NBER WORKING PAPER SERIES MEASURING CAPITAL. W. Erwin Diewert. Working Paper

Transcription

1 NBER WORKING PAPER SERIES MEASURING CAPITAL W. Erwin Diewer Working Paper 9526 hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA February 2003 Forhcoming chaper in Measuring Capial in he New Economy, edied by Carol Corrado, John Haiwanger and Dan Sichel. The auhor is indebed o Kevin Fox, Peer Hill, Ulrich Kohli, Alice Nakamura and Paul Schreyer for helpful commens. The views expressed herein are hose of he auhor and no necessarily hose of he Naional Bureau of Economic Research by W. Erwin Diewer. All righs reserved. Shor secions of ex no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi including noice, is given o he source.

2 Measuring Capial W. Erwin Diewer NBER Working Paper No February 2003 JEL No. C43, C82, D24 ABSTRACT 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. W. Erwin Diewer Deparmen of Economics Universiy of Briish Columbia Vancouver, Canada, V6N 1Z1 and NBER diewer@econ.ubc.ca

3 1 MEASURING CAPITAL by W. Erwin Diewer April 18, TABLE OF CONTENTS 1. Inroducion 2. Inflaion and he Measuremen of Economic Aciviy 3. The Fundamenal Equaions Relaing Socks and Flows of Capial 4. Cross Secion Depreciaion Profiles 5. The Empirical Deerminaion of Ineres Raes and Asse Inflaion Raes 6. Obsolescence and Depreciaion 7. Aggregaion over Vinages of a Capial Good 8. The One Hoss Shay Model of Efficiency and Depreciaion 9. The Sraigh Line Depreciaion Model 10. The Declining Balance or Geomeric Depreciaion Model 11. The Linear Efficiency Decline Model 12. The Linearly Increasing Mainenance Expendiures Model 13. A Comparison of he Twelve Models 14. 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 paper, we discuss some of he problems involved in consrucing price and quaniy series for boh capial socks and he associaed flows of services when here is general (and specific) price change in he economy. In secion 2, we discuss some of he problems ha occur when an economy is experiencing very high inflaion. Under hese condiions, i will be necessary for he naional price saisician o shoren he accouning period (or give up price measuremen alogeher). 1 The auhor is indebed o Kevin Fox, Peer Hill, Ulrich Kohli, Alice Nakamura and Paul Schreyer for helpful commens.

4 2 In secion 3, 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 4 coninues he heoreical framework ha was inroduced in secion 3. 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 5 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 6 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 7 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 8 o 11 show how he general algebra presened in secions 3 and 4 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 12 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 11. However, he main use of he analysis presened in secion 12 is o sugges a reason why acceleraed depreciaion assumpions are quie reasonable and likely o occur empirically. Secion 13 presens a graphical summary of he 12 empirical models considered in he Annex. 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 14 concludes. A daa appendix liss he Canadian daa we used in secions Inflaion and he Measuremen of Economic Aciviy Our goal in his paper is wofold: (1) o measure he price and quaniy of he sock of reproducible capial held by a producion uni (an esablishmen, a firm, an indusry or an enire economy) a a poin in ime and (2) o measure he price and quaniy of he flow of reproducible capial services uilized by a producion uni over a period of ime. In paricular, we wan o exend he procedures for measuring capial socks and flows o cover siuaions where here is general price level change or inflaion. In his secion, we shall review some of he general measuremen problems ha arise when inflaion is high. When capial flows are measured, he normal period of ime is eiher a year or a quarer. Under condiions of high inflaion, he aggregaion of homogeneous commodiy flows wihin a quarer or a year is complicaed by he fac ha he wihin period ransacions are valued a very differen prices. The recen naional income accouning lieraure explains he problem as follows: Convenional index number heory is mosly concerned wih comparisons beween poins of ime whereas, in naional accouns, price and quaniy comparisons have o be made beween discree periods of ime. Significan changes in price and quaniy flows may occur no only beween differen periods bu also wihin a single accouning period, especially one as long as a year. Indeed, he cenral problem of accouning under high inflaion is ha prices are much higher a he end of he accouning period han a he beginning. Peer Hill (1996; 11). The underlying problem is no a radiional index number problem. I sems from he use of curren value daa as inpus ino he calculaion of indirec price or quaniy measures under high inflaion. Curren accouns permi idenical quaniies of he same homogeneous produc o be valued a very differen prices during he course of he same year. Implicily, quaniies sold a higher prices laer in he year are reaed as if hey were superior qualiies when hey are no. Peer Hill (1996; 12). Under high inflaion, he moneary value of flows of goods and services a differen poins of ime wihin he same accouning period are no commensurae wih each oher because he uni of currency used as he numeraire is no sable. Adding ogeher differen quaniies of he same good valued a differen prices is equivalen, from a scienific poin of view, o using differen unis of measuremen for differen ses of observaions on he same variable. In he case of physical daa, however, i is raher more obvious ha adding quaniies measured in grams o quaniies measured in ounces is a fuile procedure. Peer Hill (1996; 32). Before he preparaion of he 1993 SNA, issues conneced wih high or significan inflaion had no been deal wih a all in inernaional recommendaions concerning naional accouns. Uneasiness especially wih he recording of nominal ineres had been ofen expressed, for insance in Europe and Norh America a he ime of wo digi inflaion and above all in counries, like in Lain America, experiencing high or hyper inflaion. In relaion wih he laer siuaions, uneasiness exended o he whole se of accouns, because, due o he significan rae of inflaion wihin each year, annual accouns in curren values could no longer be deemed homogeneous as regards he level of prices in each year. They combine inra-annual flows ha are valued a very differen prices and are no, sricly speaking, addiive. The effec of he inraannual change in he general price level can be negleced for he sake of simpliciy only when he rae of inflaion is low. When i is high, he meaning of annual accouns in curren values becomes fuzzy. André Vanoli (1998).

6 4 When inflaion is high, he aggregaion of flows from differen periods becomes very much a case of adding apples and bananas he flows a he end of he period will carry a much greaer weigh han he flows a he beginning of he period, so ha he change on average will reflec developmen a he end of he period disproporionaely. Annual naional accouns a curren prices become virually meaningless and compuaion of naional accouns a consan prices becomes very problemaic. Ezra Hadar and Soli Peleg (1998; 2). Of course, concern over he effecs of general price level change has a much longer hisory in he general cos accouning lieraure; see Baxer (1984), Tweedie and Whiingon (1984) and Whiingon (1992) for example. 3 We now discuss in more deail he accouning problems caused by high inflaion ha are referred o in he above quoaions. The basic problem is his: all discree ime economic heories and mos of index number heory assumes ha all of he ransacions of a producion uni in a homogeneous commodiy wihin he accouning period can be represened by a single price and a single quaniy. I is naural o le he single quaniy be he sum of he quaniies sold (in he case of an oupu) or he sum of he quaniies purchased (in he case of an inpu). Bu hen, if we wan he single price imes he single quaniy o equal he value of ransacions for he commodiy in he period, he single price mus equal he value of ransacions divided by he sum of quaniies purchased or sold; i.e., he single price mus equal a uni value. 4 Bu when here is subsanial inflaion wihin he accouning period, uni values give a much higher weigh o ransacions ha occur near he end of he period compared o ransacions ha occurred near he beginning; i is as if he end of period ransacions are being arificially qualiy adjused o be more valuable han he beginning of he period ransacions. The obvious soluion o his arificial implici weighing problem is o choose he accouning period o be small enough so ha he general inflaion wihin he period is small enough o be ignored. This is precisely he soluion suggesed by he index number heoris Fisher 5 and he measuremen economis Hicks: he lengh of he accouning period should be he Hicksian week : 3 The inflaion accouning lieraure exends back o Middledich: Today s dollar is, hen, a oally differen uni from he dollar of As he general price level flucuaes, he dollar is bound o become a uni of differen magniude. To mix hese unis is like mixing inches and cenimeers or measuring a field wih a rubber ape-line. Livingson Middledich (1918; ). 4 The early index number heoriss Walsh (1901; 96) (1921; 88), Fisher (1922; 318) and Davies (1924; 96) all suggesed uni values as he prices ha should be insered ino a bilaeral index number formula. Walsh nicely sums up he case for uni values as follows: Some nice quesions arise as o wheher only wha is consumed in he counry, or only wha is produced in i, or boh ogeher are o be couned; and also here are difficulies as o he single price quoaion ha is o be given a each period o each commodiy, since his, oo, mus be an average. Throughou he counry during he period a commodiy is no sold a one price, nor even a one wholesale price in is principle marke. Various quaniies of i are sold a differen prices, and he full value is obained by adding all he sums spen (a he same sage in is advance owards he consumer), and he average price is found by dividing he oal sum (or he full value) by he oal quaniies. Correa Moylan Walsh (1921; 88). 5 Essenially he same problem eners, however, whenever, as is usually he case, he daa for prices and quaniies wih which we sar are averages insead of being he original marke quoaions. Throughou his book, he price of any commodiy or he quaniy of i for any one year was assumed given. Bu wha is such a price or quaniy? Someimes i is a single quoaion for January 1 or July 1, bu usually i is

7 5 I shall define a week as ha period of ime during which variaions in price can be negleced. John R. Hicks (1946; 122). Thus i seems ha here is a simple soluion o he problem of consrucing meaningful accouning period prices and quaniies for homogeneous commodiies when here is high inflaion: simply shoren he accouning period! Hill (1996) however noes ha here are a leas hree classes of problems associaed wih he above soluion: In order o keep hese issues in perspecive, i is useful o summarise he problems creaed by coninually shorening he accouning period. 1. The compilaion of accouns for shorer ime periods requires more informaion abou he imes a which various ransacions ake place. Enquiries may have o be conduced more frequenly hereby creaing addiional coss for he daa collecors. More burdens are also placed on he respondens supplying he informaion. In many cases, hey may be unable o supply he necessary informaion because heir own inernal records and accouns do no permi hem o do so, especially when hey radiionally repor heir accouns for longer ime periods, such as a year. 2. As producion is a process which can exend over a considerable period of ime, is measuremen becomes progressively more difficul he shorer he accouning period. The problem is no confined o agriculure or foresry where many producion processes ake a year or more. The producion of large fixed asses such as large ships, bridges, power saions, dams or he like can exend over several years. The oupu produced over shorer periods of ime hen has o be measured on he basis of work in progress compleed each period. 3. Because many ransacions, especially large ransacions, are no compleed wihin he day, here are ypically many receivables and payables ousanding a any given momen of ime. They assume greaer imporance in relaion o he flows as he accouning period is reduced. This makes i more difficul o reconcile he values of differen flows in he accouns, especially if he wo paries o he ransacion perceive i as aking place a differen imes from each oher and do no record i in he same way required by he sysem. Peer Hill (1996; 34-35). Thus shorening he accouning period leads o increased coss for he saisical agency and he businesses being surveyed. Moreover, firm accouning is geared o years and quarers and i may no be possible for producion unis o provide complee accouning informaion for periods shorer han a quarer. As he accouning period becomes shorer, i is less likely ha producion, shipmen, billing and paymen for he same commodiy will all coincide wihin he accouning period. Also as he accouning period becomes shorer, work in progress will end o become ever more imporan relaive o final sales, creaing difficul valuaion problems. 6 Pu anoher way, more and more inpus will shif from being inermediae inpus (inpus ha are used up wihin he accouning period) o being durable inpus (inpus whose conribuion o producion exends over more han an average of several quoaions scaered hrough he year. The quesion arises: On wha principle should his average be consruced? The pracical answer is any kind of average since, ordinarily, he variaions during a year, so far, a leas, as prices are concerned, are oo lile o make any percepible difference in he resul, whaever kind of average is used. Oherwise, here would be ground for subdividing he year ino quarers or monhs unil we reach a small enough period o be considered pracically a poin. Irving Fisher (1922; 318). 6 There are very few price indexes for work in progress! This is o be expeced since here are very few ransacions involving parially compleed producs.

8 6 one period). In addiion o hese difficulies, here are ohers. For example, as he accouning period becomes shorer, ransacions end o become more erraic and sporadic. Many goods will no be sold in a supermarke in a paricular day or week. Normal index number heory breaks down under hese condiions: i is difficul o compare a posiive amoun of a good sold in one period wih a zero amoun sold in he nex period. A relaed difficuly is ha many commodiies are produced or demanded on a seasonal basis. If he accouning period is a year, hen here are no seasonal commodiy difficulies bu as we shoren he period from a year, we will run ino he problem of seasonal flucuaions in prices and quaniies. In many cases, a seasonal commodiy will no be available in all seasons and we again run ino he problem of comparing posiive values wih zero values in he periods when he commodiy is ou of season. Even if he seasonal commodiy does no disappear, he applicaion of sandard index number heory is no sraighforward. 7 Neverheless, even in he face of he above difficulies, i seems ha he only possible soluion o he arificial implici weighing problem ha is generaed by high inflaion is o shoren he accouning period so ha normal index number heory can be applied in order o consruc meaningful economic aggregaes. 8 In addiion o he above general problems associaed wih economic measuremen of flow variables under condiions of high inflaion, here are some addiional problems associaed wih he measuremen of capial. These addiional problems are associaed wih he sock and flow aspecs of capial. We will conclude his secion by explaining hese problems. Given an accouning period of some predeermined lengh, we can associae wih i a leas hree separae poins in ime: The beginning of he accouning period; The middle of he accouning period; and The end of he accouning period. In inerpreing he naional accouns or he accouns of a business uni, we generally hink of all flow variables as being concenraed in he middle of he period. If we follow his convenion in he conex of high inflaion, hen we require one (nominal) ineres rae o index he value of money or financial capial going from he beginning of he period o he middle of he period and we require anoher (nominal) ineres rae o index he value of money going from he middle of he period o he end of he period. Given hese wo ineres raes, we could consruc cenered user coss of capial for each ype of 7 Hadar and Peleg (1998; 5) commen on he imporance of seasonal adjusmen procedures in he conex of high inflaion: As a by-produc of he emphasis on quarerly esimaes a consan prices he seasonal adjusmen go large aenion and many resources were spen o improve he adjusmen. Diewer (1996) (1998) (1999) reviews possible approaches o he problems involved in reaing seasonal commodiies (and suggess soluions) when here is high inflaion. 8 Our discussion in he previous paragraph indicaes ha his canno be done if he economy is experiencing a hyperinflaion. Thus meaningful economic measuremen becomes impossible under very high inflaion. This is a hidden cos of inflaion ha is no discussed very much in he lieraure on he coss of inflaion.

9 7 reproducible capial, which would be he appropriae flow variables ha would mach up wih he oher flow variables in he producion accouns of he producion uni. However, in order o reduce he noaional complexiy of his annex, we do no consruc cenered user coss in wha follows. Insead, for each ype of asse, we consruc eiher a beginning of he period user cos (which measures he cos of using he asse for he period under consideraion from he perspecive of he price level prevailing a he beginning of he period) or an end of he period user cos (which measures he cos of using he asse for he period under consideraion from he perspecive of he price level prevailing a he end of he period). Of course, armed wih a knowledge of he appropriae half period ineres raes, i is easy o conver hese bookend user coss ino cenered user coss. In he following secion, we explain he fundamenal equaions relaing socks and flows of capial. 3. 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. 9 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 9 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.

10 8 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; ). In a noninflaionary environmen, 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 yield. Thus he sock value of he asse is se equal o he discouned fuure service flows 10 ha he asse is expeced o yield in fuure periods. Le he price of a new capial inpu purchased a he beginning of period be P 0. In a noninflaionary environmen, i can be assumed ha he (poenially observable) sequence of (cross secional) vinage renal prices prevailing a he beginning of period can be expeced o prevail in fuure periods. Thus here was no need o have a separae noaion for fuure expeced renal prices for a new asse as i ages. However, 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. 11 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, } 12 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, 13 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 10 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. 11 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. 12 The sequence of (cross secional) vinage renal prices {f n } is called he age-efficiency profile of he asse. 13 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.

11 9 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. 14 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. 15 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 14 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. 15 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.

12 10 which is one period old a he sar of period is characerized by he vecor f 1, f 2,, ec. 16 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: (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 16 Triple (1996; 97) used his characerizaion for capial asses of various vinages.

13 11 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 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 %. 17 This is a special case of wha has come o be known as he one hoss shay model and we shall consider i in more deail in secion 8 of his below. Noe ha equaions (5) can be rewrien as follows: 18 (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 ; 17 Böhm-Bawerk (1891; 343) wen on and consruced he sequence of vinage asse prices using his special case of equaions (5). 18 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.

14 12 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, 19 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. 20 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 = (1+r ) 1 [P 0 (1+r ) (1+i ) P 1 ] f 1 = P 1 [(1+i )/(1+r )] P 2 = (1+r ) 1 [P 1 (1+r ) (1+i ) P 2 ] f 2 = P 2 [(1+i )/(1+r )] P 3 = (1+r ) 1 [P 2 (1+r ) (1+i ) P 3 ] 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 21 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, 19 Noe ha we are implicily assuming ha he renal is paid o he owner a he beginning of period. 20 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. 21 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).

15 13 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. 22 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 ; 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. 23 However, more recenly, here is an accepance of an impued ineres charge for equiy capial as a genuine cos of producion. 24 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 22 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. 23 This lieraure is reviewed in Diewer and Fox (1999; ). 24 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.

16 14 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 7 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 paper is only a sar on he difficul problems associaed wih measuring capial inpu. 4. 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 n 25 of an asse ha is n periods old a he beginning of period as (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: 25 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 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. 26 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).

17 15 (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. 27 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 27 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 are normally used up or foregone in he various sages of asse life. Edgar O. Edwards and Philip W. Bell (1961; 175).