obsolescence, input augmentation, and growth accounting

European Economic Review 42 (1998) 1653—1665

Obsolescence, input augmentation,and growth accounting

Michael Gort!,*, Richard A. Wall"! Department of Economics, State University of New York at Buffalo, Buffalo, NY 14260, USA

" Economics & Finance, Canisius College, Buffalo, NY 14208, USA

Received 1 October 1996; accepted 20 October 1997

Abstract

The paper points to a serious conceptual inconsistency in conventional measures ofcapital used in most empirical work on economic growth. When appropriate correctionsare made, it is found that plausible alternative values for the relative magnitude ofobsolescence and physical decay imply very different rates of growth in capital. This, inturn, leads to large differences in the fraction of growth in output attributed to inputaugmenting technical change as compared with the ‘Solow residual’. ( 1998 ElsevierScience B.V. All rights reserved.

JEL classification: O47

Keywords: Capital; Technical change; Growth accounting

1. Introduction

Are the currently available measures of the net stock of physical capital usablefor the purposes for which economists put them? The purpose for capital stockdata may be simply to generate estimates for comparing capital intensity acrosswidely divergent economies or industries. For such cross-section analysis,

*Corresponding author. Tel.: #1 716 645 2121; fax: #1 716 645 2127; e-mail: [email protected].

0014-2921/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 4 - 2 9 2 1 ( 9 7 ) 0 0 1 0 8 - 6

conventional net capital stock estimates are probably suitable at least for scalarapproximations. But, if we wish to assess the sources of technological changeor the impact of investment on productivity growth, far more subtle measure-ment is required. Particularly for time series analysis, it is our view thatconventional measures of net capital as produced by the US Department ofCommerce for the United States, and by various agencies in other countries, areof very limited use for decomposing the sources of economic growth andpotentially quite misleading. While our illustrative examples are carried outwith US data, the conclusions would be substantially the same with data fromother countries.

The problem does not solely, or even primarily, rest in the usual types ofmeasurement error. For example, it is commonly agreed that estimates of netcapital depend upon somewhat tenuous guesses (in the US made mainly by theInternal Revenue Service) of useful economic lives for various classes of assets.And there are well-known problems with the price deflators used and withassumptions about capital decay functions over time (are they exponential,straight-line, etc.?). But more important than these generally recognized sourcesof error is a conceptual inconsistency that fundamentally affects time series onnet capital.

Our purpose in this paper is not to generate alternative measures of netcapital. The information necessary to carry out this task is not currentlyavailable. Instead, we have two objectives. First we explain the conceptualinconsistency referred to above. It arises from a failure to decompose deprecia-tion into its obsolescence and physical decay components. Second, we show therange of plausible effects that such decomposition has on measures of growthand productivity of the capital stock. This is done within the context of thewell-known model of Solow (1957) which yields the so-called Solow residual.Effects, however, of comparable magnitude can be expected for the empiricalapplication of most growth models currently in the literature. These effects,though they cannot be measured precisely at present, are found to be drasticwithin the range of plausible and fairly conservative underlying assumptions.

2. Inconsistencies in measures of capital

Expressing measures of capital in embodied technical efficiency units has beenanalyzed as far back as Fisher (1965). Capital investment is adjusted by anaugmentation factor that reflects advances in its physical capabilities. Augmen-tation is treated separately from the physical decay of capital, the latter assumedto be the rate of depreciation.

Unfortunately, however, depreciation (as it is conventionally measured) can-not be correctly identified with physical decay. At one extreme, physical decaymay be a primary determinant of the depreciation of, for example, milling

1654 M. Gort, R.A. Wall / European Economic Review 42 (1998) 1653—1665

machinery while at the other, obsolescence dominates depreciation of com-puters and other high-tech machinery. In many cases, the rebuilding of indus-trial machinery is expensed as maintenance rather than capitalized. Ifmaintenance expense roughly offsets physical decay of certain types of industrialmachinery, then what is the rate of depreciation really measuring?

The solution to this dilemma first requires a recognition that assumeddepreciation rates on physical capital (e.g. those of the US Department ofCommerce, Bureau of Economic Analysis) are a combination of physical decayand obsolescence. The US Department of Commerce derives its depreciationrates mainly from Internal Revenue Service estimates of useful lives, and thelatter in turn are largely a function of obsolescence.

Maintenance outlays tend to offset the effects of physical decay and, whenthey fail to do so, the explanation lies partly if not mainly in the fact that it is notworth doing so because of the decline in the economic value of an asset’s outputattributable to obsolescence. A dramatic illustration of the potential for main-tenance to prolong physical life almost indefinitely is the location of the offices ofthe Bureau de Tourisme in the French city of Rouen. The offices are located ina building erected in the early 16th Century. In this instance the historical andaesthetic values offset the effects of obsolescence, but clearly that is not the casefor most office space where vintage affects rentals per square foot. Equipment isprobably characterized by much faster technical change than buildings, buttechnical change for the latter is far from zero.

One approach to the correct measurement of the stock and growth rates ofphysical capital is to introduce obsolescence as an augmentation factor. Thispermits capital of different vintages to be expressed in a common numeraire. Incontrast, as is shown below, the conventional approach combines units ofcapital expressed in different numeraires. In a recent study, Hulten (1992), whileadjusting investment flows for quality, implicitly assumes that depreciation ratesderived from BEA data measure solely physical decay of capital stock. The pointof the analysis to follow is that if BEA rates of depreciation combine bothphysical decay and obsolescence, a correct time series measurement of the stockof physical capital requires a more complex adjustment for obsolescence. Forthis reason, growth accounting cannot ignore the decomposition of deprecia-tion, however difficult it may be to achieve.

To clarify the problem, the net capital stock Kt

is constructed using theperpetual inventory method as a weighted average of investment flows ofvarious vintages as follows:

Kt"

t+q/0

wq Iq , (1)

where the weight wq is represented as the product of physical decay andobsolescence coefficients, and Iq is an investment flow of vintage q expressed in

M. Gort, R.A. Wall / European Economic Review 42 (1998) 1653—1665 1655

economic value units of the year of acquisition of the asset. The standardprocedure is to employ the depreciation rate as a ‘discount factor’ withoutreference to the distinction between obsolescence and decay. To the extent thatexponential rates of obsolescence (a) and physical decay (d) are a reasonableapproximation, the weights in Eq. (1) can be expressed as

wq"e~(a`d)(t~q), (2)

where a#d is the rate of depreciation, and t!q is the age of investment.Combining Eqs. (1) and (2), the capital stock as conventionally measured can bewritten as

Kt"

t+q/0

e~(a`d)(t~q)Iq"e~aqt+q/0

e~d(t~q)Iq eaq . (3)

In effect, the conventional perpetual inventory capital stock, Kt, changes its

numeraire each year so that each year’s stock is expressed in efficiency units ofthat year’s vintage.

The inconsistencies in the conventional approach derive from the fact thatinvestment flows are not augmented for quality change at the rate of a, while thedepreciation rate applied to existing capital combines both physical decay at therate of d and obsolescence at the rate of a. Correcting these problems requiresa depreciation rate based solely on d for existing capital and an augmentationfactor at the rate of a for investment flows expressed as

K*t"

t+q/0

e~d(t~q)Iq ea(q~t0)"e~at0t+q/0

e~d(t~q)Iq eaq , (4)

where investment flows and the capital stock are expressed in efficiency units ofa numeraire period at t

0. The numeraire can be the initial, terminal, or any other

date in between. The rate of obsolescence is then introduced as an augmentationfactor for periods after the numeraire, and a discount factor for periods before.The critical issue is not the choice of numeraire, but of expressing the stocks ofsuccessive years in consistent efficiency units.

The relationship between the conventional discounted (Kt) and augmented

(K*t) net capital stock series combines Eqs. (3) and (4) as

Kt"e~a(t~t0)K*

t, (5)

and, in terms of growth rates,

KQt

Kt

"

KQ *t

K*t

!a . (6)

The conventional capital stock series is thus downward biased in overall sizeafter t

0and in its rate of growth over all t. Further, the magnitude of the bias


depends critically on the relative contributions of obsolescence versus physicaldecay to the overall rate of depreciation.

Stated simply, we merely assume that obsolescence is a function of vintagewhile decay is a function of age. Thus machines of an earlier vintage are lessproductive, even when new, than machines of a later vintage. But suppose thatobsolescence takes the form of changing the composition of demand andthereby reducing the intensity of use of old machines. Were old machines inthese circumstances as productive as new machines when the former were alsonew? The answer is still ‘no’ unless the change in composition of demand issimply a function of an arbitrary change in tastes.

New final goods are generally more valuable than the old, and not simply asa function of changing tastes. It is the decline in the value added by old relativeto new machines even if both were operated at capacity that explains the declinein the intensity of use of old machines. Declining intensity of use and a relativedecline in economic value per unit of output usually go hand in hand and bothare a function of vintage.

A related problem that is well documented in the production literature is thedifficulty in separating embodied from disembodied technical change in timeseries data. With a simple Cobb—Douglas model to illustrate, the productionrelation is expressed in common efficiency units as

Ot"A

0eb1tK*

tb2 ¸b3

t, (7)

where Otis output, with the usual inputs capital and labor, and where the trend

coefficient b1

measures disembodied technical change. Embodied technicalchange is subsumed in the capital input term, K*

t. Empirical estimation of

Eq. (7) using the commonly constructed discounted capital stock series isequivalent to substituting Eq. (5), giving

Ot"A

0e(b1`b2a)tKb2

t¸b3

t. (8)

The obvious problem in constructing the augmented series [Eq. (4)], correct-ing growth in the capital stock and Solow residual [Eq. (6)], or separating therelative importance of embodied versus disembodied technical change in a pro-duction function [Eq. (8)], is the need for independent information on the rate ofobsolescence.

3. Comparing results on embodiment

The fraction of economic growth attributed to input augmenting technicalchange, as compared to changes in knowledge not identifiable in the vintageof specific inputs (particularly capital goods), has ranged from negligible to


substantially all of growth. In this context, further attempts to throw light onthis question seem justified.

Two such studies, Hulten (1992) and Greenwood et al. (1992), have proceededfrom the analysis of capital goods prices. Greenwood et al. conclude thattwo-thirds of postwar productivity growth can be attributed to investment-specific technological change. Hulten’s results show a clear but more modesteffect of vintage on growth basing his estimates of vintage effects on Gordon’sstudy of durable goods prices. Gordon (1990) uses a variety of methods formeasuring quality change, including hedonic indexes for some products, e.g.computers, automobiles, trailers, and electric appliances. His method for mostdurable goods, however, entails measuring price change for identical products atsuccessive points in time. The difference between an index so derived and a priceindex that takes only minimal account of quality change (the BLS Producers’Price Index) is then the implied measure of quality change. Gordon estimatesa 2.9% annual ‘drift’ for the 1947—1983 period, and a 2.1% drift for the1972—1983 period, in the ratio of his index for all producers’ durable goods tothe official indexes. Since the latter are largely unadjusted for quality change,this implies an equivalent annual change in the productivity of producers’equipment.

In contrast to studies of capital goods prices, an alternative approach tovintage involves using large samples with pooled time series and cross-sectiondata. Two studies for manufacturing plants produced support for a strong effectfor vintage (obsolescence). Gort et al. (1993) show an average annual effect ofvintage on growth in output ranging from 2 to 4% (using their preferredmeasure of output) for manufacturing plants in the 1973—1989 period. Bahk andGort (1993) find that a 2% per year growth in output attributed, in the absenceof a vintage variable, to time-dependent disembodied technical change dis-appears completely once a variable capturing the measured change in thevintage of capital is introduced in the model. Assuming a one-third share weightfor capital in the production function, the results of these studies translate toa 6—12% annual change in the efficiency of capital — a rate vastly higher thanthat derived from Gordon’s methodology.

Recent as yet unpublished studies are also consistent with much higherestimates than Gordon’s. For example, Boskin and Lau (1991) estimate a rate ofcapital augmentation for France, West Germany and Japan of 12—15% perannum in the postwar period, and 7—9% for the United Kingdom and theUnited States. Doms (1992) estimates a depreciation rate of 7%—9% based onplant data for steel mills. Though he does not decompose depreciation intoobsolescence and physical decay, his estimate implies a probable high rate ofobsolescence.

The principal reason for a possible downward bias in Gordon’s impliedestimates of quality change is that improvements in the productivity of capitalgoods may be achieved without increases in the quantity of inputs, other than


knowledge. Gordon is well aware of this phenomenon and labels it ‘nonpropor-tionality’. In competitive markets, the prices of non-identical capital goods will,except for transitory disequilibrium, be determined by the costs of these goods— hence, by the quantities of inputs consumed in producing the capital goodsrather than by their changing productivity. Hence, BLS indexes may be biasedupwards by technical change far less than implicitly assumed in Gordon’scalculations and the difference between Gordon’s and the BLS indexes maymaterially understate the rate of technical change.

4. Quality adjustment and Solow residuals

A set of Solow residuals is derived below based on alternative assumptionsregarding rates of embodied technical change and applicable share weights ofcapital. In the absence of hard data on the relative magnitudes of obsolescenceand physical decay, our objective was to define the range of possible outcomesonce appropriate adjustments are made to express investment and capital incommon efficiency units, and once output is corrected for consistency with thenew measure of investment. The purpose, therefore, was to assess what differ-ence it makes for decomposing the sources of growth if one makes alternativeassumptions within the range of plausible values. While our analysis proceedswithin the framework of the Solow model, the issue is obviously critical for anyanalytical framework in which capital is a key input.

The standard equation for the Solow residual, also called total factor produc-tivity growth, is as follows:

AQ /A"OQ /O!wkKQ /K!w

LQ̧ /¸ , (9)

where O is output, K and ¸ are capital stock and labor input, respectively, andwkand w

Lare share weights that sum to one. A further breakdown of output into

contributions to growth by consumption and investment permits quality changeto be adjusted also in the output term through share weighted investment. Thus,we have

OQ /O"wcCQ /C#w

IIQ /I, (10)

where C is consumption and I is investment, with corresponding share weightsthat sum to one.

4.1. Growth rates, conventional measures of capital and investment

The growth rates in Table 1 are derived from private sector data provided bythe Bureau of Economic Analysis, US Department of Commerce, on value


Table 1Average annual growth of outputs and inputs, and rates of depreciation by sector, 1947—1989!

1947—1959 1960—1973 1974—1989 1947—1989

Output growthManufacturing 3.38% 4.32% 2.96% 3.17%Trade 3.48% 4.43% 3.37% 3.58%Service producing 3.72% 4.55% 3.44% 4.01%Aggregate output 3.61% 3.80% 2.94% 3.23%Aggregate consumption 3.75% 3.57% 2.93% 3.16%Aggregate investment 2.64% 5.28% 3.00% 3.68%Investment share 12.98% 13.56% 14.84% 13.86%

Labour growthUnadjusted for quality:Manufacturing 0.84% 1.33% 0.01% 0.52%Trade 0.98% 1.45% 1.67% 1.29%Service producing 1.27% 2.58% 3.32% 2.37%Aggregate labor 0.57% 1.61% 1.80% 1.24%

Adjusted for quality:Manufacturing 1.66% 1.70% 0.23% 0.94%Trade 1.42% 1.57% 1.94% 1.52%Service producing 1.98% 2.60% 3.63% 2.64%Aggregate labor 1.30% 1.86% 2.07% 1.60%

Capital depreciation ratesManufacturing 4.75% 4.72% 4.88% 4.79%Trade 5.22% 5.41% 5.83% 5.51%Service producing 3.14% 3.64% 4.35% 3.75%Aggregate capital 4.34% 4.60% 4.95% 4.71%

Capital stock growthUnadjusted for quality:Manufacturing 3.75% 4.75% 2.45% 3.63%Trade 3.81% 6.70% 5.26% 5.38%Service producing 3.50% 4.60% 3.42% 3.85%Aggregate capital 3.75% 4.55% 3.06% 3.77%

Adjusted for quality:a"50% of BEA depreciationManufacturing 6.13% 7.11% 4.89% 6.02%Trade 6.42% 9.41% 8.18% 8.13%Service producing 5.07% 6.42% 5.60% 5.72%Aggregate capital 5.89% 6.79% 5.46% 6.06%

a"100% of BEA depreciationManufacturing 8.50% 9.47% 7.33% 8.42%Trade 9.03% 12.11% 11.09% 10.89%Service producing 6.64% 8.24% 7.77% 7.60%Aggregate capital 8.09% 9.15% 8.01% 8.48%


added output, investment, and net capital stock, described more fullyin the footnote to the table. While net capital stock is based on straightline depreciation, the BEA also provides a version called ‘capital input’with depreciation derived from a beta decay function that yields a deprecia-tion rate slightly lower than straight line. Solow residuals, however, are verysimilar to those with ‘net capital stock’ and are therefore not reported to savespace.

We show estimates for three broad sectors — manufacturing, trade, and serviceproducing industries — and for the aggregate private economy over the period1947—1989, and also for three subperiods. The aggregate category includesagriculture, mining, and construction in addition to the three sectors above,while the government sector is excluded throughout. Labor input data is takenfrom Jorgenson et al. (1987), with updates through 1989 provided by Jorgenson.Quality change is determined on the basis of demographic attributes and yearsof education of the labor force.

Annual depreciation rates used in computing total capital consumptionaverage 4.7% per year for the aggregate over the entire period, ranging froma low of 3.1% in service producing industries in the first subperiod to a high of5.8% for trade in the third subperiod. These modest rates of depreciation arisepartly from inclusion of structures along with equipment in total capital. Sincestructures have much longer lives than equipment they account for a muchlarger fraction of the capital stock than of new investment. The phenomenon ofan increase in the rate of depreciation from 4.3% to 4.9% from the first to thethird subperiod is explained by a shift in the composition of investment in favorof equipment in the postwar period.

$&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

! Data on value added, investment, and capital were provided by the US Department of Commerce,Bureau of Economic Analysis. The sectors employed in this study exclude outputs and inputs forgovernments. Those included are as follows: 1. Manufacturing — durables and nondurables; 2. Trade— retail and wholesale; 3. Service producing — transportation, communications, public utilities,finance insurance and real estate, and services; 4. Other — agriculture forestry and fisheries, mining,and construction. Output is deflated Gross Product Originating (GPO) by industry which measuresthe contribution by each industry to GNP. The GPO series currently distributed by BEA dates toonly 1977 making it necessary to deflate and splice the data with the previous series, which wasavailable back to our 1947 starting point. The BEA wealth files include the following fixednonresidential private capital variables by industry (deflated): investment, gross capital stock, capitalinput, net stock, annual depreciation, and total discards. Net stock uses straight line depreciation,while capital input is derived from depreciation based on a beta decay function. Capital stockgrowth rates in this paper are based only on net stock (straight line) to save space. The results aresimilar using capital input based on beta decay. Labor input data by industry sector is found inJorgenson et al. (1987). Updated data through 1989 were provided by Jorgenson. Labor qualityadjustments were based on the following demographic attributes of the labor force: sex, age,education, employment class, occupation, and industry sector.


4.2. Solow residuals without adjustment of inputs for quality

The first part of Table 2 illustrates Solow residuals without adjustment forquality change of any inputs, excepting, of course, that:

(1) since the early 1980’s, the BEA has attempted in a limited way to adjustproducer price indexes for quality enhancement for some products and

(2) capital stock and investment series for computing equipment are adjustedfor quality rather successfully using hedonic models.

Within a framework in which output is measured by value added, the range ofplausible estimates of the weight of capital is probably bounded by 25% and33%, with the latter more consistent with Statistics of Income data on grosscorporate income and labor income in the post-1947 economy. Using USCommerce Department estimates of capital consumption, and a capital weightof 33%, we find the Solow residual for the aggregate private sector for1947—1989 to be 1.15% (Table 2). Changing the capital share appears to haveonly a modest impact on the corresponding residuals in all sectors in the periodexamined.

4.3. Solow residuals with quality-adjusted capital and labor

Quality adjustment for the capital stock, investment, and growth was carriedout consistently with Eqs. (5) and (6). Our objective, once again, is to showhow the Solow residual changes as a is set at various fractions of thecapital consumption rates shown in Table 1 (which vary by sector and timeperiod).

Table 2 also reports Solow residuals for obsolescence (a) at 50%, and 100% ofthe BEA rates of depreciation. For example, obsolescence at 50% of BEAdepreciation assumes, on the average, about a 2.35% annual change in thequality of the capital stock. The upper bound we report is obsolescence aver-aging 4.7% per annum as is the case with a set at 100% of BEA depreciation.The reader is free to set his/her own rate of obsolescence (say a at 25%) andinterpolate from Table 2 the approximate Solow residuals involved.

The Solow residual for the aggregate of the sectors, with quality change incapital and labor allowed, was computed in two ways. The first makes noquality adjustment to the investment component of output, while the seconddoes. Adjustments to output for quality change in investment were not made atthe sectoral level because it is difficult to measure the contribution of investmentin one sector to value added in other sectors. The results include Jorgenson’sadjustment for change in quality embodied in the labor input.

Focusing primarily on the residuals for the aggregate in Table 2 includingquality adjustment for the investment component of output, a capital shareweight of 25% reveals Solow residuals over the entire 1947—1989 period of


Table 2Solow residuals without and with inputs adjusted for changes in quality, 1947—1989!

1947—1959 1960—1973 1974—1989 1947—1989

(1) Inputs unadjusted for quality changewk"25%

Manufacturing 1.81% 2.14% 2.34% 1.87%Trade 1.79% 1.67% 0.80% 1.27%Service producing 1.89% 1.47% 0.10% 1.27%Aggregate 2.24% 1.46% 0.82% 1.36%

wk"33%

Manufacturing 1.57% 1.85% 2.13% 1.61%Trade 1.56% 1.23% 0.50% 0.93%Service producing 1.70% 1.30% 0.09% 1.15%Aggregate 1.98% 1.21% 0.72% 1.15%

(2) Obsolescence is 50% of BEA capital depreciation rate (quality adjusted labor included)wk"25%

Manufacturing 0.61% 1.27% 1.56% 0.95%Trade 0.81% 0.91% !0.13% 0.41%Service producing 0.97% 1.00% !0.68% 0.59%Aggregate 1.16% 0.70% 0.02% 0.51%Agg. (investment adj.) 1.43% 1.01% 0.38% 0.83%

wk"33%

Manufacturing 0.24% 0.82% 1.18% 0.53%Trade 0.40% 0.26% !0.65% !0.14%Service producing 0.71% 0.68% !0.84% 0.34%Aggregate 0.78% 0.29% !0.27% 0.14%Agg. (investment adj.) 1.05% 0.60% 0.10% 0.46%

(3) Obsolescence equals the BEA capital depreciation rate (quality adjusted labor included)wk"25%

Manufacturing 0.01% 0.68% 0.95% 0.36%Trade 0.16% 0.23% !0.86% !0.28%Service producing 0.57% 0.54% !1.22% 0.13%Aggregate 0.61% 0.11% !0.62% !0.09%Agg. (investment adj.) 1.16% 0.72% 0.11% 0.54%

wk"33%

Manufacturing !0.56% 0.03% 0.36% !0.27%Trade !0.47% !0.65% !1.62% !1.06%Service producing 0.19% 0.07% !1.57% !0.29%Aggregate 0.05% !0.50% !1.11% !0.66%Agg. (investment adj.) 0.59% 0.12% !0.38% !0.03%

! With quality adjusted inputs, two versions of the aggregate residual are computed. Aggregate:assumes no adjustment of output for investment quality; Agg. (investment adj.): investment isadjusted for quality at the same rate as capital, weighted by investment’s share of output. Theinvestment adjustment could not be made for each of the sectors individually (manufacturing, trade,service producing industries) without information on intersectoral flows of investment.


1.36%, 0.83%, and 0.54% as obsolescence ranges, respectively, from 0% to 50%to 100% of BEA depreciation. For a more plausible share weight of capital at33%, the comparable range of Solow residuals is 1.15%, 0.46% and !0.03%.

Turning now to the subperiods and industry sectors, with a at 50% of BEAdepreciation, most residuals approach zero or turn negative in the 1974—1989subperiod (except for manufacturing). In fact, for service producing industries,the Solow residual in this period turns negative with obsolescence as low as 10%of BEA depreciation, while, with a capital share of 33%, the 1974—1989 residualfor trade turns negative for a around 20%. Positive residuals for manufacturingin this period, however, are attributed to the small contribution to growth fromchanges in labor input (0.01% and 0.23% per year in Table 1). For the highestassumed rate of obsolescence (4.7% per year with a at 100% of depreciation),and with a one-third share of capital, most residuals approach zero or turnnegative for all industry sectors, in each of the three subperiods, and for1947—1989 overall.

The possibility of a zero Solow residual is consistent with results frommicro-data for the manufacturing sector obtained by Bahk and Gort (1993),though the latter study used a very different basis for measuring labor quality.Jorgenson’s method of adjusting for change in the quality of the labor input,used in the results reported here, potentially underestimates the change in laborquality. There is no allowance made for change in the quality of educationexcept insofar as it is reflected in number of years of training and in theoccupation of labor.

A larger adjustment for labor quality would obviously reduce Solow residualsreported in this paper still further. However, it is also likely that some part of thechange in labor quality is uniquely related to change in the quality of capital inthe sense that a change in production technologies requires a concurrent changein both. In that sense conventional decomposition of growth in output may beinappropriate.

Still another puzzle is the effect on growth decomposition of the understate-ment of consumer goods output. Quality change is largely unmeasured also forthe consumption sector. A correction of growth in output for change in thequality of consumer goods raises a host of questions beyond the scope of thispaper. It is worth noting, however, that while such a correction would of courseraise the rate of productivity change attributable to the combined total of allinputs, it will not necessarily change the decomposition of sources of economicgrowth.

5. Conclusions

We have shown what conceptual adjustments are necessary to most capitalstock estimates in use today so as to render them usable for decomposing


growth in output. Solow residuals and their interpretation — the decompositionof input augmenting and disembodied technical change — critically depend onthe fraction of depreciation that is explained by obsolescence, as distinct fromphysical decay of capital. The residuals decline as the rate of obsolescence ofcapital increases. For the most plausible share weight of capital (33%), theresidual is 1.15% with no adjustment for input quality. It declines by 60% forobsolescence (averaging 2.35% per year) at one-half of BEA reported deprecia-tion rates, and declines to a residual of zero if the rate of obsolescence is equal tothe BEA rate of depreciation.

Recently published research has come to sharply divergent conclusions re-garding the rate of obsolescence of capital. The importance of this empiricalquestion not only for Solow residuals but more generally for all growth ac-counting research is clear. So is its importance for the public policy implicationsof the effect of investment on technical change.

References

Bahk, B-H, Gort, M., 1993. Decomposing learning by doing in new plants. Journal of PoliticalEconomy 101, 561—583.

Boskin, M.J., Lau, L.J., 1991. Postwar economic growth in the group of five countries: A newanalysis. In: Proceedings NBER Conference on Economic Growth.

Doms, M.E., 1992. Estimating capital efficiency schedules within production functions, Discussionpaper CES 92-4. U.S. Department of Commerce, Center for Economic Studies, Washington, DC.

Fisher, F.M., 1965. Embodied technical change and the existence of an aggregate capital stock.Review of Economic Studies 32, 263—288.

Gordon, R.J., 1990. The Measurement of Durable Goods Prices. University of Chicago Press,Chicago, IL.

Gort, M., Bahk, B.-H., Wall, R.A., 1993. Decomposing technical change. Southern EconomicJournal 60, 220—234.

Greenwood, J., Hercowitz, Z., Krusell, P., 1992. Macroeconomic implications of technical change,Working paper.

Hulten, C.R., 1992. Growth accounting when technical change is embodied in capital. AmericanEconomic Review 82, 964—980.

Jorgenson, D.W., Gollop, F.M., Fraumeni, B.M., 1987. Productivity and U.S. Economic Growth.Harvard University Press, Cambridge, MA.

Solow, R.M., 1957. Technical change and the aggregate production function. Review of Economicsand Statistics 39, 312—320.


obsolescence, input augmentation, and growth accounting

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