MEASURING THE PRICE AND QUANTITY AS CAPITAL SERVICES UNDER ALTERNATIVE ASSUMPTIONS

Transcription

1 MEASURING THE PRICE AND QUANTITY AS CAPITAL SERVICES UNDER ALTERNATIVE ASSUMPTIONS by W. Erwin Diewer JUNE 2001 Discussion Paper No.: DEPARTMENT OF ECONOMICS THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, CANADA V6T 1Z1 hp://web.ars.ubc.ca/econ

2 Measuring he Price and Quaniy of Capial Services Under Alernaive Assumpions Erwin Diewer, Deparmen of Economics, Universiy of Briish Columbia, Vancouver, Canada, V6N 1Z1. Absrac. The paper revisis Harper, Bernd and Wood (1989) and calculaes Canadian reproducible capial services aggregaes under alernaive assumpions abou he form of depreciaion, he opporuniy cos of capial and he reamen of capial gains. Five differen models of depreciaion are considered: (1) one hoss shay; (2) sraigh line depreciaion; (3) declining balance or geomeric depreciaion; (4) linearly declining efficiency profiles and (5) linearly increasing mainenance profiles. The laer form of depreciaion does no seem o have been considered in he lieraure before. This model assumes ha here is a known ime profile of mainenance expendiures ha can be associaed wih each asse and he opimal ime of reiremen of he asse as well as he profile of used asse prices becomes endogenous under his specificaion. I urns ou if he mainenance profile increases linearly, hen he linearly declining efficiency profile model emerges; see (4) above. We consider 3 alernaive assumpions abou he ineres rae and he reamen of capial gains so ha we evaluae 15 models in all and compare heir differences. Following Hill (2000), we also consider he differences beween cross secion and ime series depreciaion and anicipaed ime series depreciaion (which adds anicipaed obsolescence of he asse o normal cross secion depreciaion of he asse). Finally, we follow he suggesion made by Diewer and Lawrence (2000) ha a superlaive index number formula be used o aggregae up vinages of capial raher han he usual assumpion of linear aggregaion, which implicily assumes ha he capial services yielded by each vinage of a homogeneous ype of capial are perfecly subsiuable. References. Diewer, W.E. and D.A. Lawrence (2000), Progress in Measuring he Price and Quaniy of Capial, pp in Economerics and he Cos of Capial: Essays in Honor of Dale W. Jorgenson, L.J. Lau (ed.), Cambridge MA: The MIT Press. Harper, M.J., E.R. Bernd and D.O. Wood (1989), Raes of Reurn and Capial Aggregaion Using Alernaive Renal Prices, pp in Technology and Capial Formaion, D.W. Jorgenson and R. Landau (eds.), Cambridge MA: The MIT Press. Hill, P. (2000); Economic Depreciaion and he SNA ; paper presened a he 26h Conference of he Inernaional Associaion for Research on Income and Wealh; Cracow, Poland.

3 1 Measuring he Price and Quaniy of Capial Services under Alernaive Assumpions Erwin Diewer, March 28, 2001 Deparmen of Economics, Universiy of Briish Columbia, Vancouver, Canada, V6T 1Z1. TABLE OF CONTENTS 1. Inroducion 2. The Fundamenal Equaions Relaing Socks and Flows of Capial 3. Cross Secion Depreciaion Profiles 4. The Empirical Deerminaion of Ineres Raes and Asse Inflaion Raes 5. Obsolescence and Depreciaion 6. Aggregaion over Vinages of a Capial Good 7. The One Hoss Shay Model of Efficiency and Depreciaion 8. The Sraigh Line Depreciaion Model 9. The Declining Balance or Geomeric Depreciaion Model 10. The Linear Efficiency Decline Model 11. The Linearly Increasing Mainenance Expendiures Model 12. A Comparison of he Twelve Models 13. Conclusion Appendix: Canadian Daa for Inroducion 1 Capial (I am no he firs o discover) is a very large subjec, wih many aspecs; wherever one sars, i is hard o bring more han a few of hem ino view. I is jus as if one were making picures of a building; hough i is he same building, i looks quie differen from differen angles. John Hicks (1973; v). Perhaps a more realisic moive for reading earlier wriers is no o rediscover forgoen ruhs, bu o gain a perspecive of how presen day ideas have evolved and, perhaps, by reading he original saemens of imporan ideas, o see hem more vividly and undersand hem more clearly. Geoffrey Whiingon (1980; 240). In his Appendix, we discuss some of he problems involved in consrucing price and quaniy series for boh capial socks and he associaed flows of services. We illusrae he various alernaive approaches by calculaing boh socks and service flows of reproducible capial for Canada for he years The auhor is indebed o Kevin Fox, Peer Hill, Ulrich Kohli, Alice Nakamura and Paul Schreyer for helpful commens.

4 2 In secion 2 we presen he basic equaions relaing socks and flows of capial assuming ha daa on he prices of vinages of a homogeneous capial good are available. This framework is no applicable under all circumsances 2 bu i is a framework ha will allow us o disenangle he effecs of general price change, asse specific price change and depreciaion. Secion 3 coninues he heoreical framework ha was inroduced in secion 2. We show how informaion on vinage asse prices, vinage renal prices and vinage depreciaion raes are all equivalen under cerain assumpions; i.e., knowledge of any one of hese hree sequences or profiles is sufficien o deermine he oher wo. Secion 4 discusses alernaive ses of assumpions on nominal ineres raes and anicipaed asse price changes. We specify hree differen ses of assumpions ha we will use in our empirical illusraion of he suggesed mehods. Secion 5 is a secion ha is no used in he remaining secions bu i discusses he significance of our assumpions made in he previous secion and relaes hem o conroversies in naional income accouning. In paricular, we discuss wheher anicipaed asse price decline should be an elemen of depreciaion as undersood by naional income accounans. Secion 6 discusses he problems involved in aggregaing over vinages of capial, boh in forming capial socks and capial services. Insead of he usual perpeual invenory mehod for aggregaing over vinages, which assumes perfecly subsiuable vinages of he same sock, we sugges he use of a superlaive index number formula o do he aggregaion. Secions 7 o 10 show how he general algebra presened in secions 2 and 3 can be adaped o deal wih four specific models of depreciaion. The four models considered are he one hoss shay model, he sraigh line depreciaion model, he geomeric model of depreciaion and he linear efficiency decline model. In each of hese secions, we illusrae he mehod by compuing he corresponding socks and flows using Canadian daa on wo asse classes. Secion 11 considers a fifh ype of depreciaion model, one ha is based on he assumpion ha each vinage of he asse has a specific mainenance and operaing cos requiremen associaed wih i. We show ha his ype of model can lead o he linear efficiency decline model sudied in secion 10. However, he main use of he analysis presened in secion 11 is o sugges a reason why acceleraed depreciaion assumpions are quie reasonable and likely o occur empirically. Secion 12 presens a graphical summary of he 12 empirical models considered in his Appendix. 2 Mos noably, our framework canno deal wih unique or one of a kind asses, which by definiion, do no have vinages.

5 3 Secion 13 concludes. A daa appendix liss he Canadian daa we used in secions The Fundamenal Equaions Relaing Socks and Flows of Capial Before we begin wih our algebra, i seems appropriae o explain why accouning for he conribuion of capial o producion is more difficul han accouning for he conribuions of labour or maerials. The main problem is ha when a reproducible capial inpu is purchased for use by a producion uni a he beginning of an accouning period, we canno simply charge he enire purchase cos o he period of purchase. Since he benefis of using he capial asse exend over more han one period, he iniial purchase cos mus be disribued somehow over he useful life of he asse. This is he fundamenal problem of accouning. 3 Hulen (1990) explains he consequences for accounans of he durabiliy of capial as follows: Durabiliy means ha a capial good is producive for wo or more ime periods, and his, in urn, implies ha a disincion mus be made beween he value of using or rening capial in any year and he value of owning he capial asse. This disincion would no necessarily lead o a measuremen problem if he capial services used in any given year were paid for in ha year; ha is, if all capial were rened. In his case, ransacions in he renal marke would fix he price and quaniy of capial in each ime period, much as daa on he price and quaniy of labor services are derived from labor marke ransacions. Bu, unforunaely, much capial is uilized by is owner and he ransfer of capial services beween owner and user resuls in an implici ren ypically no observed by he saisician. Marke daa are hus inadequae for he ask of direcly esimaing he price and quaniy of capial services, and his has led o he developmen of indirec procedures for inferring he quaniy of capial, like he perpeual invenory mehod, or o he accepance of flawed measures, like book value. Charles R. Hulen (1990; ). From basic capial heory, he value of an asse a he beginning of an accouning period is equal o he discouned sream of fuure renal paymens ha he asse is expeced o 3 The difficuly of impuing expenses o individual sales or even o he gross earnings of he accouning period, he monh or year, is an ever presen problem for he accounan in he periodic deerminaion of enerprise income. The longer he period for which he income is o be deermined, he smaller he relaive amoun of error. Absolue accuracy can be aained only when he venure is compleed and he enerprise erminaed. William T Crandell (1935; ). Early enerprises and parners working in he main in isolaed rading venures, needed only an irregular deerminaion of profi. Bu before he business corporaion had been very long in operaion i was eviden ha i needed o be reaed as a coninuing enerprise. For example, calculaing dividends by separae voyages was found impracical in he Eas India Company by Profi calculaion herefore became a maer of periodic esimaes in place of he known resuls of compleed venures. A.C. Lileon (1933; 270). The hird convenion is ha of he annual accouning period. I is his convenion which is responsible for mos of he difficul accouning problems. Wihou his convenion, accouning would be a simple maer of recording compleed and fully realized ransacions: an ac of primiive simpliciy. Sephen Gilman (1939; 26). All he problems of income measuremen are he resul of our desire o aribue income o arbirarily deermined shor periods of ime. Everyhing comes righ in he end; bu by hen i is oo lae o maer. David Solomons (1961; 378). Noe ha hese auhors do no menion he addiional complicaions ha are due o he fac ha fuure revenues and coss mus be discouned o yield values ha are equivalen o presen dollars.

6 4 yield. Thus he sock value of he asse is se equal o he discouned fuure service flows 4 ha he asse is expeced o yield in fuure periods. In an inflaionary environmen, i is necessary o disinguish beween he observable renal prices for he asse a differen ages or vinages a he beginning of period and fuure expeced renal prices for asses of various vinages. 5 Thus le f 0 be he (observable) renal price of a new asse a he beginning of period, le f 1 be he (observable) renal price of a one period old asse a he beginning of period, le f 2 be he (observable) renal price of a 2 period old asse a he beginning of period, ec. Then he fundamenal equaion relaing he sock value of a new asse a he beginning of period, P 0, o he sequence of cross secional vinage renal prices prevailing a he beginning of period, {f n : n = 0,1,2, } 6 is: (1) P 0 = f 0 + [(1+i 1 )/(1+r 1 )] f 1 + [(1+i 1 )(1+i 2 )/(1+r 1 )(1+r 2 )] f 2 + In he above equaion, 7 1+i 1 is he renal price escalaion facor ha is expeced o apply o a one period old asse going from he beginning of period o he end of period (or equivalenly, o he beginning of period +1), (1+i 1 )(1+i 2 ) is he renal price escalaion facor ha is expeced o apply o a 2 period old asse going from he beginning of period o he beginning of period +2, ec. Thus he i n are expeced vinage renal price inflaion raes ha are formed a he beginning of period. The erm 1+r 1 is he discoun facor ha makes a dollar received a he beginning of period equivalen o a dollar received a he beginning of period +1, he erm (1+r 1 )(1+r 2 ) is he discoun facor ha makes a dollar received a he beginning of period equivalen o a dollar received a he beginning of period +2, ec. Thus he r n are one period nominal ineres raes ha represen he erm srucure of ineres raes a he beginning of period. 8 4 Walras (1954) (firs ediion published in 1874) was one of he earlies economiss o sae ha capial socks are demanded because of he fuure flow of services ha hey render. Alhough he was perhaps he firs economis o formally derive a user cos formula as we shall see, he did no work ou he explici discouning formula ha Böhm-Bawerk (1891; 342) was able o derive. 5 Noe ha hese fuure expeced renal prices are no generally observable due o he lack of fuures markes for hese vinage fuure renals of he asse. 6 The sequence of (cross secional) vinage renal prices {f n } is ofen called he age-efficiency profile of he asse. 7 I should be noed ha Irving Fisher (1897; 365) seemed o be well aware of he complexiies ha are imbedded in equaion (1): There is no space here o discuss he heory in greaer deail, nor o apply i o economic problems. A full reamen would ake accoun of he various sandards in which income is or may be expressed, of he case in which he raes of ineres a differen daes and for differen periods does no remain consan, of he fac ha he services of capial which are discouned in is value are only expeced services, no hose which acually maerialise, and of he consequen discrepancy beween income anicipaed and income realised, of he propriey or impropriey of including man himself as a species of income-bearing capial, and so on. 8 Peer Hill has noed a major problem wih he use of equaion (1) as he saring poin of our discussion: namely, unique asses will by definiion no have vinages and so he cross secional vinage renal prices f n will no exis for hese asses! In his case, he f n should be inerpreed as expeced fuure renals ha he unique asse is expeced o generae a oday s prices. The (1+i n ) erms hen summarize expecaions abou he amoun of asse specific price change ha is expeced o ake place. This reinerpreaion of equaion (1) is more fundamenal bu we chose no o make i our saring poin because i does no lead o a workable mehod for naional saisicians o form reproducible esimaes of hese fuure renal paymens. However, in many siuaions (e.g., he valuaion of a new movie), he saisician will be forced o aemp o implemen Hill s (2000) more general model.

7 5 We now generalize equaion (1) o relae he sock value of an n period old asse a he beginning of period, P n, o he sequence of cross secional vinage renal prices prevailing a he beginning of period, {f n }; hus for n = 0,1,2,, we assume: (2) P n = f n + [(1+i 1 )/(1+r 1 )] f n+1 + [(1+i 1 )(1+i 2 )/(1+r 1 )(1+r 2 )] f n+2 + Thus older asses discoun fewer erms in he above sum; i.e., as n increases by one, we have one less erm on he righ hand side of (2). However, noe ha we are applying he same price escalaion facors (1+i 1 ), (1+i 1 )(1+i 2 ),, o escalae he cross secional renal prices prevailing a he beginning of period, f 1, f 2,, and o form esimaes of fuure expeced renal prices for each vinage of he capial sock ha is in use a he beginning of period. The vinage renal prices prevailing a he beginning of period, f 0, f 1, are poenially observable. 9 These cross secion renal prices reflec he relaive efficiency of he various vinages of he capial good under consideraion a he beginning of period. I is assumed ha hese renals are paid (explicily or implicily) by he users a he beginning of period. Noe ha he sequence of vinage asse sock prices a he beginning of period, P 0, P 1, is no affeced by general inflaion provided ha he general inflaion affecs he expeced asse inflaion raes i n and he nominal ineres raes r n in a proporional manner. We will reurn o his poin laer. The physical produciviy characerisics of a uni of capial of each vinage are deermined by he sequence of cross secional renal prices. Thus a brand new asse is characerized by he vecor of curren vinage renal prices, f 0, f 1, f 2,, which are inerpreed as physical conribuions o oupu ha he new asse is expeced o yield during he curren period (his is f 0 ), he nex period (his is f 1 ), and so on. An asse which is one period old a he sar of period is characerized by he vecor f 1, f 2,, ec. 10 We have no explained how he expeced renal price inflaion raes i n are o be esimaed. We shall deal wih his problem in secion 5 below. However, i should be noed ha here is no guaranee ha our expecaions abou he fuure course of renal prices are correc. A his poin, we make some simplifying assumpions abou he expeced renal inflaion raes i n and he ineres raes r n. We assume ha hese anicipaed vinage renal inflaion facors a he beginning of each period are all equal; i.e., we assume: (3) i n = i ; n = 1,2, We also assume ha he erm srucure of ineres raes a he beginning of each period is consan; i.e., we assume: 9 This is he main reason ha we use he vinage approach o capial measuremen raher han he more fundamenal discouned fuure expeced renals approach advocaed by Hill. 10 Triple (1996; 97) used his characerizaion for capial asses of various vinages.

8 6 (4) r n = r ; n = 1,2, However, noe ha as he period changes, r and i can change. Using assumpions (3) and (4), we can rewrie he sysem of equaions (2), which relae he sequence or profile of vinage sock prices a he beginning of period {P n } o he sequence or profile of (cross secional) vinage renal prices a he beginning of period {f n }, as follows: (5) P 0 = f 0 + [(1+i )/(1+r )] f 1 + [(1+i )/(1+r )] 2 f 2 + [(1+i )/(1+r )] 3 f 3 + P 1 = f 1 + [(1+i )/(1+r )] f 2 + [(1+i )/(1+r )] 2 f 3 + [(1+i )/(1+r )] 3 f 4 + P 2 = f 2 + [(1+i )/(1+r )] f 3 + [(1+i )/(1+r )] 2 f 4 + [(1+i )/(1+r )] 3 f 5 + P n = f n + [(1+i )/(1+r )] f n+1 + [(1+i )/(1+r )] 2 f n+2 + [(1+i )/(1+r )] 3 f n+3 + On he lef hand side of equaions (5), we have he sequence of vinage asse prices a he beginning of period saring wih he price of a new asse, P 0, moving o he price of an asse ha is one period old a he sar of period, P 1, hen moving o he price of an asse ha is 2 periods old a he sar of period, P 2, and so on. On he righ hand side of equaions (5), he firs erm in each equaion is a member of he sequence of vinage renal prices ha prevails in he marke (if such markes exis) a he beginning of period. Thus f 0 is he ren for a new asse, f 1 is he ren for an asse ha is one period old a he beginning of period, f 2 is he ren for an asse ha is 2 periods old, and so on. This sequence of curren marke renal prices for he asses of various vinages is hen exrapolaed ou ino he fuure using he anicipaed price escalaion raes (1+i ), (1+i ) 2, (1+i ) 3, ec. and hen hese fuure expeced renals are discouned back o he beginning of period using he discoun facors (1+r ), (1+r ) 2, (1+r ) 3, ec. Noe ha given he period expeced asse inflaion rae i and he period nominal discoun rae r, we can go from he (cross secional) sequence of vinage renal prices {f n } o he sequence of vinage asse prices {P n } using equaions (5). We shall show below how his procedure can be reversed; i.e., we shall show how given he sequence of vinage asse prices, we can consruc esimaes for he sequence of vinage renal prices. I seems ha Böhm-Bawerk was he firs economis o use he above mehod for relaing he fuure service flows of a durable inpu o is sock price: If he services of he durable good be exhaused in a shor space of ime, he individual services, provided ha hey are of he same qualiy which, for simpliciy s sake, we assume are, as a rule, equal in value, and he value of he maerial good iself is obained by muliplying he value of one service by he number of services of which he good is capable. Bu in he case of many durable goods, such as ships, machinery, furniure, land, he services rendered exend over long periods, and he resul is ha he laer services canno be rendered, or a leas canno be rendered in a normal economic way, before a long ime has expired. As a consequence, he value of he more disan maerial services suffers he same fae as he value of fuure goods. A maerial service, which, echnically, is exacly he same as a service of his year, bu which canno be rendered before nex year, is worh a lile less han his year s service; anoher similar service, bu obainable only afer wo years, is, again, a lile less valuable, and so on; he values of he remoe services decreasing wih he remoeness of he period a which hey can be rendered. Say ha his

9 7 year s service is worh 100, hen nex year s service assuming a difference of 5 % per annum is worh in oday s valuaion only 95.23; he hird year s service is worh only 90.70; he fourh year s service, 86.38; he fifh, sixh and sevenh year s services, respecively, worh 82.27, 78.35, of presen money. The value of he durable good in his case is no found by muliplying he value of he curren service by he oal number of services, bu is represened by a sum of services decreasing in value. Eugen von Böhm-Bawerk (1891; 342). Thus Böhm-Bawerk considered a special case of (5) where all service flows f n were equal o 100 for n = 0,1,,6 and equal o 0 hereafer, where he asse inflaion rae was expeced o be 0 and where he ineres rae r was equal o.05 or 5 %. 11 This is a special case of wha has come o be known as he one hoss shay model. We shall consider i in more deail in he conex of inflaion in secion 7 of his Appendix below. Noe ha equaions (5) can be rewrien as follows: 12 (6) P 0 = f 0 + [(1+i )/(1+r )] P 1 ; P 1 = f 1 + [(1+i )/(1+r )] P 2 ; P 2 = f 2 + [(1+i )/(1+r )] P 3 ; P n = f n + [(1+i )/(1+r )] P n+1 ; The firs equaion in (6) says ha he value of a new asse a he sar of period, P 0, is equal o he renal ha he asse can earn in period, f 0, 13 plus he expeced asse value of he capial good a he end of period, (1+i ) P 1, bu his expeced asse value mus be divided by he discoun facor, (1+r ), in order o conver his fuure value ino an equivalen beginning of period value. 14 Now i is sraighforward o solve equaions (6) for he sequence of period vinage renal prices, {f n }, in erms of he vinage asse prices, {P n }: (7) f 0 = P 0 [(1+i )/(1+r )] P 1 f 1 = P 1 [(1+i )/(1+r )] P 2 f 2 = P 2 [(1+i )/(1+r )] P 3 = (1+r ) 1 [P 0 (1+r ) (1+i ) P 1 ] = (1+r ) 1 [P 1 (1+r ) (1+i ) P 2 ] = (1+r ) 1 [P 2 (1+r ) (1+i ) P 3 ] 11 Böhm-Bawerk (1891; 343) wen on and consruced he sequence of vinage asse prices using his special case of equaions (5). 12 Chrisensen and Jorgenson (1969; 302) do his for he geomeric depreciaion model excep ha hey assume ha he renal is paid a he end of he period raher han he beginning. Varians of he sysem of equaions (6) were derived by Chrisensen and Jorgenson (1973), Jorgenson (1989; 10), Hulen (1990; 128) and Diewer and Lawrence (2000; 276). Irving Fisher (1908; 32-33) derived hese equaions in words as follows: Puing he principle in is mos general form, we may say ha for any arbirary inerval of ime, he value of he capial a is beginning is he discouned value of wo elemens: (1) he acual income accruing wihin ha inerval, and (2) he value of he capial a he close of he period. 13 Noe ha we are implicily assuming ha he renal is paid o he owner a he beginning of period. 14 Anoher way of inerpreing say he firs equaion in (6) runs as follows: he purchase cos of a new asse P 0 less he renal f 0 (which is paid immediaely a he beginning of period ) can be regarded as an invesmen, which mus earn he going rae of reurn r. Thus we mus have [P 0 f 0 ](1+r ) = (1+i )P 1 which is he (expeced) value of he asse a he end of period. This line of reasoning can be raced back o Walras (1954; 267): A man who buys a house for his own use mus be resolved by us ino wo individuals, one making an invesmen and he oher consuming direcly he services of his capial.

10 8 f n = P n+1 [(1+i )/(1+r )] P n+1 = (1+r ) 1 [P n (1+r ) (1+i ) P n+1 ] ; Thus equaions (5) allow us o go from he sequence of vinage renal prices {f n } o he sequence of vinage asse prices {P n } while equaions (7) allow us o reverse he process. Equaions (7) can be derived from elemenary economic consideraions. Consider he firs equaion in (7). Think of a producion uni as purchasing a uni of he new capial asse a he beginning of period a a cos of P 0 and hen using he asse hroughou period. However, a he end of period, he producer will have a depreciaed asse ha is expeced o be worh (1+i ) P 1. Since his offse o he iniial cos of he asse will only be received a he end of period, i mus be divided by (1+r ) o express he benefi in erms of beginning of period dollars. Thus he ne cos of using he new asse for period 15 is P 0 [(1+i )/(1+r )] P 1. The above equaions assume ha he acual or implici period renal paymens f n for asses of differen vinages n are made a he beginning of period. I is someimes convenien o assume ha he renal paymens are made a he end of each accouning period. Thus we define he end of period vinage renal price or user cos for an asse ha is n periods old a he beginning of period, u n, in erms of he corresponding beginning of period renal price f n as follows: (8) u n (1+r ) f n ; n = 0,1,2, Thus if he renal paymen is made a he end of he period insead of he beginning, hen he beginning of he period renal f n mus be escalaed by he ineres rae facor (1+r ) in order o obain he end of he period user cos u n. 16 Using equaions (8) and he second se of equaions in (7), i can readily be shown ha he sequence of end of period user coss {u n } can be defined in erms of he period sequence of vinage asse prices {P n } as follows: (9) u 0 = P 0 (1+r ) (1+i ) P 1 u 1 = P 1 (1+r ) (1+i ) P 2 u 2 = P 2 (1+r ) (1+i ) P 3 u n = P n (1+r ) (1+i ) P n+1 ; 15 This explains why he renal prices f n are someimes called user coss. This derivaion of a user cos was used by Diewer (1974; 504), (1980; ), (1992a; 194) and by Hulen (1996; 155). 16 I is ineresing ha Böhm-Bawerk (1891; 343) carefully disinguished beween renal paymens made a he beginning or end of a period: These figures are based on he assumpion ha he whole year s uiliy is obained all a once, and, indeed, obained in anicipaion a he beginning of he year; e.g., by hiring he good a a year s ineres of 100 payable on each 1 s January. If, on he oher hand, he year s use can only be had a he end of he year, a valuaion underaken a he beginning of he year will show figures no inconsiderably lower. Tha he figures should aler according as he dae of he valuaion sands nearer or farher from he dae of obaining he uiliy, is an enirely naural hing, and one quie familiar in financial life.

11 9 Equaions (9) can also be given a direc economic inerpreaion. Consider he following explanaion for he user cos for a new asse, u 0. A he end of period, he business uni expecs o have an asse worh (1+i ) P 1. Offseing his benefi is he beginning of he period asse purchase cos, P 0. However, in addiion o his cos, he business mus charge iself eiher he explici ineres cos ha occurs if money is borrowed o purchase he asse or he implici opporuniy cos of he equiy capial ha is ied up in he purchase. Thus offseing he end of he period benefi (1+i ) P 1 is he iniial purchase cos and opporuniy ineres cos of he asse purchase, P 0 (1+r ), leading o a end of period ne cos of P 0 (1+r ) (1+i ) P 1 or u 0. I is ineresing o noe ha in boh he accouning and financial managemen lieraure of he pas cenury, here was a relucance o rea he opporuniy cos of equiy capial ied up in capial inpus as a genuine cos of producion. 17 However, more recenly, here is an accepance of an impued ineres charge for equiy capial as a genuine cos of producion. 18 In he following secion, we will relae he vinage asse price profiles {P n } and he user cos profiles {u n } o depreciaion profiles. However, before urning o he subjec of depreciaion, i is imporan o sress ha he analysis presened in his secion is based on a number of resricive assumpions, paricularly on fuure price expecaions. Moreover, we have no explained how hese asse price expecaions are formed and we have no explained how he period nominal ineres rae is o be esimaed (we will address hese opics in secion 6 below). We have no explained wha should be done if he sequence of second hand asse prices {P n } is no available and he sequences of vinage renal prices or user coss, {f n } or {u n }, are also no available (we will address his problem in laer secions as well). We have also assumed ha asse values and user coss are independen of how inensively he asses are used. Finally, we have no modeled uncerainy (abou fuure prices and he useful lives of asses) and aiudes owards risk on he par of producers. Thus he analysis presened in his Appendix is only a sar on he difficul problems associaed wih measuring capial inpu. 3. Cross Secion Depreciaion Profiles Recall ha in he previous secion, P n was defined o be he price of an asse ha was n periods old a he beginning of period. Generally, he decline in asse value as we go from one vinage o he nex oldes is called depreciaion. More precisely, we define he cross secion depreciaion D 19 n of an asse ha is n periods old a he beginning of period as 17 This lieraure is reviewed in Diewer and Fox (1999; ). 18 Sern Sewar & Co. has popularized his concep of EVA, Economic Value Added. In a newspaper adverisemen in he Financial Pos in 1999, i described his new concep as follows: EVA measures your company s afer ax profis from operaions minus he cos of all he capial employed o produce hose profis. Wha makes EVA so revealing is ha i akes ino accoun a facor no convenional measures include: he cos of he operaion s capial no jus he cos of deb bu he cos of equiy capial as well. 19 This erminology is due o Hill (1999) who disinguished he decline in second hand asse values due o aging (cross secion depreciaion) from he decline in an asse value over a period of ime (ime series

12 10 (10) D n P n P n+1 ; n = 0,1,2, Thus D n is he value of an asse ha is n periods old a he beginning of period, P n, minus he value of an asse ha is n+1 periods old a he beginning of period, P n Obviously, given he sequence of period vinage asse prices {P n }, we can use equaions (10) o deermine he period sequence of declines in vinage asse values, {D n }. Conversely, given he period cross secion depreciaion sequence or profile, {D n }, we can deermine he period vinage asse prices by adding up amouns of depreciaion: (11) P 0 = D 0 + D 1 + D 2 + P 1 = D 1 + D 2 + D 3 + P n = D n + D n+1 + D n+2 + Raher han working wih firs differences of vinage asse prices, i is more convenien o reparameerize he paern of cross secion depreciaion by defining he period depreciaion rae δ n for an asse ha is n periods old a he sar of period as follows: (12) δ n 1 [P n+1 /P n ] = D n / P n ; n = 0,1,2, In he above definiions, we require n o be such ha P n is posiive. 21 depreciaion). Triple (1996; 98-99) uses he cross secion definiion of depreciaion and shows ha i is equal o he concep of capial consumpion in he naional accouns bu he does his under he assumpion of no expeced real asse inflaion. We will examine he relaionship of cross secion o ime series depreciaion in secion 6 below. 20 Of course, he objecions o he use of second hand marke daa o deermine depreciaion raes are very old: We readily agree ha where a marke is sufficienly large, generally accessible, and coninuous over ime, i serves o coordinae a large number of subjecive esimaes and hus may impar a momen of (social) objeciviy o value relaions based on prices forced on i. Bu i can hardly be said ha he secondhand marke for indusrial equipmen, which would be he proper place for he deerminaion of he value of capial goods which have been in use, saisfies hese requiremens, and ha is valuaions are superior o inra-enerprise valuaion. L.M. Lachmann (1941; ). Criicism has also been voiced abou he viabiliy of used asse marke price daa as an indicaor of in use asse values. One argumen, drawing on he Ackerlof Lemons Model, is ha asses resold in second hand markes are no represenaive of he underlying populaion of asses, because only poorer qualiy unis are sold when used. Ohers express concerns abou he hinness of resale markes, believing ha i is sporadic in naure and is dominaed by dealers who under-bid. Charles R. Hulen and Frank C. Wykoff (1996; 17-18). 21 This definiion of depreciaion daes back o Hicks (1939) a leas and was used exensively by Hulen and Wykoff (1981a) (1981b), Diewer (1974; 504) and Hulen (1990; 128) (1996; 155): If here is a perfec second hand marke for he goods in quesion, so ha a marke value can be assessed for hem wih precision, corresponding o each paricular degree of wear, hen he value-loss due o consumpion can be exacly measured John R. Hicks (1939; 176). Curren cos accounans have also advocaed he use of second hand marke daa (when available) o calculae objecive depreciaion raes: Bu as a pracical maer he quanificaion and valuaion of asse services used is no a simple maer and we mus fall back on esimaed paerns as a basis for curren cos as well as hisoric cos depreciaion. For hose fixed asses which have acive second hand markes he problem is no overly difficul. A paern of service values can be obained a any ime by comparing he marke values of differen ages or degrees of use. The differences so obained, when relaed o he value of a new asse, yield he proporions of asse value which

13 11 Obviously, given he sequence of period vinage asse prices {P n }, we can use equaions (12) o deermine he period sequence of vinage cross secion depreciaion raes, {δ n }. Conversely, given he cross secion sequence of period depreciaion raes, {δ n }, as well as he price of a new asse in period, P 0, we can deermine he period vinage asse prices as follows: (13) P 1 = (1 δ 0 ) P 0 P 2 = (1 δ 0 )(1 δ 1 ) P 0 P n = (1 δ 0 )(1 δ 1 ) (1 δ n 1 ) P 0 ; The inerpreaion of equaions (13) is sraighforward. A he beginning of period, a new capial good is worh P 0. An asse of he same ype bu which is one period older a he beginning of period is less valuable by he amoun of depreciaion δ 0 P 0 and hence is worh (1 δ 0 ) P 0, which is equal o P 1. An asse which is wo periods old a he beginning of period is less valuable han a one period old asse by he amoun of depreciaion δ 1 P 1 and hence is worh P 2 = (1 δ 1 ) P 1 which is equal o (1 δ 1 )(1 δ 0 ) P 0 using he firs equaion in (13) and so on. Suppose L 1 is he firs ineger which is such ha δ L 1 is equal o one. Then P n equals zero for all n L; i.e., a he end of L periods of use, he asse no longer has a posiive renal value. If L = 1, hen a new asse of his ype delivers all of is services in he firs period of use and he asse is in fac a nondurable asse. Now subsiue equaions (12) ino equaions (9) in order o obain he following formulae for he sequence of he end of he period vinage user coss {u n } in erms of he price of a new asse a he beginning of period, P 0, and he sequence of cross secion depreciaion raes, {δ n }: (14) u 0 = [(1+r ) (1+i )(1 δ 0 )] P 0 u 1 = (1 δ 0 )[(1+r ) (1+i )(1 δ 1 )] P 0 u n = (1 δ 0 ) (1 δ n 1 )[(1+r ) (1+i )(1 δ n )] P 0 ; Thus given P 0 (he beginning of period price of a new asse), i (he new asse inflaion rae ha is expeced a he beginning of period ), r (he one period nominal ineres rae ha he business uni faces a he beginning of period ) and given he sequence of cross secion vinage depreciaion raes prevailing a he beginning of period (he δ n ), hen we can use equaions (14) o calculae he sequence of vinage end of he period user coss for period, he u n. Of course, given he u n, we can use equaions (8) o calculae he beginning of he period user coss (he f n ) and hen use he f n o calculae he sequence of vinage asse prices P n using equaions (5) and finally, given he P n, we can use equaions (12) in order o calculae he sequence of vinage depreciaion raes, he δ n. are normally used up or foregone in he various sages of asse life. Edgar O. Edwards and Philip W. Bell (1961; 175).

14 12 Thus given any one of hese sequences or profiles, all of he oher sequences are compleely deermined. This means ha assumpions abou depreciaion raes, he paern of vinage user coss or he paern of vinage asse prices canno be made independenly of each oher. 22 I is useful o look more closely a he firs equaion in (14), which expresses he user cos or renal price of a new asse a he beginning of period, u 0, in erms of he depreciaion rae δ 0, he one period nominal ineres rae r, he new asse inflaion rae i ha is expeced o prevail a he beginning of period and he beginning of period price for a new asse, P 0 : (15) u 0 = [(1+r ) (1+i )(1 δ 0 )] P 0 = [r i + (1+ i )δ 0 ] P 0. Thus he user cos of a new asse u 0 ha is purchased a he beginning of period (and he acual or impued renal paymen is made a he end of he period) is equal o r i (a nominal ineres rae minus an asse inflaion rae which can be loosely inerpreed 23 as a real ineres rae) imes he iniial asse cos P 0 plus (1+ i )δ 0 P 0 which is depreciaion on he asse a beginning of he period prices, δ 1 P 0, imes he inflaion escalaion facor, (1+ i ). 24 If we furher assume ha he expeced asse inflaion rae is 0, hen (15) furher simplifies o: (16) u 0 = [r + δ 0 ] P 0. Under hese assumpions, he user cos of a new asse is equal o he ineres rae plus he depreciaion rae imes he iniial purchase price. 25 This is essenially he user cos formula ha was obained by Walras in 1874: Le P be he price of a capial good. Le p be is gross income, ha is, he price of is service inclusive of boh he depreciaion charge and he insurance premium. Le µp be he porion of his income represening he depreciaion charge and νp he porion represening he insurance premium. Wha remains of he gross income afer boh charges have been deduced, π = p (µ+ν)p, is he ne income. 22 This poin was firs made explicily by Jorgenson and Griliches (1967; 257): An almos universal concepual error in he measuremen of capial inpu is o confuse he aggregaion of capial sock wih he aggregaion of capial service. See also Jorgenson and Griliches (1972; 81-87). Much of he above algebra for swiching from one mehod of represening vinage capial inpus o anoher was firs developed by Chrisensen and Jorgenson (1969; ) (1973) for he geomerically declining depreciaion model. The general framework for an inernally consisen reamen of capial services and capial socks in a se of vinage accouns was se ou by Jorgenson (1989) and Hulen (1990; ) (1996; ). 23 We will provide a more precise definiion of a real ineres rae laer. 24 This formula was obained by Chrisensen and Jorgenson (1969; 302) for he geomeric model of depreciaion bu i is valid for any depreciaion model. Griliches (1963; 120) also came very close o deriving his formula in words: In a perfecly compeiive world he annual ren of a machine would equal he marginal produc of is services. The ren iself would be deermined by he ineres coss on he invesmen, he deerioraion in he fuure produciviy of he machine due o curren use, and he expeced change in he price of he machine (obsolescence). 25 Using equaions (13) and (14) and he assumpion ha he asse inflaion rae i = 0, i can be shown ha he user cos of an asse ha is n periods old a he sar of period can be wrien as u n = (r + δ n )P n where P n is he beginning of period second hand marke price for he asse.

15 13 We are now able o explain he differences in gross incomes derived from various capial goods having he same value, or conversely, he differences in values of various capial goods yielding he same gross incomes. I is, however, readily seen ha he values of capial goods are rigorously proporional o heir ne incomes. A leas his would have o be so under cerain normal and ideal condiions when he marke for capial goods is in equilibrium. Under equilibrium condiions he raio [p (µ+ν)p]/p, or he rae of ne income, is he same for all capial goods. Le i be his common raio. When we deermine i, we also deermine he prices of all landed capial, personal capial and capial goods proper by virue of he equaion p (µ+ν)p = ip or P = p/[i+µ+ν]. Léon Walras (1954; ). However, he basic idea ha a durable inpu should be charged a period price ha is equal o a depreciaion erm plus a erm ha would cover he cos of financial capial goes back much furher 26. For example, consider he following quoaion from Babbage: Machines are, in some rades, le ou o hire, and a cerain sum is paid for heir use, in he manner of ren. This is he case amongs he frame-work kniers: and Mr. Hensen, in speaking of he rae of paymen for he use of heir frames, saes, ha he proprieor receives such a ren ha, besides paying he full ineres for his capial, he clears he value of his frame in nine years. When he rapidiy wih which improvemens succeed each oher is considered, his ren does no appear exorbian. Some of hese frames have been worked for hireen years wih lile or no repair. Charles Babbage (1835; 287). Babbage did no proceed furher wih he user cos idea. Walras seems o have been he firs economis who formalized he idea of a user cos ino a mahemaical formula. However, he early indusrial engineering lieraure also independenly came up wih he user cos idea; Church described how he use of a machine should be charged as follows: No sophisry is needed o assume ha hese charges are in he naure of rens, for i migh easily happen ha in a cerain building a number of separae lile shops were esablished, each conaining one machine, all making some paricular par or working on some paricular operaion of he same class of goods, bu each shop occupied, no by a wage earner, bu by an independen mechanic, who rened his space, power and machinery, and sold he finished produc o he lessor. Now in such a case, wha would be he shop charges of hese mechanics? Clearly hey would comprise as heir chief if no heir only iem, jus he ren paid. And his ren would be made up of: (1) Ineres. (2) Depreciaion. (3) Insurance. (4) Profi on he capial involved in he building, machine and power-ransmiing and generaing plan. There would also mos probably be a separae charge for power according o he quaniy consumed. Exclude he iem of profi, which is no included in he case of a shop charge, and we find ha we have approached mos closely o he new plan of reducing any shop ino is consiuen producion cenres. No one would preend ha here was any insuperable difficuly involved in fixing a jus ren for lile shops le ou in his plan. A. Hamilon Church (1901; ). A producion cenre is, of course, eiher a mechanic, or a bench a which a hand crafsman works. Each of hese is in he posiion of a lile shop carrying on one lile special indusry, paying ren for he floor space occupied, ineres for he capial involved, depreciaion for he wear and ear, and so on, quie independenly of wha may be paid by oher producion cenres in he same shop. A. Hamilon Church (1901; 734). Church was well aware of he imporance of deermining he righ rae o be charged for he use of a machine in a muliproduc enerprise. This informaion is required no only o price producs appropriaely bu o deermine wheher an enerprise should make 26 Solomons (1968; 9-17) indicaes ha ineres was regarded as a cos for a durable inpu in much of he nineeenh cenury accouning lieraure. The influenial book by Garcke and Fells (1893) changed his.

16 14 or purchase a paricular commodiy. Babbage and Canning were also aware of he imporance of deermining he righ machine rae charge: 27 The grea compeiion inroduced by machinery, and he applicaion of he principle of he subdivision of labour, render i necessary for each producer o be coninually on he wach, o discover improved mehods by which he cos of he aricle he manufacures may be reduced; and, wih his view, i is of grea imporance o know he precise expense of every process, as well as of he wear and ear of machinery which is due o i. Charles Babbage (1835; 203). The quesion of adequae raes of depreciaion, in he sense ha hey will ulimaely adjus he valuaions o he realiies, is ofen discussed as hough i had no effec upon ulimae profi a all. Of some modes of valuing, i is said ha hey end o overvalue some asses and o undervalue ohers, bu he aggregae of book values found is nearly righ. If he managemen pay no aenion a all o he uni coss implied in such valuaions, no harm is done. Bu if he cos accounan gives effec o hese individually bad valuaions hrough a machine-rae burden charge, and if he selling policy has regard for apparen uni profis, he valuaion may lead o he wors raher han o he bes possible policy. John B. Canning (1929; ). 27 Under moderae inflaion, he difficulies wih radiional cos accouning based on hisorical cos and no proper allowance for he opporuniy of capial, he proper pricing of producs becomes very difficul. Diewer and Fox (1999; ) argued ha his facor conribued o he grea produciviy slowdown ha sared around 1973 and persised o he early 1990 s. The radiional mehod of cos accouning can be raced back o a book firs published in 1887 by he English accounans, Garcke and Fells, who suggesed allocaing he indirec coss of producing a good proporionally o he amoun of labour and maerials coss used o make he iem: In some esablishmens he direc expendiures in wages and maerials only is considered o consiue he cos; and no aemp is made o allocae o he various working or sock orders any porion of he indirec expenses. Under his sysem he difference beween he sum of he wages and maerials expended on he aricles and heir selling price consiues he gross profi, which is carried in he aggregae o he credi of profi and loss, he indirec facory expenses already referred o, ogeher wih he esablishmen expenses and depreciaion, being paricularised on he debi side of ha accoun. This mehod has cerainly simpliciy in is favour, bu a more efficien check upon he indirec expenses would be obained by esablishing a relaion beween hem and he direc expenses. This may be done by disribuing all he indirec expenses, such as wages of foremen, ren of facory, fuel, lighing, heaing, and cleaning, ec. (bu no he salaries of clerks, office ren, saionery and oher esablishmen charges o be referred o laer), over he various jobs, as a percenage, eiher upon he wages expended upon he jobs respecively, or upon he cos of boh wages and maerials. Emile Garcke and John Manger Fells (1893; 70-71). Compare his raher crude approach o cos accouning o he maserful analysis of Church! Garcke and Fells endorsed he idea ha deprecaion was an admissible iem of cos ha should be allocaed in proporion o he prime cos (i.e., labour and maerials cos) of manufacuring an aricle bu hey explicily ruled ou ineres as a cos: The iem of Depreciaion may, for he purpose of aking ou he cos, simply be included in he caegory of he indirec expenses of he facory, and be disribued over he various enerprises in he same way as hose expenses may be allocaed; or i may be deal wih separaely and more correcly in he manner already alluded o and hereafer o be fully described. The esablishmen expenses and ineres on capial should no, however, in any case form par of he cos of producion. There is no advanage in disribuing hese iems over he various ransacions or aricles produced. They do no vary proporionaely wih he volume of business. The esablishmen charges are, in he aggregae, more or less consan, while he manufacuring coss flucuae wih he cos of labour and he price of maerial. To disribue he charges over he aricles manufacured would, herefore, have he effec of disproporionaely reducing he cos of producion wih every increase, and he reverse wih every diminuion, of business. Such a resul is grealy o be deprecaed, as ending o neiher economy of managemen nor o accuracy in esimaing for conracs. The principles of a business can always judge wha percenage of gross profi upon cos is necessary o cover fixed esablishmen charges and ineres on capial. Emile Garcke and John Manger Fells (1893; 72-73). The aversion of accounans o include ineres as a cos can be raced back o his quoaion.

17 15 The above equaions relaing vinage asse prices P n, beginning of he period vinage user coss f n, end of he period vinage user coss u n and he (cross secion) vinage depreciaion raes δ n are he fundamenal ones ha we will specialize in subsequen secions in order o measure boh wealh capial socks and capial services under condiions of inflaion. In he following secion, we shall consider several opions ha could be used in order o deermine empirically he ineres raes r and he asse inflaion raes i. 4. The Empirical Deerminaion of Ineres Raes and Asse Inflaion Raes We consider iniially hree broad approaches 28 o he deerminaion of he nominal ineres rae r ha is o be used o discoun fuure period value flows by he business unis in he aggregae under consideraion: Use he ex pos rae of reurn ha will jus make he sum of he user coss exhaus he gross operaing surplus of he producion secors in he aggregae under consideraion. Use an aggregae of nominal ineres raes ha he producion secors in he aggregae migh be facing a he beginning of each period. Take a fixed real ineres rae and add o i acual ex pos consumer price inflaion or anicipaed consumer price inflaion. The firs approach was used for he enire privae producion secor of he economy by Jorgenson and Griliches (1967; 267) and for various secors of he economy by Chrisensen and Jorgenson (1969; 307). I is also widely used by saisical agencies. I has he advanage ha he value of oupu for he secor will exacly equal he value of inpu in a consisen accouning framework. I has he disadvanages ha i is subjec o measuremen error and i is an ex pos rae of reurn which may no reflec he economic condiions facing producers a he beginning of he period. The second approach suffers from aggregaion problems. There are many ineres raes in an economy a he beginning of an accouning period and he problem of finding he righ aggregae of hese raes is no a rivial one. The hird approach works as follows. Le he consumer price index for he economy a he beginning of period be c say. Then he ex pos general consumer inflaion rae for period is ρ defined as: (17) 1 + ρ c +1 /c. 28 Oher mehods for deermining he appropriae ineres raes ha should be insered ino user cos formulae are discussed by Harper, Bernd and Wood (1989) and above in Chaper 8 of Blades (2001) and in Chaper 5 of Schreyer (2001). Harper, Bernd and Wood (1989) evaluae empirically 5 alernaive renal price formulae using geomeric depreciaion bu making differen assumpions abou he ineres rae and he reamen of asse price change. They show ha he choice of formula maers (as we do laer).