Obsolescence vs. Deterioration with Embodied Technological Change

Transcription

1 Obolecence v. Deterioration with Embodied Technological Change Frank C. Wykoff* October 9, 2004 Department of Economic Pomona College 425 N. College Avenue Claremont, CA (909) (office) (909) (home) * The author i Eldon Smith Profeor of Economic, Pomona College. Thi i a reviion of the paper preented at the SSHRC International Conference on Index Number Theory and the Meaurement of Price and Productivity, June 30 July 3, 2004 in Vancouver, Britih Columbia, Canada. I appreciate comment from participant at the SSHRC conference, at the Hawaii International Conference on Economic and Buine in June 2004, and at the Global Conference on Buine and Economic in July Pleae do not quote without the author permiion. Comment are welcome.

2 Obolecence v. Deterioration with Embodied Technological Change Abtract Frank C. Wykoff Few iue in economic have been a contentiou longer than the role of capital in production and ditribution. I apply here the neoclaical model of Solow, Hall, Jorgenon and other in order to addre a central apect of today debate. When technological change i embodied in new capital, hould tatitical meaure of capital ditinguih between obolecence of old capital and deterioration? Thi quetion underlie concern about how contruct capital aggregate. I make the cae that, from the neoclaical perpective, no ditinction hould be made between obolecence and deterioration in contructing capital aggregate meaure. However, empirical evidence on thi quetion i unavailable. Core to contruction of coherent aggregate i the duality equilibrium condition that the ratio of uer-cot equal relative in-ue productive efficiencie. I alo argue that ordinarily one hould not adjut capital-ervice quantity meaure for variation in utilization. I make two argument. One, a coherent meaure of the capital input require aggregation in comparable unit. Thi require that adjutment be made both for deterioration and obolecence. Two, rational deciion by capital uer are independent of the obolecencedeterioration ditinction. The firt half of the paper deal directly with capital aggregation under embodied technical change. The econd with behavior, in repone to new technological innovation, of a repreentative rational producer who i uing old capital. fwykoff@pomona.edu Department of Economic Pomona College 425 N. College Avenue Claremont, CA (909) (office) (909) (home) 2

3 Obolecence v. Deterioration with Embodied Technological Change Introduction Several tubborn quetion continue to plague tatitical agencie that produce capital account. In thi paper I ak how they hould meaure the capital tock when technological change i embodied in new vintage of capital. Technological change embodied in new capital render exiting capital relatively obolete. It doe not, however, lower the marginal product of old capital. In meauring capital tock, doe one adjut for obolecence and deterioration in the ame way? If o, how doe one do o? If not, then i different information needed by tatitician to make the ditinction? If one wihe to ditinguih the tock of capital from the quantity of capital ervice ued in production in the current period, then what role doe utilization play in making thee ditinction? Thee quetion can be addreed from two different but interrelated perpective. One deal directly with the meaurement of capital aggregate. Four economic meaure of capital are of interet. (1) The tock of capital K t i a quantity meaure of the productive capacity of phyical capital in the relevant economic entity (country, indutry, region, firm) in time period-t. (2) The price of capital P t which i the cot of acquiring ownerhip right to one unit of the relevant capital tock in time t. Thi i often referred to a the acquiition price of capital to allow for the poibility that thi may be a market price or a hadow price. Thee firt two term refer to tock. The ditinctive feature of capital i that it i durable and provide a flow of ervice over time. Economit are intereted in price and quantity flow concept that have bearing on production deciion and procee. The econd two meaure refer to flow concept. (3) The flow of capital ervice k t i the quantity of ervice provided by the tock in produc- 3

4 ing output in period t. (4) The uer-cot of capital c t i the cot the producer incur for thee ervice. Thi uer-cot could be a market price, for example the cot of renting the capital good to produce output, or, if the capital i ued by it owner then the uer-cot i a hadow cot the owner incur to derive the ervice. Clearly, the four concept, K, P, k, and c, are interrelated. The cot of acquiring capital in period-t i related to the future equence of uer-cot that one expect to incur from the acquired aet, and the capacityquantity of capital i related to the flow of capital ervice to be provided in the future. Jorgenon in (1974) preent a model, baed on neoclaical concept of equilibrium developed by many economit, mot notably Solow (1957, 1970), Hall (1968), and Hall and Jorgenon (1963), to pecify thee interdependencie. Hulten in (1990) integrate the Solow Jorgenon model raiing different implication of variou concept of capital and technological change from other trand of literature and illutrate potential problem encountered when trying to contruct thee aggregate. Here I focu on one quetion: What are the conequence for contructing capital aggregate of technological change embodied in new aet? New, uperior vintage render old aet obolete. At the ame time, old aet deteriorate. How do thee two force, obolecence and deterioration, influence how one aggregate acro vintage? Thi pecific quetion i of immediate policy importance. International tatitical agencie, charged with meauring capital, productivity and other apect of real aggregate economic activity, would like to deign a uniform ytem of capital account to facilitate comparion of performance among countrie and region. The econd perpective on the conequence of embodied technological change deal with how economic agent repond to change in price reulting from obolecence v. change reulting from deterioration. If a new better aet model enter the market, how do producer who are uing old model repond? If an aet become le productive relative to a new aet, doe the producer repond differently than he would in repone to the introduction of better new model? Anwer to the behavioral quetion have implication for meaurement if one want meaurement of economic aggregate to reflect economic behavior. 4

5 Even though the iue raied here are not new and exiting literature ome of them, dipute about phyical capital perit even after year of conflict that at time flared up into contentiou debate. Today, interet among economit i relegated mainly to the meaurement community margin. Hitorical dipute about capital have been ideological, theoretical and tatitical. To avoid confuion in a field long rife with conflict, I firt put thi paper in context with repect to relatively recent hitorical treatment of phyical capital. I develop the neoclaical model of capital baed primarily on work by Hall, Jorgenon and Hulten building on the Solow analyi of capital in hi growth model. Next, I clarify my ue of pertinent term like depreciation, obolecence, deterioration, age, and vintage. Thee term have been ued by different author to mean different thing. Confuion can be avoided if we are clear about what we mean by thee term. I will ue thee term a ued in the economic literature on depreciation from Hotelling (1925) to Solow (1957, 1971), Hall and Jorgenon (1967), Hall (1968, 1971), Jorgenon (1974), Feldtein and Rothchild (1974), and Hulten (1990) a well a from Wykoff (1970, 2003), Oliner (1993, 1996), Hulten and Wykoff (1975, 1979, 1981, 1996) and Triplett (1996). 1 Thee definition are alo conitent with definition by the U.S. Internal Revenue Service for tax purpoe. I then examine embodied technological change in the neoclaical model that I hall refer to a the Jorgenon model of capital (or the J-model). 2 Fourth, I ugget everal trategie for meauring the quantity of capital ervice when technological change i embodied in new capital. I will how the aumption that underlie different meaure a well a conceptual implication of different meaure. Thi pave the way for analyi of producer behavior in repone to change in capital cot reulting from obolecence compared to deterioration. Thi analyi feed back on how obolecence and deterioration are treated 1 Diewert (2004), Dunn, Dome, Oliner and Sichel (2004) alo grapple with the iue of terminology. Diewert review ome very intereting arcane definition and build on ome of thee term. He alo ue ome of thee term differently; and he note that different ue of thee word appear in other literature. 2 A noted above, Jorgenon alone did not originate thi entire model. Along with Hall and Hulten, he built on the neoclaical model and epecially the eminal work of Solow. Many other alo contributed to thi ubject, but I do not intend a formal literature review here. I ue the term Jorgenon model, becaue he i the author who i mot cloely affiliated with the model I hall be analyzing in thi paper. 5

6 in meauring the capital input. It alo lead to ome obervation about the role of utilization of capital. Finally, I um up and ugget future line of reearch. Overview of the capital controvery Few iue in economic have been contentiou longer than the role of capital. Karl Marx (1867) Da Kapital till reonate with thoe who find capitalim objectionable and aggravate follower of Adam Smith. The iue mot famou flare up among academic economit in the Wet wa the famou Two Cambridge debate in the The principal protagonit, Joan Robinon for Cambridge, England and Robert Solow for Cambridge, Maachuett, fired alvo acro the Atlantic. 3 Robinon argued that marginal analyi of capital i circular, vacuou and mileading. Solow defended marginal analyi of capital a coherent, ubtantive and ueful. The conflict wa energized in part by ideological difference among Marxit and capitalit. A le exotic but important quetion in the two-cambridge debate wa: How doe one meaure capital? Robinon argued that one could compile a lit capital input, but aggregation wa not enible. Solow endored a neoclaical aggregation procedure baed on marginal analyi. Not urpriingly, little wa reolved. Eventually, each ide tired of the argument, and, a Keyneian economic turned focu away from tock toward flow, reearcher on both ide imply went their own way. Diagreement on capital tock meaurement iue continued to immer beneath the urface. It flared up in the 1970 when Dale Jorgenon and Zvi Griliche quetioned U.S. Statitical Agencie on their meaure of the capital input in the national income and product account. The agencie were defended by Edward Denion. 4 A before, the combatant tired with neither ide convincing the other. The Jorgenon model now play a 3 Several key article by Robinon and by Solow appear in the 1971 book of reading on growth edited by Harcourt and Laing. 4 The debate i in the May 1972 Survey of Current Buine, US Commerce Department. See Jorgenon, et al. (1974). Jorgenon and Griliche employed the neoclaical framework to be ued in thi paper. 6

7 major role in deign of capital account by U.S. tatitical agencie, and poibly other, o one would think the iue have been reolved. Still, a low-grade cold war ha been going on within the U.S. tatitical agencie and among American academic over cloely related iue. At the international level, even le agreement exit. One bone of contention ha been whether tatitical agencie hould meaure gro capital tock or net capital tock? A related quetion: I there one correct capital meaure or do different ue of data require different meaure? U.S. tatitical agencie have ettled on producing both gro and net capital tock etimate. Hulten (1990) criticize thi compromie arguing that gro meaure employed by the agencie are not enibly grounded in economic analyi. Today, even though capital meaurement ha become rather eoteric and interet primarily the meaurement community, concern with productivity growth make capital meaurement important to public policy. I focu here on one quetion in the debate. When technological change in embodied in new capital, hould tatitical meaure of the capital tock ditinguih between obolecence of old capital and deterioration? I upect that thi quetion underlie current dipute among academic and government economit involving the meaurement of the capital input. Jorgenon on economic depreciation To model difference in aet productivity a they age, Jorgenon (1974) develop the efficiency function. At any point in time, the efficiency function, φ, i the ratio of the inue productive efficiency of age- aet relative to new aet. It atifie the following condition: (1) ϕ 1 0 = and ϕ ϕ Jorgenon define the mortality function, m, a the decline in efficiency φ with age-: (2) m = ϕ ϕ = 0,1,2,

8 Mortality i the decline in productive efficiency in the aet cohort a it age, given time. 5 Becaue efficiency i tationary thi i alo the rate of decline in productive efficiency a an aet cohort work through it life. If the cohort ha a finite maximum life S, then the -equence terminate at S. Jorgenon illutrate the efficiency and mortality equence with everal familiar function: one-ho-hay, linear, and geometric. Harper later propoed the hyperbolic function to reflect the efficiency pattern. In the mot tractable cae, mortality occur at a contant geometric rate, δ. In thi cae, (3) ϕ = (1 δ ) ; m = δ(1 δ) = δϕ The rate of decline in efficiency of an in-ue aet i δ. Jorgenon applie a duality condition to in-ue aet by equating the ratio of the uercot of an age- aet, c to a new one, to the efficiency function: (4) ( c / c ) = ϕ 0 Duality reult from cot minimization; o, in economic terminology, the efficiency function, φ, i the age-equence of marginal rate of ubtitution to a new aet or the ratio of marginal product of age- aet to the marginal product of a new aet: (5) ϕ MRS MP / MP. 0 The duality equilibrium condition i the core of the Jorgenon model, becaue it connect concept on the price ide of capital, relative uer-cot, to the quantity ide, relative marginal product. 6 Uing duality Jorgenon link phyical lo of in-ue efficiency φ to the decline in the ervice price, uer-cot of the aet a it age, ( c/ ) t=0. The aet price p and the uer-cot c are linked through the fundamental price equation of capital theory. 5 Jorgenon doe not refer explicitly to cohort. I dicu the idea of cohort intead of individual aet below. 6 Formally, Hall refer to the price ide a the dual to the quantity ide of the neoclaical model. The primal define the quantitative choice made by producer electing the input mix to produce output and the dual i the equivalent condition on the price ide. Duality indicate that the two ide have to be coherent becaue duality i an equilibrium condition linking relative cot to relative marginal product. 8

9 , (6) S c t + p = 0, t 0 (1 r) = + The acquiition price of a new aet in period-t equal the preent dicounted value of the future tream of uer-cot. 7 If the efficiency function, φ, i tationary, o that φ,t = φ for all t, then we can ue duality to replace c,t+ with c 0,t φ in the price equation. S ϕ (7) p = c 0, t 0, t 0 (1 r) = + Duality ha linked quantitie to price becaue the efficiency function enter both quantity and price meaure. Thi create the coheion between price and quantitie impoed by the model. Economic depreciation i the decline in aet price with age holding time contant. c c (8) D = p p = + 1+ x + x t, + 1, t t, (1 r) x (1 r) x x + x + Employing duality and the definition of mortality we can write depreciation a: m (9) D = c + x t, 0, t (1 r) x x + The mortality function while defined on the primal to repreent decline in productive efficiency of phyical capital alo enter the dual via the depreciation equation. Economic depreciation i equal to the preent value of the mortality function applied to future flow of uer cot. In the geometric cae, economic depreciation i: (1 δ ) + x D t, (10) D = δ = δ p ; = δ t, t, (1 r) x x + p t, The rate of economic depreciation δ i the ame a the rate of decline of in-ue productive efficiency. Thi reult i true for the contant rate of depreciation illutration, but not the one ho hay or the traight line example. 7 Like equation (4), thi equation expree an equilibrium condition. If one aume perfect foreight, then equation (6) amount to auming that in fully operating competitive purchae and rental market the purchae price equal the preent value of the future equence of rental price. Dropping perfect foreight would require development of an expectation generating model for both c and r, and of coure both r and expected r could vary by date. We abtract from thee iue here. 9

10 Several important aumption are embedded in the Jorgenon model. Thee aumption, while trong, make the model tractable, coherent, and econometrically viable. Firt, the lo of productive efficiency i aumed to depend on age. In reality, phyical decay probably reflect intenity of ue. Perhap thi i not a major problem for the model. The unit of time need not be age in the ene of calendar time but hour of work completed. The model i the ame; jut the φ-function depend on a different metric than age. The J- model alo aume perfect foreight. Without perfect foreight, an expectation generating model may alter the interpretation of the equation linking aet price to uer cot, but they will be linked. A third aumption i that the efficiency function i tationary: φ for all- i independent of date and time. Thi aumption greatly implifie the model but i ineential to many of the reult. We could replace tationary duality with: (11) ( c / c ) = ϕ. t, 0, t t, Equation (11) allow φ,t, φ,t-1, and φ,t+1 to differ. Of coure, (12) ϕ MRS MP / MP t, t, t, 0, t A practical difficulty with uing thi verion of the model to contruct capital tock i that one need to identify aet by vintage and not imply by age. 8 Statitical agencie would have to compile vintage account, a cotly endeavor becaue vintage accounting require continual updating of the φ-equence. A fourth aumption embedded in the efficiency function i that efficiency i built right into the aet. Thi trong aumption appear to be inconitent with producer making deciion about frequency of repair, replacement, crappage, retrofitting, and o on that producer can urely make. 9 Why doe one need thi aumption and i it worth the cot in realim? A we noted earlier, the unit of efficiency need not be age but could be hour 8 Solow (1970) develop a vintage-capital model to allow for thi ituation. I, like Hall, Jorgenon, Hulten and other, rely heavily on it in what follow. 9 Feldtein and Rothchild (1974) preent many way in which efficiency can be endogenouly influenced while aet remain in ue. Of coure, a imple model i a good model if it hold up empirically, o reality i not the lat word in model aement. 10

11 or ome gauge of intenity of ue, although uch a re-interpretation create conceptual iue of it own. It alo require ignificantly different and harder to get data, but in principle would allow the producer to chooe intenity of aet ue. The Jorgenon model i an econometric model devied to ue available data. Another implication of built-in efficiencie, however, i unavoidable if we wih to contruct a capital aggregate. The marginal rate of ubtitution between age- and new aet are independent of the marginal product of other input. Thi i the ame aumption required to contruct a capital aggregate for ue in a production function. When can the production function, Q = F(K1, K2, L) be written a Q = F [ K(K1, K2), L]; i.e., when i the capital aggregator function, K(K1, K2), independent of the labor input? The anwer i that the marginal rate of ubtitution that determine the weight for K1 and K2 mut be independent of the marginal product of the labor input. Thi aumption i alo atified by the non-tationary efficiency function. According to the Jorgenon model, at equilibrium the ratio of the uer-cot between age and new aet equal the marginal rate of technical ubtitution between age- and new aet. Auming φ i inherent in the aet i equivalent to the aumption needed to obtain a capital aggregate. Of coure, thi aumption may be wrong, but then a coherent meaure of capital aggregation independent of other apect of the production proce doe not exit. 10 In thi cae, meaurement i more complex, and appropriate ue of meaure in productivity reearch or policy analyi can be more difficult, becaue ome important ummary tatitic are no longer reliable. Economic depreciation with embodied technological change I define economic depreciation here to deal with cohort of at rather than individual unit. Let economic depreciation be the decline in price of one (average) unit of an original cohort of at with age given date. The rate of economic depreciation i the rate 10 An important practical application of the model where thi aumption may be problematic i the contruction of computer tock over time when technological change i occurring in both hardware and oftware. See Diewert (2004) and Wykoff (2004). 11

12 at which thi price decline. 11 The term depreciation i often ued in other way; for example, it may refer to the amount of money et aide to prepare for replacement a an aet wear out, amortization. Depreciation of a tock of capital i functionally related to the price concept depreciation and that relationhip will be developed below. To accountant depreciation refer to a et of accounting rule for writing down an invetment for bookkeeping or tax purpoe. But thee rule may not accurately reflect economic depreciation. In the J-model depreciation refer only to the decline in price reulting from aging but not the paage of time. The effect on price of the paage of time, holding age contant, i revaluation. 12 Economic depreciation ha two component, becaue a aet age, given time, two ditinct type of effect influence price: vintage effect and aging effect. The relationhip between age, time and vintage, a pointed out by Hall (1968), impoe an important identification problem on price analyi. Vintage i defined a v = t where t i the time period, and i the cohort age. If one take the partial derivate of price with repect to age holding time contant, then vintage i changing a well: an older aet at a point in time i an earlier vintage. Similarly, the partial derivative of price with repect to time holding age contant advance vintage, becaue an aet the ame age in the next period i a newer vintage. Since thi concept can be eluive, conider the following thought experiment. Suppoe we have a room for each year that aet price are oberved and that a label on the door tate the year, ay All the price in room-2003 are price oberved of aet in the year Within each room we have boxe, one box containing the price oberved for aet of each age. We have a new box, a one-year-old box, and o on. We go to the 2005 room and open the box labeled age-4. Now, uppoe omeone ay let create a filing ytem in box-4 for each vintage in order to further ditinguih aet by vintage. You tart to pull out price in order to eparate them into vintage. Since you are in room Mathematically economic depreciation at a point in time i the partial derivative of price with repect to age, holding time contant: p(,t)/ t=0 where p(,t) i the price of an aet age- at time-t. 12 Revaluation i the partial derivative of price with repect to time, holding age contant: q(p,t)/ t =0. 12

13 and in box age-4, every aet price you pull out will be a vintage =2001! You don t need any file. In fact they make no ene becaue pecifying the age box in the date room lock in the vintage. We have only 2 degree of freedom, not 3. With thi identification contraint in mind, we turn to the two component of economic depreciation, deterioration and obolecence. Deterioration Deterioration i a relatively traight forward concept referring to the decline in price of one unit of the cohort of aet a the cohort age reulting from ue, wear and tear, and retirement from ervice. In principle, deterioration i independent of the pecific date. It reult from aet getting older and poibly le efficient. Even if all aet were oneho-hay, o that in-ue decrepitation i zero, price would fall with age becaue the cohort i uing up ome of it capacity both by being conumed and by being retired from ervice each one ho hay i getting cloer to that moment when it turn to dut. Triplett, following Griliche, call thi effect exhaution. 13 Decline in efficiency, while till in-ue, i alo poible. After all, the one-ho-hay i imaginary preciely to mock perfectionit who complain becaue part of aet breakdown at different time. Strictly peaking a decline in efficiency i a decline in the marginal product of the capital and may reflect le accuracy, lower peed, increaed operating cot, longer down-time, increaed frequency of breakdown, and o forth. Phyicit call thi entropy. For all thee reaon old aet are uually wore phyically than when they were new, and thi decline in marginal product i reflected in the price. (We are ignoring here tart-up cot and learning curve.) In other word, deterioration i the decline in price that reflect the decline in phyical prowe of the original cohort of a- 13 Triplett and Griliche are thinking in term of exhaution of a pecific aet and not a cohort of aet. Age i a proxy for ue. Data on intenity of ue i rarely available, and the concept of ue i, a Hulten (1990) argue, itelf ambiguou. The decline in marginal product with age would depend on the intenity of ue, the care of the aet, the nature of the work it doe, maintenance and repair, the quality of complementary good, and numerou other factor. 13

14 et. The ratio of the marginal product of a cohort of age-(+1) to age- aet i le than one marginal product decline with age. 14 Obolecence The concept of obolecence i more ubtle than the concept of deterioration. Obolecence i the effect on the price of an average unit of a cohort reulting from the availability of new vintage of the aet with uperior characteritic to the older vintage. Nothing phyically happen to the ued aet per e, a it did with deterioration. The in-ue marginal phyical product are not changed. A newer verion i now on the market, o the older one, while unchanged, are relatively obolete. Even if the old aet are the ame, the environment i different a better ubtitute 15 ha become available. Obolecence i a date-pecific concept it reflect unique date-pecific characteritic embodied in new aet that are not embodied in older aet. A 1993-Bordeaux i a unique vintage defined by the date the grape were grown on the hillide of Bordeaux. 16 The term vintage mean the date the grape were grown and harveted. With wine, of coure, new vintage may be better or wore than exiting vintage. The ame may be true of ome type of capital a well. If we run out of quality material and build new aet that are not a good, then like wine the older vintage may be uperior to newer one. We ve all heard the expreion, they don t make em like they ued to. Alo, claic deign can caue older vintage of certain aet to be extremely valuable the 1964 Ford Thunderbird for example. Still, a a rule new aet frequently have feature that make them better than pre-exiting verion a a reult of technological innovation embodied in the new vintage. For mot capital aet the concept of vintage i not a imple a with wine. One cannot, by definition, produce a 1993-vintage wine in 1995; however, with produced capital thi i not the cae. Jut becaue a new deign or new feature i available, one doe not have 14 In continuou time: [ Q/ K]/ <0 where Q i output, K i capital and i age. 15 The term better here mean either higher marginal product or le cotly or both. 16 Ahenfelter, Ahman, and LaLonde (1995) analyze the caue of vintage effect on Bordeaux wine. 14

15 to build all new aet with the new bet deign. Not all automobile model, for intance, produced in 2002 had all the newet characteritic available in Thi mean that our analogy of a different room for each year and a different box per age implying only one particular vintage, in one ene, doe not exactly apply. We could go to room-2004, open box-age-3 and find everal different model of car and create a file for each model in the box. Some of the model have new feature but ome may only have feature that were introduced in ome earlier vintage. Thi would appear to violate the Hall Identification Theorem baed on v = t -. It alo raie the quetion: If new aet in each year are themelve heterogeneou, how do we define a vintage-2001 car or a vintage-january Dell Computer? 17 New feature tend to enter market of complex capital good gradually over time. Very few new car have the newet feature uually only the top of the line model do. The ame i true of computer and conumer electronic product. A ucceful feature i gradually built into lower-line model over the year. According to Diewert, thi gradual adoption in the market of new good and of quality improvement i what create the new-good and quality-change biae in conumer price indexe. A Triplett explain, thi tendency of phyical-capital market to produce ome model containing old feature and ome with new feature, permit reearcher to ue hedonic method to iolate the price of quality improvement from inter-temporal change in the price of a fixedquality good. Embodied Technological Change: Two Model We now allow for embodied technological change in the Jorgenon model by pecifying vintage efficiency function. 18 Conider a imple model in which embodied technological change occur at a contant rate τ. How do we reflect thi rate of technological change in the generic efficiency and mortality function and in the pecial geometric cae? The ef- 17 Technological change ha been o rapid in computer that vintage vary by month not jut year. 18 See Hulten (1990) for a theoretical preentation of a imilar model. 15

16 ficiency function i the ratio of the marginal product of age- aet to new aet, but now with technological we mut ak to which new aet? There are now two poible way to model capital with vintage-pecific marginal product. We could re-normalize each year on that year new aet or we could normalize on new aet in a bae year. The firt model normalize the efficiency function in each year on new aet in that year, o that φ,t, = 0, 1, 2, 3,, the equence of the ratio of the marginal product of ued to new aet in each year, atifie the property that φ 0 = 1. It follow that every ued aet not embodying the new technology ha to be drawn down both for decay and technological obolecence in order to convert them into unit compatible with the new, uperior vintage in the next period. Conider the example with the contant rate of economic depreciation, δ. One can decompoe the depreciation rate δ into it two component part: deterioration γ and obolecence θ. In the primal, correponding to deterioration and obolecence in the dual, are decay and technological obolecence. Letting δ = γ + θ, 19 o that efficiency i: (13) ϕ = (1 δ) = [1 ( γ + θ)] The correponding mortality function i: (14) m = ( γ + θ)[1 ( γ + θ)] = ( γ + θ) ϕ. The mortality of an aet reult both from deterioration, decline in the aet marginal product, and from technological obolecence. It i neceary to include obolecence in the mortality function in order to convert old aet into unit comparable to technologically-advanced new aet new (uperior vintage) aet equivalent. The improvement in the quality of new aet mean a decline in the relative, though not the abolute, marginal product of aged aet. The econd model normalize on a bae vintage. Thi model explicitly recognize that new aet efficiency varie by vintage. New aet efficiency i one only in the bae pe- 19 The depreciation rate δ actually equal γ + φ γφ. It i impler to ignore the interaction term which i in any cae mall. 16

17 riod. It i le than one before the bae and greater than one after the bae. To capture the difference in new aet efficiency over time, we normalize on t = 0: 20 (15) ψ = (1 θ) t ;0 < θ < 1; t =... 1, 2,0,1, 2,3 0, t If θ =.01, then the equence of new aet productive efficiencie will be: Bae t = 0 t: ψ 0,t : (1-.01) 3, (1-.01) 2, (1-.01) 1, 1, (1-.01) -1, (1-.01) -2, (1-.01) -3, The normalization function ψ i an efficiency function only in the ene that it equal the marginal rate of ubtitution between vintage-t new aet and bae-vintage new aet. 21 Auming that decay occur at rate γ, we write the normalization function of age- aet in period-t to a new aet in the bae period a: (16) ψ = (1 θ ) t (1 γ ) (1 θ) = (1 θ) t (1 δ) ; t, t, ψ = (1 ) t t, θ ϕ The productive efficiency of an age- aet in year-t relative to a new aet in the bae year equal the product of the normalization correction for vintage and the tationary efficiency function φ. For example, if the bae period i 2002 then normalization of the 4- year-old cohort of aet in 2005 i ψ 4,3 = (1-θ) -3 (1-δ) 4 = (1 θ ) 3 ϕ4. The age-date tableau of relative efficiencie, normalized on a new bae 2000 aet i illutrated in Figure 1. Thi tableau aume no change related to paage of time per e; inter-temporal change in upply and demand may be dealt with eparately. Inflation i not relevant becaue we are talking about relative aet efficiencie not relative cot. We are uing a tableau of efficiencie, rather than price or uer cot, in order to analyze the implication for meauring the flow of capital ervice in the primal that reult from obolecence v. deterioration change in the dual. 20 We defined the rate of technological change to be τ. We could expre the normalization in term of τ. Intead of (1-θ) t we would have (1+τ) t. It i more convenient to expre the normalization in term of the rate of obolecence θ. τ = θ/(1-θ). 21 Producer making deciion in period-t deal with contemporary marginal rate of ubtitution between aet of different age and not between inter-temporal MRS between new aet in different period. 17

18 Figure 1 Tableau aet efficiencie normalized on new vintage 2000 aet* ψ t, t = date Age 0 Date (1-θ) -1 (1-θ) -2 (1-θ) -3 (1-θ) -4 1 (1-γ) (1-θ) 1-γ (1-γ) (1-θ) -1 (1-γ) (1-θ) -2 (1-γ) (1-θ) (1-γ) 2 (1-θ) 2 (1-γ) 2 (1-θ) (1-γ) 2 (1-γ) 2 (1-θ) -1 (1-γ) 2 (1-θ) -2 (1-γ) 3 (1-θ) 3 (1-γ) 3 (1-θ) 2 (1-γ) 3 (1-θ) (1-γ) 3 (1-γ) 3 (1-θ) -1 4 (1-γ) 4 (1-θ) 4 (1-γ) 4 (1-θ) 3 (1-γ) 4 (1-θ) 2 (1-γ) 4 (1-θ) (1-γ) 4 5 (1-γ) 5 (1-θ) 5 (1-γ) 5 (1-θ) 4 (1-γ) 5 (1-θ) 3 (1-γ) 5 (1-θ) 2 (1-γ) 5 (1-θ) * Where indexe age and t indexe time relative to bae Thi tableau aume contant rate of decay γ and of obolecence θ; no hift in upply and demand. We normalize on vintage 2000 new aet, ψ 0,2000 = 1. 18

19 The hitory of any cohort of aet run down a diagonal tarting at age-0 in year-t from left to right. For intance, the new period-2001 cohort of aet ha the following efficiency equence through it hitory: (, t+): (0, 2001) (1, 2002) (2, 2003) (3, 2004) ψ,t+=,v=2001 : (1-θ) -1, [(1-θ) -1 (1-γ)], [(1-θ) -1 (1-γ) 2 ], [(1-θ) -1 (1-γ) 3 ], Each cohort decay over it life and obolecence ha no effect directly on the cohort efficiency relative to the bae. Decay depend on age and i independent of vintage. 22 Thi reflect the aumption that the technology embodied in the 2001 aet remain contant a that aet cohort evolve over it hitory and that γ and θ do not interact. At any point in time, ay year 2004, aet of different age have different level of technology embodied in them reflecting a contant rate of technological change per period. Thi can be een by tracing the equence down a column. For example the 2004 efficiency equence i: : ψ,t=2004 : (1-θ) -4, [(1-γ)(1-θ) -3 ], [(1-γ) 2 (1-θ) -2 ], [(1-γ) 3 (1-θ) -1 ], (1-γ) 4, The ψ-function i not an economic efficiency function. The efficiency function i the ratio of in-ue marginal product between age- and new aet in one period. The ratio of marginal product in year-t equal the ratio of two normalization function, ψ,t /ψ 0,t. In thi econd model, the in-ue productive efficiency function i: 23 ψ (17) t, = (1 δ) = ϕ ; t, ψ 0, t Comparing equation (13) to (17), we ee that the marginal rate of ubtitution between age- and new aet, in each year, are the ame in each model. However, the model are different. The econd model ha two mortality function not one. If by mortality we mean 22 Decay rate would be vintage pecific if technological change altered decay rate. Thi elaboration of the model would require that each cohort have it own decay rate, ay γ v. Since we are making a theoretical point here, we abtract from thi elaboration. 23 Thi efficiency function i the ame a in the J-model, becaue we removed the ψ-normalization by calculating the ψ-ratio. 19

20 the difference between in-ue productive efficiencie between age- and age-(+1) aet given date, then the period-t mortality function i: 24 (18) m = ϕ ϕ = δ(1 δ) = δϕ. t, + 1 If by mortality we mean the lo of in-ue productive efficiency over the cohort hitory, then the mortality function i: (19) µ = ψ ψ = γ(1 θ) t (1 δ) t, + 1, t+ 1 t, µ = γψ = γ(1 θ ) t ϕ t, t, It i now evident that we have two different model on which to bae capital aggregation from invetment flow data. The efficiency and mortality function, equation (13) (14) are baed directly on the Jorgenon model. The efficiency function i virtually the ame, becaue δ = γ + θ. The mortality function i the ame whether tracing an aet hitory between period or comparing aet of different age at a point in time. The econd model, equation (16) (19), i different. Aet cohort relative efficiencie are normalized on a bae year. Thi normalization place the in-ue efficiency of all aet in every year in the ame unit new, bae-vintage equivalent. Aet efficiency at a point in time, in-ue productive marginal rate of ubtitution between ued and new aet, i the ratio of thee normalization function Thee ratio generate the ame tationary efficiency function a in the firt model. There are two poible mortality function in the econd model. Difference in in-ue productive efficiency at a point in time are the ame a in the J-model; but change in efficiency over an aet-cohort hitory are different. Mortality in the latter ene reult only from decay, not from obolecence. Thi µ-mortality function only reduce the productive efficiency of ued aet that reflect a decline in the abolute value of the marginal product, not the marginal product relative to next period uperior vintage aet. For reaon that will become obviou hortly I call thi model the G-model. 24 Thi mortality function i the ame a the J-model. 20

21 Before we evaluate the two econometric model, we lay out the empirical problem they are intended to olve. The quantity of aggregate capital ervice To etimate or meaure capital aggregate, tatitical agencie begin with data on current and pat invetment expenditure flow that have been collected for contruction of national income and product account. Direct obervation of quantitie of capital in each year would require annual capital cenue. To my knowledge no capital cenu ha been undertaken. 25 Four capital aggregate meaure are ueful: the quantity of capital ervice and their uer-cot, and the tock of capital, and the price of acquiring capital. 26 Let I t be nominal gro aggregate invetment expenditure in period-t. Thee total expenditure are the um of the product of the price and quantitie of individual invetment good, j=1,, n, that were produced, marketed, and old (or acquired) in period-t: n (20) I = p ι p ι t t, j t, j t t j= 1 The identity indicate that I t equal the dot product of two vector, p t and ι t. Aume that each vector can be repreented by an index. Let price index p 0,t be the period-t market price of one new, period-t invetment good in period-t dollar. Let ι 0,t be the period-t quantity of new period-t invetment good. I aume the price index i not corrected for inflation and that neither index i corrected for quality change. Heuritically, thee term p t and ι t repreent the current market price and quantity of the generic new period-t invetment good. The problem now i to derive two quantity meaure from the pat equence of price p and quantitie ι of new invetment good. One i the current flow of capital ervice, k. 25 Accountant for variou purpoe contruct book value of capital tock, but thee reflect accounting convention and not economic procee. 26 My main focu here reflect my interet in determining the quantitative implication of obolecence v. deterioration on ervice flow, o I do not want to deflect attention by raiing complexitie related to capacity-tock (or wealth tock) meaure or on purchae price and hadow price of aet. 21

22 The other i the capacity tock of capital good, K. 27 We derive two et of quantitie from the p and ι: (21) k = f( p, ι) K = F( p, ι) t t p ( p, p,..., p ) ; ι ( ι, ι,..., ι ) t t 1 t S t t 1 t S We will alo determine the relationhip between aet price and uer-cot, ince thee will play a role via duality in meauring the quantity aggregate. For convenience we aume that the flow of the quantity of period-t ervice k t and the tock of capital K t are each linear function of pat real invetment flow, ι 0,t-. 28 S S (22) k = ω ι K = w ι t, t t t, t t = 0 = 0 The weight for both quantity aggregate are vintage pecific: they depend both on age- and on vintage, v = t. Aume momentarily that thee weight are tationary. In thi cae we normalize on one new period-t invetment good by letting ω 0 and w 0 be one in each period. The firt term in the um, ι t, the quantity of new invetment good acquired in period t, i, by definition, the quantity of new period-t capital good, K 0,t. The propoed normalization on the ervice flow from one unit of newly acquired capital et ω 0 to one. Thi aume that the quantity of ervice, k t, that flow of ervice equal the quantity of new capital, K t. Thi aumption eem quite trong but I think it only acknowledge that we, a economit, ordinarily cannot oberve the number of capital ervice flowing from a unit of capital. One poible olution to obtaining flow meaure ditinct from tock involve modifying the tock for capital utilization. A Hulten (1990) point out, however, it i not clear that a capital good only provide ervice when it i actually in ue. Even if I don t it in all the chair in my office all the time, I have the option value of itting in each one whenever I want to, o chair provide ervice even when not active. Producer know product demand will fluctuate but they require the option-value of producing good when called for. I do not think we know enough to be able to meaure a unit of ervice ditinct from 27 Thi aggregate i ometime called the wealth tock. 28 Diewert (2004) argue for an index number formula rather than a linear model. He aume the new invetment flow have already been corrected for quality change. 22

23 a unit of capital. One could define the quantity of output produced by capital and et to one the quantity produced by new capital, but thi jut beg the quetion of what i the output, i.e., ervice flow. I hall argue below that, once one take into account expectation, an adjutment for utilization i rarely appropriate. Another problem remain to be reolved. Recall that ι v where v = t - i a quantity meaure derived from the flow of new invetment in period-v. If we interpret φ,t to be the marginal rate of ubtitution in between a new and a vintage-v aet that ha urvived until period-t, then we have to deal with the fact that ome of the aet acquired new in year-v have expired or been retired before year-t. There are two way of dealing with thi problem, and each ha advantage and diadvantage. Hulten (1990) propoe that a correction be made to the quantity of aet acquired in year v = t -. o that the quantity of ervice in equation (22) i baed on the quantity of urvivor from the original cohort. He ugget that we replace ι v with κ v defined a: (23) κ = t ξ, t t ι The ξ-function i a correction for retirement. κ v now replace ι v. The advantage of thi approach i that retirement i een to be a different proce than in-ue decay and may reult from different factor, but thi i alo it diadvantage. Deterioration, the decline in price (the dual) that reflect phyical decay (the primal), combine exhaution (retirement) and in-ue decay. Thi mean that the mortality function combine the two effect o that the duality condition already allow in theory for retirement. Thi bring u to the econd way of dealing with the retirement problem, the approach ued here. Define the φ-function to be the in-ue value in period-t of one unit of the original cohort of vintage-v = t- aet to one unit of the new cohort of aet in period-t. The correction for retirement i included in the φ-function itelf o that ι,t i not changed. We give thi interpretation to the efficiency function and alo to the mortality function. We are now prepared to ak what weight, ω and w, we hould ue? The J-model anwer thi quetion with the efficiency function. Figure 2 illutrate that the efficiency function 23

24 i the equence, expreed a a function, of marginal rate of technical ubtitution between age- and new capital good at time t = v +. The duality condition i conitent with the producer electing point a on the ioquant a the neceary condition for cot minimization. Since φ,t i the marginal rate of technical ubtitution between an age- and a new aet, in current production, it i the quantity of age- aet one need to provide the equivalent output of one new period-t aet the lope of the ioquant. We do not in thi analyi need to know what that output i; we only need to know the lope of the ioquant. Duality equate the lope of the iocot curve to the lope of the ioquant. Figure 2 Duality i the Cot Minimizing Condition K 0,t ϕ MRS MP / MP, t, t, t 0, t a c / c, t 0, t K, t Two Model for Capital Aggregation A natural weight for the flow of ervice from aet of different age i the marginal rate of ubtitution between the age- aet and the new aet in each period. In each model, the J-model and the G-model, the marginal rate of ubtitution are the ame, φ, o per- 24

25 hap in each cae ω = φ. However, one purpoe of the G-model would be that intead of reetting the efficiency of a new aet to one each period, we let each vintage have the efficiency that reflect it embodied technology relative to technology in the bae. In the G-model, then, the ω are vintage pecific ψ-value: ω,t = ψ,t = 0,, S. For every, ψ,t i different in each period t. In the J-model the φ weight are the ame in each period t. Thi mean that the aggregate quantity of capital ervice per year will be different in the two model, even though the ratio of the marginal product in each year will be the ame in the two model: MP /MP 0 = ψ,t /ψ 0,t = φ. Given thee ditinction between the two model, we now derive the capital aggregate for both model. We will then begin to ae their relative merit. The Jorgenon model weight the quantity of in-ue ervice by the efficiency function φ. S S (24) k = (1 ) ϕι = δ ι t t t = 0 = 0 The duality condition erve two purpoe in thi analyi, one theoretical and one practical. The theoretical purpoe i to jutify the weight of in-ue capital, becaue in theory we are uing the deciion of rational cot-minimizing economic agent to convert aet that are heterogeneou by age into new aet equivalent. The practical contribution of the duality condition olve the problem that a economit, though poibly not engineer, we do not directly oberve the relative efficiencie, lope of the ioquant. The dual, equating relative uer cot to relative efficiencie, allow u to infer the φ-equence from the ratio of uer-cot the lope of the iocot. The capacity capital tock weight in the J-model can be derived from a parallel tructure to the ervice flow. Correponding to the in-ue efficiency function φ i a capacity efficiency function Φ. An aet exchange, or capacity, duality condition correpond to inue ervice flow duality condition. (25) p t, =Φ p t, 0, t 25

26 The relative cot of acquiring an age- aet compared to the cot of acquiring a new aet in period-t equal the relative capacity of future in-ue ervice in age- aet to the capacity in new aet, Φ,t. A capacity mortality function M indicate the decline in capacity between age- and new aet: p p D + 1, t t, t, (26) M =Φ Φ = = + 1 p p 0, t 0, t The econd equality reult from duality and the lat equality reflect the definition of economic depreciation. In the unique cae in which the depreciation rate i contant δ: p t, (27) M = δ = δφ. p, t 0, t When the economic depreciation rate i contant, δ, the rate of mortality in capacity and the rate of mortality in ervice flow both equal δ. Φ i a natural weight w in the capacity tock aggregator. S (28) K = Φ ι. t, t t = 0 In the G-model the ervice flow weight, the ω, are the vintage-pecific ψ-function value: S S (29) k = ψ ι (1 θ) t (1 ) = δ ι t, t t t = 0 = 0 Duality in the G-model i: (30) ( c / c ) = ψ. t, 0,0 t, The ψ-function expree the ervice price of an age- aet in period-t relative to a new aet in the bae period t=0. The marginal rate of ubtitution between age- and new aet ervice in period-t i: ψ (31) ( c / c ) = t, = ϕ t, 0, t ψ t, 0, t 26

27 Model-G, like model-j, ha a capital tock framework that correpond to it treatment of ervice. Correponding to the in-ue normalization function ψ i a capacity normalization function Ψ. Both are vintage pecific. Capacity duality in the G-model i: p t, (32) =Ψ p t, 0,0 Relative aet acquiition cot equal relative productive capacity. Mortality i: p p + 1, t+ 1 t, t, (33) Μ =Ψ Ψ = = t, + 1, t+ 1 t, p p 0,0 0,0 S µ + x, t + x = c t, 0,0 0 (1 r) x x = + The (capacity) capital tock equation for the G-model i: S (34) K = Ψ ι. t, t t = 0 We have now preented two complete model of capital aggregation. In the Jorgenon model, the quantity of capital ervice i equation (24) and the capacity capital tock i equation (28). In the G-model, the quantity of capital ervice i equation (29) and the capital tock i equation (34). The final tep in developing the model i to derive the inter-temporal change in the flow of capital ervice, k t = k t+1 k t, and in capital tock, K t = K t+1 K t. From equation (24) we derive the change in the flow of capital ervice from period t to period t+1. S S S (35) k = ϕι ϕι = ι + m ι t t+ 1 t t+ 1 t = 0 = 0 = 0 When depreciation i contant at rate δ: S (36) k = ι δ (1 ) δ ι t t + 1 t = 0 k = ι δ k t t+ 1 t 27