BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF PRODUCTIVITY AND TECHNOLOGY

The Measurement of Productive Capital Stock, Capital Wealth, and Capital Services

Michael J. Harper, U.S. Bureau of Labor Statistics

Working Paper 128

June 1982

 

The author wishes to thank the many BLS staff members who have reviewed and commented on this paper, particularly

Dick McDonald and Leo Sveikauskas who offered detailed suggestions for its improvement. The views expressed are those of

the author and do not reflect the policies of the Bureau of Labor Statistics or the views of other BLS sfaff members.
The Measurement of Productive Capital Stock, 

Capital Wealth, and Capital Services 

by Michael J. Harper* 

Bureau of Labor Statistics 

-BWorking Paper 128 - 

June, 1982 

This paper is circulated for the information of members of the staff of the 

Bureau of Labor Statistics and other professionals interested in the work of 

the Bureau. It has not been cleared for publication and should not be quoted 

without the pemission of the author. 
.-,-.-- .-_. _ __.^__...______ -... -.-.--- . ..__. -_._ 

Abstract 

The Measurement of Productive Capital Stock, 

Capital Wealth, and Capital Services 

by: Michael J. Harper 

In order to construct measures of multifactor productivty, the Bureau of 

Labor Statistics has investigated a number of issues. This paper discusses 

several related to the vintage aggregation of capital: a primary step in the 

measurement of capital input. 

Unless a capital good's efficiency declines geometrically, its price as 

it ages will follow a different schedule than its efficiency. The price 

schedule can be calculated if we assume an efficiency schedule, a discount 

rate, and the vintage aggregation conditions. Capital services are proportional to a "productive stock" constructed from a perpetual inventory 

calculation using the age/efficiency schedule. The wealth represented by all 

assets is consistently estimated by doing a similar calculation using the 

corresponding aye/price schedule. Attempts in the literature to establish that 

age/efficfency schedules are geometric by studying used asset prices fall short 

of doing so, because very different age/efficiency schedules can generate very 

similar age/price profiles. Arguments which rationalize the use of the 

geometric assumption even when efficiency does not decay geometrically lack 

merit. All thfngs considered, there is little evfdence regarding which 

age/efficiency pattern is correct, but mounting evidence that the vintage 

aggregation conditions are badly violated. 
The Measurement of Productive Capital Stock, 

Capital 'Aealth, and Capital Services 

by Michael J. Harper* 

Bureau of Labor Statistics 

I. Introduction 

In capital theory, a careful distinction is made between declines in the 

efficiency of an asset and the depreciation of that asset. The efficiency of a 

used asset relative to a new one is defined as the marginal rate of technical 

substitution of the old asset for the new one. Depreciation is the change in 

the real price of an asset. That price reflects changes in the discounted 

stream of future services which can be expected from the asset. These. two 

concepts of decline are strictly related in theory. 

This relationship is well illustrated by a light bulb. If the bulb burns 

steadily for 1000 hours before the filament breaks, it loses no efficiency 

until its final hour. Yet, since inflation and interest are minor factors in 

. 

the 6 odd weeks of its life, Its value can be expected to decline in a straight 

line pattern. That is, after 500 hours, it should be worth one half what a 

new bulb costs (assuming changing and selling of bulbs is costless). In this 

case efficiency follows a one hoss shay pattern and price a straight line 

pattern. The efficiency function of an asset might take any conceivable form. 

Economists usually define initial efficiency as 100% and assume efficiency 

declines monotonically, approaching zero eventually. Besides the patterns 

we've mentioned, geometric decay functions, and a whole family of hyperbolic 

functions meet these criteria. Others are possible as well. 

Economists frequently assume that the relative efficiency of an asset '. 

declines geometrically over time. This assumption has several advantages. 

The author wishes to thank the many BLS staff members who have reviewed and 

commented on this paper, particularly Dick McDonald and Leo Sveikauskas who 

offered detailed suggestions for its improvement. 
-Ltinder geometric efficiency decline, depreciation is also geometric and proceeds 

at the same constant rate. Therefore, no distinction need be made in the 

measurement process. Furthermore, because of the constancy of this rate, no 

system of vintage accounting needs to be maintained. Finally, under geometric 

efficiency decline, depreciation is independent of the real own rate of 

interest. These properties are enjoyed only by the geometric form. In general 

efficiency decline and depreciation are different, their relationship depending 

on real interest rates. If one form is not geometric, neither is the other. 

Thus it is not surprising that many economists have made attempts to 

establish whether geometric decay is consistent with the behavior of the real 

prices of used capital assets. These studies generally show that depreciation 

is relatively more rapid than straight line early in the life of an asset. 

These studies also often fail to reject the geometric form of depreciation 

statistically. They often conclude that this evidence suggests that geometric 

decay is not an unrealistic assumption for measuring depreciation or for 

measuring the efficiency of the capital stock. Jorgenson (1974) marshals a 

further argument based on renewal theory. Essentially, for a capital stock 

characterized by a stable age distribution and a stable growth pattern, 

replacement requirements tend to a constant rate regardless of the underlying 

efficiency patterns. 

Counter to these views is the argument, based often on intuition, that the 

productive efficiency of assets does not decline nearly so rapidly as the 

geometric hypothesis requires. However, available studies of this type are 

vulnerable on theoretical grounds. 
-... _- .____ Mw_“--- .-... I.- -.. ._# __._... 

-3- 

The purpose of this paper is to examine the issues related to the 

computation of a vintage aggregate. A central theme is the comparison of 

various alternative mathematical forms for the age/efficiency profile which 

must be assumed in order to build a vintage aggregaate. In Section II the 

theory of capital measurement is reviewed with particular attention to the 

duality between the efficiency and price of an asset. The age/efficiency 

profile must be used to estimate an asset's productive stock, to which services 

are proportional. The age/price profile must be used to estimate a stock of 

wealth, from which depreciation may be inferred. Both services and 

depreciation rmst be estimated for each type of asset in order to distinguish 

and aggregate asset classes of differing durabilities using modern procedures 

based on the rental price of capital. The assumption of geometric decay, which 

has been used in the past in conjunction with theoretically complete efforts to 

measure capital, avoids the distinction between productive capital and capital 

wealth. However, the existing vintage aggregation model is general enough that 

such simplification is unnecessary and incorrect if efficiency does not really 

decline geometrically. 

In Section III we construct .eight different efficiency functions and then 

calculate their dual age/price profiles under a variety of interest rate, asset 

life, and discard pattern assumptions. Several observations are made which are 

important for our examination of the issues in Section IV. There we evaluate 

arguments which have been made in favor of particular efficiency functions. A 

major point raised is that the empirical studies, which claim to support 

geometric decay, fall short of doing so. That work is based on estimating 
-4- 

age/price profiles. A wide range of efficiency patterns which are concave to 

the origin are associated with convex age/price profiles. Existing empirical 

studies have by no means estimated age/price profiles well enough to identify 

the age/efficiency pattern. In addition we evaluate the renewal theory 

arguments which rationalize use of geometric decay even if assets do not decay 

geometrically. Using data for the U.S. Manufacturing sector, we demonstrate 

important problems which can result if efficiency does not in fact decay 

geometrically. We also introduce an analytic statistic, the ratio of implicit 

rent to capital cost, as a possible criterion for evaluating various efficiency 

assumptions, and, more fundamentally, for detecting when the entire vintage 

aggregation procedure is failing. 

In Section V, we conclude that the existing literature and evidence offers 

little basis for preferring any one efficiency .form. In this complicated area, 

it would seem common sense should carry as much weight as any evidence which D 

economists have yet assembled. 
______ .-__--__ _ _.____.______._.._-..._. I_- __..,_. . ~. . ---...--_-_ 2.. 

II. Depreciation and the Theory of Capital Measurement 

In this section we review a model for measuring capital services. This 

model consists of two major steps: the aggregation of past investments for a 

particular type of capital asset, and the aggregaton of different types of 

assets into an overall "measure" of capital. Regarding the first, or vintage 

aggregation step, we appeal to the duality between an asset's efficiency and 

real price in order to propose a distinction in types of vintage aggregates. 

One can construct a "productive stock" based on the age/efficiency function, 

and a "wealth stock" based on the age/price profile. This distinction 

evaporates only if we presume geometric decay. Regarding the asset aggregation 

step, we point out that the rental price formulation, which is used to weight 

differing assets, and which has previously been used only in tandem with the 

geometric assumption, can easily be used with other efficiency assumptions. 

The capital stock of .a particular type of asset is generally measured as a 

weighted average of past investments where the weights reflect the relative 

efficiencies of various capital vintages. These weights are conventionally 

normalized to one for new investment, with monotonic declines as the asset 

ages. We shall refer to this as vintage aggregation. Fisher (1965) has shown 

that capital augmenting technical change, or an equivalent assumption, is a 

precondition for consistent vintage aggregation. In one form of the 

assumption, it is evident that vintage weights must be independent of exogenous 

influences such as output demand and relative factor prices. More generally, 

Hall (1968) has“shown that some normalization rule is required in order to 

distinguish the vintage weights from exponential rates of embodied and 

disembodied technical change. Hall suggests that fixing the relative 

efficiency of assets of different ages as time passes is the most useful rule, 
-6- 

The process of constructing a vintage aggregate based on past inveslxents 

weighted by relative efficiencies is frequently called the perpetual inventory 

method. We shall refer to the resulting stock measure as the "productive 

stock" of capital. It has a clear interpretation as the answer to the 

following question. "How much new investment would be required to produce the 

same present services as the existing stock of assets?" Age/efficiency 

functions should be constructed in such a way that the real capital services, 

and implicitly the rents, generated by capital of various vintages in a given 

period are proportional to their relative efficiencies. Under these 

conditions, the productive stock is proportional to capital services. Hence it 

is the correct measure of the real capital input quantity for a particular 

asset class for productivity measurement and econometric production function 

analysis. 

In order to measure productivity or otherwise model production, it is 

necessary to construct a rental price for capital. If there were only one type 

of capital, we could use the assumption that factors of production are paid 

their marginal products, and simply divide the productive stock into total 

capital income lie.. before tax profits plus net interest plus depreciation plus 

subsidies plus indirect business taxes) to arrive at an estimate of the current 

rental price. Christensen and Jorgenson (1969) derive a more complicated 

method for the case where-there are several assets of different durabilities. 

Essentially, relatively more of current capital income should be allocated to 

less durable assets because their rapid depreciation entails more cost. In 

other words, the faster the depreciation of an asset class, the higher its 

rental price. 
-7- 

Therefore, in order b construct rental prices, it is necessary to 

estimate depreciation by type of asset. Unfortunateiy, depreciation cannot 

generally be directly inferred from the productive capital stock measure. In 

order to measure depreciation, it is necessary to recognize the duality between 

an asset's agejefficiency function and its age/price function or profile. An 

asset's market price can be expected to equal the discounted stream of services 

(rents) which it will generate. As Arrow (1964) and Hall (1968) have shown, 

the age/price profile can be derived from the age/efficiency function by 

integrating the later (weighted by a discount factor) for assets of various 

ages. Jorgenson (1974) replicates this work for the discrete case. In Section 

III we study several simulat ions of age/pr ice profiles. For now it is 

sufficient to point out that the age/price profile is different from the 

age/efficiency function unless geometric decay is assumed. 

Once an age/price profile is derived, it is possible to perform a second 

perpetual inventory calculation, weighting past investments with the age/price 

profile. We refer to the resultant measure as the "stock of capital wealth" or 

the "wealth stock". This measure can be interpreted as the real value of the 

existing stock of assets, based on the discounted stream of future services 

expected from them. Depreciation can be inferred directly from this second 

perpetual inventory calculation as the drop in value of the previous period's 

wealth stock before adding in new investment. The following exemplifies the 

distinction between the productive and wealth stocks. Inflation and 

technological change are disregarded. Suppose your house is '. 
-8- 

broken into and a five year old television, which is in excellent condition, is 

taken. Your insurance company may reimburse you for the "fair market value" of 

the item, that is, they will allow for inflation in TV prices, but will deduct 

for depreciation. If you wish to continue watching television, and lack access 

to or distrust the used television market, you may prefer %o purchase a new 

set. The chances are the new set will cost more than you were reimbursed. The 

old set should have been included in the wealth stock at its fair market 

value. The old set should have been included in the productive stock at the 

value it costs to exactly replace its services with a new set. 

Of course these distinctions between productive and wealth stocks and 

efficiency loss and depreciation are unnecessary in capital studies which 

presume geometric decay. The Christensen-Jorgenson (1969) study, and 

subsequent studies including Gollop-Jorgenson (1980) and Frameni-Jorgenson 

(1980) have made the simplifying assumption of geometric decay. However, the 

Hall (1968) or Jorgenson (1974) papers assume a general form of efficiency 

decline, and formulate rental prices in terms of the rate of depreciation as 

distinct from the rate of efficiency decline. 

In the following Section we compute and discuss a variety of age/price 

profiles for a variety of age/efficiency profiles. Then in Section IV we 

evaluate various studies and arguments which have attempted to identify the 

"correct" efficiency pattern. In the process, we introduce new evidence based 

on an experimental study of capital in the U.S. Manufacturing sector. 
III. Alternative Patterns of Decline 

In this section we generate depreciation patterns for a variety of 

patterns of efficiency decline, interest rates, and asset lives. This work 

forms the foundation for a critical comparison of the various fixed assumptions 

about decline which underlie a large number of economic studies. It also 

serves as a tool by which to evaluate the conclusions of econometric studies of 

market declines in asset prices. 

In each of our trials, exhibited in Tables 1 thru 5, we compare real 

price patterns associated with assumed efficiency patterns. These patterns 

represent most of the methods commonly used by economists. They fall into 

three major types: the hyperbolic family, double declining balance, and 

geometric. The hyperbolic methods are formulated as 

St = IL-t) / (L-Bt) (1) 

where st is the relative efficiency of a t-year old asset. One hoss shay 

?p 

(gross stock), and straight line are special cases where B = I and B = 0, 

respectively. Regardless of B, So =l and SL = 0, that is a new asset's value 

is indexed to one and an asset's value reaches 0 at the end of its life. In 

order to ensure that St is n;t less, than St-l, B must not exceed 1. For 

geometric decay, 

St= St-1 * (l-Z/L) (2) 
-lOIn other words, an asset's efficiency declines at a constant rate.' We follow 

Jorgenson's usual convention of using the double declining balance rate. 

Lastly, we define double declining balance efficiency decay as the same as 

geometric until the straight line decay rate becomes faster than the double 

declining rate. At that point, a straight line is used to ensure that an 

asset's efficiency vanishes at time t=L. This form provides us with a critical 

comparison for geometric decay, since it permits us to evaluate how heavily the 

special simplifying properties for geometric decay depend on its infinite tail. 

On each of these tables, for each efficiency pattern, we have estimated an 

age/price profile using the following formula 

oc 

Pt = 

F _ rzt S,(l-r)'t-t 

F v: Sz(l-r)z 

(3) 

where r is an assumed real discount rate which is the same in all periods. 

,* 

Summations were truncated after 200 years with negligible effects. 

Table 1, our basic case, is based on an assumed own interest rate of 4% 

and an asset life of ten years. Tables 2 and 3 are based on different interest 

rates (0% and 8%), each with the ten year life. Table 4 is based on 4% interest 

and a 50 year asset life. In Table 4, real asset prices are presented only 

every five years to permit ready comparison with Table 1. Finally, Table 5 

makes use of the same service life (10 years) and interest rate (4%) as Table 1, 

but postulates a normally distributed pattern of discards, with a mean life of 

10 years, and a standard deviation of 2 l/2 years, truncated at 5 and 15 years. 
-11- 

in the tables, decay assumptions are arrayed in order of increasingly 

rapid patterns of early efficiency decline. 'cle will refer to decline and 

depreciation patterns which are slower than straight line as concave, and 

patterns which are more rapid as convex. These terms refer to the usual plot 

of efficiency and market value against the age of an asset (see figure 1). 

We have carefully selected eight patterns of efficiency decline for study, 

which appear in the upper half of each of Tables 1 thru 5. Straight line 

depreciation is used by BEA (with a discard function) to determine capital 

consumption allowances. Double declining balance frequently appears in tax 

accounting, and one hoss shay has sometimes been used by economists to 

represent efficiency. Denison (1979) uses a.3-1 weighted average of one hoss 

shay and straight line stocks. Jorgenson and many others use geometric decay 

at a double declining balance rate. Faucett, in work done with the U.S. Bureau 

of Labor Statistics (1979) Office of Economic Growth uses hyperbolic decay with 

B values of .9 for structures and .75 for equipment. We also present results 

for B values of .5 since this is intermediate between one hoss shay (B=l) and 

straight line (B=O). Finally we present hyperbolic decay for B= -1.25. This 

particular value was chosen because it yields an efficiency function (Table 1) 

which tracks closely with geometric and double decline balance patterns, for an 

asset with a ten year life, until the seventh or eight year. 

In the lower half of each Table (1 thru 5) we present the duals to the 

upper half: the relative price functions. 

Some gener,al observations regarding the tables are in order. First, 

except for geometric decay, depreciation always proceeds faster than efficiency 

decline. This result is independent of choice of interest rate or asset life, 
. 

. 

_ Figure 1. Asset Age/~i~~~~~~~~~~~ic~-?rof~ler;------ --- ..: 

For an asset with a ten year life and four percent Aal 

interest, the solfd line represents efficiency, and the dashed line .' 

xpresents real matiet price for an asset based on hypetiolic 

efficiency decline, defined as S(t) = lo-t/(10-.5t), ie. with 

3 - .S. The dottid lfne represents both efficiency and market price 

based on geometric efficiency decline, defined as 

'S(t)=S(t-l).(~-Z/10), ie at the double declining balance rate. 

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-18- 

at least in the meaningful range. Furthermore, a wide variety of efficiency 

patterns give rise to convex price patterns, including some of the concave 

efficiency patterns. In particular, straight ?ine decay and hyperbolic decay 

with i3=.5, yield convex patterns in a?? of our trials. Even very high values 

of 3 give convex price patterns for shorter asset lives and lower interest 

rates. 

On Tables 7 thru 3, the price functions for the geometric double declining 

balance, and hyperbolic (with B= -1.25) efficiency functions are generated from 

efficiency patterns which are nearly identical for the first five years. For 

each interest rate, geometric decay gives significantly higher asset prices, 

due to lower depreciation, than the other two, even in the early years when 

efficiency patterns are the same. The differences range from 2.5 -5% in the 

first year and are as high as 35% in the fourth year. This disparity is 

remarkable, and we will argue later that the high levels of geometric 

efficiency near the end of the asset's life (still 10.7% at the assumed end of 

its life), and the nonzero levels of efficiency which extend well beyond the 

asset's life, account for this difference. 

Some additional observations are in order. In genera?, higher interest 

rates lead to slower depreciation patterns. This is true for a?? the patterns 

tested in Tables 1 thru 5, except. for geometric, which is independent of 

interest rates. This observation corresponds to the common sense notion that 

future declines in efficiency should be of less importance when higher interest 

rates deflect more weight to the early part of the efficiency pattern. 

For similar reasons, the longer service life (Table 4) produces slower 

depreciation for identical efficiency patterns and identical positive interest 
-19- 

rates (Table 1). Under positive interest, events in the distant future have 

less effect on current depreciation. Again, geometric decay is unaffected due 

to its special shape. (Actually, geometric efficiency and price patterns in 

Table 4 are slightly higher than in Table 1. This is because of our strict 

adherence to the double declfning balance rate (equation 2). Additional 

compounding, from extra subdivisions of the asset‘s life, causes this effect.) 

The observation that for nongeometric forms, depreciation tends bo become 

slower as the asset life lengthens is important. Empirical studies of 

depreciation, such as Hulten and Wykoff (19801, find more rapid depreciation, 

relative to the asset life, for equipment than for structures. Therefore, a 

single nongeometrfc form of the efficiency function can be consistent with both 

the observed slower depreciation associated with longer lived assets such as 

structures and with the relatively faster depreciation associated with 

equipment. A geometric form can produce the slow depreciation patterns 

observed for structures only by assuming a slow decay rate. This slow rate 

fmplies extraordinary levels of capital services well beyond the presumed end 

of the asset's useful life. 

Careful examination of the hyperbolic series for P.9 in Table 4 reveals a 

curious possiblity. Initially real asset prices are higher than the straight 

line efficiency pattern, but later in the life they become lower. In other 

words, the asset price curve, of the type plotted fn Figure 1, fs inflected. 

It is characterized by early concavity, and later convexity. This can be 

attributed to the fnteractfon of two effects. The fnterest rate effect 

combfned with long asset lffe (which we discussed above) dominates in early 

years. In later years, the general tendency for asset prices to take a convex 
-2oform dominates. Indeed, under zero interest rates, one hoss shay efficiency 

based asset prices take a straight line form (Table 2) while all other forms 

are convex. This tendency toward convex price patterns seems to dominate the 

interest effect for all but the longest asset lives, highest interest rates, 

and highest B values. Even then, convexity of prices occurs in the later 

stages of an asset's life. It is little wonder that researchers find convex 

prices in market data on used assets. 

One might imagine that the observations we have made about efficiency and 

price in this section would be different if we considered the separate life 

histories of each member of a cohort of investment goods. To address this 

question, we have constructed Table 5 in a manner similar to Table 1, except 

that each of the finite efficiency functions (those other than geometric) are 

constructed presuming a normally distributed pattern of discards. The discard 

pattern itself can be seen by looking at the one hoss shay column of Table 5, * 

3 

where only discards, and not relative efficiency losses, come into play. The 

asset is presumed to have a mean service life of 10 years. The distribution is 

truncated at 2 standard deviations (5 and 15 years). In columns 2-5 and 7-8, 

the same discard pattern is presumed, and, in addition, a separate efficiency 

function is calculated using the given formula for efficiency, for each 

possible lifetime between 5 and 15 years. The various efficiency functions are 

then integrated (using a discrete approximation) into a single, cohort 

efficiency function using the normal density function to weight the various 

cohort elements, 

Our observation is that Table 5 closely resembles Table I. The two are 

strikingly similar during the first five years, differing somewhat in the late 
-21- 

years as they obviously must. The major observations, such as that price 

decline is relatively faster than efficiency except in the geometric case, are 

unaffected. We note that we have not subjected geometric decay to a discard 

pattern. Dick McDonald, of the BLS Office of Research and Evaluation, has 

pointed out that the efficiency pattern for such a cohort would only be 

geometric if the distribution of discards were geometric, That is, it is 

unlikely that a cohort efficiency function would be geometric, even if members 

of the cohort have geometric efficiency patterns. Therefore there is no 

equivalence of efficiency and price patterns unless a cohert's distrihution,of 

discards is geometric. 

IV. An Evaluation of Alternate Efficiency Assumptions 

In this section we review and evaluate various arguments which have been 

put forth in support of particular mathematical forms for the efficiency 

function. Ve find reasons to prefer a concave function, or relative'ly slow 

form of decay, based on several arguments. In evaluating arguments favoring 

convex forms, particularly the geometric form, we find the evidence or 

reasoning is inadequate to support a conclusion of convex efficiency. Our 

overall assessment of the situation is that evidence is inadequate to 

conclusively establish any particular form. 

First we review the common sense argument for a concave form. Then we 

evaluate the possibility that obsolescence can account for a convex form. Next 

we compare the opinions of various major productivity measurement experts. We 

observe that the worf~ of those who have used highly concave forms can be 

challenged because of its failure to address the requirements of theory, not 

because of the concave forms. Next we evaluate several attempts at 

ascertaining the shape of the efficiency function by indirect means, and find 
-22- 

that, if anything, they point toward concave efficiency. Next we review 

defenses of geometric decay based on renewal theory, and find them groundless 

unless cohorts actually decay geometrically. Finally we introduce a new 

analytic ratio, the ratio of rent to cost, as an empirical criterion by which 

to judge the success of the capital measurement process. 

Realism 

There are a number of important productivity researchers, including 

Denison (19791, Diewert (19801, and Kendrick (19801, who have questioned the 

geometric form on the grounds that it is either unrealistic or represents an 

extreme view- If one selects a fast rate of geometric decay, as does 

Jorgenson, who generally assumes the double declining balance rate, a capital 

good must lose half of its efficiency in the first third of its life. Whether 

you consider a car, a typewriter, or a building, this seems unreasonable. 

Although some of these items may lose value this quickly, it is difficult to 

imagine how some combination of physical deterioration and increased 

maintenance costs would cut their efficiency so quickly. For example, in a 

majority of cases a two-year-old car can provide substantially the same 

transportation services as a new car even considering realiability, current 

maintenance and appearance. At the same time the older car will sell for 20% 

to 40% less because of the accumulated, though hidden, wear and tear which will 

require major maintenance two years sooner. If one selects a slower rate of 

geometric decay, as Hulten and Wykoff (1980) suggest, one is confronted with a 

substantial tail of services well beyond the end of an asset's presumed life 

span. With a concave efficiency form, one does not need to juggle these two 

unreal istic possibilities. 
-23- 

One hoss shay (gross stocks) and geometric seem to represent opposite 

extremes on the range of likely shapes. Capital goods rarely gain efficiency 

as they age, as would be implied were one hoss shay not concave enough. On the 

other hand, it would be debatable whether goods were durable enough to be 

considered capital if they yielded the majority of their services much more 

quickly than the geometric assumption requires. 

Obsolescence 

Technological obsolescence might be regarded as a reason for assuming a 

geometric deterioration pattern. The reasoning is that new capital goods are 

relatively more efficient than older ones since their design reflects advances 

in technology. This view cannot be reconciled with the vintage aggregation 

model. Another perspective on this phenomenon can be gained by considering 

technological progress instead of obsolescence. Progress is reflected in 

improved.capital goods in the form of acquisjtion prices which are relatively 

higher than those of older models. That is, if our real investment figures are 

properly measured, they reflect real improvements in capital's design which 

allow it to be more efficient in production. For example, if expenditures on 

capital remain constant while innovations are improv=ing capital, so that one 

dollar's worth provides more services, then the investment deflator should be 

declining and the real investment measure should be increasing. Thus, real 

investment figures should reflect adjustment for technology-induced quality 

change of capital goods. 

Of course;there may be measurement error in our investment deflators 

which cause them not to capture all innovation. There are well known 

difficulties involved in detecting and measuring innovations and linking them 
-L4- 

when estimating our price deflators. We should not correct for this by using a 

faster deterioration pattern. The correction should be made when separating 

investment expenditures into real quantity and price. Equivalently, we could 

change our relative efficiency normalization rule (that new goods have a value 

of one) so as to place more weight on newer vintages. However, to use a 

relatively faster deterioration pattern in order to represent improvements in 

technology would bias our capital measures toward slower growth. What is. 

essentially an improvement in new capital goods would be misrepresented as a 

loss in the productive capabilities of older goods. 

A theoretical problem arises when older capital goods become 

systematically underutilized. .For example, the electric power industry tends 

to use its oldest plants only during periods of peak demand. The annual 

service flow fran these plants is diminished because of their technological 

inferiority to new plants, even though the oldest plants may resemble one hoss 

shay's in the sense that they could still do the same job they originally did. 

A violation of the vintage aggregation assumptions is inherent in this 

scenario. In this example, utilization responds to exogenous influences 

differently depending on the age of the asset, Whether those influences 

include technology, relative prices (such as that of energy) or other external 

forces, they contradict one or more of the equivalent forms of the vintage 

aggregation conditions discussed by Fisher (1965). When reality contradicts a 

maintained hypothesis, we must either modify the hypothesis or awknowledge the 

problem. Hall (1968) has shown that we may make a correction, within the 

vintage aggregation framework, to the extent that embodied technical change, 

disembodied technical change, and efficiency contain exponential (geometric) 
-25- 

components. That is, if underutilization is systematically a function of age 

(disembodied technical change) and efficiency has a geometric component, we may 

correct for underutilization, within the vintage framework, by speeding up the 

geometric efficiency decay rate. Later in this section we present evidence 

that suggests underutilization is a function of other variables besides age. 

Yhat is more, since the finite efficiency forms we have considered do not 

contain geometric factors, they would not become geometric functions even if 

they were scaled by geometric factors. 

Our conclusion on this issue is that a geometric form cannot be justified 

by reference to the obsolescence issue. Systematic underutilization due to age 

is unlikely to be a large enough effect to make the large adjustment one would 

need to get from a concave to a convex efficiency form. Even if it were, the 

resulting function, though concave, would not be geometric, and would not have 

the special prowrties of geometric decay. Nonsystematic (function of 

variables other than age) underutilization cannot be accounted for under the 

vintage aggregation assumption. Technological effects which do not result in 

efficiency losses for used assets should be counted as improvements in new 

capital, and accounted for in the deflator for new capital goods. Thus 

consideration of technology may result in serious biases in the capital 

measure, only part of which can be represented by scaling the efficiency 

function, and none of which can be represented by assuming a geometric 

efficiency function when fully utilized goods do not decay geometrically. 

Major Productivity Studies 

Unfortunately, the capital work of productivity economists who disbelieve 

geometric decay has proven vulnerable in its failure to recognize the 
-26- 

distinction between efficiency decline and depreciation. Although each of 

these researchers reveals in his writings that he is aware of the distinction, 

none makes adequate use of it in his empirical work. Denison (1979) weights 

gross and straight 1 i ne stocks 3-l for his estimate of the productive stock of 

capital, distinguishing it from the straight line depreciation used to compute 

capital consumption in the NIPA. If he were then to use NIPA adjusted capital 

consumption allowances as an estimate of depreciation when pricing capital, he 

might have reasonably consistent measures of productive capital and j 

depreciation. Unfortunately, he commits a serious theoretical error by failing 

to count depreciation at all when determining capital's share. Kendrick (1980) 

has been disuaded from this position by Jorgenson. Since Kendrick uses gross 

stocks for productive capital, and since, implicitly, straight line 

depreciation underlies his estimate of capital's price and share, his estimates 

of the effects of capital on productivity reflect measurement practices that 

are reasonably consistent with the theory. However Kendrick does not 

distinguish assets by rental price. The BLS/Faucett (1979) stocks exhibit a 

similar rate of decline to the Denison/Kendrick ones, except that tStty are 

formulated from the hyperbolic method. This work does not address the effects 

of capital on multifactor productivity. 

A weakness of the work of Kendrick, Denison, and Faucett is that they 

aggregate capital assets directly. Jorgenson and his associates, and more 

recently Norsworthy, Harper, and Kunze (19791, distinguish between capital 

stock and capital services by weighting assets, with different rates of 

depreciation by their rental prices. However, Norsworthy and Harper (1981) 

fail to distinguish between asset efficency and real asset price. Their 
-27- 

productive capital stocks, and the depreciation element in their rental price 

equations, are both based on straight line decline patterns as computed by 

SEA. This just follows the work of BEA, as outlined by Young and Nusgrave 

(1980), which also fails to make a distinction. However, this is not a mistake 

frcm the BEA point of view since their purpose is only to measure depreciation 

and wealth, not productivity and productive capital. Jorgenson avoids this . 

theoretical problem by using geometric decay. Here both depreciation and 

efficiency decline are identical. 

Therefore, only users of the Jorgenson approach have empirical work on 

capital and productivity which is fully consistent with state of the 

art economic theory. Kendrick's work at least recognizes.the duality issue. 

Jorgenson and Kendrick stand at the two extremes regarding the speed at which 

deterioration occurs. As we have mentioned in Section II, it is unnecessary to 

assume a (possibly) unrealistic efficiency form in order to fully conform with 

existing theory. It is unclear whether it is worse to use an incorrect 

efficiency form or to ignore theory. It is clear that neither mistake is 

necessary under the maintained mticiei. 

Indirect Measures of Efficiencv 

Lack of data on the relative efficiency of various vintages in production 

has lead to attempts to observe efficiency by indirect means. Two major groups 

of such indirect attempts exist. In one group, depreciation is measured in 

used asset markets with the hope of infering efficiency by using the duality of 

efficiency decline and depreciation. In another group, investment demand 

functions are fitted for various presumed patterns of deterioration. 
-28- 

A number of researchers, including Feldstein and Foot (19711, Hall !1971), 

Jorgenson (19741, and Hulten and Wykoff il9811, have attempted to test for the 

rate and form of economic depreciation as observed In used asset markets. 

Feldstein and Foot, and Hall both reject constant rates of depreciation for 

their data on real prices. However, Jorgenson points out that the Feldstein 

and Foot work is theoretically flawed. Jorgenson also redoes the Hall 

statistical tests in a nested testing structure and fails to reject a constant 

rate of depreciation. Hulten and Wykoff test for the form of depreciation in a 

Box-Cox model which places only weak restrictions on that form. The model 

takes on one hoss shay, straight line, and geometric price patterns as special 

cases. Usually, all three special cases are statistically rejected, but 

parameter estimates come closest to those which would indicate a geometric 

form. 

On these grounds, Jorgenson and Hulten and Wykoff have concluded that the 

geometric form is probably a good approximation to asset price behavior and 

hence relative efficiency. Here our numerical experiments in the previous 

section become important. The price patterns corresponding to even concave 

forms of efficiency decline tend to be convex. Indeed, they can be virtually 

indistinguishable from the geometric form to the eye (as in Figure 1) and to 

the statistical model. Failure to reject geometric decay-against a general 

form is too weak a statistIca result to establish the superiority of geometric 

decay. 

Theoretically other specific forms would have to be rejected in carefully 

conducted head to head tests in order to establish geometric decay. Since 

existing tests even tend to reject the geometric form, geometric decay is far 
-2gfrom established. A particular criticism we have of the Hulten-yykoff Sax-Cox 

test is that it fails to recognize duality between efficiency and price. The 

three specific special cases of the age/price profile their model assumes are 

one-hoss shay, straight line, and geometric. Few who examine Tables 1 thru 5 

would think it reasonable to expect to find a one boss shay age/price profile, 

even in the extreme case of a one hoss shay age/efficiency oroiile. Yet this 

is one of the two alternatives against which Hulten and Mykoff find geometric 

to be the best. The geometric pattern is the only one of the three specific 

cases they consider with a convex age/price profile. It is little wonder that 

it gives the best results. As we have seen, most forms of efficiency lead to 

convex asset prices. All that studies of asset price data have done is verify 

the expectation of convex prices. Given the similarity between the age/price 

profiles 'depcited in Figure 1, and the stochastic forces which play a role in 

generating used asset prices, it would seem improbable that studies of used 

asset prices will ever decisively identify efficiency patterns. 

Why do radically different forms of efficiency decline give rise to 

similar depreciation forms? As we noted in the previous section, double 

declining balance and hyperbolic forms, which are faster than straight line, 

can assume patterns of efficiency decline which are remarkably similar to the 

geometric pattern, and yet give rise to much faster depreciation. The source 

of this discrepancy is clearly the failure of geometric efficiency to decline 

to zero at the end of the asset's presumed life. Therefore, the infinite tail 

on the geometric curve in Figure 1 allows geometric efficiency decay to be 

. . 

associated with significantly slower depreciation rates than those which 

characterize similar patterns which are finite. 
-3iIThus the infinite tail on the geometric pattern is associated with its 

special properties, namely the equality of efficiericy decay and depreciation, 

and the independence of depreciation from interest rates. Finite efficiency 

functions with much slower rates of decline give rise to depreciation patterns 

similar to geometric. As previously mentioned, this tail becomes particularly 

large when a slow rate of geometric decay is assumed. After their 30x-Cox 

test, Hulten and Wykoff maintain the geometric form and, in the case of 

structures, find very slow rates of decay. These slow rates imply substantial 

services in some cases 100-200 years hence. The Hul ten-Uykoff sample is 

inadequate to verify this prediction. A hyperbolic efficiency assumption,. 

which predicts slow depreciation in the early years and faster depreciation 

later and which predicts slower early depreciation as asset lives lengthen, 

would seem a much more successful interpretation of the fine Hulten-Wykoff 

dataset. 

We must conclude from this that economists interested in estimating 

efficiency from studies of used asset prices would have to carefully study the 

behavior of asset prices late in the life of a cohort in order to verify 

geometric efficiency. For this approach to be reasonable, significant 

proportions of the assets in a cohort must provide significant economic 

services well beyond the average asset life. In addition, these economist 

would need to design statistical tests which are capable of distinguishing 

remarkably similar convex forms. 

Clearly studies of depreciation are an indirect and ambiguous way to go 

about establishing relative efficiency. Even though there are almost no data 

on relative efficiency, data on price are far from adequate enough to permit 

economists to measure efficiency by using duality theory. . 
Major studies were performed by Robert Coen (1975, 1980; using the 

investment demand approach to examine various efficiency patterns. Coen 

carefully observes the distinction between efficiency decline and depreciation 

in the formulation of his investment demand model. He compares the 

predictability of investment behavior which results from using each of several 

deterioration and asset life assumptions. He tries one hoss shay, straight 

line, geometric, truncated geometric, and sum of years digits efficiency 

functions for various asset lives. Then, for 21 Z-digit manufacturing industry 

for equipment and structures, he picks the method and life which fits 

investment behavior best. In the 1975 paper, truncated geometric and straight 

line, applied to equipment, each fit best for 8 industries. For structures, 

one hoss shay fit bests for 11 industries, straight line 5 and truncated 

geometric 4. Geometric was best in only one of 42 cases. In the 1980 paper, 

truncated geometric dominated for 11 industries for equipment, and 11 for 

structures. Geometric and one hoss shay were best in only one case each. 

The Coen results are very interesting, and the 1980 revision seems to 

verify '-,he "realists" position that one hoss shay and geometric decay are 

extreme positions. Coen's work does have limitations as a way of measuring 

efficiency patterns. The specification of the investment demand model is the 

most important. Many factors affect the demand for investment besides the 

coefficients of the Cobb-Douglass production function. Inelastic factor 

substitution, biased technical change, increasing returns to scale, and 

systematic exogenous factors can all disrupt this comparison. A second problem '. 

is built into the measures of equipment and structures: no allowance for 

changing composition. A third problem is with the selection criterion which is 

to pick the best one. No efficiency pattern intermediate to one hoss shay and 

qtraiqht linp iz rpnreconted while ? fa+t mofhnrk annpqr. Also. in a comment 
-32- 

on the 1980 paper, Fabricant raises questions as to the stability of these 

estimates when the various trials are compared. Results are often obtained for 

long lives with fast efficiency decay which are similar to results obtained for 

short lives and slow efficiency decay. Thus the Coen work does not appear to 

allow decisive discrimination between functional forms of efficiency. 

Renewal Theory 

Jorgenson (1974) offers a second line of defense for geometric decay. 

According to renewal theory, a population of capital assets, characterized by a 

steady state or by a stable growth investment pattern, will tend to a constant 

rate of decline, regardless of the shape of the relative efficiency function 

for a particular asset. Therefore, under stable growth conditions, the vintage 

aggregation model can reproduce any decay pattern with some geometric function. 

Jorgenson predicts two possible problems with using the geometric ,fonn in 

practice when the assets themselves do not exhibit geometric efficiency 

declines. First of all, the geometric based capital stock estimates will be 

biased when the capital population does not conform to stable growth. Thus, by 

imposing a geometric form when it is unrealistic, we could systematically 

mismeasure capital during business cycles or slowdowns. Secondly, he points 

out that even under stable growth, efficiency decline rates do not, in general, 

equal depreciation rates. (See Jorgenson, 1974, p. 2091. 

In order to evaluate the practical consequences of these possible 

problems, we draw on some capital data development work being done at BLS by 

Harper and Rosenthal (1982). Their draft paper presents in detail the sources 

and methods used to construct these experimental data. The results are 
-33- 

regarded as unofficial by BLS, but are more than adequate to address the issues 

posed in this paper. Essentially, productive and wealth stocks in the il.S. 

Xanufacturing Sector are estimated separately for 18 types of equipment and 3 

types of structures from investment data provided by the U.S. Bureau'of 

Economic Analysis from the National Income and ?roduct Accounts (NIPA). These 

are,aggregated and presented in Table 5, for four separate efficiency 

assumptions: one hoss shay, hyperbolic (3=.5), straight line, and geometric. 

There are several observations on Table 5. First the level of the capital 

stock depends substantially on the efficiency assumption. The distinction 

between productive and wealth stocks is important in terms of levels if we, 

dismiss the geometric alternative. Secondly, the year to year growth rates are 

observed to differ by over 1 % over the course of a typical business cycle. 

Essentially, stocks starting from a smaller base are more sensitive to shocks 

in the investinent data. Third, only when we view the long term trends does the 

prediction of renewal theory, that the rates of growth will be similar, take 

force. Thus renewal theory does not appear to justify using a geometric form 

if efficiency is not in fact geometric, at least not for preparing annual 

capital measures. 

In Table 7, we present the rates of efficiency decline and depreciation 

implied by the various assumptions. The rate of efficiency decline is always 

measured against the productive stock, and the rate of depreciation against the 

wealth stock. Jorgenson's second acknowledged problem is certainly 

demonstrated. What's equally important is to observe that the rate of 

depreciation di-ffers depending on the method selected. Since the rate of 

depreciation enters the rental price formulation, an error here could bias the 

cost share weights assigned to individual assets. 
. . ‘. -. -. ._ 

is*- + "1 

1973 

197L 

1375 

1976 

7377 

1978 

1379 

1974 

1375 

1975 

1977 

1373 

1379 

STOCK 

177.2 90.: 

296.3 

394.4 

405.3 

T f5L .3 

222.2 

228.7 

L2l.T 

Lj4.4 

CL7.6 

464.2 

L62.1 

3iLLIS!lj OF 1972 33LLA?S 

117.5 115.6 SC.3 

1?4.3 13L.6 :A,. 7 j 

253.5 25L.7 197.5 

270.5 271.9 202.6 

2S2.L 293.8 212.L 

290.3 291.9 219.0 

205.5 300.4 223.9 

300.6 311.3 777 _dC. 4 

321.L --- 262). 3 2L1.3 

'=eRCENT CHANSE F73M ?REVIOlJS 

23a.a 19i.L 

245.3 196,3 

252.2 231,c 

251.8 209.2 

272.0 217.4 

YEAR 

2.37 ?.5? 2.71 2.53 2.55 

3.97 4.37 4.40 4.34 L.66 

2.96 2.7: 2.s 2.62 2.7L 

3.06 2.35 2.92 2.63 2.31 

3.70 3.71 3.73 3.31 3.32 

3.33 3.7Q 3 .37 3.35 3.93 

19C8-1965 3.29 2.04 3.13 2.79 2.?4 ', .e. L.C> 

1965-1973‘ 3.65 5.36 3.91 L.f‘O L.03 s.17 

1973-7978 3.31 3.23 3.33 3.31 3.34 3.33 

NOTE: INCLUCES EQ'JfPmENT AN0 STRUCTURES OYLY 

SOURCE: 3UREAU 3: LXZOR STATISTICS 

OF?;CE OF ?ROOUCTiViTY AN0 TECHNZLOSY 

EXPERIMENTAL 3ATA 

62.7 -, r-.: 

123.i ; L7 G . : 

777.5 290.7 

132.2 296.1 

3 -. j? 

5.07 

2.54 

2.61 

3.37 

3.33 

2.3 ‘9 

t.58 

2. SF 

2.68 

3.31 

3.33 

2.66 

3.08 

3.33 

JANUARY, 1082 
, 

. 

. - 

e’.. - 

194; 

7965 4.23 7.60 

1966 L-25 7.73 

1967 C.16 7.72 

1963 1x8 7.59 

7969 4.05 7;jo 

1970 C.OL 7.77 

1371 L-07 7.75 

1972 4.15 7.3s 

1373 6.21 7.32 

197c L.26 7.97 

1975 

1376 

1977 

1978 

1579 

ONC 3gSS Si(AY 

3.5C 

c.23 

4.22 

4.19 

L.75 

L.13 

0.46 

7.97 

2.03 

am 

2.22 

3.51 

3.32 9.57 

7.a1 11.33 

7.7L 11.33 

7.6L 11.30 

7.57 11.31 

7.53 11.31 

7.50 11.33 

7.67 11.37 

7.78 ll.L? 

7.85 11.56 

7.33 11.6L 

7.35 11.57 

7.90 11 .dO 

7.97 11.95 

3.04 12.12 

Z.lL 12.32 

N2TC’ w. INCLUOES EGlJiP!fENT 1N3 jT?.UCTU.SES 3NLY 

S3URCE: 3USZAU 3F LA30R STATISTICS 

3FiICE 3F F2OOUCTiVITY AND TZC9NOLOGY 

EXPE?i!lENTAL OA.TA 

3.21 11.12 9.90 

9.77 12.37 70.36 

9.7s 12.91 11.02 

9.70 12.93 11.20 

0.68 12.96 17.31 

9.69 12.95 Il.24 

9.70 

9.75 

9.35 

9.02 

9.97 

12.9~ 

12.35 

13.35 

13.16 

13.25 

11.22 

ll:lG 

11.17 

ii.23 

11.33 

9 -3 

?&7 

10.17 

10.30 

10.LL 

1 3.32 17.L6 

13*&Z 11.37 

13.65 11.69 

13.35 ?1.8E 

lL.05 12.25 

em ?A:,: 0; 

DECLINE 

D"?. -a 
-36- 

Tables 6 and 7 provide defense to our contention that the choice of 

mathematical form of efficiency loss is important. It might be argued that by 

suitable adjustments to presumed asset lives, just about any level of capital 

stock can be achieved. For example, geometric decay might be assumed to occur 

at l/L instead of 2/L where L is the estimated useful life. Where such 

reasoning fails is that an erroneous mathematical form of efficiency, even if 

it achieves an accurate estimate of the level and growth rate of the productive 

stock, will lead to an erroneous estimate of the stock of wealth and 

depreciation. Besides damaging these measures, such an error would damage the 

asset aggregation process, which depends on the estimate of depreciation. Thus 

there is not a free tradeoff between mathematical form and presumed life. 

In Table 8 we present estimates of the growth rate of Tornquist indexes of 

all capital assets, including the nondepreciable ones, together with 

"composition effects", based on comparing Tornquist and direct aggregates of 

productive stocks. Our observations are similar to those for Table 6. Table 8 

will permit the reader to assess the overall implications of these issues for a 

final measure of capital input. 

Clearly the renewal theory argument is of no value. It makes no sense to 

impose the implications (geometric decay) of a stable growth rate for 

investment if investment does not exhibit such a pattern. After all, the whole 

idea is to measure capital. Defense of any choice of efficiency function, 

including a geometric one, must rest entirely on whether it is right. 

The Ratio of Rent to Cost 

The Harper Rosenthal (1982) work provides a new criterion by which to 

evaluate relative efficiency functions, and, for that matter, the validity of 
: ;;;-3j 

1935-73 

1973-73 

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-4 ‘Ythe vintage aggregation assumption. Jorgenson and his colleagues construct a 

reflta? price as a means of allocating capita? income to assets. In order to 

ensure that this happens, a rate of return is constructed from NIPA capital 

income data in such a way that this allocation occurs exactly. Such a method 

allowing for taxes and inflation, yields an estfmate of the internal rate of 

return on capital. In earlier literature on the user cost, or rental price, of 

capital, a discount rate usually assumed to be a market interest rate is used. 

Of course nobody believes that capital markets are in equilibrium every year as 

the user cost model assumes. But if they were in equilibrium, if our vintage 

aggregation procedure faithfully represented the implicit rental values of 

capital, and if perfect competition and foresight existed, it might be 

reasonable to predict that the rents implied by the rental price equations and 

by market interest rates would equal current capital costs. 

Using the ijloody AAA bond rate, and the rental price equations described in 

Harper and Rosenthal (19821, we estimate the implicit rental value of capital 

by summing the products of rental prices and productive stocks. We then divide 

this implicit rent into the actual current dollar capital income reported in 

the NIPA. As we have argued, this "ratio of rent to costs" should be about one 

if all the assumptions we make are really correct. Harper and Rosenthal (1982) , 

discuss the possible relationship between this ratio and "Tobin's Q", or the 

ratio of market value to book value of the stock market. Essentially, Tobin ‘5 

Q may reflect investor's discounted expectations of future values of the rat io 

of rent to cost. 

In Table 9'.we present estimates, for the various efficiency assumptions, 

from 1948-1979. Tiiis ratio appears to confirm the view that existing capital 
_. -.----.-. --ye. -. 

i?50 .57 .57 

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i?52 1.33 1 .Cl .?9 

iF53 i .lJL ; .30 . 95 

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i.955 1.13 1.05 4 i .b '3 

1955 .33 ‘7 .53 

1957 .39 :;7 .3!. 

1353 1.36 1.27 1.21 

1959 1;79 1.25 1. i3 

1350 1 .a ;2 1.20 1.73 

1351 1 .33 1.2L 1.15 

1362 1.20 1.03 1.02 

1353 1.33 .93 .93 

136L 1.31 .92 . 37 

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1952 . 39 .33 :7!, 

1959 .3L . 30 .77 

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i372 .?? .89 .s> 

1373 .72 .65 . 62 

197L .39 . 1L . 14 

i?75 .35 . LO .A0 

1376 .32 .80 .75 

1377 .b3 .53 . 55 

1973 .c9 .ca . L6 

1973 .jL l ji . 33 

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-4ostocks experienced a large loss of value beginning in 1973, the year of the 

Arab Oil embargo. After the 1974-5 recession, during which rents nearly 

vanished, it would appear that the existing stock of capital remained incapable 

of generating rents consistent with any of the efficiency assumptions. 

Frcm this we must judge that the vintage aggregation assumption, based on 

any presumed fixed efficiency pattern, has failed miserably since 1972. ihe 

assumpton that capital services are proportional to stocks would appear wrong. 

Some downward adjustment in capital services appears to be indicated. 

Regarding the earlier years, it would appear that conventional methods are 

fairly accurate from 1952-1972. If we accept the reasoning that the ratio of 

rent to cost should approximate one, we can compute the root mean square 

differences from one for each efficiency assumption as a criterion for 

evaluating their performance. These statistics have been presented for the 

entire time period, and for the period during which the model seems reasonably 

well behaved. tie observe that the one hoss shay assumption appears to predict 

rents which tend to be higher than capital income, whil,e the geometric form 

predicts rents which are almost always lower than capital income. 

Gle recognize that this test is limited by data and assumptions, but we 

take comfort anyway that it appears to demonstrate that one hoss shay and rapid 

geometric decay are opposite extremes when confronted with actual capital cost 

data. 
-41- 

V. Summary and Conclusions 

In this study we have examined how real capital input should be measured 

in light of the economic theory of replacement and depreciation and in light of 

evidence on the patterns and rates of deterioration and depreciation. 

Our first conclusion is that evidence on the correct form of decay is 

lacking. The lack of direct evidence on rental values as an asset ages is the 

major problem. Indirect approaches to the question cannot be regarded as 

conclusive. Worse, it is unclear how direct or indirect methods could be 

dramatically improved. Direct approaches are burdened by the lack of rental 

markets. When rental markets do exist, there are long term leases which 

obscure time relationships and factor in tax incentives. Indirect approaches 

face sensitive statistical identification problems while dealing with a 

variable, price, which is affected by all kinds of things besides the 

efficiency function we wish to measure. 

Our second conclusion is that the correct specification for efficiency is 

important. These first two conclusions echo the assessment of Diewert (1980, 

p-478). We have found additional evidence that an incorrect specification will 

distort the relationship between efficiency and depreciation. Neither renewal 

theory, discard patterns, obsolescence, nor empirical evi.dence can justify the 

use of an incorrect mathematical form of the efficiency function, because even 

if productive stocks are adequately estimated, other variables, such as wealth 

and depreciation, will be seriously biased. 

However, the contributions of this paper are by no means negative. 

First of all, we have distinguished two concepts of capital stock, productive 

and wealth, which have distinct and interesting interpretations. We have 

argued that this distinction follows from the Hall (1968) and Jorgenson 
-42- 

(1974) papers, and is logically necessary within the framework of that model, 

if we are to elect a nongeometric efficiency assumption. We have done so 

empirically for four alternative efficiency assumptions in the U.S. 

Manufacturing sector. 

Secondly, we have identified a particular mathematical form, hyperbolic 

with B=.5, which is associated with an age/price profile extremely similar to 

the geometric pattern. Besides having the convex shaped price pattern found in 

the Hulten-\/ykoff data, this form has a concave shaped efficiency pattern which 

many researchers would consider more realistic. This "realism" stems from its 

gradual rate of efficiency decline in early years coupled with elimination of 

the large troublesome tail found on a geometric form which declines slowly in 

the early years. The choice of a precise value for B seems to be no more 

arbitrary than the choice of a rate of decline for the geometric form. Such a 

choice must be made to yie1d.a model which is as realistic as possible. A high 

. 

value of B tends to generate a concave price function, while if B=O we have 

straight line efficiency decline which may be too fast. However, in spite of 

identifying B=. 5 as an appealing alternative, we emphasize our conclusion that 

evidence on which form is correct is lacking. Such a choice rests largely on 

scientifically indefensible grounds, one's instincts regarding realism. 

Thirdly, we have identified a new statistic, the ratio of rent to costs, 

which, with proper refinement, could become one criterion by which to 

discriminate various efficiency assumptions, and to identify when the 

traditional vintage aggregation procedures fail. In our test data, two 

observations were made. First, one hoss shay and geometric appear to be '. 

extreme views on the high and low side, as some suspect based on intuition. 
-43- 

Secondly, the vintage aggregation procedure seems to be a failure at 

representing capita7 input during recessions, and in the post-1973 period, as 

many have argued. Again,. we caution that conclusions based on this s%atistic 

are sensitive to many assumptions. However, the statistic does seem to affirm 

the need to adjust our capital stock estimates. Failure to address the 

problems imposed on our capital stock measures by the vintage aggregation 

conditions seems to leave us with capital measures, multifactor productivity 

measures, and models of production which are obsolete. 
-44 

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. 

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, 

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