Download Measuring Capital: Durability and Depreciation and more Schemes and Mind Maps Economic Theory in PDF only on Docsity!
4 The Measurementof Capital
Charles R. Hulten
The measurementof capital is one of the nastiestjobs that economists have set to statisticians. (Hicks 1981b,204)
The theory of capital is one of the most difficult and contentious areas of economic theory. From Karl Marx to the Cambridge controversies, there has been an ongoing disagreement among economists as to what capital is and how it should be measured.’ Economists have variously defined capital as con- gealed labor, as deferred consumption, as the “degree of round-a-boutness,” as a stock of durable commodities, or as a flow of factor services. There is also disagreement about whether capital can be aggregated into a single mea- sure, and, even within the relatively hospitable confines of neoclassical theory, exact aggregation is known to be problematic. This presents the practical economist with something of a dilemma since many interesting economic problems require a measure of capital. How, for example, are we to understand the process of economic growth if we cannot agree on how to measure one of the potentially most important factors influ- encing that process? What can we say about such important issues as the pro- ductivity slowdown of the 1970s and why growth rates differ across countries? These issues are too important to ignore, and estimates of capital, income, and wealth, however imperfect, must somehow be developed in order to get on with the larger tasks at hand.
Charles R. Hulten is a professor of economics at the University of Maryland and research associateof the National Bureau of Economic Research. The author would like !o thank Dale Jorgenson,Frank Wykoff, Ingmar F’rucha,and RobertM. Schwabfor their valuable commentson earlier drafts of this chapter.Judy Xanthopoulosprovided invaluable researchassist;lnce.
120 Charles^ R. Hulten
The Conference on Research in Income and Wealth and, more generally, the National Bureau of Economic Research have been at the forefront of the development process. Many of the 50-odd volumes of the Studies in Income and Wealth series are devoted, in whole or in part, to issues of capital mea- surement. These studies, by such pioneers as Kuznets, Goldsmith, Stigler, and Kendrick, have laid the conceptual foundation for many of the measure- ment procedures used today; they provide statistical series that are still in use. It is therefore fitting that the commemoration of the fiftieth anniversary of the conference should include an essay on the current state of the art of capital measurement. I undertake this task with the recognition that the subject is too large to be easily encompassed by a single essay. I have therefore chosen to limit my focus largely to depreciable assets used in the business sector, although the discussion will sometimes stray across this boundary and many of the results discussed will be applicable to other sectors and other types of capital. I will also allocate the bulk of my space to a sketch of the theory of capital measure- ment. This choice reflects, in part, the historical objective of the conference in bringing together measurement theory and practice. However, it also re- flects the too often ignored need for theoretical consistency in the construction of data as, for example, when capital stocks are estimated using one assump- tion about depreciation and estimates of capital income are based on another assumption. The chapter is organized into two major parts. The first outlines the theory of capital measurement and is divided into six sections. The first three sec- tions cover measurement and valuation of a single homogeneous type of cap- ital, while the following section extends the analysis to the case of many cap- ital goods. The final two sections deal with the issues of quality change and capacity utilization. The second part of the paper examines some practical issues in the measurement of capital. The scope and nature of existing esti- mates and procedures are reviewed, and then critiqued in light of the theory of the preceding sections.
4.1 Applied Capital Theory Two aspects of capital (including human capital) differentiate it from a pri- mary input like labor: capital is a produced means of production, and capital is durable.2 The first aspect is the primary source of the Cambridge contro- versy in pure theory, but the latter causes much of the actual difficulty in mea- suring capital. Durability means that a capital good is productive for two or more time periods, and this, in turn, implies that a distinction must be made between the value of using or renting capital in any year and the value of owning the capital asset. This distinction would not necessarily lead to a measurement problem if the
122 Charles R. Hulten
tional to surface area, so that older blocks are proportionately smaller than new blocks. In this case, old and new capital can be thought of as differing by a constant + and the aggregate K can be seen as a physically homogeneous entity. Much the same can be said of light bulbs, since older vintages shine as brightly as new ones (until they fail), and a homogeneous K can be formed by assigning C$a value of one for all surviving vintages and adding up past in- vestment. The efficiency sequence can be determined, in both examples, from the nature of the good itself-dry ice is homogeneous, and thus old and new units are prefect substitutes up to some constant +. If the same were true of all capital, then the measurement problem would be reduced to determining the relative technological “size” of new and used capital. Unfortunately, most capital does not accommodate this kind of measurement because older ma- chines are typically neither physically smaller nor dimmer than their newer counterparts. Nevertheless, such machines may be less efficient because of increased downtime, higher maintenance requirements, or reduced speed or accuracy, or they may embody less advanced technology than new ma- chines. The possiblity that older vintages of capital may be less productive suggests that the C$sequence might more usefully be defined in terms of the production process itself. The $3 could be thought of as relative marginal products, and the resulting K may be interpreted as the ability of the surviving vintages (I,,.. , I,-,)^ to produce output.^ This^ approach does not rule out the “dry ice” case of inherent productivity differences, but does allow for the possibil- ity that relative efficiency is a matter of economic choice and that different technologies may imply different +‘s for the same type of capital. Or, in other words, the capital aggregation depends on the nature of the technology and on market behavior. This link between aggregate capital and the production function was devel- oped by Leontief (1947a, 1947b), Solow (1960), and Fisher (1965). The basic issue involves the conditions under which different vintages of capital and technology can be collapsed into an aggregate production function defined with respect to an aggregate measure of capital. It is assumed that each vin- tage of capital can be combined with labor via its own production function to produce output
(2) Q,,^ = f”(L ,,“, Iv)^ v^ =^ t, t-^ 1, t-2,^... , t-T, where Q,,, is the output produced by capital of vintage I, and L,,, is the homo- geneous labor applied to that capital. The production functions are allowed to differ in order to incorporate the possibility of technical change, that is, old machines are installed with the technology prevailing in year u. Output from all vintages is assumed to be homogeneous and aggregate output is thus the sum of the Q,,,, that is,
123 The Measurementof Capital
Q, = CQ,,, = ~f”(L,w 1,). Y Y
The aggregation problem is to write (4) as
(4) Q:^ = HL,, KU,,... , I,-,>I,
where L, = CL,,“, Q; is the maximum output that can be produced assuming
labor is optimally allocated among vintages, and K(*) is independent of L. Necessary and suffic:ient conditions for this capital aggregation are given by the Leontief theoreni, which- states that the marginal rate of substitution be- tween any pair of i;iputs within the aggregate must be independent of the inputs outside the capital aggregate:
2 apiary
aL, aQ*iar, 1 1
= 0 for all v,s = t,... , t-T,
or
a apiaI,
aL, afslar, 1 1
=0whenf;=f!;v, 5 =t, ...,t-T.
Fisher (1965) shows that, under constant returns to scale, this condition re- quires that differences between vintages must be expressible as
(7) fYL,,“r^ 1”) = J-a,,,,^ b,-“Lb
That is, the technology must be such that the difference between the produc- tivity of old and new capital is a fixed constant depending only on vintage. Or, as Hall (1971) puts it: “In vintage production functions with constant re- turns, the basic theorem of capital aggregation establishes that a capital aggre- gate exists if and only if the marginal product of capital of age r at time t has the fixed ratio... to the marginal product of new capital at time t” (242). In our notation, this amounts to
aQ*/al, _
__ - 4+-“,
aQ*ial,
v = t, t-l,....
Thus, formal aggregation theory leads us back to the perpetual inventory method of capital aggregation. Old capital enters the production process as if it were equivalent to a smaller amount of new capital-as in the case of dry ice. There is little reason to believe that real-world technologies exhibit the sep- arability required by the Leontief conditions. Moreover, even if aggregation over vintages were possible, there is no guarantee that the aggregated produc- tion function (4) would be a valid representation of the technology of an entire industry or industrial sector. Further conditions are required for aggregation
125 The Measurement of Capital
(11) r=^ l,...,T-1.
As with the one-hoss shay form, T completely determines the efficiency pat- tern. The popularity of the straight-line pattern reflects the widely used con- vention, borrowed from depreciation accounting, that assets should be amor- tized in equal increments over a useful life. Geometric decay is the third widely used pattern. In this form, productive capacity decays at a constant rate 6, that is,
(12) implying
(13) 4, = 1, +, = (l-6), 4, = (1-S)’ ,... ) +, = (1-6)T ,...
The geometric form is widely used in theoretical expositions of capital theory because of its simplicity. But, while it enjoys empirical support from studies of used capital prices, it is nexrertheless regarded by some (e.g., Harper 1982) as empirically implausible because of the rapid loss of efficiency in the early years of asset life (c:.g., 34% of an asset’s productivity is lost over four years with a 10% rate of depreciation). Moreover, assets are (implausibly) never retired, so that the efficiency sequence is no longer a function of the useful life T. However, 6 is frequently derived from published estimates of T using the double declining balance formula, 6 = 2/T, obtained from tax accounting, although other declining balance formulae are also used.B We have thus far taken the date of retirement T to be the same for all assets in a given cohort (all assets put in place in a given year).9 However, there is no reason for this to be true, and the theory is readily extended to allow for different retirement dates. A given cohort can be broken into components, or subcohorts, according to date of retirement and a separate T assigned to each. Each subcohort can then be characterized by its own efficiency sequence $(I), which depends among other things on the subcohort’s useful life Ti. The con- tribution to total capital at time t made by a cohort of vintage v is the sum over the subcohorts of that vintage
(14) CC&p.
The stock of capital at time I is then equal to
(15) K,= c #,“II”+... + c #“,I:) +.... I I
Letting w!J = It)lI, be the weight of the ith subcohort in vintage v invest- ment, this can be written as
126 Charles R. Hulten
When the subcohort weights o are stationary over time, that is, independent of v, (16) reduces to (1). In this case, the efficiency weight 4. in (1) can be interpreted as the average efficiency of the investment in the cohort, and it thus captures both in-place loss of efficiency and efficiency loss due to retire- ment. The average efficiency function of a cohort can be quite different from the
individual efficiency functions $(I). Every 4”) can have the one-hoss shay
form, while scan be such that the average efficiency decline is geometric.
This point has important consequences for the measurement of 4: as noted,
the intuition that suggests that assets decay according to the one-hoss shay
pattern is based on the observation of the individual 4”‘. But equation (16)
implies that the extension of this intuition to an entire cohort of assets may involve a fallacy of composition in which each asset in the cohort follows one pattern but the cohort as a whole follows a different pattern. Two final points should be noted before leaving the subject of efficiency functions. First, the early literature on capital measurement distinguished be- tween net and gross capital stock. The net stock is defined as our (1) or (16) (we will largely ignore the distinction throughout the rest of this part of the paper). The gross stock is defined by
(17) K;^ =^ I,^ +^ I,-,^ -I-^ t^ I,wT, in the special case when all assets are assumed to be retired at the same point in time, or by the more general form
(18) Kf^ =^ RJ,^ +^ i&I,-,^ t^..^ +^ flTZ,-T, when retirements are distributed over time and f2, is the (stationary) propor- tion of assets surviving to time 7. Estimates of gross capital stock are commonly published along with esti- mates of the net stock (e.g., U.S. Department of Commerce 1987), and gross stocks are used in some analyses of productivity change.‘O However, it is clear from the separability condition of (18) that $,-” is defined with respect to relative marginal products, so it is the “net” measure of capital that is consist- ent with the production function Q, = F(L,K). That is,the net stock K,, along with labor L,, produces gross output Q,, and the gross stock of capital is con- sistent with the production function only when the efficiency sequence is one- hoss shay, (9). But, in this case, net and gross stocks are the same, and the argument in favor of the gross capital stock is really an argument that the net stock must be one-hoss shay regardless of empirical evidence about the 4’s. Finally, it is important to emphasize a point made by Feldstein and Roths-
child (1974): there are limitations to the use of any perpetual inventory method based on the procedures for estimating 4 discussed in this section. For example, we have assumed that firms are not free to retire old capital as economic conditions dictate, maintenance and repair activities do not influ- ence the $‘s, and a higher rate of utilization does not cause asset efficiency to
128 Charles R. Hulten
is the following symmetry between prices and quantities: I,,S = 4,1v,oand
PE = 4JYo.
This symmetry implies that the rental price of vintage v capital is 4, times
the rental price of new capital. The asset price, P:,,, in (21) can therefore be written in terms of the relative efficiency sequence and the rental price of new assets:
This expression links asset valuation to asset efficiency. It has been derived in the case in which rental markets exist, but is also valid for the case in which capital is utilized by its owner. Indeed, (22) can be “solved” to obtain an expression of the implicit rent in terms of the other variables of (22).
(23) PfT = [r-~,,~t(l+p,,~)~,,~lP~,~~s = 0, 1, 2,. ,
where
is the expected “inflation” in the vintage asset price occurring between years t andt + 1,and
(25) 8,,$= - k^ - I],
is the rate of decline in the asset price with age s (or, more accurately, the decline in price as vintage v capital becomes like vintage v - 1 capital). Equation (23) thus has a straightforward interpretation: when assets are owner utilized, the equilibrium value of the implicit rental must cover the real oppor- tunity cost of an investment of value P!,, as well as the loss in asset value as the asset ages. In practice, elaborations of this formula, based on Jorgenson (1963) and Hall and Jorgenson (1967), are used to impute a value of the rental price and thus the value of the marginal product of capital. The term 6 deserves attention in its own right, since it can be shown to be the rate of economic depreciation. Hicks (1946) defines income as the maxi- mum amount that can be spent during a period while maintaining capital val- ues intact; economic depreciation is then defined as the sum of money, in constant dollars, that needs to be set aside in order to maintain that capital value in real terms. In our notation, the Hicksian definition of depreciation is equivalent to P:,, - P:,,+ ,. This in turn implies that depreciation is equal to 8,,,P:,, by (25), which leads to the conclusion that the variable 6 is the Hicksian rate of economic depreciation. When p # 0, a revaluation adjustment is nec- essary but essentially the same interpretation carries over. Following Jorgenson (1973), equation (25) can also be used to link eco-
129 The Measurementof Capital
nomic depreciation to changes in asset efficiency. Rearranging terms in (25) yields
(26) w., = p:s- p:s+I = c
m(4,+, - 4,+,+,) p:+,,o
7=0 (l+r)‘+’^ ’
for an asset of age s. This expression states that Hicksian economic deprecia- tion is the present value of the rental income loss due to the efficiency decay
4 S+T - 4s+s+, occurring^ in each year in the future^ (7 =^ 0,^ 1, 2,.. .). In
other words, depreciation occurs because the efficiency pattern is shifted one year for every year the asset ages. It is the shift in the enh-e efficiency pattern that leads to a decline in asset value. Equation (26) shows that economic depreciation (a price effect) and effi- ciency decay (a quantity effect) are not independent concepts. One cannot select an efficiency pattern independently of the depreciation pattern and maintain the assumption of competitive equilibrium at the same time. And, one cannot arbitrarily select a depreciation pattern independently from the observed path of vintage asset prices Pi (suggesting a strategy for measuring depreciation and efficiency). Thus, for example, the practice of using a straight-line efficiency pattern in the perpetual inventory equation in general commits the user to non-straight-line pattern of economic depreciation. I This framework is useful for revealing what economic depreciation is, but it is also useful for revealing what it is not. Depreciation is not the replacement
cost of the efficiency units used up in any year, that is, X(+, - 4,+ ,)P:o, be-
cause P:, is not generally equal to c$,P:,,,unless decay is geometric. This can be seen intuitively by considering a one-hoss shay asset with a lo-year useful life. The efficiency lost between years 8 and 9 is zero, by definition, so the cost of replacing the loss units is also zero. However, the decline in the price of the asset is certainly not zero, since the asset is almost at the point of retire- ment. As a result, Hicksian depreciation occurs because the efficiency pattern has shifted, despite the retention of asset efficiency. A parallel confusion arises over the valuation of the capital stock. Recall that K defined in (1) can be thought of as the number of efficiency units em- bodied in the existing stock-that is, the amount of new capital that must be purchased in order to yield the same productive capacity as the existing vin- tages of capital. It is thus natural to think of the value of the stock as the cost of purchasing these equivalent efficiency units: Pf,,K,. However, this is not the case. The value of the stock is the asset value of the separate pieces of the stock, that is, the amount that would be obtained from selling each piece of capital at its market price:
This is the wealth associated with the stock K,. It is not the same as Pi,oK,, except when deprec,ation fellows the geometric pattern (again, because in
(^131) The Measurementof Capital
where each K’ is itself an aggregate over individual investment vintages (I assume, here, that the conditions for vintage aggregation discussed in sec. 4.1.1 are satisfied for each type of capital). A necessary condition is that the marginal rate of substitution between each type of capital be independent of the amount of labor used:
2 aQtaP;
aL, aQtaq 1 1
=O, i,j=l,..., N.
Under this restriction, the aggregator function K(s) determines the nature of the capital aggregate, and the measurement of aggregate capital thus becomes a matter of discovering the form of K(e). This can be done by direct estimation of F’(m)and K(s), which obviates the need for constructing the capital aggre- gate, or by Divisia indexing procedures. The Divisia index is constructed by weighting the growth rate of each type of capital by its share in total capital income, S; = PFK, / Z PPKI, and sum- ming the result: l
This can be shown to be related to the logarithmic differential of the produc- tion function F(s) when rental prices are proportional to marginal products. As shown in Hulten (1973), the existence of a linearly homogeneous aggre- gator function K(s) allows this expression to be integrated to obtain the “level” of the aggregate capital in each year (with one time period arbitrarily normal- ized at one.)” The Divisia index is formulated in continuous time and is therefore not generally applicable to economic data. In practice, a discrete approximation to (3 1) is used in which the continuous growth rates are replaced by the differ- ence in natural logarithms,’ In E; - In Kf- L, and the continuous shares by the arithmetic average (L/2)(S; t S:- ,). The result is the discrete time Tomqvist- translog index of capital.18 When rental prices are proportional to marginal products (e.g., under cost minimization) and when the production function has the homogeneous translog form, the Tomqvist-translot index of capital is exact (Diewert 1976). This approach provides an internally consistent, but restrictive, procedure for aggregating capital. A problem arises, however, when the number of asset types N is very large. The Tomqvist-translog approach requires that a capital stock and a rental price be calculated for each type of capital, which in turn requires an investment series, asset prices, and efficiency sequences for each type of asset. This is a difficult requirement when the number of assets gets even moderately large, and it is impossible for the thousands (if not millions) of varieties of capital actually used in production. The enormous variety of capital assets virtually insures that some types of
132 Charles R. Hulten
capital will be treated as homogeneous even though they are not. Categories like “commercial buildings” and “machine tools” come to be regarded (out of necessity) as homogeneous for the purpose of measurement, despite the fact that they include quite diverse types of capital. In such cases, the quantity of the pseudohomogeneous good is found by adding up current dollar values of each component good and deflating the result to the price level prevailing in some base year. Data on current investment expenditures are relatively easy to obtain, but finding a plausible price index for the deflation process is an- other matter. The price index problem is greatly simplified if individual asset prices move together (i.e., are proportional). In this case, the price levels differ only by a constant, so that units of quantity can, in principle, be redefined to make asset prices identical. This is the case in which the Hicks aggregation theorem ap- plies and a capital stock can be calculated provided that the aggregate effi- ciency sequence is the same for each component of the aggregate. The theory of hedonic prices provides another solution to the problem of excessive variety. In this framework, individual capital goods are viewed as bundles of characteristics rather than as discrete physical entities. For ex- ample, different types of personal computers may be classified with respect to speed, memory size, graphics capability, and so on. The “inputs” to the pro- duction function (1) are then the amount of each characteristic rather than the amount of each physical good. The hedonic approach is particularly useful when there are many varieties of capital embodying a few characteristics- that is, when there are many bands and/or options that can be reduced to far fewer characteristic dimensions. Under certain conditions, hedonic techniques can be used to estimate the “prices” associated with different characteristics. These prices can be used to deflate the total dollar expenditure on a group of pseudohomogeneous capital goods or to deflate the components individually. But, while this is an appeal- ing approach, it is greatly limited by the fact that capital goods are purchased as physical units and the prices of component characteristics are not directly observable. Furthermore, the shadow prices of the individual characteristic tend to be complicated functions of all other characteristics and not just pa- rameters as with physical goods, so estimation is often difficult. In the final analysis, the great diversity and variety of the capital stock vir- tually insures that simple adding-up procedures will occur at some level of disaggregation. Simple, and usually ad hoc, deflation procedures will inevi- tably be used for some portion of investment, and the use of more sophisti- cated translog and hedonic techniques may reduce aggregation bias but will not eliminate it. But, as Griliches (1971) has noted in his survey of hedonic methods, “half a loaf is better than none.”
4.1.5 Embodied Technical Change An important variant on the heterogeneity problem deserves attention in its own right: different vintages of capital may differ in quality because they em-
134 Charles^ R. Hulten
Any number of combinations of (6, y, A) can yield a given ((Y, p), and there is thus an identification problem. This problem also occurs on the price side in identifying the separate effects of depreciation (the change in vintage asset price with respect to age, t-u), obsolescence (the change with respect to vintage, v), and inflation (the change with respect to time, t). Assuming constant rates of growth, the price of a vintage v asset at time t can be shown to equal
(34) p11.1=^ &p+y,re- (Y+61s
This implies that the trend in efficiency decay and obsolescence cannot be identified using data on used asset prices. Hall suggests the following procedure to solve his identification problem: “As we have seen, if our framework is restricted to consideration of the @- ciency of capital in use, the trend is ambiguous, and it would be senseless to try to estimate it. An alternative to this view is to suppose that embodied technical change, far from being a mystery, can be explained in terms of changes in the observed charucreristics of capital goods. By characteristics, we mean size, weight, power, and other information of an engineering nature” (1971, 258; emphasis added). This approach brings us back to hedonics as a solution to the quality problem, but this time in response to quality change over time rather than asset diversity at any point in time. It is worth noting, here, that Hall’s suggestion has been implemented for the computer compo- nent of equipment investment in the U.S. National Income and Product Ac- counts, and Gordon (1989) has extended this to 17 types of producers’ durable equipment. I The hedonic approach captures differences in quality that are revealed in price differentials. In competitive equilibrium, prices will tend to be driven into equality with marginal costs, implying that only those quality differences that are associated with cost differentials will be picked up by hedonic meth- ods. This, in turn, implies that hedonic techniques will capture only part of the embodied change in the index +. The use of hedonic prices to deflate investment expenditures is a complete solution to the embodiment problem only under restrictive assumptions.
4.1.6 Capital Stocks versus Capital Flows Capital stock estimates are widely used in econometric and growth account- ing analyses of production. However, the production function Q = F(K, L) is conventionally interpreted as a relationship between theflow of output and the J?OWof input services. We have thus far ignored the distinction between capital stocks and flows and must now consider the problem of converting estimates of the latter into a flow equivalent. The minimalist solution to this problem is to assume that capital flows are proportional to stocks, so that the one is a perfect surrogate for the other. In this case, capital utilization-defined as the ratio of the flow to the stock-is
135 The Measurement of Capital
assumed to remain constant over time and, in particular, over the business cycle. However, while convenient, proportionality is clearly a dubious as- sumption, since published estimates of utilization tend to vary over the cycle. An alternative approach is to multiply the estimated capital stock by an estimate of capital utilization. But, while this solves the problem of introduc- ing variation in stock estimates over the business cycle, it merely converts the problem from one of measuring capital services (given the capital stock) to one of estimating utilization. If the flow of capital services cannot be mea- sured, then estimation of the ratio of services to stock is also problematic. Ambiguity about the exact nature of capital services is at the center of the problem. What, exactly, is a capital “service”? Is a chair in “service” only when it is occupied? Or, does the availability of the chair for potential occu- pancy count for something too? If so, are potential services equivalent to ac- tual services? And, how do we assess the decorative value of the chair if it adds to the office ambience? In the same vein, is an office building utilized only during business hours, or is it utilized all the time to keep out thieves and inclement weather? In both cases, the services (whatever they are) cannot readily be observed because they are not easily defined. The measurement of such services, or of capital utilization, is thus problematic. An alternative approach is to dispense with the notion of capital service altogether and to analyze production from the standpoint of capital stocks alone. This is the approach taken in the recent literature on temporary equilibrium, in which the production function is inter- preted as a relationship between the flow of output and a flow of variable labor input applied to a quasi-fixed stock of capital. Because the stock is taken as fixed in the short run, short-run fluctuations in demand can only be accom- modated by changes in the amount of labor used in production. The capacity of the capital stock is defined with respect to the cost-minimizing level of output for the given amount of capital, and the optimal level of capacity oc- curs when actual output is at the cost-minimizing level. Capacity utilization, in this sense, is increased when more labor is applied to the fixed amount of capital.*O There are two concepts of rental price in the temporary equilibrium frame- work. The ex ante rental price is defined as the implicit (or possibly explicit) rent that is expected to be paid in each future period. The ex ante equilibrium condition is given by the analog of (19)
where the rental price, Pf+,,T is now the expected cost in year t + r given that demand is at its expected level, and r is now the rate of interest expected to prevail in future periods. The actual, or ex post, rental price, Zf+,,7, is the gross quasi-rent realized from the capital stock when labor is adjusted to meet
137 The Measurementof Capital
This may change in coming years, but it seems safe to say that these models have yet to have an impact on current national income and accounting prac- tice, which relies on the more conventional measurement framework outlined above.
4.2 Practical Problems in the Measurement of Capital
The bits and pieces of theory presented in the preceding sections provide a practical framework for measuring capital stocks. The principal options are to look for a direct estimate of the capital stock, K,, or to adjust book values for inflation, mergers, and accounting procedures, or to use the perpetual inven- tory method. This last option requires an estimate of the value of investment spending, P’I, a quality-adjusted investment deflator, PI, an efficiency se- quence 4, a.ndpossibly a retirement distribution. Any of these procedures can be implemented at any level of industrial or asset detail for which the neces- sary data exists. The discussion of section 4.1 reviews the conceptual difficulties with the various procedures. Statisticians involved in the actual estimation of capital stocks are, however, aware that the conceptual problems are only part of the problem. Dozens, if not hundreds, of “small” practical problems also cause headaches: Should estimates be assembled on a company or establishment basis? By industry of use or industry of ownership? Using data on investment expenditures or investment shipments? According to which industry and asset classification? The various practical issues that must be addressed are too numerous and detailed to be dealt with in a relatively brief survey article. We will focus, instead, on three of the central problem areas of the perpetual inventory method: the estimation of investment in current dollars by industry and asset,
i
the development of suitable investment-good deflators, and the estimation of efficiency sequences and retirement distributions. The following sections deal with these topics in turn, and a final assessment is offered in the conclusion.
4.2.1 Investment Data Data on the current dollar value of U.S. investment are available from a variety of sources. The principal ones include: the U.S. National Income and Product Accounts (NIPA), the Bureau of Economic Analysis’s (BEA) plant i and equipment survey (P&E), and the investment data underlying the BEA capital stock studies (CSS). These data and others (like those from the input- output studies) are based on different classification systems and different de- grees of coverage and must be interpreted accordingly. The NIPA equipment data, for example, are based on deliveries of investment goods, while the P&E and CSS data are largely based on investment expenditures. The P&E data, however, are collected on a company basis while the CSS data refer to estab- lishments. The choice between alternative investment series depends primarily on how
138 Charles R. Hulten
the series will be used and not on inherent differences in data quality. Studies of the financial structure of firms or industries, for example, require company- based investment data since the decisions of interest are generally made at the company level of organization. Studies of productivity change, on the other hand, require establishment-level data since technology and production deci- sions are generally implemented at the establishment level within the com- pany.23Similarly, studies of the distribution of wealth may require data on the ownership of capital, while studies of production require data on the utiliza- tion of capital. Leased capital should be attributed to the owner in the first type of study, but attributed to the user in the second. There is thus no uniquely correct source of investment data. Furthermore, the choice among competing investment series depends on the desired level of asset and industry detail. There is a relative abundance of data for the econ- omy as a whole, but the choice is far more limited at lower levels of aggrega- tion. The BEA capital stock studies (Gorman et al. 1985; U.S. Department of Commerce 1987) provide the most extensive “official” investment data set for the United States: estimates of fixed nonresidential private investment (in cur- rent and constant dollars) are provided at the two-digit SIC industry level of detail; estimates are also provided for residential capital (by legal form of organization), durable goods owned by consumers (by type of good), fixed nonresidential government capital (by type of government and type of equip- ment and structure), and fixed nonresidential capital (by legal form of orga- nization). The study also presents separate estimates of nonresidential capital for 22 types of producers’ durable equipment and 14 types of nonresidential structures, and a cross-classification by two-digit industry and type of capital is available. This impressive degree of detail requires data from many sources. Table A of the 1987 BEA study lists no fewer than 21 such sources. And, this covers only new nonresidential fixed investment. This multiplicity of sources is re- quired in order to achieve the desired industry detail and to obtain sufficiently long investment series. The length of the investment series is an issue be- cause, under the perpetual inventory method, the capital stocks at any point in time are the weighted sum of past investments. The investment series used in the perpetual inventory method must therefore span the years for which the efficiency weights are positive, or at least span the time period in which the weights are large enough to affect significantly the capital stock. This problem can be illustrated by the case of geometric depreciation. With a constant rate of depreciation 6, the perpetual inventory equation (1) can be written as (38) K, = I, + (l--6)1,-, +... + (1-@‘-“I, + (1 -Z)‘-“+lK “-,.
That is, the capital stock at time t is the efficiency-weighted sum of investment back to year v, plus the remaining efficiency of the capital stock of time u - 1. By making the investment series sufficiently long, that is, making u suffi-
