Empirical studies of depreciation.
Jorgenson, Dale W.
The purpose of this paper is to survey empirical research on
depreciation and its applications. As a framework for the survey I
employ the econometric model of asset prices introduced by Hall [1971]
that has been the primary vehicle for this research for the past two
decades. Since 1985 research on asset prices has been used successfully
to generate constant quality price indices for investment in computers;
these indices are included in the U.S. National Income and Product
Accounts. The challenge facing economic statisticians now is to employ
asset price information effectively while making badly needed revisions
in the treatment of depreciation in the national accounts.
Empirical research on constant quality price indices focuses on the
relationship between the prices of assets and their
characteristics--processing speed and memory capacity in the case of
computers. By contrast, empirical research on depreciation centers on
the relationship between the price of an asset and its age, so that
prices of assets of different ages or vintages must be analyzed. Since
the connection between research on constant quality price indices and
depreciation is not obvious, I describe Hall's "hedonic"
model of asset prices, which comprehends both, in the next section.
My second objective is to summarize empirical research on
depreciation, beginning with the landmark studies of Hulten and Wykoff
[1981a; 1981b; 1981c] for the Office of Tax Analysis of the Department
of the Treasury. As a consequence of the rapid assimilation of the
results of Hulten and Wykoff, depreciation has been transformed from one
of the most contentious and problematic areas in economic measurement to
one of the best understood and most useful. Empirical research has
generated the information needed for revising the treatment of
depreciation in the U.S. national accounts.
My third objective is to illustrate the use of information on asset
prices in constructing an integrated system of income, product, and
wealth accounts. For this purpose I describe a system of vintage
accounts introduced by Christensen and Jorgenson [1973]. This system
includes accumulation equations that generate the perpetual inventory of
assets required for the wealth accounts and asset pricing equations that
produce the information on depreciation needed for income and product
accounts. A system of vintage accounts is the key to the successful
integration of measures of income and product with a measure of wealth.
My overall conclusion is that the U.S. National Income and Product
Accounts have failed to provide internally consistent measures of
capital stocks and depreciation. Top priority should be assigned to
improving the national accounts by incorporating information now
available from empirical research on depreciation. Fortunately, this
task can be facilitated by using concepts and methods already familiar
to economic statisticians from their work on constant quality price
indices.
The following section outlines Hall's econometric model of
asset prices, focusing on the version of this model employed by Hulten
and Wykoff. I also consider an alternative version recently introduced
by Oliner [1993]. In section II, I summarize empirical research on
depreciation, including studies of asset prices and retirements. Studies
of both types are required for constructing a system of vintage asset
prices. In section III, I outline the role of depreciation in an
integrated system of income, product and wealth accounts. The system of
vintage accounts originated by Christensen and Jorgenson provides a
natural framework for incorporating the results of empirical research on
depreciation. Section IV concludes the paper.
I. ECONOMETRIC MODELS
To measure depreciation one needs a set of prices and quantities of
investment goods. Investment represents the acquisition of capital
goods, for example, a certain number of computers with a given
performance. The price of acquisition of a durable good is the unit cost
of acquiring it. For instance, the price of acquisition of a computer is
the unit cost of purchasing the machine. Implicit in these definitions
is the assumption that prices and quantities are measured in units of
constant quality.
Capital services are defined in terms of the use of an investment
good for a specified period of time. For example, computers can be
leased for days, months, or years. The rental price of capital services
is the unit cost of renting an investment good rather than purchasing
it, so that the rental price of a computer is the unit cost of using a
machine for a specified period of time. Depreciation is the component of
unit cost associated with the aging of assets. This component can be
isolated by comparing prices of assets of different ages. The objective
of empirical research on depreciation is to construct a system of asset
prices for this purpose.
I begin the description of a system of vintage accounts with
quantities of investment goods. I refer to investment goods acquired at
different points of time as different vintages of capital. A system of
vintage accounts can be generated by the perpetual inventory method.
This method is based on the assumption that the quantity of capital is
proportional to the initial level of investment. The constants of
proportionality are given by the relative efficiencies of different
vintages of capital. An important simplifying assumption is that
relative efficiencies of different vintages of an investment good depend
only on the age of the investment good and not on the time at which it
was acquired.
The quantity of capital input is the flow of capital services into
production. Since the capital services provided by a given investment
good are proportional to the initial investment, the services provided
by different vintages at the same point in time are perfect substitutes.
Under perfect substitutability the flow of capital services is a
weighted sum of past investments; the weights correspond to the relative
efficiencies of the different vintages of capital. A system of vintage
accounts containing data on investments of every age in every period is
essential for the measurement of depreciation.
We turn next to the price data required for a vintage accounting
system. The rental price of capital services is the price of capital
input. Under perfect substitut-ability the rental prices for all
vintages of capital are proportional to a single rental price with
constants of proportionality given by the relative efficiencies of the
different vintages. The price of acquisition of a capital good is the
sum of present values of future rental prices of capital services,
weighted by relative efficiencies of the capital good in future time
periods. To measure rental prices, a system of vintage accounts
containing data on prices of capital goods of every age in every period
is required.
Durable goods decline in efficiency with age, and thus require
replacement in order to maintain productive capacity. This is the
quantity interpretation of the intuitive notion of "maintaining
capital intact." Similarly, the price of a durable good declines
with age, resulting in depreciation that reflects both the current
decline in efficiency and the present value of future declines in
efficiency. Depreciation provides the price interpretation of
"maintaining capital intact." Both concepts are used in a
system of vintage accounts.(1)
Price and quantity indices of capital services in each period can
be constructed for each durable good given a full set of vintage price
and quantity data. With less complete information, a simplified set of
price and quantity indices can be constructed from empirical estimates
of the relative efficiencies of investment goods of different ages. This
is the point at which empirical research on depreciation can be
particularly useful. As an illustration, consider a system of vintage
accounts introduced by Christensen and Jorgenson [1973]. This system is
based on the assumption that the decline in efficiency of assets with
age is geometric.
To construct this simplified accounting system, estimate capital
stock at the end of each period [K.sub.t] as a weighted sum of
investments of age v at time t {[A.sub.t-v]}
[Mathematical Expression Omitted]
This is the capital accumulation equation, relating the stock of
assets to past investments, required for a perpetual inventory of
assets. With a constant rate of decline in efficiency [delta],
replacement [R.sub.t] is proportional to capital stock:
[R.sub.t]=[delta][K.sub.t-1].
Similarly, the price of acquisition of new investment goods
[p.sub.A,t] is a weighted sum of the present values of future rentals
{[p.sub.K,t]} This is the capital asset pricing equation, relating the
asset price to the values of future capital services, needed for a
vintage price system. With a constant rate of decline in efficiency the
rental price becomes
[p.sub.k,t] = [p.sub.A,t-1][r.sub.t] +
[delta][p.sub.A,t]-([p.sub.A,t]-[p.sub.A,t-1]),
where [r.sub.t] is the rate of discount. Depreciation [p.sub.D,t] is
proportional to the acquisition price:
[p.sub.D,t]= [delta] [p.sub.A,t]
Finally, the acquisition price of investment goods of age v at time
t, say PA t v iS
[p.sub.A,t,v] = [(1 - [delta]).sup.v] [p.sub.A,t]
The vintage price system consists of the acquisition prices
{[p.sub.A,t,v]} while the vintage quantity system consists of the
quantities of past investments {[A.sub.t-v]}. The price and quantity of
capital services can be derived from these two systems of vintage
accounts, together with depreciation and replacement of capital
goods.(2)
Under the assumption that the decline in efficiency of a durable
good is geometric, the vintage price system required for construction of
price and quantity indices for capital input depends on the price for
acquisition of new capital goods [p.sub.A,t]. At each point in time the
prices for acquisition of capital goods of age v, say {[p.sub.A,t,v]},
are proportional to the price for new capital goods. The constants of
proportionality decline geometrically at the rate 6. The rate of decline
can be treated as an unknown parameter in an econometric model and
estimated from a sample of prices for the acquisition of capital goods
of different vintages.
I obtain an econometric model for vintage price functions by taking
logarithms of the acquisition prices {[p.sub.A,t,V]} and adding a random
disturbance term:
ln[p.sub.A,t,v]=ln[p.sub.A,0]+ln(1 -[delta])v + ln (1 + [g]) t +
[[epsilon].sub.t,v] =[[alpha].sub.0]+[[beta].sub.v]v+[[beta].sub.t]t+[[epsilon].sub.t,v], (t=1,2...T; v=0,1...),
where the rate of decline in efficiency [delta] and the rate of
inflation in the price of the asset [g] are unknown parameters and
[[epsilon].sub.t,v] is an unobservable random disturbance.
Assume that the disturbance term has an expected value equal to
zero and constant variance, say [[sigma].sup.2], so that (2.) These
vintage accounts are used to generate an integrated system of income,
product and wealth accounts for the U.S. by Christensen and Jorgenson
[1973]. This system is extended to the industry level by Jorgenson
[1980] and implemented by Fraumeni and Jorgenson [1980]. Fraumeni and
Jorgenson [1986] have incorporated the results of the empirical research
on depreciation summarized in section II, below.
E([[epsilon].sub.t,v]) = 0, V([[epsilon].sub.t,v]) =
[[sigma].sup.2] (t = 1,2...T; v = 0,1...),
and that disturbances corresponding to distinct observations are
uncorrelated:
C([[epsilon].sub.t,v] [[epsilon].sub.t',v]) = 0,(t[not equal
to]t', v[not equal to]v')
Under these assumptions the unknown parameters of the econometric
model can be estimated by linear regression methods.
The econometric model for vintage price functions can be
generalized in several directions. First, the age of the durable good v
and the time period t can enter nonlinearly into the vintage price
function. Hall [1971] has proposed an analysis of variance model for the
vintage price function. In his model each age is represented by a dummy
variable equal to one for that age and zero otherwise. Similarly, each
time period can be represented by a dummy variable equal to one for that
time period and zero otherwise.
Hall's analysis of variance model for vintage price functions
can be written as
ln[p.sub.A,t,v]=[[alpha].sub.0]+[[beta].sub.v']
[D.sub.v]+[[beta].sub.t'] [D.sub.t]+[[epsilon].sub.t,v'] (t =
1,2...T; v = 0,1...),
where [D.sub.v] is a vector of dummy variables for age v and
[D.sub.t] is a vector of dummy variables for time t; [[beta].sub.v] and
[[beta].sub.t] are the corresponding vectors of parameters. In the
estimation of this model dummy variables for one vintage and one time
period can be dropped in order to obtain a matrix of observations on the
independent variables of full rank.
An alternative approach to nonlinearity has been proposed by Hulten
and Wykoff [1981b]. They transform the prices of acquisition
{[p.sub.A,t,v]}, age v, and time t by means of the Box-Cox
transformation, obtaining
[Mathematical Expression Omitted]
where the parameters [[theta].sub.p], [[theta].sub.v], and
[[theta].sub.t] can be estimated by nonlinear regression methods from
the model for vintage price functions:
[Mathematical Expression Omitted]
The model giving the logarithm of an asset price as a linear function
of age v and time period t is a limiting case of the Hulten-Wykoff model
with parameter values
[[theta].sub.p] = 0, [[theta].sub.v] = 1, [[theta].sub.t] = 1.
A third approach to nonlinearity is taken by Oliner [1993]. He
proposes to augment the linear model by introducing polynomials in age
and time. For example, a quadratic model can be represented by
ln[p.sub.A,t,v] = [[alpha].sub.0] + [[beta].sub.1,v] +
[[beta].sub.2,v] [v.sup.2] +[[beta].sub.1,t] + [[beta].sub.2,t]
[t.sup.2] + [[beta].sub.v,t] v t + [[epsilon].sub.t,v] (t = 1,2...T; v =
0,1...).
This has the advantage of flexibility in the representation of time
and age effects, but economizes on the number of unknown parameters.
A further generalization of the econometric model of vintage price
functions has been proposed by Hall [1971]. This is appropriate for
durable goods with a number of different models that are perfect
substitutes in production. Each model is characterized by a number of
technical characteristics that affect relative efficiency. We can
express the price for acquisition of new capital goods at time zero as a
function of the characteristics:
ln [p.sub.A,0] = [[alpha].sub.0] + [[beta].sub.c]' C,
where C is a vector of characteristics and pc is the corresponding
vector of parameters.
An econometric model of prices of new capital goods makes it
possible to correct these prices for quality change. Changes in quality
can be incorporated into price indices for capital goods by means of the
"hedonic technique" originated by Waugh [1929] in dealing with
the heterogeneity of agricultural commodities. This approach was first
applied to capital goods in an important study of automobile prices by
Court [1939]. A seminal article by Griliches [1961] revived the hedonic
methodology and applied it to postwar automobile prices. Chow [1967]
first applied this methodology to computer prices in research conducted
at IBM.(3)
Cole, Chen, Barquin-Stolleman, Dulberger, Helvacian, and Hodge
[1986] reported the results of a joint project conducted by the Bureau
of Economic Analysis of the Department of Commerce and IBM to construct
a constant quality price index for computers. Triplett [1986] discussed
the economic interpretation of constant quality price indices in an
accompanying article. Subsequently, the Bureau of Economic Analysis
[1986] described the introduction of constant quality price indices for
computers into the U.S. National Income and Product Accounts. A more
detailed report on the Bureau of Economic Analysis-IBM research on
computer processors is presented by Dulberger [1989], who employs speed
of processing and main memory as technical characteristics in modeling
the prices of processors. An extensive survey of research on hedonic
price indices for computers is given by Triplett [1989].(4)
Hall's [1971] methodology provides a means for determining
both a quality-corrected price index for new capital goods and relative
efficiencies for different vintages of capital goods. Substituting the
"hedonic" model of prices of net assets into the econometric
model of vintage price functions:
ln[p.sub.A,t,v] = [[alpha].sub.0] + [[beta].sub.v]' [D.sub.v]
+ [[beta].sub.t]' [D.sub.t] + [[beta].sub.c'] C +
[[epsilon].sub.t,v], (t = 1,2...T; v = 0,1...)
The unknown parameters of this model can be estimated by linear
regression methods. Hall's methodology has been applied to prices
of mainframe computers and computer peripherals by Oliner [1993; 1995a].
In Oliner's applications the chronological age of assets is
replaced by the "model" age, that is, the time that a model
has been available from the manufacturer.
An alternative approach is to substitute observations on the price
of new capital goods [p.sub.A,0] into the econometric model, so that the
dependent variable is the difference between the logarithms of new and
used assets:
ln[p.sub.,t,v] - ln[p.sub.A,0] = [[alpha].sub.0] +
[[beta].sub.v]' [D.sub.v] + [b]t]'[D.sub.t] +
[[epsilon].sub.t,v], (t = 1,2...T; v = 0,1...).
This makes it possible to separate the modeling of the vintage price
function and the quality-corrected price index of new capital goods.
Hulten, Robertson, and Wykoff [1989], Wykoff [1989], and the Office of
Tax Analysis [1990; 1991a; 1991b] employ this approach.
A further decomposition of the econometric model of asset prices
has been suggested by Biorn [1992b]. In order to isolate the vintage
effect from the other determinants of asset prices, Biorn proposes to
substitute observations on the prices of new capital goods in each time
period [p.sub.A,t] into the econometric model:
ln[p.sub.A,t,v] - ln[p.sub.A,t] = [[alpha].sub.0] +
[[beta].sub.v]' [D.sub.v] + [[epsilon].sub.t,v], (t = 1,2... T; v =
0,1 ... ).
This decomposition is implicit in the vintage price model originated
by Terborgh [1954]. The decomposition of vintage price functions is
discussed in more detail by Biorn [1992a; 1992b].
The concepts and methods employed in developing constant quality
price indices for the U.S. National Income and Product Accounts can also
be used in measuring depreciation. A constant quality price index can be
constructed by focusing on a cross section of prices on assets with
different characteristics. This approach is used in constructing price
indices for computers in the U.S. National Income and Product Accounts.
Alternatively, prices of assets of different vintages can be analyzed to
obtain estimates of depreciation suitable for introduction into the U.S.
national accounts. Finally, these two objectives can be combined in
Hall's "hedonic" model of asset prices. This model
provides a unified framework for modeling asset prices for these
different purposes.
II. STUDIES OF DEPRECIATION
To illustrate the econometric modeling of vintage price functions I
present a model implemented by Hulten and Wykoff [1981b] for eight
categories of assets in the United States. Their study includes
tractors, construction machinery, metalworking machinery, general
industrial equipment, trucks, autos, industrial buildings, and
commercial buildings. In 1977 investment expenditures on these
categories amounted to 55 percent of spending on producers' durable
equipment and 42 percent of spending on nonresidential structures.(5)
In the estimation of econometric models of vintage price functions,
the sample of used asset prices is "censored" by the
retirement of assets. The price of acquisition for assets that have been
retired is equal to zero. If only observations on surviving assets are
included in a sample of used asset prices, estimates of depreciation are
biased by excluding observations on assets that have been retired. In
order to correct this bias Hulten and Wykoff [1981b] multiply the prices
of surviving assets of each vintage by the probability of survival,
expressed as a function of age.
Vintage price functions for commercial and industrial buildings are
summarized in Table I. For each class of assets the rate of economic
depreciation is tabulated as a function of the age of the asset. The
natural logarithm of the price is regressed on age and time to obtain an
average rate of depreciation, which Hulten and WyRoff refer to as the
best geometric rate. The square of the multiple correlation coefficient (R2) is given as a measure of the goodness of fit of the geometric
approximation to the fitted vintage price function for each asset.
Vintage price functions are estimated with and without the correction
for censored sample bias.
The first conclusion that emerges from Table I is that a correction
for censored sample bias is extremely important in the estimation of
vintage price functions. The Hulten-Wykoff study is the first to employ
such a correction. The second conclusion reached by Hulten and Wykoff
[1981b, 387] is that "a constant rate of depreciation can serve as
a reasonable statistical approximation to the underlying Box-Cox rates
even though the latter are not geometric [their italics]. This result,
in turn, supports those who use the single parameter depreciation
approach in calculating capital stocks using the perpetual inventory
method."
After 1973 energy prices increased sharply and productivity growth
rates declined dramatically at both aggregate and sectoral levels. Baily
[1981] attributes part of the slowdown in economic growth to a decline
in relative efficiencies of older capital goods resulting from higher
energy prices. Hulten, Robertson, and Wykoff [1989] test the stability
of vintage price functions during the 1970s. Wykoff [1989] analyzes
price data for four models of business-use automobiles collected from a
large leasing company, applying the Hulten-Wykoff methodology.
Hulten, Robertson, and Wykoff [1989, 255] have carefully documented
the fact that the relative efficiency functions for nine types of
producers' durable equipment were unaffected by higher energy
prices: "While depreciation almost certainly varies from year to
year in response to a variety of factors, we have found that a major
event like the energy crises, which had the potential of significantly
increasing the rate of obsolescence, did not in fact result in a
systematic change in age-price profiles." They also conclude that
"the use of a single number to characterize the process of economic
depreciation [of a given type of asset] seems justified in light of the
results."
Table II presents rates of economic depreciation derived from the
best geometric approximations of Hulten and Wykoff [1981b] for
thirty-four categories of nonresidential business assets and one
category of residential assets. These rates were extended to include
public utility and residential property by Jorgenson and Sullivan
[1981]. Hulten and Wykoff compare the best geometric depreciation rates
presented in Table 1 with depreciation rates employed by the Bureau of
Economic Analysis [1977] in estimating capital stock. The Hulten-Wykoff
rate for equipment averages 0.133, while the Bureau's rate averages
0.141, so that the two rates are very similar. The Hulten-Wykoff rate
for structures is 0.037, while the Bureau's rate is 0.060, so that
these rates are substantially different.
Table 1
Rates of Economic Depreciation
Without Censored Without Censored
Sample Correction Sample Correction
Age Commercial Industrial Commercial Industrial
5 2.85 2.99 2.66 2.02
10 2.64 3.01 1.84 1.68
15 2.43 3.04 1.48 1.50
20 2.30 3.07 1.27 1.39
30 2.15 3.15 1.02 1.25
40 2.08 3.24 0.88 1.17
50 2.04 3.34 0.79 1.11
60 2.02 3.45 0.72 1.06
70 2.02 3.57 0.66 1.03
Best Geometric
Rate 2.47 3.61 1.05 1.28
[R.sup.2] 0.985 0.997 0.971 0.995
Source: Hulten and Wykoff [1981], Table 5, p. 387; commercial corresponds to
office, industrial corresponds to factory.
The distribution of retirements used by Hulten and Wykoff [1981b]
to correct for censored sample bias are based on the Winfrey [1935] S-3
curve with Bureau of Economic Analysis [1977] lifetimes. These lifetimes
are taken, in turn, from Bulletin F, compiled by the Internal Revenue
Service and published in 1942. Between 1971 and 1931 the Office of
Industrial Economics conducted forty-six studies of survival
probabilities, based on vintage accounts for assets reported under the
Asset Depreciation Range System introduced in 1962. The results of
twenty-seven of these studies have been summarized by Brazell, Dworin,
and Walsh [1989]. These results provide estimates of the distribution of
useful lives based on the actual retention periods for the assets
examined. A very important objective for future research on vintage
price functions is to incorporate information from the Office of
Industrial Studies research into corrections for sample selection bias.
Before 1981 the tax law linked tax depreciation to retirement of
assets. Between 1981 and 1986 the Accelerated Cost Recovery System severed the link between tax depreciation and economic depreciation
altogether. The Tax Reform Act of 1986 re-instituted tax depreciation
based on economic depreciation.(6) Under the 1986 Act the Office of Tax
Analysis was mandated by the Congress to undertake empirical studies of
economic depreciation, including "the anticipated decline in value
over time,"(7) and report the results in the form of a useful life.
This is the lifetime for straight-line depreciation that yields the same
present value as economic depreciation. For this purpose the Office of
Tax Analysis [1990; 1991a; 1991b] conducted major surveys of used asset
prices and retirements for scientific instruments, business-use
passenger cars, and business-use light trucks and analyzed the results.
Oliner [1995b] conducts an extensive survey of used asset prices
and retirement patterns for machine tools with the assistance of the
Machinery Dealers National Association. He compares his results with
those from previous empirical studies of economic depreciation for this
industry, including those of Beidleman [1976] and Hulten and Wykoff
[1981b]. Finally, he compares estimates of depreciation and capital
stock with those of the Bureau of Economic Analysis's 1987 Capital
Stock Study. Oliner's study, like those of the Office of Tax
Analysis, combines information on used asset prices and retirements for
the same or similar populations of assets. This is an important advance
over previous studies based on the vintage price approach.
Oliner [1993] collects and analyzes used asset prices for IBM
mainframe computers and estimates constant-quality price change and
economic depreciation simultaneously. Previous studies of computer
prices, such as the studies surveyed by Triplett [1989], were limited to
constant quality price change. The primary data source for computer
prices used by Oliner is the Computer Price Guide, published by Computer
Merchants, Inc. The data on retirement patterns were obtained from data
on the installed stock of IBM computers tabulated by the International
Data Corporation. Oliner [1995a] conducts a similar study of computer
peripherals--large and intermediate disk drives, printers, displays, and
card readers and punches. The prices of used assets are based on the
Computer Price Guide and estimates of retirement patterns are inferred
from the duration of price listings.
An alternative to the vintage price approach is to employ rental
prices rather than prices of acquisition to estimate the pattern of
decline in efficiency.(8) This approach is employed by Malpezzi, Ozanne,
and Thibodeau [1987] to analyze rental price data on residential
structures and by Taubman and Rasche [1969] to study rental price data
on commercial structures. While leases on residential property are very
frequently one year or less in duration, leases on commercial property
are typically for much longer periods of time. Since the rental prices
are constant over the period of the lease, estimates based on annual
rental prices for commercial property are biased toward the
one-hoss-shay pattern found by Taubman and Rasche; Malpezzi, Ozanne, and
Thibodeau find rental price profiles for residential property that
decline with age.
A second alternative to the vintage price approach originated by
Meyer and Kuh [1957] is to analyze investment for replacement purposes.
Coen [1980] compares the explanatory power of alternative patterns of
decline in efficiency in a model of investment behavior that also
includes the price of capital services. For equipment he finds that
eleven of twenty-one two-digit manufacturing industries are
characterized by geometric decline in efficiency, three by sum of the
years' digits and seven by straight-line. For structures he finds
that fourteen industries are characterized by geometric decline, five by
straight-line and two by one-hoss-shay patterns. Hulten and Wykoff
[1981c, 110] conclude that: "The weight of Coen's study is
evidently on the side of the geometric and near-geometric forms of
depreciation."
Alternative approaches for analyzing investment for replacement
purposes are employed by Pakes and Griliches [1984] and Doms [1994].
Pakes and Griliches relate profits for U.S. manufacturing firms to past
investment expenditures. The weights on investments of different ages
can be interpreted as the relative efficiencies of these assets. Doms
includes a weighted average of past investment expenditures in a
production function. Treating the weights as unknown parameters to be
estimated, he obtains estimates of relative efficiencies of assets.
While Pakes and Griliches find patterns of relative efficiencies that
rise and then decline, Doms obtains relative efficiencies that decline
geometrically.
Empirical research on depreciation is now available for the
principal categories of assets included in the U.S. National Income and
Product Accounts. The most extensive body of research is that of Hulten
and Wykoff [1981a; 1981b; 1981c], Hulten, Robertson, and Wykoff [1989],
and Wykoff [1989]. However, important additional studies have been
completed by the Office of Tax Analysis [1990; 1991a; 1991b] and Oliner
[1993; 1995a; 1995b]. Finally, estimates of retirement distributions
required for correcting sample selection bias have been completed by the
Office of Industrial Economics and summarized by Brazell, Dworin, and
Walsh [1989]. Taken together, these empirical studies of depreciation
provide the information needed to revise the treatment of depreciation
in the national accounts.
III. APPLICATIONS
The measurement of depreciation has been an important objective of
research at the Bureau of Economic Analysis for several decades.
National income, net of depreciation, is included in the U.S. National
Income and Product Accounts. Stocks of depreciable assets are estimated
as a component of national wealth. The culmination of this research was
the magisterial Bureau of Economic Analysis's [1987] study, Fixed
Reproducible Tangible Wealth in the United States, 1925-85, updated in
1993 to cover the period 1925-89. Perhaps surprisingly, the
Bureau's estimates of capital stocks and depreciation do not
incorporate the results of the extensive empirical literature on vintage
price functions that has been summarized in section II.(9)
The Bureau employs measures of capital stocks for equipment and
structures with relative efficiencies that are constant over the
lifetime of each capital good. This produces a measure of gross capital
stock. Depreciation based on the straight-line method is used to produce
a measure of net capital stock. The first issue is whether the
Bureau's [1987; 1993] studies are internally consistent. This issue
arises because patterns of decline in efficiency of assets are used in
estimating both depreciation and capital stocks, but these patterns can
be the same or different. Obviously, internal consistency requires that
the same relative efficiencies be used in both sets of estimates.(10)
If we consider economic depreciation under the assumption that
assets do not decline in efficiency until the end of their useful lives,
we can simplify the analysis by assuming that rates of return
{[r.sub.t+s]} and prices of capital services {[p.sub.k,t+[tau]]} are
constant to obtain depreciation on an asset of age v:
[TABULAR DATA OMITTED]
Under the Bureau's assumption that relative efficiencies of
assets of different ages are constant over the lifetimes of assets,
economic depreciation declines geometrically. This reflects the time
discounting of the retirement of an asset.
The Bureau's straight-line estimates of depreciation obviously
reflect a different pattern of relative efficiencies than that employed
in estimating capital stock, so that its measurements of depreciation
and capital stock are internally inconsistent. This issue was first
analyzed in detail by Jorgenson and Griliches [1972a, esp. pp. 81-87].
Denison [1972, esp. pp. 101-109] defends the Bureau's use of
straight-line depreciation. However, he fails to respond to the
criticism that straight-line depreciation is inconsistent with the
relative efficiencies of capital goods of different ages that the Bureau
employs in measuring capital stock.
The logical inconsistencies in the Bureau's methodology arise
from the definition of depreciation. Denison's [1972, 104-105]
defense of the straight-line formula is based on the "capital input
definition" of depreciation that he originally introduced in 1957.
This definition was subsequently adopted in the U.S. National Income and
Product Accounts. The capital input definition is described by Young and
Musgrave [1980, 32] as follows: "Depreciation is the cost of the
asset allocated over its service life in proportion to its estimated
service at each date." Within the vintage accounting framework
presented in section I, Denison's capital input definition of
depreciation allocates the cost of an asset over its lifetime in
proportion to the relative efficiencies of capital goods of different
ages.(11)
Young and Musgrave [1980, 33-37] contrast the Denison definition
with the "discounted value definition" employed in the vintage
accounting system outlined in section I. Among the advantages for the
capital input definition claimed by Denison [1957, 240] and by Young and
Musgrave [1980, 33] is that this definition avoids discounting of future
capital services. In fact, discounting can be avoided in the measurement
of depreciation only if the decline in the efficiency of capital goods
is geometric. The Bureau of Economic Analysis's assumption that
efficiency is constant over the lifetime of an asset requires
discounting, as I have already demonstrated. Only if relative
efficiencies decline geometrically does economic depreciation coincide
with Denison's capital input definition. But this definition then
implies declining balance rather than straight-line depreciation.
The second issue that arises in the measurement of depreciation is
the incorporation of up-to-date information from empirical studies like
those summarized in section II. The available empirical evidence on
efficiency functions for different classes of assets, growing out of the
work of Hulten and Wykoff [1981a; 1981b; 1981c], is very extensive. This
evidence includes the results of econometric modeling of vintage price
functions, like those of Hulten, Wykoff, and Robertson [1989], Wykoff
[1989], Office of Tax Analysis [1990; 1991a; 1991b], and Oliner [1993;
1995a; 1995b]. It also includes studies of retirement patterns like
those developed by the Office of Industrial Economics and summarized by
Brazell, Dworin, and Walsh [1989]. Both types of information are needed
for the measurement of depreciation.
The Bureau of Economic Analysis [1987; 1993] employs only
information about retirements in measuring depreciation. This is a
consequence of applying the "capital input definition" of
depreciation under the assumption that relative efficiencies of assets
are constant over the lifetime of each capital good. This assumption is
contradicted by the empirical evidence on depreciation I have reviewed
in section II. The incorporation of this evidence into the U.S. National
Income and Product Accounts requires that the Bureau's definition
of depreciation be replaced by the "discounted value
definition" presented in section I.
A vintage accounting system for prices and quantities of investment
goods, such as that originated by Christensen and Jorgenson [1973],
provides an internally consistent framework for measuring capital stocks
and depreciation. Christensen and Jorgenson use this framework in
constructing an integrated system of income, product and wealth
accounts. They distinguish between two alternative measures of economic
performance, identifying the production approach with the production
possibility frontier employed by Jorgenson and Griliches [1967]. This
approach is implemented by means of a production account with an
accounting identity between the values of outputs and inputs. These data
are used to allocate the growth of output between the growth of capital
and labor inputs and productivity growth.
Christensen and Jorgenson have also described a welfare approach to
the measurement of economic performance, based on a social welfare
function like that employed by Jorgenson and Yun [1991a]. This approach
can be implemented by means of an income and expenditure account with an
accounting identity between income net of depreciation and the sum of
saving and consumption. Data from this account can be used to allocate
growth of income between the growth of current consumption and the
growth of future consumption through saving. Both production and income
and expenditure accounts are essential components of the U.S. National
Income and Product Accounts and accounting systems that employ the
United Nations [1968] System of National Accounts.
Saving is linked to the asset side of the wealth account through
capital accumulation equations for each asset like those presented in
section I. These equations provide a perpetual inventory of assets
accumulated at different points in time. Prices for different vintages
of investment goods are linked to the prices of capital services through
a parallel set of asset pricing equations, like those presented in
section I. The complete system of vintage accounts gives stocks of
assets of each age and the corresponding asset prices for each period.
The stocks can be cumulated to obtain quantities of assets, while the
prices can be used to value the stocks and derive rental prices for
services of the assets and measures of depreciation.
Hulten [1992] has formalized the welfare and production approaches
to economic performance by means of a model of optimal economic growth.
This model includes a production function with output as a function of
capital and labor inputs. Output is divided between consumption, which
contributes to the welfare of a representative consumer, and saving,
which contributes to future consumption through capital accumulation.
Income net of depreciation measures the welfare resulting from
intertemporal optimization of consumption in Hulten's model of
growth, as in a similar model proposed by Weitzman [1976]. This measure
of welfare summarizes the stream of present and future consumption.
Hulten shows that gross product is the measure of output
appropriate for separating the growth of output between productivity
growth and the growth of capital and labor inputs, while national income
net of depreciation is appropriate for allocating the growth of income
between consumption growth and contributions to the growth of future
consumption through saving. A complete system of accounts includes a
production account, an income and expenditure account, and a wealth
account. Measures of depreciation employed in all three accounts can be
generated in an internally consistent way from the system of vintage
accounts for assets outlined in section I.
The income account of an integrated system of national accounts can
be used in measuring welfare. Hulten [1992] points out that the
Haig-Simons definition of taxable income requires that capital cost
recovery for tax purposes must equal economic depreciation. Capital cost
recovery for tax purposes has differed from economic depreciation
whenever capital consumption allowances and the investment tax credit
have been used to provide tax incentives for private investment, as in
the Accelerated Cost Recovery System during the period 1981-1986. Under
U.S. tax law capital recovery is based on the original acquisition price
of an asset rather than current replacement cost. During periods of high
inflation, the original acquisition price and the current replacement
cost have diverged substantially.
Making use of the asset classification scheme of the Bureau of
Economic Affairs's [1987] capital stock study for individual
industries, Jorgenson and Yun [1991b] map the economic depreciation
rates for the thirty-five asset categories given in Table II, into the
Bureau's 1987 asset classification scheme to incorporate additional
information about asset lives. Jorgenson and Yun [1991b] employ data on
economic depreciation and capital cost recovery under U.S. tax law to
summarize the tax burden on capital income. For this purpose they employ
the marginal effective tax rate introduced by Auerbach and Jorgenson
[1980]. An effective tax rate represents the complex provisions of tax
law in terms of a single ad valorem rate.
TABLE II
Rates of Economic Depreciation: Business Assets
Asset Depreciation Rate
Producer Durable Equipment
1. Furniture and fixtures 0.1100
2. Fabricated metal products 0.0917
3. Engines and turbines 0.0786
4. Tractors 0.1633
5. Agricultural machinery 0.0971
6. Construction machinery 0.1722
7. Mining and oil field machinery 0.1650
8. Metalworking machinery 0.1225
9. Special industry machinery 0.1031
10. General industrial equipment 0.1225
11. Office, computing and accounting machinery 0.2729
12. Service industry machinery 0.1650
13. Electrical machinery 0.1179
14. Trucks, buses and truck trailers 0.2537
15. Autos 0.3333
16. Aircraft 0.1833
17. Ships and boats 0.0750
18. Railroad equipment 0.0660
19. Instruments 0.1500
20. Other equipment 0.1500
Nonresidential Structures
21. Industrial buildings 0.0361
22. Commercial buildings 0.0247
23. Religious buildings 0.0188
24. Educational buildings 0.0188
25. Hospital and institutional buildings 0.0233
26. Other nonfarm buildings 0.0454
27. Railroad structures 0.0176
28. Telephone and telegraph structures 0.0333
29. Electric light and power structures 0.0300
30. Gas structures 0.0300
31. Other public utility structures 0.0450
32. Farm structures 0.0237
33. Mining exploration, shaft and wells 0.0563
34. Other nonbuilding structures 0.0290
Residential Structures
35. Residential structures 0.0130
Source: Jorgenson and Sullivan [1981], Table I.
The production account of an integrated system of income, product,
and wealth accounts can be used in measuring productivity. The estimates
of depreciation by Jorgenson and Yun [1991b] have been incorporated into
price and quantity indices of capital services for productivity
measurement by Jorgenson [1990]. The underlying estimates of capital
stocks and rental prices are classified by four asset
classes--producers' durable equipment, nonresidential construction,
inventories, and land--and three legal forms of organization--corporate
and noncorporate business and nonprofit enterprises. This study is based
on annual data for the period 1947-1985 for an average of as many as 156
components of capital services for each of thirty-five industries. These
data incorporate investment data from the Bureau of Economic
Analysis's [1987] study of U.S. national wealth.
In constructing data on capital input for each of the thirty-five
industrial sectors, Jorgenson [1990] combines price and quantity data
for different classes of assets and legal forms of organization by
expressing sectoral capital services, say {[K.sub.i]}, as a translog
function of its 156 individual components, say {[K.sub.ki]}. The
corresponding index of sectoral capital services is a translog quantity
index of individual capital services:
[Mathematical Expression Omitted] where weights are given by
average shares of each component in the value of sectoral property
compensation:
[Mathematical Expression Omitted]
(i = 1,2 ... n; k = 1,2 ... p),
and
[Mathematical Expression Omitted]
(i = 1,2...n; k = 1,2...p).
The value shares are computed from data on capital services
{[K.sub.ki]}and the rental price of capital services {[Mathematical
Expression Omitted]}, cross-classified by asset class and legal form of
organization.
Internal consistency of a measure of capital input requires that
the same pattern of relative efficiency is employed in measuring both
capital stock and the rental price of capital services. The decline in
efficiency affects both the level of capital stock and the depreciation
component of the corresponding rental price. Estimates of capital stocks
and rental prices that underlie the data presented by Jorgenson [1990]
are based on geometrically declining relative efficiencies. The same
patterns of decline in efficiency are used for both capital stock and
depreciation, so that the requirement for internal consistency of price
and quantity indices of capital services is met.
The Bureau of Labor Statistics [1983, 57-59] also employs relative
efficiency functions estimated by Hulten and Wykoff. However, the Bureau
of Labor Statistics does not utilize the geometric relative efficiency
functions fitted by Hulten and Wykoff. Instead, it has fitted a set of
hyperbolic functions to Hulten and Wykoff's relative efficiency
functions. Consistency is preserved between the resulting estimates of
capital stocks and rental prices by implementing a system of vintage
price accounts for each class of assets. This set of accounts includes
asset prices and quantities of investment goods of all ages at each
point of time. The Bureau of Labor Statistics [1983, 57-59] shows that
measures of capital services based on hyperbolic and geometric relative
efficiency functions are very similar.
The Bureau of Labor Statistics [1991] expands annual productivity
estimates reported for private business, private nonfarm business, and
manufacturing to include measures of capital services that differ among
fifty-seven industries. For each industry, capital service prices and
quantities are estimated for seventy-two types of depreciable assets.
The Economic Research Service [1991] publishes official productivity
estimates for agriculture that include prices and quantities of capital
services, based on the approach of Ball [1985]. The methodology for
depreciable assets, except for breeding livestock, is similar to that
employed by the Bureau of Labor Statistics. Data on prices and
quantities of capital services for breeding livestock have been
constructed by the vintage accounting approach developed by Ball and
Harper [1990].(12)
I conclude that the U.S. National Income and Product Accounts fail
to provide internally consistent measures of capital stocks and
depreciation. Top priority should be given to replacing the
"capital input definition" of depreciation by the
"discounted value definition" presented in section I. This has
important implications for improving the income, product, and wealth
measures now included in the U.S. national accounts. Second, estimates
of depreciation should be enhanced by incorporating the information from
studies of vintage price functions that are summarized in section II.
These two tasks can be accomplished by utilizing concepts and methods
already familiar to economic statisticians as a result of the
introduction of constant quality prices for computers into the national
accounts.
IV. CONCLUSION
Hall's [1971] "hedonic" model of vintage asset
prices, outlined in section I, has proved to be an indispensable guide
to empirical research. The applications surveyed here are based on the
pioneering studies of Hulten and Wykoff [1981a; 1981b; 1981c], who have
constructed vintage price functions covering a sizable proportion of
U.S. investment expenditures. Alternative methodologies, such as the
rental price approach and the modeling of investment for replacement
purposes, have been superseded by the vintage price approach with a
correction for sample bias originated by Hulten and Wykoff.
I conclude that the results of empirical research on depreciation
should be incorporated into the U.S. National Income and Product
Accounts. The research of Hulten and Wykoff has been used successfully
in two important applications of depreciation--the empirical description
of the U.S. tax system for depreciable assets by Jorgenson and Yun
[1991b] and the measurement of capital services and rental prices for
studies of productivity by Jorgenson [1990] and the Bureau of Labor
Statistics [1991]. Information from the research summarized in section
II should be combined with the detailed data on investment flows by
class of asset and industry produced by the Bureau of Economic Analysis
[1987; 1993] to improve measures of U.S. national income, product and
wealth.
While the Bureau of Economic Analysis [1987; 1993] provides
estimates of capital stocks and economic depreciation, the two sets of
estimates are logically inconsistent since they are based, implicitly,
on different assumptions about the relative efficiencies of assets of
different ages. Gross capital stocks are based on unchanging efficiencies throughout the lifetime of an asset. This implies a
declining balance method of depreciation that reflects the time
discounting of retirements. However, the Bureau of Economic
Analysis's measures of depreciation are based on the straightline
method. The first step toward removing these inconsistencies is to adopt
the discounted value definition of depreciation presented in section I.
My overall conclusion is that top priority in future applications
of empirical studies of depreciation should be assigned to developing
internally consistent measures of national income and wealth. An equally
important priority is to utilize the results of empirical studies of
depreciation growing out of the work of Hulten and Wykoff. The
appropriate conceptual framework for both of these important tasks is
provided by the system of vintage price and quantity accounts introduced
by Christensen and Jorgenson. This vintage accounting system is the key
to integrating income and product accounts with wealth accounts and
incorporating relative efficiencies of assets of different ages based on
econometric estimates of vintage price functions.
(1.) The system of vintage accounts summarized here is discussed in
greater generality by Christensen and Jorgenson [1973] and Jorgenson
[1980]. The model of capital as a factor of production that underlies
this system is discussed by Diewert [1980], Hulten [1990], Jorgenson
[1973; 1980; 1989], and Triplett [1995]. (3.) Surveys of the hedonic
technique are given by Triplett [1975;1987; 1990]. (4.) Gordon [1989]
has presented an alternative constant quality price index for computers.
This was incorporated into Gordon's [1990] study of constant
quality price indices for all components of producers' durable
equipment in the U.S. national accounts. (5.) Hulten and Wykoff [1981b]
estimated vintage price functions for structures from a sample of 8066
observations on market transactions in used structures. These data were
collected by the Office of Industrial Economics of the U.S. Department
of the Treasury in 1972 and were published in Business Building
Statistics [1975]. They estimated vintage price functions for equipment
from prices of machine tools collected by Beidleman [1976] and prices of
other types of equipment collected from used equipment dealers and
auction reports of the U.S. General Services Administration. (6.)
Detailed histories of U.S. tax policy for capital recovery are presented
by Brazell, Dworin, and Walsh 11989] and Jorgenson and Yun [1991b]. (7.)
Joint Committee on Taxation [1986,103]. (8.) Hulten and Wykoff [1981c]
summarize studies of economic depreciation completed prior to their own
study Vintage price functions have provided the most common methodology
for such studies. (9.) The methodology for constructing estimates of
depreciation and the corresponding capital stocks is described by the
Bureau of Economic Analysis [1987; 1993]. (10.) The Bureau's
methodology is discussed in greater detail by Hulten and Wykoff [1981c]
and Wykoff [1989]. The incorporation of retirement patterns described in
section II, below, results in a decline m efficiency with age after
retirements begin. However, this does not affect my conclusion that the
Bureau's methodology is internally inconsistent. (11.) The capital
input definition has been categorically rejected by economists outside
the Bureau of Economic Analysis. See the comments on Denison [1957] by
Kuznets [1957] comments on Denison [1972] by Jorgenson and Griliches
[1972b], and comments on Young and Musgrave [1980] by Faucett [1980].
(12.) Boskin et al. [1989a, 1989b] successfully employed the vintage
accounting approach in measuring capital stocks and depreciation for the
government sector of the U.S. economy. Jorgenson and Fraumeni [1989]
apply this approach to the measurement of stocks and depreciation of
human capital.
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