CHAPTER 2 MEASURING MULTI-FACTOR PRODUCTIVITY WHEN RATES OF RETURN ARE EXOGENOUS

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1 CHAPTER 2 EASURING ULTI-FACTOR PRODUCTIVITY WHEN RATES OF RETURN ARE EXOGENOUS Paul Schreyer. Inroducon: Gross Operang Surplus and he Remuneraon of Capal Offcal sascs do no normally provde drec observaons on he prce and volume of capal servces. Wha s avalable from he naonal accouns s a resdual measure of gross operang surplus (GOS): a measure ofen nerpreed as profs from normal busness acvy, ncludng mxed ncome whch s he ncome of self-employed persons. Thus, he naonal accouns provde he researcher wh daa accordng o he followng accounng deny: () P wl GOS where P s he sum of curren-prce oupu n he economy, P [ P,P2,... P ] denoes he vecor of prces and [, 2,... ] denoes he vecor of quanes of oupu. To smplfy noaon, we use P for he nner produc beween P and. Noe, however, ha P normally he quanes n are no drecly measured. Oupu s defned and measured as valueadded, and prces are defned and measured a basc prces,.e., hey exclude all produc axes bu nclude subsdes on producs. The erm wl s he remuneraon of labour, wh wage componen w and volume componen L, wh he value and prce componens measured drecly. For smplcy, wll be assumed here ha mxed ncome s eher zero or s spl up beween he labour and he GOS componens. Thus, he wo sdes of () represen he oal producon and he oal ncome sdes of he naonal accouns. The naonal accouns provde no gudance as o he facors of producon ha are remuneraed hrough GOS. Fxed asses ceranly are among hese facors, bu here could be ohers oo. The busness leraure has dscussons abou he mporance of nangble asses, and here are good reasons o argue ha such asses accoun a leas for par of GOS. Whle hs may appear a mnor pon, calls no queson an assumpon ofen made by analyss of producvy and growh, namely ha GOS exacly represens he remuneraon of he fxed asses recognsed n he Sysem of Naonal Accouns (SNA), or he value of he servces of hese. The conac nformaon for he auhor s: Paul Schreyer, OECD Sascs Drecorae, Paul.Schreyer@OECD.org. Opnons expressed n hs paper are hose of he auhor and do no necessarly reflec vews of he OECD or s ember counres. The auhor hanks Erwn Dewer, Chuck Hulen and Alce Nakamura for helpful commens. Specal hanks go also o ahlde as (Unversy of Valenca and IVIE) for useful dscussons on he subjec. Caon: Paul Schreyer (200), easurng ul-facor Producvy when Raes of Reurn Are Exogenous, chaper 2, pp n W.E. Dewer, B.. Balk, D. Fxler, K.J. Fox and A.O. Nakamura (200), PRICE AND PRODUCTIVITY EASUREENT: Volume 6 -- Index Number Theory. Trafford Press. Also avalable as a free e-publcaon a and Alce Nakamura, 200. Permsson o k o, or copy or reprn, hese maerals s graned whou resrcon, ncludng for use n commercal exbooks, wh due cred o he auhors and edors.

2 * Le u * * * [u,u 2,...u N ] denoe a vecor of user coss for N ypes of capal servces and le K [K, K2, K, KN ] denoe he correspondng vecor of he quanes of capal servce flows. The assumpon ypcally made s: (2) u * K GOS, where u * K denoes he nner produc of he prce and quany vecors:.e., where * N * u K u. In oher words, s assumed ha remuneraon of capal servces exacly K exhauss gross operang surplus. Emprcally, he equaly u * K GOS s obaned by choosng wha s hough o be an approprae value for he ne rae of reurn on asses, whch s par of he user coss. 2 Wh hs formulaon, he rae of reurn s assumed o adjus endogenously. Ths seup s conssen wh compeve behavour on produc and facor markes and a producon process ha exhbs consan reurns o scale. Under hese condons, () can be resaed as * (3) P wl u K, snce hese condons ensure ha he gross operang surplus corresponds exacly o he remuneraon of he asses ncluded n K; hence f only fxed asses are assumed o be n K, hs s equvalen o assumng ha GOS corresponds o he remuneraon of fxed asses. Noe ha hs seup also depends on he followng beng rue: he se of asses [ K,K2,...K N] s complee;.e., all asses are observed by he offcal sascans who comple he naonal accouns and here are only he saed fxed asses; he ex-pos rae of reurn on each asse (mplcly observed by he naonal accounans as par of GOS) equals s ex-ane rae of reurn, whch s he economcally relevan par n he user cos of capal servces; here are no resdual profs (or losses) such as mgh arse n he presence of marke power, or wh non-consan reurns o scale, or owng o he avalably of publcly avalable or any oher uncouned or mscouned capal asses. Several quesons arse when some of he above condons do no hold. For example, when here s ndependen nformaon abou he raes of reurn o capal servces, here s no guaranee ha he sum of labour remuneraon and he observed capal remuneraon wll equal measured oal value added a curren prces. How should mul-facor producvy (FP) be concepually defned, compued and nerpreed? How should growh accounng exercses be carred ou? How should measures of echncal change be defned and evaluaed? These are he quesons explored n hs paper. In he res of hs paper, a preference s expressed for a smple 2 In a smple connuous-me formulaon, he user cos or renal prce of an asse (Jorgenson and Grlches, 967) s gven by u q (r δ d q / d) where q () s he purchase prce of a new asse of ype, r() s he ne rae of reurn, δ s a rae of deprecaon, and dq d s he rae of change of q. 4

3 FP measure ha s conssen wh ndex number radons. Such a measure canno be nerpreed as capurng only, or all, echncal change Why GOS ay Dffer from Remuneraon of Capal Ths paper generalzes he formulaon of he ncome-producon relaonshp (3) by allowng for and ulzng ndependen measures of capal remuneraon, u [ u,u2,... un ] ha may no sasfy condon (2). Under hese crcumsances, equaon (3) s replaced by (4) P wl GOS u K, where he erm denoes he dfference obaned by subracng from curren-prce oupu boh he remuneraon of asses ncluded n K, 4 u K, and he value of he labour npu;.e., s he observed curren prce oupu mnus observed facor paymens. C wl u K s used as shorhand for observed facor paymens. Hence gross operang surplus can be spl no a componen ha reflecs observable facor remuneraon plus a resdual wh several possble nerpreaons. In prncple, here s no resrcon on he sgn of. However, f he sgn were negave over an exended perod of me, hs would mply susaned losses. Snce hs seems economcally mplausble, n wha follows, he non-negavy of s assumed. 5 Four possble reasons for nonzero values of are consdered n hs paper. odels of shor-run dsequlbrum over he busness cycle provde a frs possble heorecal jusfcaon for he exsence of nonzero values of. 6 A second possbly s ha reflecs he exsence of pure profs as a consequence of he presence of decreasng reurns o scale combned wh margnal cos prcng for oupus, or of ncreasng reurns o scale and a posve mark-up over margnal coss. If reurns o scale are he key source of non-zero values of, hen he sze of wll depend on he degree of compeon n oupu markes: free marke enry and compeon would be expeced o drve mark-ups and prces o a level where oal revenues jus cover oal coss, mplyng 0. The Lucas-Romer model of endogenous growh (Romer, 990) provdes a hrd possble jusfcaon for non-zero values. Accordng o hs model, a he frm level, reurns o scale are consan, bu a he aggregae level here are ncreasng reurns o scale due o exernales. A fourh possbly s ha reflecs he exsence of unobserved npus and hence reflecs a measuremen problem. Ths suaon could arse f no all of he capal npus ha gve rse o operang surplus are recognsed n he naonal accouns. In conras o he second 3 Aspecs of he nerpreaon and dervaons ha follow buld on Jorgenson and Grlches (967), Fuss and cfadden (eds.) (978), Dewer and Nakamura (2007) and Harper e al. (989). 4 As he conex makes clear, he symbol s also used somemes o denoe he number of oupu goods. 5 In he emprcal par of he paper, s posve for he four counres revewed (Canada, France, Japan, Uned Saes) for mos years over If were negave over an exended perod of me, hs would cas doub on he measures for he remuneraon of capal, and n parcular on he choce of he exogenous raes of reurn. 6 These nclude models of me-varyng capacy ulsaon of he sor nvesgaed by Bernd and Fuss (986) and Hulen (986). The cyclcaly of producvy measures and he relaon of hese o echncal change are deal wh by Basu and Kmball (997). They fnd srong effecs of varable capacy ulsaon on measures of producvy. 5

4 nerpreaon, n hs case we would expec o reman posve even n he longer run because rue oal coss are hgher han wha he observed asses would jusfy and would cover hese Producon Technology and Producer Behavour We le Z() denoe a feasble se of npus and oupus n perod. We furher assume ha here s a oal cos funcon TC ha shows he mnmum coss of producon, gven a vecor of quanes [,2,... ] for he oupus and gven a correspondng se of npu prces. Inpus comprse labour L, N ypes of observed capal servces K,K2,... KN and one unobserved asse D. The correspondng prces are he wage rae, w, he user coss of capal, u [ u,u2,... u ], and he prce of he unobserved npu D, φ. The oal cos funcon s defned as: (5) TC[, w,u,, ] mn { wl u K φd :(,L,K,D) belongs o Z() } φ. L,K,D The cos funcon s early homogenous n npu prces and non-decreasng, bu no necessarly early homogenous n he vecor of oupus [,... ]. Thus, here s no assumpon of consan reurns o scale. However, producers are assumed o mnmse oal cos, so ha acual coss equal mnmum coss ( wl u K φd TC(, w,u, φ, ) ). Furhermore, producers are assumed o face compeve facor markes so ha Shephard s (970) condons for opmaly for facor npus apply: TC (6a) L ; w TC (6b) K,, K, N ; u TC (6c) D. φ On he oupu sde, mperfec produc markes are allowed for wh he sole spulaon ha oupu prces are proporonal o margnal coss. No explc assumpon s made abou he knd of mperfec compeon ha prevals or concernng wheher producers are prof maxmsng or no. All ha s needed s a relaonshp beween prces and margnal coss so ha f he prce of oupu s P and f / μ s a produc-specfc, me-varyng mark-up facor, producer behavour on he oupu sde s descrbed by (7) P μ TC /,...,. Nex, we follow he leraure (e.g., Panzar 989) and defne he local elascy of cos wh respec o scale as 7 Non-observed npus and her k o measured FP growh and echncal change have been nvesgaed by Basu e al. (2003). They nroduce unobserved nangble organsaonal capal ha hey ake as complemenary o observed nvesmen n nformaon echnology. Unlke he presen model, however, hers s a general equlbrum seng ha res o accoun no only for he unobserved nangble npus bu also for her unobserved producon. 6

5 TC (8) ε. Hence, ε > 0 ndcaes he percenage change n oal cos for a gven percenage change n all oupus. The nverse of hs parameer can readly be nerpreed as a measure of local reurns o scale for he producon un. For nsance, ε > mples ha a one percen rse n he quany of each of he oupus ncreases oal coss by more han one percen, whch s anamoun o a suaon of decreasng reurns o scale. Smlarly, ε < and ε correspond o ncreasng and consan reurns o scale, respecvely. 8 (9) Gven (7), he measure of he cos elascy defned n (8) can be furher ransformed: ε P μ TC TC P μ P P TC where μ P μ TC P μ. P In (9), μ s he economy-wde nvered average mark-up facor a weghed average of ndusryspecfc mark-ups wh smple oupu shares as weghs. Expresson (9) can be rearranged as (0) P ( ε / μ) TC. Thus, he value of oal oupu revenues equals oal coss, adjused by a mark-up facor μ and ε, he parameer for he scale elascy. The equales n (9) can now be combned wh he naonal accouns nformaon menoned earler. In parcular, was poned ou ha gross operang surplus s defned as he dfference beween he value of oupu and labour ncome: GOS P wl. Usng he resul P ( ε / μ)tc n (0), from (4), one obans () GOS TC( ε / μ) wl. Recall ha he dfference beween GOS and observed capal ncome has been labelled : GOS u K. Usng he expresson () for GOS and akng no accoun he defnon of TC now allows us o derve a relaon for ha can readly be nerpreed: 8 As we operae wh a mul-produc cos funcon, a dsncon needs o be made beween general economes of scale and produc-specfc economes of scale. The former reaed here deals wh changes n coss when all oupus are changed by he same proporon. The laer deals wh changes n coss as one parcular oupu s ncreased whle holdng all oher oupus consan. For he laer form of economes of scale, see Panzar (989). 7

6 (2a) GOS u K P wl u K ε μ ε μ ε μ or alernavely (2b) TC wl u K TC TC (TC wl u K) TC φd from (0) μ φd P usng (0). ε P from (4) ε μ TC TC φd usng (5) Expressons (2a) and (2b) show how he dfference beween GOS from he naonal accouns and he sum of paymens o observed facors reflecs mark-ups and reurns o scale (capured by ε / μ ) and he nfluence of unobserved capal npus (capured by φ D ). The expressons n (2) wll be nsrumenal for he dscusson n he followng secons. 4. Techncal Change In an envronmen of consan reurns o scale, Hcks-neural echncal change can be defned eher as a shf of he producon funcon over me (an oupu-based measure) or as a shf of he cos funcon over me (an npu-based measure). Here producer behavour has been descrbed by way of a cos funcon, so we shall use he npu-based approach o derve measures of echncal change. One mporan advanage of he cos-based measure s ha no assumpons abou prof or revenue maxmsaon need o be made for he oupu markes. If here were an assumpon of consan reurns o scale, and compeve markes, he choce of he npu-based producvy measure would smply be a maer of convenence, wh no consequences for resuls. However, for he momen we have mposed no such a-pror condon, and he npu-based measure wll n general be dfferen from he oupu-based measure, as wll be shown n secon Techncal change s measured here as a downward shf over me of he oal cos funcon. To derve an analycal expresson, TC s dfferenaed oally and echncal change s hen defned as he negave of he paral dervave of he cos funcon wh respec o me: (3) TC TC TC d TC TC d w d w d TC d d N TC d u u d TC d φ. d φ 8

7 To nerpre (3), consder s pars n urn. On he rgh-hand sde, frs here s a Dvsa-ype N TC d oupu quany change ndex,, ha aggregaes he growh raes of he d quanes of ndvdual oupus. To fnd a compuable expresson for he growh rae of oupu, use (7) and (9) o oban: (4a) TC d d ε TC d TC d P μ P P μ P μ P d TC d d d. P μ P TC d TC d P μ P d d Thus, he oupu aggregae resembles a radonal oupu aggregae wh revenue shares as weghs, bu he laer are now correced for he relave mark-ups μ / μ and he scale facor ε. ovng on o he erms n brackes on he rgh hand sde of (3), can be seen ha hese measure he dfference n he growh rae of oal coss and he growh raes of he varous ypes TC d w N TC d u TC d φ of npu prces. In fac, s a Dvsa ndex w d u d φ d of npu prces. Ths s apparen by nvokng he opmaly condons for facor npus (6a)-(6c) and hen nserng hem no he above expresson whch now becomes wl d w φ φ N u K d u D d TC d TC d TC d. oreover, by consrucon, he dfference beween he Dvsa ndex of oal coss and he Dvsa ndex of npu prces s he Dvsa ndex of npu quanes. The erm n brackes on he rgh hand sde of (3) can be rewren as (4b) d TC d TC d w φ N TC d u TC d w d u d φ d wl d L N TC d uk TC d K d φd d D. TC d Hence, he heorecal ndex (3) becomes: TC P μ φ (5a) ε d wl d L N u K d K D d D P μ d TC d TC d TC d. Turned around, he growh accounng form of (5a) s: P D d D TC (5b) μ d wl d L N uk d K φ ε P d TC d. μ TC d TC d Expresson (5b) delvers an explc formula for he change n aggregae npus and oupus. If here were no unobserved facor D, and f mark-up facors and he local scale elascy were known, (5b) could readly be mplemened. However, wh an unobserved facor D, hngs are more complcaed. We sar wh a proposal for a compuable FP measure and follow wh a dscusson of s nerpreaon. 9

8 5. Dervng Compuable easures There are essenally hree sraeges for he mplemenaon of expresson (5b): () o nroduce addonal, and ypcally resrcve, hypoheses abou he sze or naure of he unknown varables unl an expresson emerges ha s boh compuable and ha offers a (seemngly) clear nerpreaon of producvy growh; () o say away from nvokng addonal hypoheses, and defne a compuable measure of producvy growh whle allowng for he fac ha may reflec more han pure echnology shfs; or () mpose he assumpons needed o apply economerc mehods o esmae or correc for he unknown facor and consruc esmaes of he concepually correc aggregaes of oupus, npus and producvy. We dscard he hrd possbly smply because s no a praccal way for sascal offces when hey have o compue and publsh perodc and easly reproducble sascal seres. We do, however, acknowledge ha hs economerc opon may be an mporan one for more research-orened, one-off projecs. As such may also delver useful nsghs concernng he emprcal mporance of he unobserved facor. Smlarly, o assess some of he choces among non-paramerc mehods as descrbed below, economerc sudes (such as Paque and Robdoux, 200 or Olvera-arns e al., 996) can be very useful. 5. Apparen ul-facor Producvy We frs follow avenue () and propose a measure of apparen mul-facor producvy. Then, n he followng subsecon we consder sraegy (). For he purpose a hand, le here be an aggregaor X ha combnes he quanes of he observable npus K and L. Specfcally, defne (6) d X d N u K C d K d wl d L C d as a Dvsa quany ndex of observable npus, nong ha he weghs correspond o he shares of each npu n oal observable npus, as C wl u K. Nex, defne he rae of apparen mul-facor producvy growh (AFP) as he dfference beween a Dvsa quany ndex of oupu and he quany ndex of observable npus as specfed above n (6): (7a) d d X AFP. d d The Dvsa oupu ndex n (7a) s a radonal one,.e., an average of raes of change d P for ndvdual oupus, each weghed by s revenue share: d. Noe d P d ha hs Dvsa oupu ndex dffers from he more general oupu growh ndex denfed n (5). The growh accounng equaon ha corresponds o (7a) s: (7b) d d N u K C d K d wl d L C d AFP 20

9 where, n conjuncon wh (5b), can be shown ha: (7c) TC εμ φ P d D d D d X AFP P μ d TC d d. Accordng o (7b), he drec growh conrbuon of observed capal npus and labour s gven by he rae of change n hese varables weghed by her respecve average shares n observed coss C. The producvy erm AFP reflecs he hree facors shown n (7c): pure echncal change or he shf of he cos funcon, a erm ha capures he effecs of mark-ups and nonconsan reurns, and a erm ha capures he effecs of he non-observed varable D. Consder he followng specal cases: If here s no unobserved npu (D0), he hrd erm n (7c) dsappears and AFP capures echncal change plus a erm ha reflecs he non-consan reurns and mark-ups a resul smlar o he one developed by Denny, Fuss and Waverman (98). AFP wll exacly correspond o echncal change f here are consan reurns o scale ( ε ) and f he same markup facor apples hroughou he economy ( μ μ ). If he volume change of he unobserved npu equals he volume change of observed npus, he hrd erm dsappears also and AFP reflecs only echncal change and he effecs of non-consan reurns and mark-ups. We conclude ha, whaever he exac naure of he unobserved facor D, he AFP compuaon wll capure pure echncal change, he growh conrbuons of unobserved asses and scale effecs, and also he dsrbuon of mark-ups. Wh he excepon of he mark-ups ha can be a consequence of marke power, hese effecs are echnology-relaed and could be consdered analycally meanngful expressons of producvy growh. These effecs are now pah ndependen hey vary wh he levels and growh raes of observed and non-observed npus, and he laer depend n urn on prces of npus and oupus as well as on mark-up sze. The conrbuon of producvy change o oupu growh s gven by AFP. Clearly, he nerpreaon of AFP has o be kep n mnd: reflecs he combned effecs of echncal change, of non-observed npus, of non-consan reurns o scale and of devaons from perfec compeon n produc markes. In oher words, AFP s a rue resdual or a non-heorec producvy measure. Bu for many praccal purposes, wll fulfl s role as a mul-faceed measure of producvy growh. 9 We noe n passng ha AFP could also serve as a useful measure of producvy growh when echncal change s of a more general naure, and no necessarly Hcks-neural. If one wans o exend he analyss, an addonal analycal sep could be aken o decompose AFP no s echncal change componen and oher effecs. However, hs requres nvokng paramerc mehods of esmaon f one does no wan o mpose compeve behavour on oupu produc markes. 9 As can be seen from he ls of asses n our emprcal mplemenaon, one mporan asse ha s lef ou s land, whch s no consdered a produced asse by he naonal accouns. Ths asse presumably does no grow much so he las erm n (7c) s lkely o be negave n OECD counres, ponng o a downward bas of AFP as a measure of echncal change. 2

10 5.2 Invokng Addonal Assumpons Ths subsecon follows he approach () oued a he sar of he secon: addonal hypoheses are nvoked o deal wh he possble presence of unobserved npus, non-consan reurns o scale and mark-ups. Each se of hypoheses s desgned so as o lead o a correc measure of FP n he sense ha reflecs Hcks neural echncal change f he hypoheses hold. In addon, consderaon s gven o how, under he assumed crcumsances, he pragmac AFP measure would fare. I s of neres, for example, wheher s use would mply an upward or downward bas for measurng echncal change and when he values would concde wh hose for he convenonal FP measure Assumng no unobserved npu, common mark-up facors and CRS If one assumes ha here are no unobserved npus (D 0), and a common mark-up facor n he dfferen oupu markes ( μ μ ), he only possbly o explan a dfference beween oal coss of observed measures and GOS are he combned effecs of a posve markup and non-consan reurns. In hs case, he mark-up/reurns o scale rao s gven by μ, so s deermned by he rao / P where corresponds o he dfference ε P beween non-labour ncome (GOS) and he sum of observed capal coss and where P s he sum of revenues. If emprcal nformaon exss on he average mark-up facor, μ, can be used o deermne ε. Alernavely, nformaon may exs on he average degree of reurns o scale n he economy. If no, an addonal assumpon has o be nvoked ypcally ha of consan reurns o scale ( ε ). Havng defned away D, one fnds ha oal coss equal observed coss, or TC C. In hs case, he growh accounng equaon (5b) can be expressed as (8a) P d wl d L N uk d K FP P d C d, wh C d (8b) TC FP. Here, TC s he modfed cos funcon ha apples under he condons D 0, μ μ and ε. Expresson (8b) ndcaes ha FP correcly races echncal change provded he assumpons D 0, μ μ and ε are accurae. Under he saed assumpons, s easy o see d d X ha FP, where hs s he defnon of AFP gven n (7a). Thus, f he d d assumpons above hold, he rue producvy measure FP gven n (8b) concdes wh he resul obaned from applyng an AFP measure Assumng proporonaly of D and K, absence of mark-up facors and CRS A second possbly s o allow for an unobserved facor (D > 0) bu o mpose margnal cos prcng (hence μ for, K, ) and consan reurns o scale. Ths s equvalen o assumng fully compeve oupu markes, and means ha (0) mples ha P TC. Wh hs seup, follows ha he enre dfference beween GOS and he sum of observed paymens o capal s denfed wh paymens o he unobserved npu: φd. To evaluae (5b) for hs suaon, an addonal assumpon s needed. One possbly would be o assume ha he 22

11 rae of change of he unobserved npu D equals ha of he observed capal npus: d D d K d K N u K d K, where d d d s a Dvsa quany ndex of observed fxed u K d asses. Under hese condons, he growh accounng equaon (5b) can be wren as (9a) (9b) P d P d TC2 FP2. wl d L u K d K d D FP2 P d P d P d wl d L u K d K FP2, wh P d P P d Now FP2 gven n (9b) races he shf of he cos funcon TC2 correcly as long as he assumpons hold. The measured growh conrbuon of he observed capal npus mers wl u K furher dscusson. I s easly verfed ha so ha he wegh ha now P P P u K GOS aaches o he observed capal npus,, equals he share of GOS n oal P P P producon, whch n urn s he complemen o he labour share n oal ncome. Thus, he ncome of he unobserved facor D s dsrbued across he observed capal npus, and (9a) can be rewren as: d wl d L GOS d K (9c) FP2, d P d P d usng he expresson for (d / d) n he paragraph ha follows equaon (7a). Equaon (9c) bears a srong resemblance o a model wh endogenous ne raes of reurn as descrbed below n secon In boh cases, he overall rae of growh of capal servces, d K d, eners wh he same wegh one mnus he labour share n oal ncome. Of course, n he endogenous * model, he growh rae of observed capal servces, d K / d, wll n general be dfferen from d K / d n he presen case snce each asse s user cos erm s based on an endogenous raher han an exogenous rae of reurn. Noneheless, as wll be apparen from he emprcal secon, he wo FP measures race each oher que closely, a leas for he four counres examned and for he me perod used n he emprcal par of hs paper. Suppose he above assumpons are rue bu an AFP measure s appled. Wha would be he resug bas wh regard o he rue FP2 measure? Afer some manpulaons, can be shown ha d K d X (20) AFP FP2. P d d Thus, AFP wll oversae FP2 f he growh rae of observed capal asses and by assumpon he growh rae of he unobserved asse exceeds he growh rae of all observed npus. In he emprcal examples presened n secon 6, hs s he case and AFP urns ou o be conssenly hgher han FP2. 23

12 5.2.3 Defnng away mark-ups and unobserved npus and assumng CRS These are he assumpons nvoked when FP compuaons rely on endogenous raes of reurn: oupu markes are aken as compeve ( μ ;, K, ), here are no unobserved facors (D 0) and here are consan reurns o scale. The endogenous approach goes back o Chrsensen and Jorgenson (969), and has been appled n many subsequen sudes of producvy growh, ncludng many carred ou by naonal sascal offces (e.g., BLS 2003). Ths s he mos wdely-used mehodology bu also he one ha requres he mos resrcve se of assumpons: he assumpons needed o jusfy he use of endogenous raes of reurn 0. In addon o he above assumpons, here mus be perfec ancpaon of asse prce changes and deprecaon. Ths mples ha P wl u * K. The growh accounng model (5b) becomes * N (2a) P d wl d L u K d K P d P d P d FP3, where (2b) TC3 FP3, and where TC3 denoes a cos funcon wh observed npus only. If he addonal resrcons hold, measured producvy change corresponds o he shf of a cos funcon TC3 wh CRS and only observed npus. As alluded o above, here s smlary wh (9c) because (2a) can be re-wren as (22) d * wl d L GOS d K * * d K N u FP3 where d P d P d K d K, * d u K d so ha he conrbuon of capal asses s he produc of he rae of growh of observed capal servces and he share of GOS n oal oupu or cos. If he above assumpons are correc, and f an endogenous rae of reurn s used, he TC3 evaluaon of AFP* would yeld he correc resul snce AFP * FP3 n hs case. We have marked AFP* wh an asersk here o draw aenon o he fac ha AFP s based on a capal measure ha reflecs endogenous raes of reurn. If AFP s compued on he bass of exogenous raes, would clearly dffer from FP3. Ths s also borne ou n he emprcal example below. However, no a-pror saemen can be made as o he sgn of hs dfference Assumng no mark-ups, no unobserved npu and decreasng reurns o scale: an npu-based measure Ths consues ye anoher possbly for deag wh he dfference beween revenues and observed facor paymens: he unobserved facor s defned away (D 0) as well as markups of prces over margnal coss ( μ ;, ), bu he producon echnology s assumed 0 The endogenous rae of reurn s compued by choosng ha ne rae of reurn ha jus equalzes he sum of user coss of observed asses wh non-labour ncome (GOS for smplcy). Usng he same noaon for user coss as n N foonoe 2, hs means ha GOS q (r* δ dq d)k. 24

13 o exhb decreasng reurns o scale ( ε > ). Then, he enre dfference beween GOS and observed asse renal paymens s ascrbed o he effecs of margnal cos prcng under decreasng reurns o scale: ( ε ) TC and TC C. Under hese crcumsances, he reurns o scale parameer can be compued as ε / TC. Gven a value for ε, (5b) can be rewren as (23a) (23b) ε P d P d wl d L TC d uk TC d K d N FP4 TC4 FP4, where and where TC4 s a cos funcon wh decreasng reurns o scale and wh observed facor npus only. Some more dscusson s useful here. Frs, because all coss are observed, TC C and (23a) can be wren as d d X (23c) ε FP4. d d We noe n passng ha he same growh accounng equaon and/or producvy measure FP4 could have been derved from a model wh a consan reurns o scale cos funcon for observed and unobserved npus, bu wh he added assumpon ha he quany of he unobserved npu s posve and fxed. The unobserved npu hen acs as he addonal cos facor ha s equvalen o a decreasng reurns o scale echnology. If he assumpons above are correc, how does FP4 relae o AFP? I s easly d esablshed ha under hese crcumsances, AFP FP4 ( ε ). If reurns o scale are d decreasng ( ε > ), and f he quany of oupu ncreases ( d / d > 0 ), AFP wll be smaller han FP4, snce AFP capures boh he effecs of pure echncal change and non-consan reurns. Ths s borne ou n he emprcal secon Assumng no mark-ups, no unobserved npu and decreasng reurns o scale: an oupu-based measure I s well known ha a producon echnology wh non-consan reurns o scale gves rse o several producvy measures (see, for example, Balk 998). In parcular here are dfferences beween oupu-based measures of echnology such as he shf of a producon funcon or of a revenue funcon over me and npu-based measures of echnology such as he shf of a cos funcon or of an npu dsance funcon over me. In he secons above, he analyss has been based on a cos funcon,.e., an npu-based measure. To nroduce an alernave and oupubased measure of echncal change, we shall consder a revenue funcon and s shf over me. As n secon 5.2.4, we assume ha here s no unobserved npu and ha here are no mark-ups. As a consequence, he value of s enrely deermned by he decreasng reurns o scale and oal coss equal observed coss (snce ( ε ) TC where TC C). The dea s based on Dewer and Nakamura (2007) who nroduce an unknown varable no a cos funcon o deal wh decreasng reurns o scale. 25

14 To derve he oupu-based producvy measure, consder he revenue funcon 2 R, defned so as o show maxmum revenues gven a vecor of npus and a vecor of oupu prces: (24) R(P,L,K,) max{ P : (,L,K) belongs o Z() }. Dewer (983) frs used a revenue funcon o defne a heorecal producvy ndex, albe n dscree me. We follow hs approach and defne he connuous-me equvalen as he paral dervave of he revenue funcon wh respec o me: oal dfferenaon of R yelds he followng oupu-based measure of echncal change: R d R R d P (25) R d L d P d L d N R K d K d To derve a compuable measure of he oupu-based producvy measure, an addonal assumpon has o be nroduced: revenue-maxmsng behavour on he par of producers. Then, observed revenues equal maxmum revenues: P R. If n addon frms are prce akers, one ges R / P. I s hen sraghforward o oban a compuable expresson for he elascy of revenues wh respec o oupu prces: (26) R P P R P. P Noe ha he assumpons of revenue maxmsaon and prce akng on oupu markes were no necessary for he dervaon of he npu-based measure n secon Thus, he oupu-based producvy sasc requres dfferen assumpons han he npu-based sasc. (27) Now defne he Dvsa decomposon of oal revenues no a prce and a quany ndex: d R d P d P P d P d P. d The frs wo expressons on he rgh hand sde of (25) are equvalen o a Dvsa quany ndex d R P d P P d d of oupus: d. P d P d d To fnd compuable expressons for he npu elasces of he revenue funcon, we nvoke he prof-maxmsng behavour of producers. Ths mples ha hey solve a max R(P,L,K, ) wl u K. The frs order condons for maxmsaon problem of he knd { } L,K R / L w R / R / K uk / R a maxmum are and K u (, N). Consequenly, R / L wl / R and (, N). Then, he hrd and fourh expresson on he rgh-hand sde of (25) can be rewren as. 2 The concep of a revenue funcon s due o Samuelson (953-54). 26

15 (28) R L d L d N R K d K d wl d L R d C R C R wl C d X. d d L d N uk d K R d N uk C d K. d Bu C / R TC / P / ε and he fnal compuable expresson for he oupu sde-based producvy measure n (25) s R d d X (29) FP5. d ε d The k o AFP s readly esablshed: can be shown ha d d X d X d X (30) FP5 AFP. d d ε d ε d Thus, FP5 wll exceed AFP f he quany ndex of npus grows a a posve rae as can be observed n he counry examples n secon A noe on ncreasng reurns o scale There s no reason o beleve ha reurns o scale may no be locally ncreasng; hence hs case mus be reaed as well. Suppose ha ε <. Unless he case of < 0 s allowed, mplyng connung losses for producers, ncreasng reurns o scale mus go ogeher wh posve mark-ups over margnal coss. Thus, n order o have > 0 under ncreasng reurns o scale, μ mus be posve, bu also less han uny for a leas one produc. Then, P ( μ / ε) f we assume ha here s no unobserved npu (D 0). Under hese assumpons, he growh accounng and producvy equaon (5b) akes he form: ε P (3) d μ d X μ FP P d d 6 Whle (3) s a vald measure for he shf n a cos funcon, TC6, gven ncreasng reurns o scale and whou unobserved npus, s apparen ha wh he observable nformaon on prces, quanes and facor remuneraon (3) sll canno be compued. Alhough ε / μ s known, here s no enough nformaon o deduce values for produc-specfc mark-up facors μ. Exraneous nformaon abou mark-ups s requred o compue FP6. Whle such nformaon somemes s avalable, hs canno be expeced on an ongong, mely and comprehensve bass. For example, Olvera-arns e al. (996) esmaed mark-up raos for 4 OECD counres by ndusry and repor esmaes of he ypcally posve mark-ups. Bu one-off sudes are quckly oudaed. Also, ndusry-level mark-up esmaes are frequenly confned o manufacurng ndusres, leavng uncovered mporan areas of he servce secor. Overall, would no seem praccal for a sascal offce o rely on mark-up esmaes for purposes of producvy sascs. For he same reason, we are no n a poson o compue emprcal resuls for FP6.. 27

16 6. Emprcal Implemenaon Afer he heorecal dervaons n secon 5, we shall now move on o emprcal consderaons. Several quesons arse. One concerns ndex numbers: how should he connuous-me formulae be ranslaed no dscree ndex number formulae o accommodae he fac ha daa observaons come n dscree form? A second queson relaes o how exacly some of he varables should be measured, n parcular capal servces and user coss of capal. Fnally, we wsh o compare he varous producvy measures o ge a sense of he mporance of choces of assumpons. 6. Choce of Index Number Formulae Concernng he ndex number ssue, our approach has been one of approxmang he connuous-me Dvsa ndces n he heorecal par of he paper by Törnqvs-ype ndces for he presen emprcal par. We are aware of he mehodologcal shorcomngs of hs procedure: hs dscree approxmaon s essenally an arbrary choce, 3 no rgorously backed up by heory. A more horough procedure would have been o sar ou wh dscree formulaons for he cos and revenue funcons and hen derve he approprae ndex number formulae ogeher wh he producvy measure. 4 However, we feel ha he heorecal advanages of a full dervaon n dscree me are ouweghed by he algebrac complcaons ha such an approach brngs along wh ncludng all he neracon erms whch would add lle o he message delvered n he presen paper whle makng he exposon much less readable. For he purpose a hand hen, we chose he followng Törnqvs-ype approxmaons o he above Dvsa-ype formulaons of he varous producvy ndces: (32a) 2 P P P P d d (32b) N 2 2 X X K K C K u C K u L L C L w C L w d d X (32c) / X X AFP (32d) / / AFP FP 3 Tha nearly all common ndex number formulae can be consdered as dscree approxmaons o he Dvsa ndex has already been shown by Frsch (936). For a more recen saemen, see Dewer (980) or Balk (2005). 4 Examples are provded by Balk (998, secon 3.7). 28

17 (32e) N 2 2 / K K P K u P K u L L P L w P L w FP2 (32f) N * * 2 2 / K K P K u P K u L L P L w P L w FP3 (32g) ( ) ε ε 2 / X X 4 FP (32h). X X FP5 2 / ε ε 6.2 easurng Oupus and Inpus The emprcal producvy measures developed n he presen paper all relae o he oal economy. Ths reflecs daa consrans raher han a preferred choce whch would have been o lm compuaons o he corporae or busness secor. Neher capal npu measures nor hours worked are easly avalable n such a secoral breakdown and calculaons reman a he aggregae level, n e wh he daa avalable from he OECD Producvy Daabase Oupus Value-added has been measured a basc prces,.e., excludng axes on producs bu ncludng produc subsdes, because hs valuaon consues he economcally relevan varable from a producer perspecve. Tme seres on value-added and ne ndrec axes were aken from he OECD Annual Naonal Accouns. A second adjusmen o aggregae value-added s also requred o manan conssency beween npu and oupu daa: capal npu n he OECD Producvy Daabase s lmed o non-resdenal, fxed asses n scope and consequenly, he value-added produced wh resdenal asses should be excluded from producvy calculaons. Thus, oal value-added s correced for he producon of owner-occupers. 6 Noe ha boh adjusmens (valuaon of oupu a basc prces and excluson of he producon of owner-occupers of dwelgs) have mmedae consequence for he sze of he endogenous rae of reurn as compued n FP3 and for he weghs ha aach o capal and labour n FP2. AFP, on he oher hand, s nfluenced The need for hs excluson and possble consequences for he measuremen of he endogenous raes of reurn were poned ou o me by ahlde as (Unversy of Valenca). 29

18 by hese adjusmens only o he exen ha hey bear on he volume growh rae of oupu. oreover, curren-prce value-added does no ener he AFP compuaon because labour and capal weghs are deermned ndependenly of he oupu measure. Ths s a dsnc advanage n he presence of he AFP approach Inpus Labour npu s measured as oal hours worked n he economy a dffcul ask, especally a he nernaonal level. Even so, hs remans an mperfec measure: no accoun s aken of dfferences n he value of hours of persons wh dfferen skll and experence levels. A more approprae ndex of labour npu would wegh dfferen ypes of hours worked by her correspondng shares n overall compensaon. The mos mporan measuremen ssues are descrbed n a noe avalable on he se of he OECD Producvy Daabase. Capal npus are derved wh he perpeual nvenory mehod. The esmaon of capal servce flows sars wh denfyng hose asses ha correspond o he breakdown currenly avalable from he OECD/Eurosa Naonal Accouns quesonnare, augmened by nformaon on nformaon and communcaon echnology asses. Only non-resdenal gross fxed capal formaon s consdered for seven ypes of asses or producs: Producs of agrculure, meal producs and machnery (IT hardware; communcaons equpmen; oher); ranspor equpmen; non-resdenal consrucon; oher producs (sofware; oher). Invesmen. For each ype of asse, a me seres of curren-prce nvesmen expendure and he correspondng prce ndces are assembled sarng wh 960. For many counres, hs nvolves a ceran amoun of esmaon, n parcular for he perod Such esmaes are ypcally based on naonal accouns daa pror o he nroducon of SNA93, or on relaonshps beween dfferen ypes of asses ha are esablshed for recen perods and projeced backwards. For purposes of exposon of he mehodology, he curren prce nvesmen seres for asse ype n year are denoed by (,2,..., 7) and he correspondng prce ndex s denoed by. IN Prce ndces are normalsed o he reference year 995 where. q, 0 Prce ndces should be consan qualy deflaors ha reflec prce changes for a gven nvesmen good. Ths s parcularly mporan for hose ems ha have seen rapd qualy change such as nformaon and communcaon echnology (ICT) asses. For nsance, observed prce changes of compuer boxes had o be qualy-adjused o perm comparson of dfferen vnages. Schreyer (2000) used a se of harmonsed ICT deflaors o conrol for some of he dfferences n mehodology. 8 We follow hs approach and assume ha he raos beween ICT and non-ict asse prces evolve n a smlar manner across counres, usng he Uned Saes as he benchmark. Alhough no clam s made ha he harmonsed deflaor s necessarly he q, 0 7 For example, he avalable OECD naonal accouns daa do no allow us o sngle ou he producon of he owner-occuped dwelgs ndusry only he paren aggregae wh real esae, renng and busness acves s avalable. For purposes of he presen compuaons, an assumpon had o be made ha he producon of owneroccupers accouns for one hrd of he enre ndusry. Obvously, hs nroduces a poenal bas n hose compuaons ha depend on hs adjusmen. 8 Wyckoff (995) was one of he frs o pon ou ha he large dfferences ha could be observed beween compuer prce ndces n OECD counres were lkely much more a reflecon of dfferences n sascal mehodology han rue dfferences n prce changes. In parcular, hose counres ha employ hedonc mehods o consruc ICT deflaors end o regser a larger drop n ICT prces han counres ha do no. 30

19 correc prce ndex for a gven counry, he possble error due o usng a harmonsed prce ndex s smaller han he bas arsng from comparng capal servces based on naonal deflaors 9. Producve socks. Gven prce and volume seres for nvesmen goods, for each of he (supposedly) homogenous asse ypes, a producve sock T τ τ τ,0 τ τ (33) S (IN / q )h F,,, 7. S has been consruced as follows: In hs expresson, he producve sock of asse a he begnnng of perod s he sum over all pas nvesmens for hs asse, where curren prce nvesmen n pas perods, τ,0 has been deflaed wh he purchase prce ndex of new capal goods, q. T represens he maxmum servce lfe of asse ype. Because pas vnages of capal goods are less effcen han new h τ ones, an age effcency funcon has been appled. I descrbes he effcency me profle of an asse, condonal on s survval and s defned as a hyperbolc funcon of he form used by he Uned Saes Bureau of Labor Sascs (BLS 983), h ( T τ) /( T βτ). Capal goods of he same ype purchased n he same year do no generally rere a he same momen. ore lkely, here s a reremen dsrbuon around a mean servce lfe. In he presen calculaons, a normal dsrbuon wh a sandard devaon of 25 percen of he average servce lfe s chosen o represen he probably of reremen. The dsrbuon was runcaed a an assumed maxmum servce lfe of.5 mes he average servce lfe. The parameer τ IN τ s he cumulave value of hs dsrbuon, descrbng he probably of survval over a cohor s lfe span. The followng average servce lves are assumed for he dfferen asses: 7 years for IT equpmen; 5 years for communcaons equpmen, oher equpmen and ranspor equpmen; 60 years for non-resdenal srucures; 3 years for sofware; and 7 years for he remanng producs. The parameer β n he age-effcency funcon was se o 0.8. Servce lves and parameer values were specfed followng BLS pracces. User coss of capal. In a fully funconng asse marke, he purchase prce of an asse wll equal he dscouned flow of he value of servces ha he asse s expeced o generae n he fuure. Ths equlbrum condon s used o derve he renal prce or user cos expresson for asses. Le 0 q, denoe he purchase prce n year of a new (zero-year old) asse of ype, and le u τ, τ be he renal prce ha hs asse s expeced o fech n perod τ (frs subscrp o he rgh) when he asse wll be of age τ (second subscrp o he rgh). Wh r as he nomnal dscoun rae vald a me, he asse marke equlbrum condon for a new asse (age zero) s: ( τ ), 0 τ τ τ 0, ) (34a) q u ( r. F τ 9 See Schreyer e al. (2003) for deals. There s a dffculy wh he harmonsed deflaor ha should be noed. From an accounng perspecve, adjusng he prce ndex for nvesmen goods for any counry mples an adjusmen of he volume ndex of oupu. In mos cases, such an adjusmen would ncrease he measured rae of volume oupu change. A he same me, effecs on he economy-wde rae of GDP growh appear o be relavely small (see Schreyer (2002) for a dscusson). 3

20 Ths formulaon mples ha renals are pad a he end of each perod. To solve hs expresson for he renal prce, he prce for a one year old asse n perod s compued as ( τ ) q, u τ τ τ 0 2, ( r) u,0 q, 0 ( r) q, or u, 0 q,0 ( r) q,, 0,0, 0 (34b) u q (r d ζ d ζ ). and hen subraced from he expresson above o oban, 0, 0, 0 whch can be ransformed no Ths s he user cos formulaon 20 appled n he presen paper, where he rae of deprecaon of asse has been defned as d, s q,s / q, s and he rae of prce change of he same asse s gven by ζ q, s / q,s. Noe ha he dfferen varables n he user cos equaon are expecaons because hey nvoke knowledge abou prce changes n fuure perods. These expecaons govern he renal prce. The Sysem of Naonal Accouns ha capal sock daa should e no s based on ex-pos prces, observed n he conex of acual ransacons. Would he use of user cos expressons such as hose dscussed above be n conradcon wh he prncples of naonal accouns? The answer s no. The presence of expecaons does no make he user cos erm less real : ransacons are concluded a hs prce, even f wh hndsgh (ex pos) he expecaons underlyng he ransacons may urn ou o be wrong. Ths s mos apparen when one hnks of a case where capal goods are rened: he observed renal prce characerses he ransacon and s he relevan marke prce, ypcally dependen on expecaons on he sde of he lessor and he lessee. Nobody would challenge usng such observed prces n he naonal accouns. If renal prces are no observable, values have o be mpued, and he expresson above ndcaes how hs can be done on he bass of economc heory. Impuaons are numerous n he naonal accouns, and n hs sense, he mpuaon of user coss would no consue an excepon. Thus, s no he presence of an expeced varable as such ha s a ssue. The real ssue from a capal and producvy measuremen vewpon s wheher he realsed, bu unobserved, margnal producvy of fxed asses s beer approxmaed by an ex-ane or an ex-pos measure of user coss. 2 On hs maer, Bernd (990) pons ou ha: f one wans o use a measure of capal o calculae acual mulfacor producvy growh, hen heory ells us que clearly ha we should wegh he varous radonally measured capal npus by her realsed margnal producs, no her expeced margnal producs. Ths means ha n choosng capal servce prce weghs, one should employ shadow values or ex pos raes of reurn, and no he ex ane raes of reurn ha are approprae n he nvesmen conex. Whle we concur wh Bernd s saemen ha for purposes of producvy measuremen, realsed margnal producs are he approprae weghs, hs does no mean ha ex pos raes of reurn are always he preferred approxmaon o realsed margnal producvy. Suppose ha a 20 Jorgenson and Yun (200) show how ax consderaons ener he user cos of capal and how hey affec measured economc performance. Ths s one of he projecs for expanson of he OECD Producvy Daabase. A presen, however, hese parameers are no consdered n our se of user coss and capal measures. 2 The dsncon beween ex-ane and ex-pos user coss has been dscussed by Bernd and Fuss (986), Harper e al. (989), Dewer (200), Bernd (990) n hs dscusson of Hulen (990) and Hll and Hll (2003). 32

21 capal asse s rened by a producer a a gven, pre-agreed renal prce o be pad by he end of he perod. The lessee of he asse wll use n hs producon process as planned regardless of he ex-pos renal prce. Therefore, he margnal producvy of he asse n he producon process would bes be approxmaed by he ex-ane renal prce ha s he prce a whch he renal ransacon acually ook place. Take anoher case of an owner/producer and suppose ha here has been nvesmen a he begnnng of he perod n e wh he ex-ane user cos. Now le here be a change n marke condons ha leads o a modfcaon of expecaons and of user coss. If capal s fully flexble and can be adjused connuously, wll be adjused n e wh he new user cos erm. Bu he user cos erm s governed by expecaons, even hough he expecaons may have changed. I s only when capal canno be adjused ha he ex-pos user cos erm would furnsh he preferred approxmaon o he realsed margnal producvy of an asse. Ths s he case ha Bernd (990) and Bernd and Fuss (986) have n mnd and reles on quas-fxy of capal n he producon process. Thus, here s no general concluson ha ex-pos user cos measures should always be preferred o ex-ane ones for measurng and aggregang capal npu. There s anoher concepual dffculy wh ex pos user coss: he compuaon of he realsed raes of reurn s commonly done by choosng a rae of reurn so ha he ensung user cos and oal value of capal servces jus exhauss he measured gross operang surplus avalable from he naonal accouns. Ths compuaon reles, however, on he assumpon ha here s only one ex-pos rae of reurn ha apples o all asses. Whle equalsaon of raes of reurn across asses s a naural assumpon n an ex-ane conex, s much harder o jusfy n an ex-pos conex, especally gven saes of dsequlbrum. Essenally amouns o mposng an equlbrum condon o mplemen an (ex-pos) measure ha was specfcally chosen on he grounds ha capures he naure of a suaon of dsequlbrum. Dewer (200) also pons ou ha whle he ex-pos measure (of he nomnal rae of reurn) s wdely used n emprcal research, s subjec o measuremen error and may no reflec he economc condons facng producers a he begnnng of he perod. A praccal argumen agans he use of an ex-pos rae s ha s calculaon requres nformaon on he level of he producve capal sock a curren prces (or alernavely on he wealh sock a curren prces). However, levels of capal socks end o be less relable sascs han her raes of change, especally when long hsorcal nvesmen seres have o be esmaed. Ths problem does no arse when user coss and nomnal raes of reurn are of an ex-ane naure and herefore are exogenous varables. On he oher hand, ex-pos raes of reurn are of neres as such, and sraghforward o compue. In sum hen, here s no clear concluson on hs maer. In he presen work, preference s gven o an ex-ane approach, manly because allows us o develop capal servce measures ndependenly from measures of labour compensaon, gross operang surplus and mxed ncome n he naonal accouns. Exogenous ne rae of reurn. To compue he ne rae of reurn, followng a suggeson of Dewer (200), he sarng pon s he consan value for he expeced real neres rae rr. The consan real rae s compued by akng a seres of annual observed nomnal raes (an unweghed average of neres raes wh dfferen maures 22 ) and deflang hem by he 22 These are he average bank rae, he bank rae on prme loans, long-erm governmen bond yelds, shor-erm governmen bond yelds, he neres rae on a 90 day bank fxed depos, and he reasury bll rae. 33

22 consumer prce ndex. The resug seres of real neres raes s averaged over he perod ( ) o yeld a consan value for rr. The expeced nomnal neres rae for every year s hen compued as r ( rr)( p ) where p s he expeced value of an overall deflaor, he consumer prce ndex. To oban a measure for p, he expeced overall nflaon, we consruc a 5-year cenered movng average of he rae of change of he consumer prce ndex 2 p CPI, where s 2 s CPI s he annual percenage change of he consumer prce ndex. Ths equals he expeced rae of overall prce change and, by mplcaon, he nomnal ne rae of reurn. Expeced asse prce changes, anoher elemen n he user cos equaon, are derved as a smoohed seres of acual asse prce changes: a 5-year cenered movng average fler s used. Deprecaon raes have been compued usng he defnon gven above, d q / q. So, he rae of deprecaon for a new asse equals one mnus he rao of he, 0,, 0 marke prce for a year old asse over he marke prce for a new asse. The marke prce for a new asse can be observed drecly, bu he prce for a one-year old asse mus be compued usng he asse marke equlbrum condon (34), he age-effcency funcon h and he dscoun rae. 6.3 Resuls Tables -4 summarse emprcal resuls for Canada, France, he Uned Saes and Japan. They show he raes of change of oupu (GDP) and labour npu as well as he volume changes of capal servces alernavely based on exogenous and endogenous raes of reurn as well as he varous FP measures. The frs observaon s ha movng from an endogenous o an exogenous rae of reurn leads o a rse n he observed measure of capal npu a leas n he case of he counres consdered and for he perod a hand. Also, labour and capal shares urn ou o be que dfferen when based on oal coss raher han oal revenue. The second panel n each able revews resuls for he fve alernave FP measures presened n he ex above. I s mmedaely apparen ha he dfferen opons each assocaed wh a parcular se of assumpons abou marke srucures or producon echnology can lead o consderable varaon n he resug FP measures, France beng a noceable excepon. Unless a-pror knowledge abou echnology and marke srucure are avalable, wll be dffcul o choose beween he dfferen opons. Also, every dfferen FP measure mples a dfferen message abou he relave conrbuon of capal servces o oupu growh. For all four counres examned, measured producvy growh urns ou o be slowes when based on endogenous rae models (FP3) or when assumng proporonaly beween capal npu and an unobserved facor (FP2). The oupu-based producvy measure ha allows for decreasng reurns o scale (FP5) s generally he fases-growng em n each counry, followed by he npu-based producvy measure wh decreasng reurns o scale (FP4). However, a smple geomerc average of he fve specfc FP measures yelds a me seres ha s very close o he smple AFP measure. In he absence of a-pror nformaon on mark-ups, reurns o scale, or unobserved asses, he choce of a measure ha s close o he average of he dfferen opons may be a reasonable one. Ths s one of our conclusons. 34