This PDF is a selection from a published volume from the National Bureau of Economic Research. Volume Title: Measuring Capital in the New Economy

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1 This PDF is a selecion from a published volume from he Naional Bureau of Economic Research Volume Tile: Measuring Capial in he New Economy Volume Auhor/Edior: Carol Corrado, John Haliwanger and Dan Sichel, ediors Volume Publisher: Universiy of Chicago Press Volume ISBN: Volume URL: hp:// Conference Dae: April 26-27, 2002 Publicaion Dae: Augus 2005 Tile: Issues in he Measuremen of Capial Services, Depreciaion, Asse Price Changes, and Ineres Raes Auhor: W. Erwin Diewer URL: hp:// Chaper pages in book: ( )

2 12 Issues in he Measuremen of Capial Services, Depreciaion, Asse Price Changes, and Ineres Raes W. Erwin Diewer 12.1 Inroducion In his chaper, 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 are general and asse-specific price changes in he economy. 1 In secion 12.2, 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, bu i is a framework ha will allow us o disenangle he effecs of general price change, asse-specific price change, and depreciaion. Secion 12.3 coninues he heoreical framework ha was inroduced in secion We show how informaion on vinage asse prices, vinage renal prices, and vinage depreciaion raes are all equivalen under cerain assumpions; ha is, knowledge of any one of hese hree sequences or profiles is sufficien o deermine he oher wo. Secion 12.4 discusses alernaive ses of assumpions on nominal ineres raes and anicipaed asse price changes. We specify hree differen ses W. Erwin Diewer is professor of economics a he Universiy of Briish Columbia and a research associae of he Naional Bureau of Economic Research. The auhor is indebed o Carol Corrado, Kevin Fox, John Haliwanger, Peer Hill, Ning Huang, Ulrich Kohli, Alice Nakamura, Paul Schreyer, Dan Sichel, and Frank Wykoff for helpful commens. This research was suppored by a SSHRC research gran. None of he paries acknowledged here are responsible for any errors or opinions expressed in he paper. A longer version of he presen paper, including he daa used, is available as Deparmen of Economics Discussion Paper 04-10, Universiy of Briish Columbia, Vancouver, Canada. 1. We cover some of he same issues discussed in he recen paper by R. J. Hill and Hill (2003). However, Hill and Hill did no deal wih he problems associaed wih adjusing nominal ineres raes for general inflaion. 479

3 480 W. Erwin Diewer of assumpions ha we will use in our empirical illusraion of he suggesed mehods. Secion 12.5 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 12.6 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 12.7 o show how he general algebra presened in secions 12.2 and 12.3 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 secion 12.11, we show how hese models differ empirically by compuing he corresponding socks and flows using Canadian daa on wo asse classes. The deails of he compuaions and he daa used may be found in Diewer (2004). Secion shows how our framework can be modified o model he reamen of some forms of inangible capial, such as invesmens in research and developmen. Secion concludes wih some observaions on how saisical agencies migh be able o use he maerial presened in his chaper 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 labor 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. 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 equal o he discouned fuure service flows 2 ha he asse is expeced 2. 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. Al-

4 Measuremen of Capial Services and Asse Price Changes 481 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, in his no-general-inflaion case, here is 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 a he beginning of period and fuure expeced renal prices for asses of various ages. 3 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 wo-period-old asse a he beginning of period, and so on. 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 renal prices for asses of age n prevailing a he beginning of period, { f n ; n 0, 1, 2,...} 4 is (1) P f 1 i 0 0 f (1 i 1 )(1 i ) f r (1 r )(1 r ) In he above equaion, 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 i 1 2 ) is he renal price escalaion facor ha is expeced o apply o a wo-period-old asse going from he beginning of period o he beginning of period 2, ec. Thus, he i n are expeced raes of price change for used asses of varying ages n 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 r 1 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, and so on. Thus, he r n are one-period nominal ineres raes ha represen he erm srucure of ineres raes a he beginning of period. 5 hough 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, p. 342) was able o derive. 3. Noe ha hese fuure expeced renal prices are no generally observable due o he lack of fuures markes for hese fuure period renals of he asses of varying ages. 4. The sequence of (cross-secional) vinage renal prices ( f n ) is called he age-efficiency profile of he asse. 5. 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 used versions of he same asse in he markeplace during he curren period, and so he cross-secional renal prices f for asses of age n in period will no exis for hese asses! In his case, he f n n should be

5 482 W. Erwin Diewer We now generalize equaion (1) o relae he sock value of an n-periodold 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 f 1 i n n f (1 i n 1 )(1 i ) f n r (1 r )(1 r ) Thus, older asses discoun fewer erms in he above sum; hus, as n increases by one, we have one less erm on he righ-hand side of equaion (2). However, noe ha we are applying he same price escalaion facors (1 i ), (1 i )(1 i ),..., o escalae he cross-secional renal prices prevailing a he beginning of period, f, f 1 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 renal prices prevailing a he beginning of period for asses of various ages, f, f,... are poenially 0 1 observable.6 These cross-secion renal prices reflec he relaive efficiency of he various vinages of he capial good ha are sill in use 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 asse sock prices for various ages a he beginning of period, P, P 0 1,... is no affeced by general inflaion provided ha he general inflaion affecs he expeced asse raes of price change i and he nominal ineres raes r n n in a proporional manner. We will reurn o his poin laer. The physical produciviy characerisics of a uni of capial of each age are deermined by he sequence of cross-secional renal prices. Thus, a brand-new asse is characerized by he vecor of curren renal prices for asses of various ages, f, f, f 0 1 2,..., which are inerpreed as physical conribuions o oupu ha he new asse is expeced o yield during he curren period (his is f ), he nex period (his is f 0 1 ), and so on. An asse ha is one period old a he sar of period is characerized by he vecor f, f 1 2,..., and so on.7 We have no explained how he expeced renal price raes of price 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 compleely objecive 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 P. Hill s (2000) more general model. In secion below, we apply a varian of he expeced renals inerpreaion of our equaions o value inangible capial. 6. This is he main reason ha we use his escalaion of cross-secional renal prices approach o capial measuremen raher han he more fundamenal discouned fuure expeced renals approach advocaed by Hill. 7. Triple (1996, p. 97) used his characerizaion for capial asses of various vinages.

6 Measuremen of Capial Services and Asse Price Changes 483 change i n are o be esimaed. We shall deal wih his problem in secion 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 raes of renal price change for fuure periods i n and he ineres raes r n. We assume ha hese anicipaed specific price change escalaion facors a he beginning of each period are all equal; ha is, we assume (3) i n i ; n 1, 2,... We also assume ha he erm srucure of (nominal) ineres raes a he beginning of each period is consan; ha is, 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 equaion (2), which relaes he sequence or profile of sock prices of age n a he beginning of period {P n } o he sequence or profile of (cross-secional) renal prices for asses of age n a he beginning of period { f n }, as follows: (5) P f 1 i 0 0 f 1 r 1 2 P f 1 i 1 1 f r P f 1 i 2 2 f r P f 1 i n n f n r 1 i 1 r 1 i 1 r 1 i 1 r f f f 1 i 2 1 r 3 1 i 3 1 r 3 1 i 4 1 r 3 f 3... f 4... f i f 1 i n 2 3 f n r 1 r On he lef-hand side of equaions (5), we have he sequence of period- asse prices by age 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 wo 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 renal prices by age of asse ha prevails in he marke (if such markes exis) a he beginning of period. Thus, f is he ren for a new asse, f 0 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 wo 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, and so on, and hen hese fuure expeced renals are discouned back o he beginning of period using he nominal discoun facors (1 r ), (1 r ) 2,

7 484 W. Erwin Diewer (1 r ) 3, and so on. Noe ha given he period- expeced asse inflaion rae i and he period- nominal discoun rae r, we can go from he (crosssecional) sequence of vinage renal prices { f n } o he (cross-secional) sequence of vinage asse prices {P n } using equaions (5). We shall show below how his procedure can be reversed; ha is, we shall show how, given he sequence of cross-secional asse prices, we can consruc esimaes for he sequence of cross-secional renal prices. Böhm-Bawerk (1891, p. 342) considered a special case of equaions (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 percen. 8 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 Noe ha equaions (5) can be rewrien as follows: 9 1 i 1 r (6) P 0 f 0 P 1 ; 1 i 1 r P 1 f 1 P 2 ; 1 i 1 r P 2 f 2 P 3 ;... 1 i 1 r P n f n P n 1 ;... The firs equaion in equaions (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 0,10 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 Böhm-Bawerk (1891, p. 343) wen on o consruc he sequence of vinage asse prices using his special case of equaions (5). 9. Chrisensen and Jorgenson (1969, p. 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, p. 10), Hulen (1990, p. 128) and Diewer and Lawrence (2000, p. 276). Irving Fisher (1908, pp ) also derived hese equaions in words. 10. Noe ha we are implicily assuming ha he renal is paid o he owner a he beginning of period. 11. Anoher way of inerpreing say he firs equaion in equaions (6) runs as follows: he purchase cos of a new asse P less he renal 0 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) (1 r 0 ) (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, p. 267).

8 Measuremen of Capial Services and Asse Price Changes 485 Now i is sraighforward o solve equaions (6) for he sequence of period- cross-secional renal prices, { f n }, in erms of he cross-secional asse prices, {P }: n 1 i 1 r (7) f 0 P 0 P 1 (1 r ) 1 [P 0 (1 r ) (1 i )P 1 ] 1 i 1 r f 1 P 1 P 2 (1 r ) 1 [P 1 (1 r ) (1 i )P 2 ] 1 i 1 r f 2 P 2 P 3 (1 r ) 1 [P 2 (1 r ) (1 i )P 3 ] 1 i 1 r... f n P n 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 renal prices by age n{ f } o he sequence of asse prices by age n{p n n }, while equaions (7) allow us o reverse he process. Equaions (7) can be derived from elemenary economic consideraions. Consider he firs equaion in equaions (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 expeced ne cos 12 of using he new asse for period is P [(1 i 0 )/(1 r )] P. 1 The above equaions assume ha he acual or implici period renal paymens f n for asses of differen ages 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 endof-period- 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-ofperiod- renal price f n as follows: (8) u (1 r n ) 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-period user cos u. n 12. This explains why he renal prices f n are someimes called user coss. This derivaion of a user cos was used by Diewer (1974, p. 504; 1980, pp ; 1992b, p. 194) and by Hulen (1996, p. 155).

9 486 W. Erwin Diewer 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 asse prices by age {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-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-period benefi (1 i )P 1 is he iniial purchase cos and opporuniy ineres cos of he asse purchase, P (1 r 0 ), leading o an end-of-period- ne cos of P (1 r 0 ) (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. 13 However, more recenly here is an accepance of an impued ineres charge for equiy capial as a genuine cos of producion. 14 In he following secion, we will relae he asse price profiles {P } and n 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 12.7). We have no explained wha should be done if he sequence of secondhand asse prices {P n } is no available and he sequences of vinage renal prices or user coss, { f } or {u n 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 oward risk 13. This lieraure is reviewed in Diewer and Fox (1999, pp ). 14. Sern Sewar and Co. has popularized he idea of charging for he opporuniy cos of equiy capial and has called he resuling income concep economic value added (EVA).

10 Measuremen of Capial Services and Asse Price Changes 487 on he par of producers. Thus, he analysis presened in his chaper is only a sar on he difficul problems associaed wih measuring capial inpu 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 n15 of an asse ha is n periods old a he beginning of period as (10) D P P n n 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 1 Obviously, given he sequence of period- cross-secion asse prices {P n }, we can use equaions (10) o deermine he period- sequence of declines in asse values by age, {D n }. Conversely, given he period- crosssecion depreciaion sequence or profile, {D n }, we can deermine he period- asse prices by age n by adding up amouns of depreciaion: (11) P D D D P D D D P D D D n n n 1 n 2... Raher han working wih firs differences of asse prices by age, i is more convenien o reparameerize he paern of cross-secion depreciaion by defining he period- depreciaion rae n for an asse ha is n periods old a he sar of period as follows: (12) 1 P n n 1 P n D n ; n 0, 1, 2,... P n In he above definiions, we require n o be such ha P n is posiive.16 Obviously, given he sequence of period- asse prices by age n, {P n }, we 15. This erminology is due o P. Hill (1999), who disinguished he decline in secondhand 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, pp ) uses he cross-secion definiion of depreciaion (calling i deerioraion) 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 price change. We will examine he relaionship of cross-secion o ime series depreciaion in secion 12.5 below. 16. This definiion of depreciaion daes back o Hicks (1939, p. 176) a leas and was used exensively by Edwards and Bell (1961, p. 175), Hulen and Wykoff (1981a,b), who call i deerioraion, Diewer (1974, p. 504), and Hulen (1990, p. 128; 1996, p. 155).

11 488 W. Erwin Diewer can use equaions (12) o deermine he period- sequence of cross-secion depreciaion raes, { n }. Conversely, given he cross-secion sequence of period- depreciaion raes, { n }, as well as he price of a new asse in period, P 0, we can deermine he period- asse prices by age as follows: (13) P 1 (1 0 )P 0 P (1 2 )(1 0 1 )P 0... P n (1 0 )(1 1 )... (1 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 ha is one period older a he beginning of period is less valuable by he amoun of depreciaion P and hence is worh (1 0 0 )P 0 0, which is equal o P 1. An asse ha is wo periods old a he beginning of period is less valuable han a one-period-old asse by he amoun of depreciaion P 1 1 and hence is worh P (1 2 )P, which is equal o (1 1 1 )(1 1 )P using 0 0 he firs equaion in equaions (13), and so on. Suppose L 1 is he firs ineger ha is such ha is equal o one. Then P L 1 n equals zero for all n L; ha is, 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 formulas for he sequence of he end-of-period- user coss by age n, {u }, in erms of he price of a new asse a he beginning of period, P, n 0 and he sequence of cross-secion depreciaion raes, { }: n (14) u 0 [(1 r ) (1 i )(1 0 )] P 0 u (1 1 )[(1 r 0 ) (1 i )(1 )] P 1 0 u (1 n )... (1 0 )[(1 r n 1 ) (1 i )(1 )] P ;... n 0 Thus, given P (he beginning-of-period- price of a new asse), i 0 (he nominal rae of new asse price change 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 n ), hen we can use equaions (14) o calculae he sequence of he end-of-period user coss for period, he u. Of course, given he u n n, we can use equaions (8) o calculae he beginning-of-period user coss (he f n ) and hen use he f o calculae he sequence of asse prices by age P n n using equaions (5), and finally, given he P n, we can use equaions (12) in order o calculae he sequence of depreciaion raes for asses of age n a he beginning of period

12 Measuremen of Capial Services and Asse Price Changes 489, he n. 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 user coss by age of asse, or he paern of asse prices by age of asse canno be made independenly of each oher. 17 I is useful o look more closely a he firs equaion in equaions (14), which expresses he user cos or renal price of a new asse a he end of period, u, in erms of he depreciaion rae 0 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 0 )] P 0 [r i (1 i ) 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 ha can be loosely inerpreed 18 as a real ineres rae) imes he iniial asse cos P plus (1 i 0 ) P 0 0, which is depreciaion on he asse a beginning-of-period prices, P 1 0, imes one plus he expeced rae of asse price change, (1 i ). 19 If we furher assume ha he expeced rae of asse price change i is 0, hen equaion (15) furher simplifies o: (16) u 0 [r 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. 20 This is essenially he user cos formula ha was obained by Walras (1954, pp ) in However, he basic idea ha a durable inpu should be charged a period 17. This poin was firs made explicily by Jorgenson and Griliches (1967, p. 257); see also Jorgenson and Griliches (1972, pp ). 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, pp ; 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, pp ; 1996, pp ). 18. We will provide a more precise definiion of a real ineres rae laer. 19. This formula was obained by Chrisensen and Jorgenson (1969, p. 302) for he geomeric model of depreciaion, bu i is valid for any depreciaion model. Griliches (1963, p. 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). 20. 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 (r n )P, where P n n n is he beginning of period- secondhand marke price for he asse.

13 490 W. Erwin Diewer price ha is equal o a depreciaion erm plus a erm ha would cover he cos of financial capial goes back o Babbage (1835, p. 287) and ohers. 21 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 (1901, pp. 734 and 907 8) in paricular gave a very modern exposiion of he ingrediens needed o consruc user coss or machine rens. 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 or purchase a paricular commodiy. Babbage (1835, p. 203) and Canning (1929, pp ) were also aware of he imporance of deermining he righ machine rae charge. 22 The above equaions relaing asse prices by age n, P, beginning-ofperiod user coss by age n, f, end-of-period user coss, u n n n, and he (crosssecion) depreciaion raes 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 expeced asse raes of price change i The Empirical Deerminaion of Ineres Raes and Raes of Asse Price Change We consider iniially hree broad approaches 23 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: 21. Solomons (1968, pp. 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. 22. 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, pp ) argued ha his facor conribued o he grea produciviy slowdown ha sared around 1973 and persised o he early 1990s. The radiional mehod of cos accouning can be raced back o a book firs published in 1887 by he English accounans Garcke and Fells (1893, pp ). Their raher crude approach o cos accouning should be compared o he maserful analysis of Church. Garcke and Fells (pp ) endorsed he idea ha depreciaion was an admissible iem of cos ha should be allocaed in proporion o he prime cos (i.e., labor and maerials cos) of manufacuring an aricle bu hey explicily ruled ou ineres as a cos. The aversion of accounans o including ineres as a cos can be raced back o he influence of Garcke and Fells. 23. Oher mehods for deermining he appropriae ineres raes ha should be insered ino user cos formulas are discussed by Harper, Bernd, and Wood (1989) and in chaper 5 of Schreyer (2001). Harper, Bernd, and Wood evaluae empirically five alernaive renal price formulas using geomeric depreciaion bu making differen assumpions abou he ineres

14 Measuremen of Capial Services and Asse Price Changes 491 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 for 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, p. 267) and for various secors of he economy by Chrisensen and Jorgenson (1969, p. 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. This approach (incorrecly in our view) ransforms pure profis (or losses) ino a change in he opporuniy cos of financial capial. 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, defined as (17) 1. c Le he producion unis under consideraion face he real ineres rae r. Then, by he Fisher (1896) effec, he relevan nominal ineres rae ha he producers face should be approximaely equal o r, defined as follows: (18) r (1 r )(1 ) 1. The Ausralian Bureau of Saisics assumes ha producers face a real ineres rae of 4 percen. This is consisen wih long-run observed economywide real raes of reurn for mos Organizaion for Economic Cooperaion and Developmen (OECD) counries, which fall in he 3 o 5 percen range. We shall choose his hird mehod for defining nominal ineres raes and choose he real rae of reurn o be 4 percen per annum; ha is, we assume ha he nominal rae r is defined by equaion (18) wih he real rae defined by c 1 rae and he reamen of asse price change. They show (as we will laer) ha he choice of formula maers.

15 492 W. Erwin Diewer (19) r.04, assuming ha he accouning period chosen is a year. 24 We urn now o he deerminaion of he asse expeced raes of price change, 25 he i, which appear in mos of he formulas derived in he preceding secions of his chaper. There are hree broad approaches ha can be used in his conex: Use acual ex pos raes of price change for a new asse over each period. Assume ha each asse rae of price change is equal o he general inflaion rae for each period. Esimae anicipaed raes of asse price change for each period. In wha follows, we will compue cross-secional user coss using Canadian daa on invesmens for wo broad classes of asses (nonresidenial consrucion and machinery and equipmen) for four differen ses of assumpions abou depreciaion or he relaive efficiency of asses by age. We will underake hese compuaions in an inflaionary environmen and make each of he hree ses of assumpions abou he asse inflaion raes lised above for each of he four depreciaion models, giving welve models in all ha will be compared. If he various models give very differen resuls, his indicaes ha he saisical agency compuing capial socks and service flows under inflaion mus choose is preferred model wih some care. When we assume ha he rae of price change for each asse is equal o he general inflaion rae defined by equaion (17), he equaions presened earlier simplify. Thus, if we replace 1 i wih 1 and 1 r wih (1 r )(1 ), equaions (5), which relae he period- asse prices by age np o he renal prices f, become n n (20) P f f f f r 1 r 1 r P f f 1 r 2 2 P f 1 n n f 1 r n r f r f r f 1 n 2 3 f n r Noe ha only he consan real ineres rae r appears in hese equaions. If we replace 1 i wih 1 and 1 r wih (1 r )(1 ), equaions 24. If we are in a high-inflaion siuaion so ha he accouning period becomes a quarer or a monh, hen r mus be chosen o be appropriaely smaller. 25. These are someimes called revaluaion erms in user cos formulas.

16 Measuremen of Capial Services and Asse Price Changes 493 (14), which relae he end-of-period user coss u n o he depreciaion raes, become n (21) u 0 (1 )[(1 r ) (1 0 )]P 0 (1 )[r 0 ]P 0 u 1 (1 )(1 0 )[(1 r ) (1 1 )]P 0 (1 )(1 )[r 0 ]P u (1 n )(1 )... (1 0 )[(1 r n 1 ) (1 )]P n 0 (1 )(1 )... (1 0 )[r n 1 ]P. n 0 Now use equaions (8) and 1 r (1 r )(1 ) and subsiue ino equaions (21) o obain he following equaions, which relae he beginning-of-period user coss f n o he depreciaion raes n : (22) f 0 (1 r ) 1 [r 0 ]P 0 f 1 (1 r ) 1 (1 0 )[r 1 ]P 0... f n (1 r ) 1 (1 0 )... (1 n 1 )[r n ]P 0. Noe ha only he consan real ineres rae r appears in equaions (22), bu equaions (21) also have he general inflaion rae (1 ) as a muliplicaive facor. As menioned above, in our hird class of assumpions abou raes of asse price change, we wan o esimae anicipaed raes of asse price change and use hese esimaes as our i in he various formulas we have exhibied. Unforunaely, here are any number of forecasing mehods ha could be used o esimae he anicipaed asse raes of price change. We will ake a somewha differen approach han a pure forecasing one: we will simply smooh he observed ex pos new asse raes of price change and use hese smoohed raes as our esimaes of anicipaed raes. 26 A similar forecasing problem arises when we use ex pos acual consumer price index inflaion raes (recall equaions [17] and [18] above) in order o generae anicipaed general inflaion raes. Thus, in our hird se of models, we will use boh smoohed asse inflaion raes and smoohed general inflaion raes as our esimaes for anicipaed raes. In our firs class of models, we will use acual ex pos raes in boh cases. Before we proceed o consider our four specific depreciaion models, we 26. Unforunaely, differen analyss may choose differen smoohing mehods, so here may be a problem of a lack of reproducibiliy in our esimaing procedures. Harper, Bernd, and Wood (1989, p. 351) noe ha he use of ime series echniques o smooh ex pos asse inflaion raes and he use of such esimaes as anicipaed price change daes back o Epsein (1977).

17 494 W. Erwin Diewer briefly consider in he nex secion a opic of some curren ineres: namely, he ineracion of (foreseen) obsolescence and depreciaion. We also discuss cross-secion versus ime series depreciaion Obsolescence and Depreciaion We begin his secion wih a definiion of he ime series depreciaion of an asse. Define he ex pos ime series depreciaion of an asse ha is n periods old a he beginning of period, E n, o be is secondhand marke price a he beginning of period, P n, less he price of an asse ha is one period older a he beginning of period 1, P 1 n 1 ; ha is, (23) E P 1 n n P n 1 ; n 0, 1, 2,... Definiion (23) should be conrased wih our earlier definiions in equaion (10), which defined he cross-secion amouns of depreciaion for he same asses a he beginning of period, D P P. n n n 1 We can now explain why we preferred o work wih he cross-secion definiion of depreciaion, equaion (10), over he ime series definiion, equaion (23). The problem wih equaion (23) is ha ime series depreciaion capures he effecs of changes in wo hings: changes in ime (his is he change in o 1) 27 and changes in he age of he asse (his is he change in n o n 1). 28 Thus, ime series depreciaion aggregaes ogeher wo effecs: he asse-specific price change ha occurred beween ime and ime 1 (asse revaluaion due o general inflaion and asse-specific price change) and he effecs of asse aging (depreciaion). Thus, he ime series definiion of depreciaion combines ogeher wo disinc effecs. The foregoing definiion of ex pos ime series depreciaion is he original definiion of depreciaion, and i exends back o he very early beginnings of accouning heory. 29 However, wha has o be kep in mind ha he early auhors who used he concep of ime series depreciaion were implicily or explicily assuming ha prices were sable across ime, in which case, ime series and crosssecion depreciaion coincide. P. Hill (2000, p. 6) and R.J. Hill and Hill (2003, p. 617) 30 recenly argued ha a form of ime series depreciaion ha included expeced obsolescence was o be preferred over cross-secion depreciaion for naional accouns purposes. Since he depreciaion raes n defined by equaion (12) are cross- 27. This change could be capured by eiher P P 1 n n or P P 1. n 1 n This change could be capured by eiher P P or P 1 n n 1 n P 1. n See, for example, Maheson (1910, p. 35) and Hoelling (1925, p. 341). 30. We agree in general wih P. Hill (2000) and R. J. Hill and Hill (2003) ha expeced obsolescence should be added o cross-secional depreciaion o form an overall depreciaion charge. However, Hill and Hill assumed ha here was no general inflaion in heir exposiion, so some clarificaion is needed o deal wih his complicaion.

18 Measuremen of Capial Services and Asse Price Changes 495 secion depreciaion raes and hey play a key role in he beginning and end of period- user coss f and u n n defined by equaions (14), (21), and (22), i is necessary o clarify heir use in he conex of P. Hill s poin ha hese depreciaion raes should no be used o measure depreciaion in he naional accouns. Our response o he Hill criique is wofold: Cross-secion depreciaion raes as we have defined hem are affeced by anicipaed obsolescence in principle, bu Hill is correc in arguing ha cross-secion depreciaion will no generally equal ex pos ime series depreciaion or anicipaed ime series depreciaion. Before discussing hese wo poins in deail, i is necessary o discuss he concep of obsolescence in a bi more deail. Wykoff, in his discussion of his chaper (chap. 12 commen in his volume), akes a narrow echnological definiion of obsolescence. In his view, an asse can only become obsolee if a new model of he asse becomes available ha can deliver a leas he service flow of he old asse a a lower price. In his view, if here is no echnological change embodied in he new asse, hen by definiion here is no obsolescence. However, i is possible o define obsolescence more broadly and include he effecs of changes in he economy ha reduce he demand for he asse s services o such an exen ha is real price falls. 31 In wha follows, we will use he second, broader concep of obsolescence. One more poin mus be considered a his poin. If here is echnological obsolescence due o a new and improved model of he asse being made available, hen we assume ha he price of he new model has been (somehow) qualiy adjused so ha he qualiy-adjused price is measured in quaniy unis ha are comparable o he older models. Now consider he firs bulleed poin. Provisionally, we define anicipaed obsolescence as a siuaion where he expeced new asse rae of price change (adjused for qualiy change) i is negaive. 32 For example, everyone 31. This broader definiion goes back o Church (1917) a leas: Even hough a machine is used fairly and uniformly as conemplaed when he rae of depreciaion was fixed here is anoher influence ha may shoren is period of usefulness in an unexpeced way. The progress of he echnical ar in which i is employed may develop more efficien machines for doing he same work, so ha i becomes advisable o scrap i long before i is worn ou. The machine becomes obsolee and he loss of value from his cause is called obsolescence. Again, unless he machine is of a very generalized ype, such as an engineer s lahe, anoher ype of misforune may overake i. If i is a machine ha can only be used for cerain definie kinds of work or some special aricle, as for example many of he machines used in auomobile and bicycle manufacure, i may happen ha changes in demand, or in syle, make he manufacure of ha special aricle no longer profiable. In his case, unless he machine can be ransformed for anoher use, i is a dead loss (192 93). 32. Paul Schreyer and Peer Hill noed a problem wih his provisional definiion of anicipaed obsolescence as a negaive value of he expeced asse inflaion rae: i will no work in a high-inflaion environmen. In a high-inflaion environmen, he nominal asse inflaion rae

19 496 W. Erwin Diewer anicipaes ha he qualiy-adjused price for a new compuer nex quarer will be considerably lower han i is his quarer. 33 Now urn back o equaions (5), which define he profile of vinage asse prices P n a he sar of period. I is clear ha he negaive i plays a role in defining he sequence of vinage asse prices, as does he sequence of vinage renal prices ha is observed a he beginning of period, he f n. Thus, in his sense, crosssecional depreciaion raes cerainly embody assumpions abou anicipaed obsolescence. Thus, for an asse ha has a finie life, as we move down he rows of equaions (5), he number of discouned renal erms declines, and hence asse value declines, which is Griliches s (1963, p. 119) concep of exhausion. If he cross-secional renal prices are monoonically declining (due o heir declining efficiency), hen as we move down he rows of equaions (5), he higher renal erms are being dropped one by one so ha he asse values will also decline from his facor, which is Griliches s concep of deerioraion. Finally, a negaive anicipaed asse inflaion rae will cause all fuure period renals o be discouned more heavily, which could be inerpreed as Griliches s concep of obsolescence. 34 Thus, all of hese explanaory facors are embedded in equaions (5). Now consider he second bulleed poin: ha cross-secion depreciaion is no really adequae o measure ime series depreciaion in some sense o be deermined. Define he ex ane ime series depreciaion of an asse ha is n periods old a he beginning of period, n, o be is secondhand marke price a he beginning of period, P n, less he anicipaed price of an asse ha is one period older a he beginning of period 1, (1 i ) P ; i.e., n 1 (24) P (1 i n n )P n 1 ; n 0, 1, 2,... Thus, anicipaed ime series depreciaion for an asse ha is periods old a he sar of period, n, differs from he corresponding cross-secion de- i will generally be posiive, bu we will require his nominal rae o be less han general inflaion in order o have anicipaed obsolescence. Thus, our final definiion of anicipaed obsolescence is ha he real asse inflaion rae i defined laer by equaion (28) be negaive; see he discussion immediaely preceding equaion (30). 33. Our analysis assumes ha he various vinages of capial are adjused for qualiy change (if any occurs) as hey come on he marke. In erms of our Canadian empirical example o follow, we are assuming ha Saisics Canada correcly adjused he published invesmen price deflaors for machinery and equipmen and nonresidenial consrucion for qualiy change. We also need o assume ha he form of qualiy change affecs all fuure efficiency facors (i.e., he f n ) in a proporional manner. This is obviously only a rough approximaion o realiy: echnical change may increase he durabiliy of a capial inpu, or i may decrease he amoun of mainenance or fuel ha is required o operae he asse. These changes can lead o nonproporional changes in he f. n 34. However, i is more likely ha wha Griliches had in mind was Hill s second poin: ha ime series depreciaion will be larger han cross-secion depreciaion in a siuaion where i is negaive.

20 Measuremen of Capial Services and Asse Price Changes 497 preciaion defined by equaion (10), D P P n n n 1, in ha he anicipaed new asse rae of price change, i, is missing from D n. However, noe ha he wo forms of depreciaion will coincide if he expeced asse rae of price change i is zero. We can use equaions (12) and (13) in order o define he ex ane depreciaion amouns in erms of he cross-secion depreciaion raes n. n Thus, using definiions (24), we have (25) n P n (1 i )P n 1 n 0, 1, 2,... P n (1 i )(1 n )P n using equaion (12) [1 (1 i )(1 )]P n n (1 )(1 1 )... (1 2 n 1 ) using equaion (13) [1 (1 i )(1 )]P n 0 (1 )(1 1 )... (1 2 n 1 )[ i n (1 )]P. n 0 We can compare he above sequence of ex ane ime series depreciaion amouns n wih he corresponding sequence of cross-secion depreciaion amouns: (26) D P P n n n 1 n 0, 1, 2,... P n (1 n )P n using equaion (12) [1 (1 )]P n n (1 )(1 1 )... (1 2 n 1 )[ ]P n 0 using equaions (13). Of course, if he anicipaed rae of asse price change i is zero, hen equaions (25) and (26) coincide, and ex ane ime series depreciaion equals cross-secion depreciaion. If we are in he provisional expeced obsolescence case wih i negaive, hen i can be seen comparing equaions (25) and (26) ha D for all n such ha D 0; ha is, if i n n n is negaive (and 0 n 1), hen ex ane ime series depreciaion exceeds cross-secion depreciaion over all in-use vinages of he asse. If i is posiive so ha he renal price of each vinage is expeced o rise in he fuure, hen ex ane ime series depreciaion is less han he corresponding cross-secion depreciaion for all asses ha have a posiive price a he end of period. This corresponds o he usual resul in he vinage user cos lieraure where capial gains or an ex pos price increase for a new asse leads o a negaive erm in he user cos formula (plus a revaluaion of he cross-secion depreciaion rae). Here we are resricing ourselves o anicipaed capial gains raher han he acual ex pos capial gains, and we are focusing on depreciaion conceps raher han he full user cos. This is no quie he end of he sory in he high-inflaion conex. Naional income accounans ofen readjus asse values a eiher he beginning

21 498 W. Erwin Diewer or end of he accouning period o ake ino accoun general price level change. A he same ime, hey also wan o decompose nominal ineres paymens ino a real ineres componen and anoher componen ha compensaes lenders for general price change. Recall equaion (17), which defined he general period- inflaion rae, and equaion (18), which relaed he period- nominal ineres rae r o he real rae r and he inflaion rae. We rewrie equaion (18) as follows: 1 r (27) 1 r. 1 In a similar manner, we define he period- anicipaed rae of real asse price change i as follows: (28) 1 i Recall definiion (23), which defined he ex ane ime series depreciaion of an asse ha is n periods old a he beginning of period, n. The firs erm in his definiion reflecs he price level a he beginning of period while he second erm in his definiion reflecs he price level a he end of period. We now express he second erm in erms of he beginning-ofperiod- price level. Thus, we define he ex ane real ime series depreciaion of an asse ha is n periods old a he beginning of period, n, as follows: P n 1 1 i 1 (29) P (1 i n n ) n 0, 1, 2,... 1 P n P (1 i n )(1 n ) using equaion (12) 1 [(1 ) (1 i )(1 )(1 n )] using equaion (28) 1 (1 0 )(1 1 )... (1 n 1 ) [1 (1 i )(1 n )]P 0 using equaions (13) (1 0 (1 1 )... (1 n 1 )[ n i (1 n )]P 0. The ex ane real ime series depreciaion amoun n defined by equaions (29) can be compared o is cross-secion counerpar D n, defined by equaions (25) above. Of course, if he real anicipaed asse inflaion rae i is zero, hen equaions (29) and (25) coincide, and real ex ane ime series depreciaion equals cross-secion depreciaion. We are now in a posiion o provide a more saisfacory definiion of expeced obsolescence, paricularly in he conex of high inflaion. We now P n