# Decomposing technical change.

I. Introduction

Within the conventional neoclassical framework, a distinction is sometimes made between product-augmenting and factor-augmenting technical change. A parallel distinction is commonly made between embodied and disembodied technical change with the former associated with factor, and the latter with product, augmentation. Disembodied change is commonly assumed to arise from increases in the stock of knowledge, independently of the characteristics of the inputs used, while embodied change relates to increases in the efficiency of inputs, that is, labor skills or the productivity of physical capital.

Unfortunately, this distinction is ambiguous. Changes in the efficiency of the inputs used are usually accompanied - indeed made possible - by increases in knowledge. And conversely, increases in the stock of knowledge often favor some inputs more than others, including the capital goods of one vintage relative to those of another.

Notwithstanding the ambiguity, the concept of embodiment has intuitive appeal and this partly explains the focus on decomposing the sources of technical change that followed Solow's [6] seminal paper. But by the late 1960s, a reader of the literature might have concluded that such decomposition was impossible. For with merely time series data on inputs and output, product-augmenting and factor-augmenting technical change are empirically indistinguishable.

In an important paper, Hall [3] showed that with data on used equipment prices and the interest rate, embodied technical change and the deterioration function can, in principle, be calculated. However, the paucity of data on the price of used capital goods has allowed little progress in this direction. The new Longitudinal Research Database created by the U.S. Bureau of the Census now permits still another approach to the estimation of "embodied" technical change associated with capital, both physical and human. In addition, it casts light on a perplexing problem that has plagued econometric estimates of production relations based on changes in inputs and output as distinct from levels of both.

This new body of information consists of time series and cross-section data for individual manufacturing plants for the period 1972 to 1986. The time series permit us to derive indexes of the vintage of capital for each plant. This, in turn, allows us to estimate the effects of vintage of capital on productivity from strictly cross-section data. And since these effects are estimated at a common point in time, temporal shifts in productivity divorced from vintage are excluded by definition.

Moreover, we are also able to distinguish between "new" plants - that is, plants without endowments of capital accumulated in earlier periods - from "old" plants. The analysis of data for old plants allows the test of a hypothesis, and yields an explanation, of why estimates based on changes in inputs and output generally lead to very different coefficients from those based on levels of the variables. The latter is an issue with important policy implications given that most investment in developed economies takes the form of expansion - that is, changes in inputs - for existing (old) plants.

The remainder of the paper is divided into six sections. In section II we present our principal model and the definitions of variables in our production function. Section III reports the estimates for technical change in the context of levels of inputs and output for new plants. Section IV discusses the implications of measuring production relations for changes in inputs and output as distinct from levels while section V presents estimates for old plants based on changes in the relevant variables. Section VI compares the results for new and old plants while section VII is a brief summary of principal conclusions.

II. Model for Measuring Technical Advance and Definition of Variables

We start with a general model

[Mathematical Expressions Omitted]

Where [O.sub.[tau]] is output, [A.sub.[tau]] is a shift parameter that is assumed to affect the productivity of all vintages of capital and all labor skills symmetrically, [e.sup.a[tau]] is disembodied technical change at rate a, [L.sup.[tau]] is labor, [Q.sup.[tau]] is human capital, and [SYMBOL NOT CONVERTED] is a vector of investment streams.

[Mathematical Expressions Omitted]

Where [tau] is the vintage year in which investment is measured, and [gamma] is the age of the plant.

As is commonly hypothesized, we expect each successive vintage of investment to be more productive than the last so that,

[Mathematical Expressions Omitted]

We next assume a standard production function approach of substituting an aggregate capital stock variable for the vector of investments and take due account of the effect of embodiment by measuring the average vintage of the stock. Accordingly, we have

O = [Ae.sup.a[tau]] F [(K.sub.v.e.sup.kv], L, Q) (4)

Where K is the sum of investments of various vintages, the subscript v is the weighted average vintage of the stock with weights based on the investment of each vintage relative to K, and [e.sup.kv] is an index of productivity enhancement at constant rate k from "embodied" effects of vintage (the subscripts of [tau] are omitted). The capital stock term of the production function, [K.sub.v.e.sup.kv], is thus converted into efficiency units based on average vintage.

The resulting model differs in several respects from production functions that are commonly estimated. First, human capital (labor skills) enters as a separate argument in the production function rather than as an adjustment to the measure of labor input. Second, capital is composed of gross investment streams, rather than net investment, so that the effect of vintage (that is, obsolescence plus decay) is estimated within the framework of the model. In this respect, our approach accords with that of Prucha and Nadiri [5], though we do not follow them in their assumption that the depreciation rate is endogenous. It contrasts with the conventional method of inferring obsolescence and deterioration from assumed economic lives and decay functions.

Our conceptual framework makes no distinction between the accumulation of knowledge and changes in the physical attributes of capital associated with vintage as long as new knowledge is uniquely related to vintage. Similarly, if new knowledge is uniquely related to labor skills, no distinction is made between the two. Changes in the shift parameter, [A.sub.[tau], disappear within a crosssection framework and only interplant variations in "disembodied" technical change remain.

Differences across plants in blueprint technology, and in the knowledge associated with it, are almost certainly uniquely related to either labor skills or the vintage of physical capital. What, then, is there left of disembodiment in the context of a cross-section model? It appears that only the effects of organizational capital, largely in the form of firm-specific information, remain unaccounted.

We next turn to a more detailed discussion of the variables in equation (4).

Physical Capital and Vintage

The stock of capital in equation (4) is the sum of deflated gross investments from the year following the birth of the plant to the year in which output is measured.(1) Obsolescence is then measured directly via the production function through estimates of the effect of vintage on output.

The effects of vintage arise from obsolescence + (physical decay - maintenance outlays). If, however, as is plausible, maintenance outlays roughly offset the effects of physical decay at least on current production (if not also on earnings), the principal source of difference in the relative efficiency of capital of different vintages is obsolescence. The implied depreciation rate then, correctly measured, becomes roughly the dual of capital augmenting technical change.

The foregoing indicates that the assumptions necessary to construct a net capital stock require implicitly a measure of embodied technical change of capital. And if physical decay roughly equals maintenance outlays, then obsolescence is all that needs to be measured to transform gross into net stocks.

Vintage was measured as the weighted average of the years of the investment stream for each plant, with weights based on the ratio of the annual investment for each plant to its total investment over the relevant period. By definition, a higher average indicated more recent vintage. Thus vintage measured (inversely) the average age of physical assets.

Since the productivity of an asset has a lower bound of zero, in principle, only non-retired assets should be included in the computation of average vintage. Otherwise, a systematic relation between the stock of retired assets and average vintage might lead, in the context of a production function, to distortions in the coefficients of both physical capital and vintage. However, since the period over which average vintage was computed was limited to 1973-86, retirements of assets from the relevant investment streams (as our tests showed) were not large enough to distort the estimates significantly and were, therefore, ignored.

Excluded from the model is circulating capital (that is, inventories). This is justified since inventory accumulation is, at least partly, unintended and is also a function of expected future rather than merely current output.

Labor and Human Capital

Our labor variable was intended to approximate pure labor independently of human capital (labor quality) and was thus measured by the number of employees for each plant. The index for the average amount of human capital associated with the labor input was simply the average wage rate for each plant.(2) In effect, we assumed that all plants have equal access to the labor market and that differences in average wages must reflect differences in human capital rather than variations in wage rates for identical classes of labor.

Our chosen measure is implicitly based on a definition of human capital as any attribute of labor that increases its productivity. Across plants at a point in time, we assume that average wage rates reflect primarily differences in the composition of the work force with respect to what Becker [1] has called general (as distinct from firm specific) human capital - that is, human capital the returns to which are probably captured by the employee. Hence, ad hoc measures of labor quality based on education, occupation, or demographic attributes are rendered unnecessary.

Differences in average wage rates across plants in the same industry were far too large to permit a conclusion that they reflected regional variations in wages. More specifically, for most industries, the highest average wage for any plant was roughly three times the lowest, and the standard deviations were typically between 20 and 30 percent of the mean wage. This represents a far greater variation than can plausibly be attributed to such factors as unionization, historical peculiarities, or regional differences in wages.

Even more decisive, if historical accident or unionization were important explanatory variables for the dispersion in average wages across plants, one would expect the variation to be larger for old than for new plants. All new plants can choose their location and, therefore, at the outset face common labor markets. In fact, the dispersion in average wages was larger for new than for old plants. This reflects the role of competing technologies with substantial trade-offs between human capital and other inputs rather than unionization or regional variations in wages.

As a final check, we divided all plants into the nine census geographic regions to assess to what extent variations in average wages were attributable to regional influences on wage rates. We found that, generally, one could not predict the regional pattern of high and low wages for one industry from the observed pattern for another industry.

Output

The dependent variable, output, was proxied alternatively by deflated shipments and by deflated value added.(3) Both shipments and value added have deficiencies as measures of output. While shipments ignore variations in purchases from other plants, value added is subject to error in measuring cost of materials and to error from inconsistencies in the valuation of inventories over time. The choice between them as to which is the best measure of changes in output depends partly on the set of industries examined. For this reason, results are generally presented with both variables as alternatives.

Conceptually, when shipments are used as the proxy for output, one might argue that materials inputs should appear on the right side of the equation. However, materials inputs are so large a fraction of shipments and, in consequence, so highly correlated with the latter at least in the context of cross-section data, that the introduction of this variable tends to dominate and obscure other relationships.

III. Estimates of Embodied Technical Change Based on Data for New Plants

From the standpoint of our analysis, the central distinction between "new" and "old" plants is not their chronological age but whether, at the starting date for analysis, there are initial endowments of physical capital that originate from earlier investments. If there are none, or if they are minor, we classify the plants as "new." The relevance of this distinction rests in our hypothesis (developed more fully later) about interactions between new investment and the initial capital stock. These interactions lead to non-separability of the relation between new inputs and changes in output and, hence, to unstable and misleading coefficients in the context of a production function.

Interactions across successive investment streams also occur for new plants with no initial endowments of capital. But, as explained later, for new, in contrast to old, plants interaction effects are far more likely to be proportional to cumulated new investment. First, across new plants, the stock of capital is far more homogeneous in its age composition. Second, for new plants investment streams of contiguous years are frequently elements of an integrated investment plan.

New plants were defined as plants born in 1973 or later while old plants were those in existence in 1972 (the initial year for the available data set). New plants were in fact considerably younger than old plants and, further, the bulk of their capital outlays were made within several years of their birth.

The Econometric Model

The production function, equation (4), is now expressed in Cobb-Douglas form as equation (5):

O = [Ae.sup.at] [L.sup.[beta].sub.1.Q.sup.[beta].sub.2] [(K.sub.v.e.sup.kv).sup.[beta].sub.3] (5) The customary Cobb-Douglas specification was modified by the inclusion of an index of human capital as a separate argument in the production function. Taking logs, simplifying, substituting [[beta].sub.o] for log ([Ae.sup.a[tau]]) and noting that [[beta].sub.4] = [[beta].sub.3]k, we have the empirical specification in equation (6):

log [O.sup.jt] = [[beta].sub.0t] + [[beta].sub.1t] log [L.sub.jt] + [[beta].sub.2t] log

[Q.sub.jt]

+ [[beta].sub.3t] log [K.sub.jt] + [[beta].sub.4t] [V.sub.vt] + [u.sub.jt]

(6) where the variables O,L,Q,K and V are defined as before and each variable is measured for plant j in time t. Note that the average vintage variable appears in linear rather than log form as a direct result of the specification in equation (5).

Results

Table I presents the results for equation (6) using the two alternative proxies for output, shipments and value added. For the analysis of new plants, two sets of industries were selected: one comprising 41 manufacturing industries and a subset of 32 industries. For the larger set, we included all industries with at least 16 new plants in 1982 (excepting only NEC industries and several that might not be considered primarily in manufacturing, e.g., publishing). For the subset of 32, the cutoff was 20 new plants.(4)

The terminal peak for each plant was defined as the year with the highest value of shipments in the period 1984-86. The cross-sections are estimated with plant data for the peak year only. The intention behind utilizing a "terminal peak" was to minimize measurement errors associated with variations across plants in capacity utilization.

In addition, for each alternative the results are shown for all plants born between 1973 and 1986 and then narrowed to those born at least three years before the terminal peak for each plant. Limiting the data to plants born at least three years prior to the terminal peak had the purpose of allowing sufficient time for capital goods to be fully phased in and, hence, for the estimates to correspond to die production frontier.

Turning to the results, the coefficients for labor, human capital, and physical capital are highly stable across the eight set of estimates, the median values for the three coefficients being .64, .61, and .33, respectively. There is considerably more volatility from sample to sample for the coefficients for vintage, with the median value being .04. The [R.sup.2] and t values, considering that cross-section data for a highly diverse group of industries were used, are all very high.

The principal contributions of the econometric model is that it enables the measurement of the effects on output of embodied technical change, and of human capital separate from "pure labor." To focus first on the latter, we observe from Table I that, particularly for estimates with value added as the dependent variable, the elasticity of output with respect to human capital is roughly the same as that for pure labor.

Given our definitions, a one percent change in human capital (measured by the average wage) must have the same effect on total costs as a one percent change in the labor input (measured by number of employees). Accordingly, the same coefficients for the two variables mean that the marginal products per dollar of expenditures are the same for the two inputs. The consistency of this result with an optimal input allocation rule is an outcome one might have expected from data drawn from an industry with a homogeneous output. It is surprising given the variety of industries and technologies from which the plants were drawn, as well as the enormous range of plant sizes encompassed by the samples.

[TABULAR DATA OMITTED]

The most frequent estimate for vintage yielded a coefficient that indicated a four percent change in output for each one-year change in the average vintage of the stock of capital. This is indeed a high value given that gross returns to capital have a weight of roughly one-third in total inputs for manufacturing industries (as measured by capital's share of gross compensation to capital plus labor, and using Statistics of Income data for 1972-86). Note also the consistency of the share of capital with the .33 median elasticity in the Cobb-Douglas model. Thus a 4 percent change in output attributed solely to embodied technical change for physical capital implies about a twelve percent change in the efficiency of capital goods from a one-year change in average vintage (k = [[beta].sub.4]/[[beta].sub.3] = .04/.33).

While there was some instability in the estimated coefficients for vintage, such estimates can only be viewed as rough approximations of average rates of change attributable to the age of capital goods. Not only is each year's investment composed of large numbers of specific capital goods, but the functional composition of capital goods (for example, structures versus equipment or office equipment versus transportation equipment) undoubtedly changes across vintages. Thus, the derived measures are meaningful only as approximations or scalar magnitudes rather than as point estimates. Our cross-sectional estimates of embodied technical change are considerably higher than estimates by Hulten [4] derived from ratios of quality adjusted to unadjusted equipment prices.

A dummy variable model was specified to assess the differences between estimated coefficients for each industry and those for the aggregate sample. Very few of the industry dummies were significant - that is, industry estimates for the coefficients did not deviate significantly from the estimates for the combined sample. Thus, the aggregate results in Table I are justifiable.

IV. Modeling Production Relations for Changes in Inputs and Output

The average proportion of capital expenditures in U.S. manufacturing that is spent on existing plants, as distinct from those under construction, has been estimated by Gort and Boddy [2] to exceed 90 percent of the total. At first glance, this fact seems puzzling. The addition of new capital goods to a production process already in place and incorporating old assets must be restricted in the kinds and combinations of inputs that can efficiently be added. Why then do firms choose to give up the flexibility and consequent economies associated with new plants of best practice technology?

There are three plausible explanations for investment in old establishments. First, their expansion may entail a shorter gestation period than creation of new plants. Second, scale economies may preclude the creation of new plants for small increases in output. And third, total input requirements for a given increase in output may be smaller for old plants because of interactions between old and new inputs. It is this third explanation that is the focus of our attention.

There are two principal ways in which such interactions may occur. First, new employees may learn from older ones thereby reducing adjustment costs. Second, new physical assets may interact with old ones by modifying them, or at least changing the way in which old assets are used. In this way, new capacity could be created with lower inputs of physical capital than required when starting from a zero base.

It is our hypothesis that increments in output entail different production function coefficients from those implicit in levels of output and inputs and that this, in turn, is a consequence of interactions. Were it otherwise - that is, in the absence of interactions - new inputs of capital (investment) on old plants could be viewed as separable levels of capital just as increases in output could similarly be viewed as the level of new output.

Gort and Boddy [2] modeled interaction effects through a simple multiplicative term - a procedure that made sense for the electric power industry they studied since the interaction took largely the form of addition of generating equipment to old structures, or of modifications of boilers for existing steam turbogenerators. The assumption of a symmetrical effect of new investment across all old capital goods as implied by a multiplicative term is, however, much too simple to capture the technological interactions observable in most industries. Indeed, interactions are difficult to model since they are likely to vary across plants within an industry as well as across industries.

Interactions occur across all vintages of investment. The fact that one year's investment may be composed of structures while the next year's is composed of equipment housed by the structures, means that growth in output cannot be expected to respond in a consistent way to a single year's investment outlays. Outlays over several years are likelier to reflect a balanced investment plan than those for a single year and, hence, (holding technology constant) are likelier to produce a proportional relation between growth in output and cumulated investment. But a balanced investment program still does not dispose of interactions that take the form of modifications of old assets made possible by new technology. Nor does it take account of differences in returns to new investment from interactions arising because of large differences across plants, at any point in time, in the size of the initial stock of capital.

It is plausible, however, that interaction effects associated with the stock of old assets existing at the outset decline as a function of time, relative to the separable output effects of new investment. Old plants vary not only in the magnitude of their initial capital endowments but also in the age of their old capital. Consequently when new investment is still small relative to old investment, interactions between new and old capital will produce unsystematic and, hence, unpredictable effects in the context of cross-section analysis. The problem, therefore, reduces itself to one of finding a subsequent point in time at which interaction effects across plants become sufficiently systematic that they can be measured.

Consider equation (7) for old plants

[Mathematical Expression Omitted] (7) where [tau] is the vintage year (with [tau] = O the base year), [O.sub.[tau]] is defined as output, [Mathematical Expression Omitted] the vector of current vintage year and previous investments such that [Mathematical Expression Omitted] [K.sub.[omicron]] the initial capital stock, [L.sub.[tau]] is labor, and [Q.sub.[tau]] is labor quality. The hypotheses concerning embodied technical change can be summarized as follows:

[Mathematical Expression Omitted]

Equation (8) (i) shows, as before, the greater productivity of more recent vintages of investment. The effect of interactions is shown in equation (8) (ii) by the positive (if any) contribution of more recent vintage investment, denoted by [tau], to the marginal productivity of past investments denoted by [tau] - 1 and [tau] - j. Interaction effects will be larger between investments of shorter time lapse between vintages (i < j) because old assets become progressively less adaptable to new capital. Finally, with obsolescence, the productivity of the initial capital stock, equation (8) (iii), declines over time. While not reflected in the above equations, the relative effect on output of interactions with [K.sub.[omicron]] declines over time for still another reason. As the sum of new investments grows over time, their separable effects on output, and the interactions across the new investments, grow in importance relative to the effects of [K.sub.[omicron]].

If one assumes that both vintage and interaction effects are of no consequence for investment (our null hypothesis), the production function for vintage year [tau] can be expressed as

[O.sub.[tau]] = [h.sub.[tau]]([K.sub.[tau]],[L.sub.tau]],[Q.sub.tau]] (9) where [K.sub.[tau]] is the capital stock aggregated from the investment vector and the initial stock. Now define [delta difference][O.sub[tau]] = [O.sub.[tau]] - [O.sub.[omicron]], where [O.sub.[omicron]] = [h.sub.[omicron]]([K.sub.[omicron]],[L.sub.[omicron]],[Q.sub.[omicron]]) under the null hypothesis stated above (that is, with no embodiment). The increment to output relative to the base year level should then be a separable production function(5) expressible as

[delta difference][O.sub.[tau]] = [h.sub.[tau]]([delta difference][K.sub.[tau]],

[delta difference][L.sub.[tau]],[Q.sub.[tau]]) 10 where [delta difference][K.sub.[tau]] = [K.sub[tau]] - [K.sub.[omicron]] and [delta difference][.sub.[tau]] = [L.sub[tau]] - [L.sub.[omicron]] measure the increments to the capital stock and labor force, respectively.

The role of human capital in equation (10) is similar to that of a technology index. One would expect that growth in physical output depends not only on growth in physical inputs, but also on the functional relationship between the existing level of human knowledge and increments to physical capital and labor. Thus, [Q.sub.[tau]], is the level of human capital reached by vintage year [tau] which can be utilized by the increment to the labor force in the production of additions to output.

Equation (10), if it holds, implies that the coefficients of the production function (ignoring economies of scale) are the same whether one estimates the relation for increments to output or for levels of output (equation (9)).

Vintage effects, when included in an empirical specification, permit a test to determine whether productivity is greater for more recent additions to the capital stock. If interactions initially have an unsystematic effect, this obscures the production relation of changes in physical capital to increments in output for vintage years soon after the base year (that is, the start of the period examined). However, as the time elapsed from the base year increases, the production relation for increments to output approaches that for a specification in terms of levels rather than changes. A change in capital variable can thus be assumed to capture the "levels" effect of a balanced investment plan and, in addition, the systematic component of the impact of interactions on productivity.

V. Estimates of Embodiment for Old Plants

Using cross-section data, we again estimate a modified Cobb-Douglas production function, this time for changes in output and in labor and capital inputs for old plants. The model is written as follows:

[Mathematical Expression Omitted]

where [delta difference] log [O.sub.jt] is the percentage change in output for the jth plant for time t relative to the initial period of 1972, [delta difference] log L is the percentage change in pure labor, and [delta difference]A log K is the percentage change in gross capital. Percentage changes standardize units to control for size effects across plants, and, in a sense, also standardize the observations for differences in initial factor proportions.

Log Q measures the level of human capital available to the increment in the labor force, and V is, as before, a measure of the weighted average vintage of investment expenditures for each plant. Output is again measured by the (deflated) value of shipments or, altematively, value added, labor by total employees, and labor quality or human capital by the average wage rate for each plant. The weights for vintage are, of course, the annual investment expenditures for the period over which changes in capital inputs are measured.

For each regression, the initial capital stock for each plant is simply its deflated gross assets in 1972,(6) and the terminal capital stock is obtained by adding to the initial value cumulated (deflated) gross capital expenditures plus the capitalized value of the change in rentals of assets. Errors associated with the measurement of initial stocks, for which data on annual investment streams are lacking, can be expected to reduce greatly the goodness of fit of our model.

Our objective was to test the implications of equation (8) that the power of interactions gradually declines over time. Equation (11) was therefore estimated consecutively for each year. According to the null hypothesis of no embodiment in the form of vintage or interaction effects, [[beta].sub.4t] = O and [[beta].sub.3t] > O, respectively, in the years immediately following 1972. A positive and stable measured effect for the [delta difference] log K variable would indicate relatively weak interaction effects, and, hence, a separable production relation for changes in output and capital input.

We now turn to results for the empirical model in equation (11), for plants in 15 industries.(7) Table II is presented with shipments as the proxy for output. The same estimates but with value added as the dependent variable yielded very similar, though somewhat more erratic, results with lower values of [R.sup.2]. For economy of space, the latter are not reported in detail.

The results show the consecutive changes in coefficients for old plants for the increments in output and inputs from 1972 to the levels for each successive year. In general, there is strong support for the conclusion that for an extended period, interactions with the initial capital stock do not permit estimation of a separable relation between change in capital inputs and the change in output. It takes roughly twelve years for the relative effect of interactions to decline to a level that permits one to estimate a stable coefficient for the change in the log of K.

Over time, the coefficient for [delta difference] log K increases from near zero and insignificance in the early 1970s to a significant positive elasticity of above 0.3 by 1984. Initially, a systematic relation between growth in capital and in output is obscured by the unpredictable effects of interactions, given large variations across plants in the age and size of the endowments of capital at the outset of the period. But as the ratio of cumulated investment to initial capital rises, interaction effects become more systematic and the coefficient for [delta difference] log K measureable.

Technical change embodied in capital is shown most directly by V. The insignificant results for V in earlier years were to be expected. If the effects of increments in capital are obscured by interactions, it is likely that so will the effects of changes in the vintage of capital. Moreover, since it is the vintage of post-1972 capital that was measured, sufficient time had to elapse for there to be enough dispersion in vintage to detect an effect.

Accordingly, the coefficient of V did not become significantly positive until 1982, but remained reasonably stable thereafter averaging .04 for the five year interval 1982-86. The approximately 4 percent increase in output for every one year change in vintage is substantially the same as that observed earlier for new plants. Thus the high rate of embodied technical change observed for new plants is confirmed with data for old plants.

The consistently rising negative value of the intercept as one moves from 1973 to 1986 is explained by the construction of the capital variable. While V measured changes in the vintage of post-1972 capital, no allowance was made for the progressive obsolescence of the initial (1972) stock of capital. Hence, for all plants, terminal year capital was systematically overstated by an increasing amount for each successive year. Thus the rising negative intercept appears to capture the obsolescence rate for old capital.

Table II gives us some insight into the effect of interactions between new and old inputs of physical capital. It is problematic, however, insofar as year-to-year changes may represent observations for less than capacity utilization and, hence, may not correctly measure the production frontier. This is especially a problem for the labor input and may explain the instability of the coefficient for log of Q in Table II. As is well known, firms retain skilled labor during contractions in output. The resulting change in the composition of labor, with its consequent change in the average wage, is likely to lead to some distortion in the coefficients for L and Q. An illustration of this phenomenon is reflected in the non-significant results for Q during the 1981-82 recession.

[TABULAR DATA OMITTED]

VI. Comparing Production Relations for New and Old Plants

The next objective was to derive estimates for all the variables in equation (11) that correspond to the production frontier and this required that we measure changes in output and inputs between points approximating capacity utilization. Production relations involving changes, as distinct from levels, of output and inputs are likely to be especially sensitive to the assumption of capacity utilization and that condition seems best approximated at output peaks.

For an empirical approximation of capacity utilization, the first peak was the higher of the 1972 or 1973 value of shipments. The terminal peak again was the year with the highest value of shipments in the period 1984-86. The cross-section data are based on the peak to peak change in logs for O, L, and K for each plant. In the overwhelming majority of cases, the highest rate of capacity utilization by almost any criterion did occur within those intervals.

The resulting estimates, with both shipments and value added as proxies for output, were also compared with the coefficients for new plants. The latter were derived from equation (6) but limited now to the same 15 industry sample as used for old plants. The estimates are shown in Table III.

[TABULAR DATA OMITTED]

Before proceeding with the comparison of old and new plants, some characteristics of the results for old plants might be noted. As compared with the average coefficients reported in Table II for consecutive years, the coefficient for labor declines markedly. Those for physical capital and vintage rise though the order of magnitude remains roughly the same as before. In estimating an industry dummy variable model as for new plants in section III, most of the industry dummies proved non-significant thereby rendering the coefficients for the aggregate more meaningful as average estimates.

Comparing the results in Table III we find:

(a) As to be expected, the [R.sup.2] values for new plants are much higher than for old plants - a fact attributable in large part to far superior data for the capital variable for new plants.

(b) For new plants, the coefficient for human capital (Q) is substantially higher than for old plants. The efficiency with which new plants use human capital appears to be their single most important advantage over old plants. Once technological options are limited by a large amount of old physical assets, the ability to substitute human capital for other inputs appears to be severely restricted.

(c) The coefficient for labor is of roughly the same magnitude for old and new plants.

(d) Especially important is the considerable stability in the coefficients for K when estimates based on changes in inputs and output for old plants are compared with levels for new plants. However, while these coefficients are generally of the same order of magnitude, the coefficients remain slightly higher for old plants. This may suggest some continued impact of interactions even after a period of as much as 14 years past the point at which the initial capital stock was measured.

(e) V continues to be much more sensitive than K to choice of sample and proxy for output. However, the higher estimates for old than for new plants are consistent with what we know about capital expenditures. A larger proportion of capital outlays are devoted to structures rather than to equipment for new than for old plants. Structures are generally assumed to be associated with much lower rates of obsolescence (hence, embodied technical change).

VII. Summary

There is renewed interest in the recent literature in measurement and measurement errors associated with output growth. The standard approach, such as in Hulten[4], is to compute Solow residuals by decomposing output growth into the contributions of input quantities, quality adjustments, and the relative proportions of embodied and disembodied technical change. This paper provides an alternative approach to measuring the vintage effects of additions to capital utilizing cross-sections of manufacturing industries at the plant level made possible by the Census LRD database.

The principal results of the paper are briefly summarized:

1. We first specified a production function with human capital as a separate argument and with embodied technical change proxied by a variable that measures the average vintage of the stock of capital.

2. The coefficients of the production function were first estimated with cross section data for roughly 2150 new manufacturing plants in 41 industries, and for subsets of this sample. An augmented Cobb-Douglas specification was used. The results proved fairly stable across varying samples of plants and with respect to alternative measures of output.

3. Substantively, it was found that the elasticity of output with respect to human capital was approximately the same as it was with respect to pure labor. Embodied technical change of capital produced an average 4 percent increase in output for each one year change in average vintage.

4. It was pointed out that most investment in a developed economy is made on old rather than on new plants. An important question, therefore, concerns the separability of the relation between new inputs (that is, changes in the level of inputs) and changes in the level of output. A model was specified with interactions between new investment and initial endowments of capital. Interaction effects were predicted to decline in importance as a function of time.

5. Using a sample of roughly 1400 old plants in 15 industries, it was found that interactions between new investment and initial endowments of capital were, for a long interval of time, too unsystematic to permit measurement of a coefficient for capital. After twelve to fourteen years of cumulative investment, a systematic relation between changes in the level of inputs and changes in the level of output became measurable. Moreover, the coefficient for changes in the capital input for old plants proved to be of approximately the same magnitude as that for level of capital for new plants.

6. Comparing new and old plants over the "long-run," the estimates of embodied technical change of capital and of the elasticity of output with respect to number of employees (pure labor) proved very similar for the two types of plants. Differences between new and old plants in the elasticity of output with respect to human capital remained very large, however, and appear to be an important difference between the two sets of plants.

References

[1.] Becker, Gary. Human Capital. The University of Chicago Press, 1964. [2.] Gort, Michael, and Raford Boddy. "Vintage Effects and the Time Path of Investment in Production Relations," in The Theory and Empirical Analysis of Production, Studies in Inconte and Wealth, Vol. 31, National Bureau of Economic Research. New York: Columbia University Press, 1967, pp. 395-422. [3.] Hall, Robert E., "Technical Change and Capital from the Point of View of the Dual." Review of Economic Studies, January 1968, 35-46. [4.] Hulten, Charles R., "Growth Accounting When Technical Change is Embodied in Capital." American Economic Review, September 1992, 964-80. [5.] Prucha, Ingmar R., and M. Ishaq Nadiri. "Endogenous Capital Utilization and Productivity Measurement in Dynamic Factor Demand Models: Theory and an Application to the U.S. Electrical Machinery Industry." Working paper, 1990. [6.] Solow, Robert M., "Technical Change and the Aggregate Production Function." Review of Economics and Statistics, August 1957, 312-20.

(*) The project was carried out with assistance from the ASA/NSF/CENSUS fellowship program. The authors are, of course, solely responsible for the conclusions and methods of analysis used. (1.) Investments were deflated by the implicit price deflator for capital expenditures in all manufacturing combined, based on unpublished Bureau of Economic Analysis data. To the cumulative total of gross capital expenditures we added the capitalized value of the changes in rentals of fixed assets. The rate of capitalization was derived from the ratio of gross fixed assets to the sum of net income before taxes plus interest paid plus depreciation, as reported for 1972-86 in U.S. Internal Revenue Service, Statistics of Income. (2.) To facilitate the interpretation of the coefficients of change in human capital, as well as to correct for possible biases arising from the fact that in some of our cross-sections the observations do not relate to identical points in time, average wage rates were deflated. The deflator was the Consumer Price Index and was intended simply to correct for the average rate of inflation in the economy. (3.) Output deflators were also drawn from unpublished BEA data at the 4-digit industry level. (4.) Within the sets of industries for both new and old plants, only plants that satisfied the following criteria were chosen: (a) a continuous history in the same industry, from birth for new plants and from 1972 for old, until 1986, (b) a primary industry specialization ratio of at least 50%. This gave us about 2150 new plants for the 41 industries and roughly 1900 for the 32. The period chosen, 1972-86, was determined by the time interval for which panel data were available. A list of industries and number of plants in each will be provided upon request. (5.) The mathematical conditions necessary for exact separability of output and input levels into functions based on increments is, to our knowledge, unsolved. (6.) The deflators were derived from ratios of gross capital stocks at historical cost to stocks at constant cost as reported in Bureau of Economic Analysis, U.S. Department of Commerce, Fixed Reproducible Tangible Wealth in the United States, 1925-85, 1987. (7.) For the analysis of old plants, and for comparisons of old and new plants, a further subset of 15 industries was employed primarily to keep the size of the database manageable. These consisted generally of the largest among the original 41 industries but with selection based on broad representation across the industrial spectrum. Our sample consisted of about 1400 old plants in the 15 industries, and about 1250 new plants.

Within the conventional neoclassical framework, a distinction is sometimes made between product-augmenting and factor-augmenting technical change. A parallel distinction is commonly made between embodied and disembodied technical change with the former associated with factor, and the latter with product, augmentation. Disembodied change is commonly assumed to arise from increases in the stock of knowledge, independently of the characteristics of the inputs used, while embodied change relates to increases in the efficiency of inputs, that is, labor skills or the productivity of physical capital.

Unfortunately, this distinction is ambiguous. Changes in the efficiency of the inputs used are usually accompanied - indeed made possible - by increases in knowledge. And conversely, increases in the stock of knowledge often favor some inputs more than others, including the capital goods of one vintage relative to those of another.

Notwithstanding the ambiguity, the concept of embodiment has intuitive appeal and this partly explains the focus on decomposing the sources of technical change that followed Solow's [6] seminal paper. But by the late 1960s, a reader of the literature might have concluded that such decomposition was impossible. For with merely time series data on inputs and output, product-augmenting and factor-augmenting technical change are empirically indistinguishable.

In an important paper, Hall [3] showed that with data on used equipment prices and the interest rate, embodied technical change and the deterioration function can, in principle, be calculated. However, the paucity of data on the price of used capital goods has allowed little progress in this direction. The new Longitudinal Research Database created by the U.S. Bureau of the Census now permits still another approach to the estimation of "embodied" technical change associated with capital, both physical and human. In addition, it casts light on a perplexing problem that has plagued econometric estimates of production relations based on changes in inputs and output as distinct from levels of both.

This new body of information consists of time series and cross-section data for individual manufacturing plants for the period 1972 to 1986. The time series permit us to derive indexes of the vintage of capital for each plant. This, in turn, allows us to estimate the effects of vintage of capital on productivity from strictly cross-section data. And since these effects are estimated at a common point in time, temporal shifts in productivity divorced from vintage are excluded by definition.

Moreover, we are also able to distinguish between "new" plants - that is, plants without endowments of capital accumulated in earlier periods - from "old" plants. The analysis of data for old plants allows the test of a hypothesis, and yields an explanation, of why estimates based on changes in inputs and output generally lead to very different coefficients from those based on levels of the variables. The latter is an issue with important policy implications given that most investment in developed economies takes the form of expansion - that is, changes in inputs - for existing (old) plants.

The remainder of the paper is divided into six sections. In section II we present our principal model and the definitions of variables in our production function. Section III reports the estimates for technical change in the context of levels of inputs and output for new plants. Section IV discusses the implications of measuring production relations for changes in inputs and output as distinct from levels while section V presents estimates for old plants based on changes in the relevant variables. Section VI compares the results for new and old plants while section VII is a brief summary of principal conclusions.

II. Model for Measuring Technical Advance and Definition of Variables

We start with a general model

[Mathematical Expressions Omitted]

Where [O.sub.[tau]] is output, [A.sub.[tau]] is a shift parameter that is assumed to affect the productivity of all vintages of capital and all labor skills symmetrically, [e.sup.a[tau]] is disembodied technical change at rate a, [L.sup.[tau]] is labor, [Q.sup.[tau]] is human capital, and [SYMBOL NOT CONVERTED] is a vector of investment streams.

[Mathematical Expressions Omitted]

Where [tau] is the vintage year in which investment is measured, and [gamma] is the age of the plant.

As is commonly hypothesized, we expect each successive vintage of investment to be more productive than the last so that,

[Mathematical Expressions Omitted]

We next assume a standard production function approach of substituting an aggregate capital stock variable for the vector of investments and take due account of the effect of embodiment by measuring the average vintage of the stock. Accordingly, we have

O = [Ae.sup.a[tau]] F [(K.sub.v.e.sup.kv], L, Q) (4)

Where K is the sum of investments of various vintages, the subscript v is the weighted average vintage of the stock with weights based on the investment of each vintage relative to K, and [e.sup.kv] is an index of productivity enhancement at constant rate k from "embodied" effects of vintage (the subscripts of [tau] are omitted). The capital stock term of the production function, [K.sub.v.e.sup.kv], is thus converted into efficiency units based on average vintage.

The resulting model differs in several respects from production functions that are commonly estimated. First, human capital (labor skills) enters as a separate argument in the production function rather than as an adjustment to the measure of labor input. Second, capital is composed of gross investment streams, rather than net investment, so that the effect of vintage (that is, obsolescence plus decay) is estimated within the framework of the model. In this respect, our approach accords with that of Prucha and Nadiri [5], though we do not follow them in their assumption that the depreciation rate is endogenous. It contrasts with the conventional method of inferring obsolescence and deterioration from assumed economic lives and decay functions.

Our conceptual framework makes no distinction between the accumulation of knowledge and changes in the physical attributes of capital associated with vintage as long as new knowledge is uniquely related to vintage. Similarly, if new knowledge is uniquely related to labor skills, no distinction is made between the two. Changes in the shift parameter, [A.sub.[tau], disappear within a crosssection framework and only interplant variations in "disembodied" technical change remain.

Differences across plants in blueprint technology, and in the knowledge associated with it, are almost certainly uniquely related to either labor skills or the vintage of physical capital. What, then, is there left of disembodiment in the context of a cross-section model? It appears that only the effects of organizational capital, largely in the form of firm-specific information, remain unaccounted.

We next turn to a more detailed discussion of the variables in equation (4).

Physical Capital and Vintage

The stock of capital in equation (4) is the sum of deflated gross investments from the year following the birth of the plant to the year in which output is measured.(1) Obsolescence is then measured directly via the production function through estimates of the effect of vintage on output.

The effects of vintage arise from obsolescence + (physical decay - maintenance outlays). If, however, as is plausible, maintenance outlays roughly offset the effects of physical decay at least on current production (if not also on earnings), the principal source of difference in the relative efficiency of capital of different vintages is obsolescence. The implied depreciation rate then, correctly measured, becomes roughly the dual of capital augmenting technical change.

The foregoing indicates that the assumptions necessary to construct a net capital stock require implicitly a measure of embodied technical change of capital. And if physical decay roughly equals maintenance outlays, then obsolescence is all that needs to be measured to transform gross into net stocks.

Vintage was measured as the weighted average of the years of the investment stream for each plant, with weights based on the ratio of the annual investment for each plant to its total investment over the relevant period. By definition, a higher average indicated more recent vintage. Thus vintage measured (inversely) the average age of physical assets.

Since the productivity of an asset has a lower bound of zero, in principle, only non-retired assets should be included in the computation of average vintage. Otherwise, a systematic relation between the stock of retired assets and average vintage might lead, in the context of a production function, to distortions in the coefficients of both physical capital and vintage. However, since the period over which average vintage was computed was limited to 1973-86, retirements of assets from the relevant investment streams (as our tests showed) were not large enough to distort the estimates significantly and were, therefore, ignored.

Excluded from the model is circulating capital (that is, inventories). This is justified since inventory accumulation is, at least partly, unintended and is also a function of expected future rather than merely current output.

Labor and Human Capital

Our labor variable was intended to approximate pure labor independently of human capital (labor quality) and was thus measured by the number of employees for each plant. The index for the average amount of human capital associated with the labor input was simply the average wage rate for each plant.(2) In effect, we assumed that all plants have equal access to the labor market and that differences in average wages must reflect differences in human capital rather than variations in wage rates for identical classes of labor.

Our chosen measure is implicitly based on a definition of human capital as any attribute of labor that increases its productivity. Across plants at a point in time, we assume that average wage rates reflect primarily differences in the composition of the work force with respect to what Becker [1] has called general (as distinct from firm specific) human capital - that is, human capital the returns to which are probably captured by the employee. Hence, ad hoc measures of labor quality based on education, occupation, or demographic attributes are rendered unnecessary.

Differences in average wage rates across plants in the same industry were far too large to permit a conclusion that they reflected regional variations in wages. More specifically, for most industries, the highest average wage for any plant was roughly three times the lowest, and the standard deviations were typically between 20 and 30 percent of the mean wage. This represents a far greater variation than can plausibly be attributed to such factors as unionization, historical peculiarities, or regional differences in wages.

Even more decisive, if historical accident or unionization were important explanatory variables for the dispersion in average wages across plants, one would expect the variation to be larger for old than for new plants. All new plants can choose their location and, therefore, at the outset face common labor markets. In fact, the dispersion in average wages was larger for new than for old plants. This reflects the role of competing technologies with substantial trade-offs between human capital and other inputs rather than unionization or regional variations in wages.

As a final check, we divided all plants into the nine census geographic regions to assess to what extent variations in average wages were attributable to regional influences on wage rates. We found that, generally, one could not predict the regional pattern of high and low wages for one industry from the observed pattern for another industry.

Output

The dependent variable, output, was proxied alternatively by deflated shipments and by deflated value added.(3) Both shipments and value added have deficiencies as measures of output. While shipments ignore variations in purchases from other plants, value added is subject to error in measuring cost of materials and to error from inconsistencies in the valuation of inventories over time. The choice between them as to which is the best measure of changes in output depends partly on the set of industries examined. For this reason, results are generally presented with both variables as alternatives.

Conceptually, when shipments are used as the proxy for output, one might argue that materials inputs should appear on the right side of the equation. However, materials inputs are so large a fraction of shipments and, in consequence, so highly correlated with the latter at least in the context of cross-section data, that the introduction of this variable tends to dominate and obscure other relationships.

III. Estimates of Embodied Technical Change Based on Data for New Plants

From the standpoint of our analysis, the central distinction between "new" and "old" plants is not their chronological age but whether, at the starting date for analysis, there are initial endowments of physical capital that originate from earlier investments. If there are none, or if they are minor, we classify the plants as "new." The relevance of this distinction rests in our hypothesis (developed more fully later) about interactions between new investment and the initial capital stock. These interactions lead to non-separability of the relation between new inputs and changes in output and, hence, to unstable and misleading coefficients in the context of a production function.

Interactions across successive investment streams also occur for new plants with no initial endowments of capital. But, as explained later, for new, in contrast to old, plants interaction effects are far more likely to be proportional to cumulated new investment. First, across new plants, the stock of capital is far more homogeneous in its age composition. Second, for new plants investment streams of contiguous years are frequently elements of an integrated investment plan.

New plants were defined as plants born in 1973 or later while old plants were those in existence in 1972 (the initial year for the available data set). New plants were in fact considerably younger than old plants and, further, the bulk of their capital outlays were made within several years of their birth.

The Econometric Model

The production function, equation (4), is now expressed in Cobb-Douglas form as equation (5):

O = [Ae.sup.at] [L.sup.[beta].sub.1.Q.sup.[beta].sub.2] [(K.sub.v.e.sup.kv).sup.[beta].sub.3] (5) The customary Cobb-Douglas specification was modified by the inclusion of an index of human capital as a separate argument in the production function. Taking logs, simplifying, substituting [[beta].sub.o] for log ([Ae.sup.a[tau]]) and noting that [[beta].sub.4] = [[beta].sub.3]k, we have the empirical specification in equation (6):

log [O.sup.jt] = [[beta].sub.0t] + [[beta].sub.1t] log [L.sub.jt] + [[beta].sub.2t] log

[Q.sub.jt]

+ [[beta].sub.3t] log [K.sub.jt] + [[beta].sub.4t] [V.sub.vt] + [u.sub.jt]

(6) where the variables O,L,Q,K and V are defined as before and each variable is measured for plant j in time t. Note that the average vintage variable appears in linear rather than log form as a direct result of the specification in equation (5).

Results

Table I presents the results for equation (6) using the two alternative proxies for output, shipments and value added. For the analysis of new plants, two sets of industries were selected: one comprising 41 manufacturing industries and a subset of 32 industries. For the larger set, we included all industries with at least 16 new plants in 1982 (excepting only NEC industries and several that might not be considered primarily in manufacturing, e.g., publishing). For the subset of 32, the cutoff was 20 new plants.(4)

The terminal peak for each plant was defined as the year with the highest value of shipments in the period 1984-86. The cross-sections are estimated with plant data for the peak year only. The intention behind utilizing a "terminal peak" was to minimize measurement errors associated with variations across plants in capacity utilization.

In addition, for each alternative the results are shown for all plants born between 1973 and 1986 and then narrowed to those born at least three years before the terminal peak for each plant. Limiting the data to plants born at least three years prior to the terminal peak had the purpose of allowing sufficient time for capital goods to be fully phased in and, hence, for the estimates to correspond to die production frontier.

Turning to the results, the coefficients for labor, human capital, and physical capital are highly stable across the eight set of estimates, the median values for the three coefficients being .64, .61, and .33, respectively. There is considerably more volatility from sample to sample for the coefficients for vintage, with the median value being .04. The [R.sup.2] and t values, considering that cross-section data for a highly diverse group of industries were used, are all very high.

The principal contributions of the econometric model is that it enables the measurement of the effects on output of embodied technical change, and of human capital separate from "pure labor." To focus first on the latter, we observe from Table I that, particularly for estimates with value added as the dependent variable, the elasticity of output with respect to human capital is roughly the same as that for pure labor.

Given our definitions, a one percent change in human capital (measured by the average wage) must have the same effect on total costs as a one percent change in the labor input (measured by number of employees). Accordingly, the same coefficients for the two variables mean that the marginal products per dollar of expenditures are the same for the two inputs. The consistency of this result with an optimal input allocation rule is an outcome one might have expected from data drawn from an industry with a homogeneous output. It is surprising given the variety of industries and technologies from which the plants were drawn, as well as the enormous range of plant sizes encompassed by the samples.

[TABULAR DATA OMITTED]

The most frequent estimate for vintage yielded a coefficient that indicated a four percent change in output for each one-year change in the average vintage of the stock of capital. This is indeed a high value given that gross returns to capital have a weight of roughly one-third in total inputs for manufacturing industries (as measured by capital's share of gross compensation to capital plus labor, and using Statistics of Income data for 1972-86). Note also the consistency of the share of capital with the .33 median elasticity in the Cobb-Douglas model. Thus a 4 percent change in output attributed solely to embodied technical change for physical capital implies about a twelve percent change in the efficiency of capital goods from a one-year change in average vintage (k = [[beta].sub.4]/[[beta].sub.3] = .04/.33).

While there was some instability in the estimated coefficients for vintage, such estimates can only be viewed as rough approximations of average rates of change attributable to the age of capital goods. Not only is each year's investment composed of large numbers of specific capital goods, but the functional composition of capital goods (for example, structures versus equipment or office equipment versus transportation equipment) undoubtedly changes across vintages. Thus, the derived measures are meaningful only as approximations or scalar magnitudes rather than as point estimates. Our cross-sectional estimates of embodied technical change are considerably higher than estimates by Hulten [4] derived from ratios of quality adjusted to unadjusted equipment prices.

A dummy variable model was specified to assess the differences between estimated coefficients for each industry and those for the aggregate sample. Very few of the industry dummies were significant - that is, industry estimates for the coefficients did not deviate significantly from the estimates for the combined sample. Thus, the aggregate results in Table I are justifiable.

IV. Modeling Production Relations for Changes in Inputs and Output

The average proportion of capital expenditures in U.S. manufacturing that is spent on existing plants, as distinct from those under construction, has been estimated by Gort and Boddy [2] to exceed 90 percent of the total. At first glance, this fact seems puzzling. The addition of new capital goods to a production process already in place and incorporating old assets must be restricted in the kinds and combinations of inputs that can efficiently be added. Why then do firms choose to give up the flexibility and consequent economies associated with new plants of best practice technology?

There are three plausible explanations for investment in old establishments. First, their expansion may entail a shorter gestation period than creation of new plants. Second, scale economies may preclude the creation of new plants for small increases in output. And third, total input requirements for a given increase in output may be smaller for old plants because of interactions between old and new inputs. It is this third explanation that is the focus of our attention.

There are two principal ways in which such interactions may occur. First, new employees may learn from older ones thereby reducing adjustment costs. Second, new physical assets may interact with old ones by modifying them, or at least changing the way in which old assets are used. In this way, new capacity could be created with lower inputs of physical capital than required when starting from a zero base.

It is our hypothesis that increments in output entail different production function coefficients from those implicit in levels of output and inputs and that this, in turn, is a consequence of interactions. Were it otherwise - that is, in the absence of interactions - new inputs of capital (investment) on old plants could be viewed as separable levels of capital just as increases in output could similarly be viewed as the level of new output.

Gort and Boddy [2] modeled interaction effects through a simple multiplicative term - a procedure that made sense for the electric power industry they studied since the interaction took largely the form of addition of generating equipment to old structures, or of modifications of boilers for existing steam turbogenerators. The assumption of a symmetrical effect of new investment across all old capital goods as implied by a multiplicative term is, however, much too simple to capture the technological interactions observable in most industries. Indeed, interactions are difficult to model since they are likely to vary across plants within an industry as well as across industries.

Interactions occur across all vintages of investment. The fact that one year's investment may be composed of structures while the next year's is composed of equipment housed by the structures, means that growth in output cannot be expected to respond in a consistent way to a single year's investment outlays. Outlays over several years are likelier to reflect a balanced investment plan than those for a single year and, hence, (holding technology constant) are likelier to produce a proportional relation between growth in output and cumulated investment. But a balanced investment program still does not dispose of interactions that take the form of modifications of old assets made possible by new technology. Nor does it take account of differences in returns to new investment from interactions arising because of large differences across plants, at any point in time, in the size of the initial stock of capital.

It is plausible, however, that interaction effects associated with the stock of old assets existing at the outset decline as a function of time, relative to the separable output effects of new investment. Old plants vary not only in the magnitude of their initial capital endowments but also in the age of their old capital. Consequently when new investment is still small relative to old investment, interactions between new and old capital will produce unsystematic and, hence, unpredictable effects in the context of cross-section analysis. The problem, therefore, reduces itself to one of finding a subsequent point in time at which interaction effects across plants become sufficiently systematic that they can be measured.

Consider equation (7) for old plants

[Mathematical Expression Omitted] (7) where [tau] is the vintage year (with [tau] = O the base year), [O.sub.[tau]] is defined as output, [Mathematical Expression Omitted] the vector of current vintage year and previous investments such that [Mathematical Expression Omitted] [K.sub.[omicron]] the initial capital stock, [L.sub.[tau]] is labor, and [Q.sub.[tau]] is labor quality. The hypotheses concerning embodied technical change can be summarized as follows:

[Mathematical Expression Omitted]

Equation (8) (i) shows, as before, the greater productivity of more recent vintages of investment. The effect of interactions is shown in equation (8) (ii) by the positive (if any) contribution of more recent vintage investment, denoted by [tau], to the marginal productivity of past investments denoted by [tau] - 1 and [tau] - j. Interaction effects will be larger between investments of shorter time lapse between vintages (i < j) because old assets become progressively less adaptable to new capital. Finally, with obsolescence, the productivity of the initial capital stock, equation (8) (iii), declines over time. While not reflected in the above equations, the relative effect on output of interactions with [K.sub.[omicron]] declines over time for still another reason. As the sum of new investments grows over time, their separable effects on output, and the interactions across the new investments, grow in importance relative to the effects of [K.sub.[omicron]].

If one assumes that both vintage and interaction effects are of no consequence for investment (our null hypothesis), the production function for vintage year [tau] can be expressed as

[O.sub.[tau]] = [h.sub.[tau]]([K.sub.[tau]],[L.sub.tau]],[Q.sub.tau]] (9) where [K.sub.[tau]] is the capital stock aggregated from the investment vector and the initial stock. Now define [delta difference][O.sub[tau]] = [O.sub.[tau]] - [O.sub.[omicron]], where [O.sub.[omicron]] = [h.sub.[omicron]]([K.sub.[omicron]],[L.sub.[omicron]],[Q.sub.[omicron]]) under the null hypothesis stated above (that is, with no embodiment). The increment to output relative to the base year level should then be a separable production function(5) expressible as

[delta difference][O.sub.[tau]] = [h.sub.[tau]]([delta difference][K.sub.[tau]],

[delta difference][L.sub.[tau]],[Q.sub.[tau]]) 10 where [delta difference][K.sub.[tau]] = [K.sub[tau]] - [K.sub.[omicron]] and [delta difference][.sub.[tau]] = [L.sub[tau]] - [L.sub.[omicron]] measure the increments to the capital stock and labor force, respectively.

The role of human capital in equation (10) is similar to that of a technology index. One would expect that growth in physical output depends not only on growth in physical inputs, but also on the functional relationship between the existing level of human knowledge and increments to physical capital and labor. Thus, [Q.sub.[tau]], is the level of human capital reached by vintage year [tau] which can be utilized by the increment to the labor force in the production of additions to output.

Equation (10), if it holds, implies that the coefficients of the production function (ignoring economies of scale) are the same whether one estimates the relation for increments to output or for levels of output (equation (9)).

Vintage effects, when included in an empirical specification, permit a test to determine whether productivity is greater for more recent additions to the capital stock. If interactions initially have an unsystematic effect, this obscures the production relation of changes in physical capital to increments in output for vintage years soon after the base year (that is, the start of the period examined). However, as the time elapsed from the base year increases, the production relation for increments to output approaches that for a specification in terms of levels rather than changes. A change in capital variable can thus be assumed to capture the "levels" effect of a balanced investment plan and, in addition, the systematic component of the impact of interactions on productivity.

V. Estimates of Embodiment for Old Plants

Using cross-section data, we again estimate a modified Cobb-Douglas production function, this time for changes in output and in labor and capital inputs for old plants. The model is written as follows:

[Mathematical Expression Omitted]

where [delta difference] log [O.sub.jt] is the percentage change in output for the jth plant for time t relative to the initial period of 1972, [delta difference] log L is the percentage change in pure labor, and [delta difference]A log K is the percentage change in gross capital. Percentage changes standardize units to control for size effects across plants, and, in a sense, also standardize the observations for differences in initial factor proportions.

Log Q measures the level of human capital available to the increment in the labor force, and V is, as before, a measure of the weighted average vintage of investment expenditures for each plant. Output is again measured by the (deflated) value of shipments or, altematively, value added, labor by total employees, and labor quality or human capital by the average wage rate for each plant. The weights for vintage are, of course, the annual investment expenditures for the period over which changes in capital inputs are measured.

For each regression, the initial capital stock for each plant is simply its deflated gross assets in 1972,(6) and the terminal capital stock is obtained by adding to the initial value cumulated (deflated) gross capital expenditures plus the capitalized value of the change in rentals of assets. Errors associated with the measurement of initial stocks, for which data on annual investment streams are lacking, can be expected to reduce greatly the goodness of fit of our model.

Our objective was to test the implications of equation (8) that the power of interactions gradually declines over time. Equation (11) was therefore estimated consecutively for each year. According to the null hypothesis of no embodiment in the form of vintage or interaction effects, [[beta].sub.4t] = O and [[beta].sub.3t] > O, respectively, in the years immediately following 1972. A positive and stable measured effect for the [delta difference] log K variable would indicate relatively weak interaction effects, and, hence, a separable production relation for changes in output and capital input.

We now turn to results for the empirical model in equation (11), for plants in 15 industries.(7) Table II is presented with shipments as the proxy for output. The same estimates but with value added as the dependent variable yielded very similar, though somewhat more erratic, results with lower values of [R.sup.2]. For economy of space, the latter are not reported in detail.

The results show the consecutive changes in coefficients for old plants for the increments in output and inputs from 1972 to the levels for each successive year. In general, there is strong support for the conclusion that for an extended period, interactions with the initial capital stock do not permit estimation of a separable relation between change in capital inputs and the change in output. It takes roughly twelve years for the relative effect of interactions to decline to a level that permits one to estimate a stable coefficient for the change in the log of K.

Over time, the coefficient for [delta difference] log K increases from near zero and insignificance in the early 1970s to a significant positive elasticity of above 0.3 by 1984. Initially, a systematic relation between growth in capital and in output is obscured by the unpredictable effects of interactions, given large variations across plants in the age and size of the endowments of capital at the outset of the period. But as the ratio of cumulated investment to initial capital rises, interaction effects become more systematic and the coefficient for [delta difference] log K measureable.

Technical change embodied in capital is shown most directly by V. The insignificant results for V in earlier years were to be expected. If the effects of increments in capital are obscured by interactions, it is likely that so will the effects of changes in the vintage of capital. Moreover, since it is the vintage of post-1972 capital that was measured, sufficient time had to elapse for there to be enough dispersion in vintage to detect an effect.

Accordingly, the coefficient of V did not become significantly positive until 1982, but remained reasonably stable thereafter averaging .04 for the five year interval 1982-86. The approximately 4 percent increase in output for every one year change in vintage is substantially the same as that observed earlier for new plants. Thus the high rate of embodied technical change observed for new plants is confirmed with data for old plants.

The consistently rising negative value of the intercept as one moves from 1973 to 1986 is explained by the construction of the capital variable. While V measured changes in the vintage of post-1972 capital, no allowance was made for the progressive obsolescence of the initial (1972) stock of capital. Hence, for all plants, terminal year capital was systematically overstated by an increasing amount for each successive year. Thus the rising negative intercept appears to capture the obsolescence rate for old capital.

Table II gives us some insight into the effect of interactions between new and old inputs of physical capital. It is problematic, however, insofar as year-to-year changes may represent observations for less than capacity utilization and, hence, may not correctly measure the production frontier. This is especially a problem for the labor input and may explain the instability of the coefficient for log of Q in Table II. As is well known, firms retain skilled labor during contractions in output. The resulting change in the composition of labor, with its consequent change in the average wage, is likely to lead to some distortion in the coefficients for L and Q. An illustration of this phenomenon is reflected in the non-significant results for Q during the 1981-82 recession.

[TABULAR DATA OMITTED]

VI. Comparing Production Relations for New and Old Plants

The next objective was to derive estimates for all the variables in equation (11) that correspond to the production frontier and this required that we measure changes in output and inputs between points approximating capacity utilization. Production relations involving changes, as distinct from levels, of output and inputs are likely to be especially sensitive to the assumption of capacity utilization and that condition seems best approximated at output peaks.

For an empirical approximation of capacity utilization, the first peak was the higher of the 1972 or 1973 value of shipments. The terminal peak again was the year with the highest value of shipments in the period 1984-86. The cross-section data are based on the peak to peak change in logs for O, L, and K for each plant. In the overwhelming majority of cases, the highest rate of capacity utilization by almost any criterion did occur within those intervals.

The resulting estimates, with both shipments and value added as proxies for output, were also compared with the coefficients for new plants. The latter were derived from equation (6) but limited now to the same 15 industry sample as used for old plants. The estimates are shown in Table III.

[TABULAR DATA OMITTED]

Before proceeding with the comparison of old and new plants, some characteristics of the results for old plants might be noted. As compared with the average coefficients reported in Table II for consecutive years, the coefficient for labor declines markedly. Those for physical capital and vintage rise though the order of magnitude remains roughly the same as before. In estimating an industry dummy variable model as for new plants in section III, most of the industry dummies proved non-significant thereby rendering the coefficients for the aggregate more meaningful as average estimates.

Comparing the results in Table III we find:

(a) As to be expected, the [R.sup.2] values for new plants are much higher than for old plants - a fact attributable in large part to far superior data for the capital variable for new plants.

(b) For new plants, the coefficient for human capital (Q) is substantially higher than for old plants. The efficiency with which new plants use human capital appears to be their single most important advantage over old plants. Once technological options are limited by a large amount of old physical assets, the ability to substitute human capital for other inputs appears to be severely restricted.

(c) The coefficient for labor is of roughly the same magnitude for old and new plants.

(d) Especially important is the considerable stability in the coefficients for K when estimates based on changes in inputs and output for old plants are compared with levels for new plants. However, while these coefficients are generally of the same order of magnitude, the coefficients remain slightly higher for old plants. This may suggest some continued impact of interactions even after a period of as much as 14 years past the point at which the initial capital stock was measured.

(e) V continues to be much more sensitive than K to choice of sample and proxy for output. However, the higher estimates for old than for new plants are consistent with what we know about capital expenditures. A larger proportion of capital outlays are devoted to structures rather than to equipment for new than for old plants. Structures are generally assumed to be associated with much lower rates of obsolescence (hence, embodied technical change).

VII. Summary

There is renewed interest in the recent literature in measurement and measurement errors associated with output growth. The standard approach, such as in Hulten[4], is to compute Solow residuals by decomposing output growth into the contributions of input quantities, quality adjustments, and the relative proportions of embodied and disembodied technical change. This paper provides an alternative approach to measuring the vintage effects of additions to capital utilizing cross-sections of manufacturing industries at the plant level made possible by the Census LRD database.

The principal results of the paper are briefly summarized:

1. We first specified a production function with human capital as a separate argument and with embodied technical change proxied by a variable that measures the average vintage of the stock of capital.

2. The coefficients of the production function were first estimated with cross section data for roughly 2150 new manufacturing plants in 41 industries, and for subsets of this sample. An augmented Cobb-Douglas specification was used. The results proved fairly stable across varying samples of plants and with respect to alternative measures of output.

3. Substantively, it was found that the elasticity of output with respect to human capital was approximately the same as it was with respect to pure labor. Embodied technical change of capital produced an average 4 percent increase in output for each one year change in average vintage.

4. It was pointed out that most investment in a developed economy is made on old rather than on new plants. An important question, therefore, concerns the separability of the relation between new inputs (that is, changes in the level of inputs) and changes in the level of output. A model was specified with interactions between new investment and initial endowments of capital. Interaction effects were predicted to decline in importance as a function of time.

5. Using a sample of roughly 1400 old plants in 15 industries, it was found that interactions between new investment and initial endowments of capital were, for a long interval of time, too unsystematic to permit measurement of a coefficient for capital. After twelve to fourteen years of cumulative investment, a systematic relation between changes in the level of inputs and changes in the level of output became measurable. Moreover, the coefficient for changes in the capital input for old plants proved to be of approximately the same magnitude as that for level of capital for new plants.

6. Comparing new and old plants over the "long-run," the estimates of embodied technical change of capital and of the elasticity of output with respect to number of employees (pure labor) proved very similar for the two types of plants. Differences between new and old plants in the elasticity of output with respect to human capital remained very large, however, and appear to be an important difference between the two sets of plants.

References

[1.] Becker, Gary. Human Capital. The University of Chicago Press, 1964. [2.] Gort, Michael, and Raford Boddy. "Vintage Effects and the Time Path of Investment in Production Relations," in The Theory and Empirical Analysis of Production, Studies in Inconte and Wealth, Vol. 31, National Bureau of Economic Research. New York: Columbia University Press, 1967, pp. 395-422. [3.] Hall, Robert E., "Technical Change and Capital from the Point of View of the Dual." Review of Economic Studies, January 1968, 35-46. [4.] Hulten, Charles R., "Growth Accounting When Technical Change is Embodied in Capital." American Economic Review, September 1992, 964-80. [5.] Prucha, Ingmar R., and M. Ishaq Nadiri. "Endogenous Capital Utilization and Productivity Measurement in Dynamic Factor Demand Models: Theory and an Application to the U.S. Electrical Machinery Industry." Working paper, 1990. [6.] Solow, Robert M., "Technical Change and the Aggregate Production Function." Review of Economics and Statistics, August 1957, 312-20.

(*) The project was carried out with assistance from the ASA/NSF/CENSUS fellowship program. The authors are, of course, solely responsible for the conclusions and methods of analysis used. (1.) Investments were deflated by the implicit price deflator for capital expenditures in all manufacturing combined, based on unpublished Bureau of Economic Analysis data. To the cumulative total of gross capital expenditures we added the capitalized value of the changes in rentals of fixed assets. The rate of capitalization was derived from the ratio of gross fixed assets to the sum of net income before taxes plus interest paid plus depreciation, as reported for 1972-86 in U.S. Internal Revenue Service, Statistics of Income. (2.) To facilitate the interpretation of the coefficients of change in human capital, as well as to correct for possible biases arising from the fact that in some of our cross-sections the observations do not relate to identical points in time, average wage rates were deflated. The deflator was the Consumer Price Index and was intended simply to correct for the average rate of inflation in the economy. (3.) Output deflators were also drawn from unpublished BEA data at the 4-digit industry level. (4.) Within the sets of industries for both new and old plants, only plants that satisfied the following criteria were chosen: (a) a continuous history in the same industry, from birth for new plants and from 1972 for old, until 1986, (b) a primary industry specialization ratio of at least 50%. This gave us about 2150 new plants for the 41 industries and roughly 1900 for the 32. The period chosen, 1972-86, was determined by the time interval for which panel data were available. A list of industries and number of plants in each will be provided upon request. (5.) The mathematical conditions necessary for exact separability of output and input levels into functions based on increments is, to our knowledge, unsolved. (6.) The deflators were derived from ratios of gross capital stocks at historical cost to stocks at constant cost as reported in Bureau of Economic Analysis, U.S. Department of Commerce, Fixed Reproducible Tangible Wealth in the United States, 1925-85, 1987. (7.) For the analysis of old plants, and for comparisons of old and new plants, a further subset of 15 industries was employed primarily to keep the size of the database manageable. These consisted generally of the largest among the original 41 industries but with selection based on broad representation across the industrial spectrum. Our sample consisted of about 1400 old plants in the 15 industries, and about 1250 new plants.

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Author: | Wall, Richard A. |
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Publication: | Southern Economic Journal |

Date: | Jul 1, 1993 |

Words: | 7224 |

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