa. 17

19 Table 1. SUMMARY OF THE CHARACTERISTICS OF THE SIMULATIONS * 10 frms, one perod α ( 1 α ) = 0 * Idencal Q ( A L ) K Q and K are generaes as random varables. L s generaed hrough he producon funcon. These are physcal daa. * Q s assumed o dffer across frms; s specfc o he frm; A s he same across frms and normalzed o 1 K 0 * Elasces: α = ; 1 α = wh a random error o avod mulcollneary * Value daa: Frms se prces as a mark-up on un labor coss,.e., p = ( 1 + µ )w L / Q, where µ = s he same across frms * Wage rae: w = w s he same across frms * Prof rae: r = r = 0.10 s he same across frms * Oupu n value erms: V = pq * Capal sock n value erms: J = (V w L ) / r = (V wl ) / 0. 1 * Labor share (value erms): a = ( wl ) / V = 1 /( 1 + µ ) ; Capal share: = ( 1 a ) = µ /(1 + µ ) = 0.25 ; for each frm, Mean of s (range ) a Rae of echncal progress and TFPG * Oupu of he 10 frms grows a dfferen raes, bu Qˆ = Kˆ * Same rae of echncal progress, ϕ = α Qˆ Lˆ ), assumed o be.5% = 0.25( Qˆ Lˆ ) ( * Growh of employmen Lˆ = Qˆ ( ϕ / α ) * Elasces (physcal erms): α = ; 1 α = * Average shares (value erms) are as above * True rae of echncal progress (frm level): ϕ = Qˆ 0.25Lˆ 0.75Kˆ * TFPG: TFPG = Vˆ 0.75Lˆ 0.75Ĵ * Qˆ, Lˆ, Kˆ as before Increasng Reurns o Scale * Elasces α = 0. 3, 1 α = Degree of reurns o scale = 1.2 * Value daa calculaed as before and mark-up µ s also

20 () Cross-Frm Esmaon of he Producon Funcon In he frs example, daa n physcal uns were generaed for 10 frms for one perod under he assumpon ha hey all have dencal Cobb-Douglas consan-reurns-oscale producon funcons gven by Q 0 α ( 1 α ) K = ( A L ) (8) where Q s he number of uns of homogeneous oupu, generaed as a random varable; K s he number of dencal machnes whch are specfc o he parcular ndusry, also generaed as a random varable; L s he level of labor npu, generaed hrough he producon funcon; A 0 akes he same value for all frms and was normalzed o uny. The parameers α and ( 1 α) are he oupu elasces of labor and capal, respecvely, and are consruced o ake values of and The oupu elasces were delberaely chosen o be he converse of he facor shares found n he NIPA. In order o generae he moneary values, each frm ses prces as a mark-up on un labor coss,.e., p = ( 1 + µ )w L / Q (9) The mark-up (µ ) s he same for all frms and akes a value of one hrd, so (1+µ) = The wage rae s he same across frms, he same as he prof rae r, whch akes a value of The value of he capal sock was calculaed resdually hrough he accounng deny as J ( V wl ) / r, where V s value added, consruced as V = p Q for each frm usng equaon (9). The values of he facor shares are drecly calculaed usng hese value daa. Labor s share s calculaed as 13 To preven perfec mulcollneary, a small random varable was added o hese and, where necessary, oher varables used n he smulaon. 19

21 a = ( wl ) / V and capal s share as 1 a ). I should also be noed ha ( a = 1 /(1 + µ ), and so akes a value of 0.75 for each frm, wh a small varaon due o he error erm added. The mean value of labor s share for he10 frms s (wh a range of The mean value of he capal-oupu raos n value erms J / V ) s 2.57 ( wh a range of 2.24 o These values are very close o wha are observed emprcally, and are he resul of a roughly consan rae of prof and consan facor shares. As J / V = (1 a) / r, where ( 1 a) s capal s share and, as noed above, s approxmaely equal o 0.25 and he prof rae s 0.10, he capal-oupu rao wll no dffer much from As we are dealng wh ndvdual frms and we desgn he smulaons, we know boh he physcal daa and he values, snce we know he prces. Bu le us assume ha he prces are unknown o he researcher, as s usually he case, because he oupu and capal socks for dfferen frms are aggregaed n he NIPA usng value measures. Consequenly, V and J (n consan prces, alhough snce we only have one perod, he dsncon beween curren and consan prces does no arse) were aken as proxes for Q and K. These value daa were hen used o esmae a cross-frm producon funcon. The resuls of he esmaon are: lnv = lnL lnJ R 2 = (478.77) (136.40) (45.41) s.e.r. = Ths gves a remarkably close f o he Cobb-Douglas producon funcon, whch s o be expeced gven he mehod used o consruc he daa. However, some of Douglas and hs colleagues early sudes, whch used real, as opposed o smulaed, cross-sae daa, also found very close sascal fs. 14 The sum of he esmaed 14 For example, Douglas (1976, p.906) repors he followng resuls for a producon funcon based on Amercan cross-secon sudes, 1904, 1909, 1914 and Year α β α + β 20

22 coeffcens s 1.00 and hs s no sgnfcanly dfferen from uny (he value of he - rao esng hs hypohess s 0.02). Wh he close correspondence beween he supposed oupu elasces and facor shares calculaed from he daa (0.750; and 0.250; 0.256), s lle wonder ha such resuls could be nerpreed as provdng evdence n favor of compeve markes and dsprovng he Marxan argumen, as Douglas (1976, p.914) clamed. Ths s no whsandng he fac ha facors are no pad her margnal producs n physcal erms n our smulaon daa. Compeon could force frms o be x-effcen so ha frms do hre he facors of producon up o he pon where her physcal reurns equal her facor rewards n erms of he commody produced. Ths would deermne he opmal L / Q, whch s used n he mark-up prcng equaon. However, usng value daa would sll gve esmaes of he oupu elasces equal o 1 /(1 + µ ) and µ /( 1 + µ ), respecvely. However, should be emphaszed ha he esmaed oupu elasces are, of course, no he same as he rue oupu elasces of he mcro producon funcon. In oher words, he rue oupu elascy of labor s 0.25, bu he esmae usng value daa s The goodness of f s dependen upon he degree of varaon n he mark-up. Wh dencal mark-ups, he f s exac (and esmaon s no possble because of perfec mulcollneary). Indeed, s he consan mark-up ha s solely responsble for generang he spurous Cobb-Douglas. To demonsrae hs, he physcal values of he hree seres Q, L and K were nex generaed as random numbers. V and J were (32.5) 0.31 (15.5) (31.5) 0.34 (17.0) (30.5) 0.37 (18.5) (38.0) 0.25 (12.5) 1.01 Average 0.66 (33.0) 0.32 (16.0) 0.98 Noes: -values n parenheses. Toal number of observaons:

23 calculaed as before. The esmaon yelded a very good f o he Cobb-Douglas wh he values of he oupu elasces he same as before (he resul s no repored here). Ths does no necessarly mean ha we are posulang ha ha oupu s acually a random funcon of he npus. However, when one consders he complex producon processes of any modern frm, here may be some ndvdual pars of he process subjec o fxed coeffcens, whereas ohers subjec o dfferng elasces of subsuon, o say nohng of dfferences beween plans n manageral and echncal effcences. Thus, he randomness may smply be a reflecon of he severe msspecfcaon error nheren n specfyng he mcro-producon funcon as a Cobb- Douglas. Bu he mporan pon o noe s ha even n hs case, where here s no well defned mcro- producon funcon, he use value added daa wll gve he mpresson ha here exss a well-behaved aggregae Cobb-Douglas producon funcon. () Rae of Technologcal Progress and Toal Facor Producvy Growh In order o calculae he growh of oal facor producvy, we need he growh raes of oupu, capal and labor. We assumed ha oupu of he 10 hypohecal frms grows a dfferen raes (we only have one sngle perod), bu, for exposonal purposes only, he seres were consruced such ha he growh rae of he physcal capal-oupu rao s zero (.e., oupu and capal grow a he same rae) for all frms. I was also assumed ha each frm experences he same rae of echncal progress ( ϕ ), 0.5% per annum, equal o ϕ = α Qˆ Lˆ ( ). Ths s due o he fac ha he underlyng producon funcons are Cobb-Douglas and ha he growh of oupu equals he growh rae of capal. Hence, he growh rae of employmen for each frm was consruced as Lˆ = Qˆ ( ϕ / α ), where ϕ = 0.5%, as noed above. The oupu elasces of labor and capal n physcal erms are agan 0.25 and 0.75, he average value shares are (wh a range from o 0.795) and 0.255, and he aggregae shares are also and snce each frm has he same mark-up, whch means ha he labor 22

24 share of each frm s he same and f we aggregae over frms, he aggregae share comes o be abou he same. The rae of echncal progress usng he physcal smulaed daa would be calculaed by he nvesgaor unaware of s value (alhough we know by consrucon, 0.5%) for each hypohecal frm separaely usng he sandard growh accounng equaon, ha s, ϕ Qˆ alˆ (1 a ) Kˆ where a and ( 1 a ), he facor shares of labor and capal, are assumed o equal he oupu elasces α and ( 1 α ), 0.25 and 0.75, respecvely. As he rae of echncal progress s he same for each frm, we can alk abou he rae of echncal progress beng 0.5% per annum; even n he case where we assume ha he physcal oupus of he varous frms are no homogeneous. However, le us assume, once agan, ha he ndvdual prces of he varous frms are no avalable and so s no possble o exrac daa on he physcal uns of oupu. All ha can be used n emprcal work, as s usually rue n pracce, s he consan prce value of oupu and of he capal sock. The growh of oal facor producvy s gven by: (10) TFPG Vˆ alˆ (1 a ) Ĵ (11) where now he shares are 0.75 and 0.25, respecvely. The unweghed mean rae of oal facor producvy growh of he ndvdual frms s 1.49% per annum, whch, no surprsngly, s almos dencal o he rae of oal facor producvy growh obaned by aggregang he value daa over all 10 frms and usng hese n equaon (11) (1.48% per annum). Thus, he use of physcal daa yelds echncal progress accounng, on average (he unweghed mean), for ϕ = 0.25( Qˆ Lˆ ), ha s, 25% of labor producvy growh, wh a very small dfference beween frms due o he small random elemen 23

25 nroduced for he reasons noed above. 15 On he oher hand, he use of value daa for each of he 10 frms gves a mean value of he rae of oal facor producvy ha s TFPG = 0.75(Vˆ Lˆ ), or 75% percen he growh of labor producvy, wh a range from 80 per cen o 70 per cen. And he fgure usng he aggregae daa (.e., usng he aggregae values of oupu, labor and capal) s 74 percen. The reason for he marked dfference beween hese values and he rue rae of echncal progress s ha labor s share of oupu n value erms s 0.75, whle he rue oupu elascy of he frms producon funcons s I s worh nong ha TFPG n equaons (2) and (3) can be wren as TFPG Vˆ a Lˆ (1 a growh rae of labor producvy and )Ĵ a qˆ + (1 a ) ĵ, where qˆ = Vˆ Lˆ denoes he ĵ s he growh rae of capal producvy. Ths s a weghed average of he growh raes of labor and capal producvy. Therefore, could be argued ha TFPG s an aggregae measure of producvy growh. Ths nerpreaon faces, however, he problems dscussed n hs subsecon, namely, ha he fgure compued s no equvalen o he rue rae of echncal progress. Consequenly, he use of value daa produces a sgnfcanly dfferen esmae of he rae of echncal progress, compared wh he rue value obaned usng physcal daa. Even wh well-defned underlyng Cobb-Douglas producon funcons expressed n physcal erms, he use of value daa as a proxy for oupu can gve very msleadng esmaes of he rae of echncal progress. = Vˆ Ĵ () Increasng reurns o scale and oal facor producvy growh. Wha happens f he ndvdual frms are subjec o ncreasng reurns o scale when physcal daa are used? To examne hs queson, we frs esmaed he cross-frm producon funcons usng value daa when he mcro producon funcons exhb he 15 Ths s because here s no growh n he physcal capal-oupu rao, ϕ α( Qˆ Lˆ ) and α, he physcal oupu elascy of labor, s equal o Hence he rae of echncal progress equals one quarer of he growh of labor producvy. 24

26 same degree of ncreasng reurns o scale. The daa for he npus n physcal erms were he same as hose used n he prevous smulaon, wh he excepon ha now he ' ' elasces were mulpled by 1.20, so α = 0.30 and β = Ths represens a subsanal degree of reurns o scale and resuls n a value of oupu ha s sgnfcanly larger han when consan reurns o scale are mposed. The value daa were calculaed he same way as before, wh a mark-up once agan of 1/3. Esmang he unresrced Cobb-Douglas producon funcon gves a resul ha s vrually dencal o ha for consan reurns o scale, and repored above, excep for a change n he value of he nercep. Consequenly, we do no repor he resuls here. The esmaes of he puave oupu elasces are once agan very close o he observed (value) facor shares and sum o uny, hereby erroneously suggesng ha he producon process of he varous frms are subjec o consan reurns o scale. The reason for hs seemngly paradoxcal resul s ha he calculaon of value added s gven by pq = V = (1 + µ ) wl and as nomnal wages and he level of employmen are he same as before, so s he consan prce measure of value added, alhough he prce per un s now lower (here s no nflaon n he smulaed daa). Recall ha secon 3 showed ha wh value daa, esmaon of he producon funcon would yeld elasces equal o he facor shares. Nex, we calculaed he rae of echncal progress and he growh of oal facor producvy. For comparably wh he consan reurns o scale case, he growh raes of physcal oupu, capal and labor were he same as before. The rae of echncal progress was calculaed usng he physcal daa as ' ' ' ϕ = Qˆ α Lˆ β Kˆ (12) ' where α = 1. 2α and β = 1.2(1 α ). I can be seen ha he rae of echncal progress calculaed usng equaon (12) wll dffer from he 0.5% n he case of consan reurns o scale. In fac, wll be on average lower, gven he larger weghs of he growh of he facor npus. The rae of echncal progress, calculaed usng equaon (12) for each frm now vares consderably across frms (for reasons of space we do no repor he 25

27 full resuls). 16 The unweghed mean s (concdenally) 0.00 per cen per annum, wh a range of ± 0.4 percenage pons per annum. 17 On he oher hand, he growh raes of oal facor producvy of he ndvdual ndusres, calculaed usng equaon (11) and value daa, are agan all approxmaely 1.5% per annum. Ths s because he shares of labor and capal n value erms are once agan 0.75 and 0.25, and he growh raes of V, L and J are he same as before. 18 Thus, he use of value daa can gve a very msleadng esmae of he rue rae of echncal progress. The use of value daa erroneously ascrbes he effec of ncreasng reurns on ncreasng he effcency of he facors of producon o he rae of echncal progress. (v) Summary Of course, he acual fgures are arbrary as hey are dependen upon he assumed raes of growh of physcal oupu, capal and employmen. Neverheless, we can draw some conclusons from hese smulaons. The growh of oal facor producvy depends crucally on he weghs aached o he growh of capal and labor. The growh accounng approach assumes ha facors are pad her margnal producs and hence he echnologcally deermned oupu elasces wll equal he facor shares. However, when value daa s used we have shown ha he facor shares wll always equal he puave oupu elasces and boh are deermned by 1/(1+µ) (labor) and µ/(1+µ) (capal), where 1+µ., wll be recalled, s he mark-up. The esmaes of he oupu elasces usng value daa wll almos 16 Ths s because whle he growh raes of Q and K are he same beween frms, employmen growh raes dffer and so he change n weghng causes he rae of echncal change now o dffer across frms. 17 As we canno sum across he physcal quanes, we canno calculae a meanngful average rae of echncal progress, as he ndvdual raes canno be unambguously or unquely weghed. Neverheless, we dd calculae he unweghed mean. 18 Wh a consan mark-up of 1.33, he shares wll be always 0.75 and 0.25, regardless of he echncal condons of producon (e.g., he degree of reurns o scale). 26

28 ceranly dffer from he rue ones (always assumng ha here s a welldefned mcro-producon funcon n physcal erms). Where s possble o compare he rue growh rae of echncal progress wh he growh of oal facor producvy n value erms (namely a he frm level here), he wo values wll probably dffer markedly. In general, s no possble o recover he physcal quanes (of boh oupu and capal) from he value daa hrough he ndvdual prces, and so resor s made o value daa wh poenally very msleadng resuls. Where he physcal daa can be nferred, can only be done a a very low level of aggregaon. I requres each oupu and capal good o be measured separaely n physcal erms. The problems posed by he accounng deny are ndependen from, and n a sense more fundamenal han, eher he aggregaon problem or he Cambrdge Capal Theory Conroverses. In hese smulaons, we have found ha here s a very close sascal f o he esmaon of he producon funcon. However, wh me-seres daa, as opposed o cross-secon daa, he sandard errors are ofen much larger and he coeffcen of he capal sock ofen has he wrong sgn,.e., negave. The reason for hs s sraghforward. We know ha he accounng deny V w L + r J w L + r J, where V,, L, and J are he aggregae values and w and r are he average values, mus hold. If here s a consan mark up (and hence aggregae consan shares), dfferenang he deny and hen negrang gves V = B w 0 a ( 1 a ) r L a J ( 1 a ) (13) In logarhmc form hs becomes 27

29 lnv = ln B + [ a ln w + (1 a )ln r ] + a ln L + (1 a )ln J (14) 0 Neverheless, n nearly all esmaons of producon funcons, he expresson n brackes (.e., he weghed sum of he logarhms of he wage and he prof raes) s proxed by a lnear me rend. However, n pracce, hs erm shows a pronounced cyclcal flucuaon and so he use of a lnear me rend ofen nduces a sgnfcan msspecfcaon. I should also be noed ha n he neoclasscal schema, where s mplcly assumed ha physcal quanes can be adequaely proxed by value erms, f perfec compeon prevals and facors (a he aggregae level) are pad her margnal producs, hen from he dual, TFPG = aŵ + (1 a ) rˆ. Consequenly, here s no heorecal jusfcaon for mposng a lnear me rend o capure he puave rae of echncal progress. Cobb and Douglas (1928) n her semnal paper dd use me-seres daa and concdenally obaned a close sascal f wh facor shares equal o he oupu elasces was concdenal because he resuls were no robus; droppng he las few observaons radcally alered he esmaes and no me rend was ncluded. (McCombe 1998b; Felpe and Adams 2005). I was recognzed early on n he 1930s ha he use me-seres daa ofen gave mplausble resuls and hs may be he reason why Douglas and hs colleagues changed o he almos exclusve use of cross-sae and cross-ndusry daa n her numerous esmaons of producon funcons durng he 1930s A REINTERPRETATION OF GROWTH ACCOUNTING EXERCISES I should be clear by now ha all esmaons of aggregae producon funcons do s o approxmae he NIPA accounng deny, and as such hey are exercses whch, alhough algebracally correc, hey sand whou heorecal foundaon. Wha abou 19 a ( 1 a ) a ( 1 a ) The cross-seconal Cobb-Douglas s equvalen o V = B0 w r L J. The erm w a ( 1 a ) r shows relavely lle varaon when he daa s drawn from he same counry for frms, ndusres or saes. 28