Model A

Whereas in the short run all the parameters in equation (2) can be assumed to be constant, the rate of disembodied productivity change λ and the factor weights δ and (1 − δ) are likely to change in the longer run. The parameter λ is a “catch-all” parameter that represents the multitude of factors that explain why output tends to grow faster than measured inputs; thus, it would be surprising if λ did not change as these factors evolve. This is well recognized in the economic literature. What is less well recognized, but possibly even more important in the present context, is that δ may also change over time and affect the high-employment equilibrium labor share as production techniques and the pattern of production become more, or less, labor-intensive. Such a change in δ could help to explain the evolution of the labor share over the past three decades. The labor share remained roughly constant in most major industrial countries throughout the second half of the 1950s and the whole of the 1960s, despite a doubling or tripling of the ratio of capital to labor. Equation (6) indicates that such a development is consistent with firm equilibrium and with a constant δ only if the production function is of the Cobb-Douglas type (β = 0). But if β is zero and δ is constant, then all of the change in the labor share experienced in the 1970s and early 1980s should be viewed as a move away from equilibrium because the equilibrium labor share would be constant. A plausible alternative is that β is different from zero and that the labor share during the first period was constant because the effect of the increase in the ratio of capital to labor was offset by the effect of the change in δ. A different rate of change for the ratio of capital to labor in the second period could then have led to a different evolution of the labor share.

Allowing λ and δ to change without using many degrees of freedom is not an easy task. For λ, in particular, it is difficult to impose any a priori restriction, so that one cannot avoid a systematic search for statistically significant shifts, despite the cost in terms of degree of freedom. For this reason, this parameter will hereafter be written as λ_t. For δ, the change should be fairly gradual, and over the plausible range of variation one can assume a simple linear function of time ($\delta ={\delta }_0+{\delta }_{1}t$).

Even with this restriction on the way δ can change over time, the number of parameters in equation (2) is too large for reliable econometric estimation from a single equation. With periods of observation limited to 20 to 30 years, there is simply too much multicollinearity among the main variables. Frequently, this problem is solved by jointly estimating the production function and the demand for labor. In the present context, however, this method has to be modified for two reasons.

First, the demand for labor corresponding to a CES production function with λ_t, and $\delta ={\delta }_0+{\delta }_{1}t$ is as follows:⁶

It is apparent that there are two trend elements in equation (9) and that the equation would therefore fail to contribute anything to the estimation of either δ₁ or λ_t.

Second, the demand for labor is derived under the assumption that labor is paid its marginal product. Most studies do take into account that this assumption is plausible only in the longer run by considering equation (9) as a long-term demand for labor. The lagged value of lnL is then added to the right-hand side of equation (9) to reflect the gradual adjustment of the actual demand for labor to its longer-run equilibrium value. The problem is that this specification does not differentiate between the adjustment of L to Y over the cycle, which is often rapid, and the adjustment of L to w/p, which may be quite slow.⁷ Furthermore, because the cyclical movement in Y is usually the dominant factor, the estimated coefficient of adjustment may exaggerate the rapidity with which the equilibrium between the real wage rate and the marginal product of labor is re-established. Thus, imposing equation (9) on the data is nearly equivalent to imposing the constraint that deviations of the real wage rate from the equilibrium value corresponding to the amount of employed labor cannot last more than a few years. Therefore, the change in the labor share of income experienced in most industrial countries over the past 10 to 15 years would have to be viewed as a phenomenon fully warranted by factors such as changes in production techniques and in the relative amounts of labor and capital within firms. Such an assumption would hardly be appropriate in the context of the present study.⁸

The approach adopted here to solve the problem of multi-collinearity is to use the share equation (6) rather than the normal demand-for-labor equation. The major advantage of equation (6) is that the rate of disembodied productivity change (λ_t) does not enter into it, but the relative weight of capital (δ₀ + δ₁t) does. Thus, equation (6) is a powerful tool to obtain an estimate of δ₀ and δ₁. Because it is also derived under the assumption that labor is paid its marginal product, equation (6) will be assumed to hold only on a cyclically adjusted basis, rather than in each phase of the cycle. Furthermore, it will only be assumed to have held during 1955–69 for the European countries and during 1955–73 for the United States and Canada, periods for which there is no reason to expect that the real wage rate was out of equilibrium.⁹ For Japan, equation (6) will also be assumed to have held during 1955–73, even though the extremely low labor share during this period suggests that labor was possibly paid less than its marginal product as a result of an implicit social consensus that a high profit rate was the best way to rebuild the capital stock.

After taking account of the considerations discussed above, we obtain the two functions that make up model A:

where

• $S_L^{\prime }$ = the trend-through-peaks value of S_L adjusted downward so that the average of $S_L^{\prime }$ is equal to the average of S_L

• L’ = the trend-through-peaks value of L adjusted downward so that the average of L’ is equal to the average of L.

Again, equation (11) is imposed on the data only through 1969 in European countries and through 1973 in the United States, Canada, and Japan.

Once the parameters of the model have been estimated from equations (10) and (11), the labor share and the real wage consistent with high-employment equilibrium can be derived from the modified versions of equations (7) and (8):

The most noticeable difference between model A and the simpler model discussed earlier is that model A recognizes that a trend increase in the real wage rate that exceeds the increase in labor productivity at high employment ($\overline{Y}/\overline{L}$) is warranted when there is a reduction in the capital weight δ (that is, when δ₁ < 0). Such a situation arises when the pattern of production is shifting toward less capital-intensive industries. An evolution in the opposite direction would call for a growth of the real wage rate that is below the growth of $\overline{Y}/\overline{L}$.

Model B

Weak separability is broadly accepted as a realistic assumption for raw materials (N), but whether it is a realistic assumption for energy (E) is open to debate. Berndt and Wood (1979) and others have argued that in many instances energy and capital must be viewed as complements. In other words, once entrepreneurs have optimized the energy efficiency of their capital stock on the basis of the relative prices of capital and energy, they are largely unable to change K/L without changing E/L. Under such conditions, the relevant production model would involve the following two-level production function:

where

• Y* = value added corresponding to L, K, and E

• K* = a composite variable reflecting the joint input of capital and energy.

With E-K complementarity, a marked increase in the relative price of energy, as in 1973–74 and in 1979–80, would lead entrepreneurs to increase their demand for labor and decrease their demand for both energy and capital. Assuming that the functional form for the Y* level remains as assumed in equation (10), the labor share and the real wage rate consistent with high-employment equilibrium would still be determined by equations (12) and (13), respectively, but after substituting K* for K The labor share would now be the share of labor income in Y*, and the real wage rate would be defined in terms of p*, the deflator of Y*. In equations (10) and (11), $S_L^{\prime }$ would have to be replaced by $S_L^{\prime *}$ (the cyclically adjusted share of labor income in Y*), and Y would have to be replaced by Y*. An increase in the price of energy leading to a decline in K*/L would shift the distribution of incomes corresponding to Y* against labor if β > 0, and in favor of labor if β < 0.

Because the practical relevance of E-K complementarity is still in doubt, we will derive estimates of the warranted real wage rate under each of the polar assumptions: E-K complementarity (model B), and weak separability of energy (model A). In model B the composite variable K* will be derived by using the linear approximation to the CES functional form (as for output in equation (2), but without the time trend); that is,

Measurement of L and K

There is no generally accepted way to measure the flows of labor and capital services. In this paper, we can only provide a brief analysis of the measurement problems and an explanation of what we did. As in most other studies, the present analysis uses series on man-hours worked¹⁰ and on the gross capital stock¹¹ as proxies for the unavailable series on the flows of labor and capital services. As is well known, this procedure requires that an additional variable be inserted in the production function to pick up cyclical movements in the intensity of use of labor and capital.¹² The reason is that fluctuations in output corresponding to unanticipated changes in aggregate demand do not immediately result in corresponding changes either in the number of man-hours worked or in the level of capital stock. Initially, unexpected variations in demand lead to changes in the intensity of use of labor and capital; that is, the amounts of services obtained from a given number of man-hours and from a given capital stock vary. Gradually, however, the number of man-hours and the level of capital stock are changed, and their intensity of use is brought back to normal. As in Artus (1977), we will assume that the cyclical variable, to be denoted as D and introduced with a coefficient of 1, is a lagged function of the actual rate of change of output, net of the expected long-run rate of change (μ). More specifically, it will be assumed that

where the notation [ ]_L indicates a geometrically distributed lag operator. The expected long-run rate of change of output will be approximated by a ten-year lagged moving average of ln(Y/Y₋₁).

Even with D in the production function, there are still various measurement problems that must be considered. For L, the main problem is that the available data on man-hours do not reflect the level of education and technical expertise of the work force. In the present context, however, it is likely that the problem is not too severe because those changes occur only gradually. Even when demographic and economic developments lead to a marked increase in the growth of the work force and to a decline in its mean age, there is no strong reason to assume that the average level of education and technical expertise is affected. Although new entries in the labor force tend to be younger and better educated, they also have less on-the-job experience.¹³ Thus, the measurement error in L can be assumed to be highly collinear with a simple time trend. It could lead to a bias in the econometric estimates of the trend coefficients (λ_t and δ₁) but the estimates of the other coefficients would be unaffected. Even more important, the estimates of ${\overline{S}}_L$ and $\left(\overline{w/p}\right)$ would also be unaffected.

For K, the remaining problems are more severe. All of them are results of the fact that, currently, the only practical method to obtain an estimate of the gross capital stock is the rather mechanical one of cumulating past investment flows, net of discards. This method, if applied carelessly, may lead to a growing measurement error that, this time, would not necessarily be collinear with the trend rates already in the model. Three considerations are especially important.

First, the cumulation of real investment flows does not take proper account of the fact that a piece of machinery bought in year t + 1 embodies more technical knowledge and is thus more efficient than one bought in year t. The reason is that the price series used to deflate the investment flows tend to exaggerate the amount of inflation because the price increases that reflect the efficiency increases are not properly separated from the price increases that reflect inflation. When there is a decline in the growth of investment, as after 1973, it is likely that this decline will be followed by a temporary decline in the growth of the average efficiency of the capital stock because of the temporary reduction in the proportion of relatively new equipment in the total stock of equipment. Such a decline in the growth of the average efficiency of the capital stock will not be reflected in the capital stock series obtained by cumulating investment flows. To reduce this kind of measurement error, the capital stock series used in this study were adjusted by an efficiency scalar that is a function of the mean age of the capital stock. (A detailed description of how this adjustment was carried out can be found in the Appendix.)

Second, voluntarily or as a result of government regulation, firms purchase equipment that produces products that are omitted from the measured gross national product (GNP). In particular, firms purchase equipment to reduce environmental pollution. As long as the proportion of such investment in total investment is constant, the problem is not severe because the rate of growth of the “productive” capital stock is unaffected. In the United States, Canada, and Japan, however, new government regulations led to an upward shift of that proportion in the late 1960s and early 1970s. The data published by the U.S. Department of Commerce indicate that the proportion of pollution-abatement investment in total U.S. investment in manufacturing rose from about 1 percent in the mid-1960s to about 8 percent in the mid-1970s, then declined gradually to around 5 percent in the late 1970s and early 1980s.¹⁴ A series for expenditure on pollution abatement was derived from these data, and it was subtracted from the U.S. investment series to get a measure of “productive” investment. No precise data are available for Canada and Japan, but, as a rough adjustment for the jump in pollution-abatement expenditures, the investment figures for these two countries were reduced by 5 percent from 1969 onward. If anything, this adjustment is probably on the low side.

Third, the major unanticipated structural changes that took place in the aftermath of the 1973–74 and 1979–80 oil price increases are likely to have caused the premature obsolescence of part of the capital stock. Capital is heterogeneous and specialized. Sudden changes in the structure of demand faced by firms and the relative costs of using specific energy-intensive equipment leave some equipment without any economic value even though it may be relatively new. Even in less extreme cases, there may be an incentive for firms to speed up the replacement of some equipment by more energy-efficient equipment. Of course, a one-time loss of equipment does not lead to a permanent decline in the capital stock because normal obsolescence would, in any case, eventually lead to the discard of this equipment. Unless premature obsolescence is taken into account, however, the capital stock can be seriously overestimated in the first few years that follow the demand or supply shock.

The extent of this phenomenon in the aftermath of the 1973–74 and 1979–80 oil price increases is still an unsettled issue. Economists such as Baily (1981) take the large decline in the market value of corporations relative to the replacement cost of tangible assets, Tobin’s q, as an indication that a large part of the capital stock (perhaps 20 percent) was prematurely discarded just after the first wave of oil price increases, and that presumably as much was discarded after the second wave. Others, such as Bosworth (1982), compare the historical cost valuation of gross stocks derived from surveys conducted by the U.S. Bureau of the Census with the results derived from the perpetual inventory method of valuation with a fixed-discard pattern and conclude that only a small part of the capital stock (perhaps 2 to 3 percent) was prematurely discarded. Both methods have important weaknesses. Tobin’s q is likely to reflect many factors that have little if anything to do with the effective size of the gross capital stock and the flow of services that can be derived from it. The historical cost valuation of gross stocks derived from surveys is notoriously unreliable. The estimates of gross capital stock used in the present study assume that 10 percent of the existing capital stock was prematurely retired during 1974–76, and that the same proportion was prematurely retired during 1980-82. Because this estimate is highly tentative, a sensitivity analysis was carried out with a 5 percent and a 15 percent estimate.

Measurement of L

To obtain an estimate of the high-employment labor input in manufacturing (L) that is consistent with the definition of high employment for the whole economy, we use the method developed in Artus and Turner (1978). This method is based on the estimation of the following simple equation relating the actual labor input in manufacturing (L) to a nonlinear time trend and to the unemployment rate in the whole economy (U):

At cyclical peaks, the value of L is calculated as

where U is the unemployment rate corresponding to a situation of high employment in the whole economy, and â₄ is the estimated value of a₄.

In between peaks, the value of L is calculated by fitting log-linear trends between the successive peak values of L obtained from equation (18). After the last cyclical peak, the assumed growth rate of L is an extrapolation of the estimated rate between the last two observed cyclical peaks. The extrapolated figures are adjusted, when necessary, for changes in demographic factors and in the length of the normal work week.

The most difficult task is to estimate the high-employment rate U; that is, the rate where labor shortages become widespread because the residual unemployment is due to the normal turnover in the labor market and to regional and skill mismatches between labor supply and demand. In this study, the estimate of U is derived from the “Beveridge curve,” a graphical presentation of the inverse relationship between unemployment and vacancies.¹⁵ More specifically, U is defined as the unemployment rate where the curve becomes nearly vertical, with large increases in vacancies associated with only small reductions in the unemployment rate. Allowance is made for shifts in the curve and, therefore, for shifts in U that reflect changes in the amount of frictional and mismatch unemployment. Over the past decade, however, some of the countries considered here experienced few, if any, years in which the number of vacancies was high, so that it is difficult to draw complete Beveridge curves for this period. In such cases, we examine whether the unemployment rates corresponding to relatively low numbers of vacancies have changed from the 1960s to the 1970s and early 1980s, and we assume that the vertical part of the curve has shifted by the same amount as the part corresponding to relatively low numbers of vacancies. For Italy, where there are no data on vacancies, U was derived by using a more ad hoc method (see the Appendix).

Table 2 presents the main results related to the estimation of L. A striking result is the large value of â₄ for Japan, which reflects the effect of labor-sharing arrangements and the apparent greater ease with which the service sector absorbs the increase in labor force during periods of slow growth in the manufacturing sector. Another striking result is the marked reduction in the growth rate of L in Japan and in European countries during the past decade and a half. This reduction reflects a marked increase in the value of U during this period (see the Appendix), as well as an acceleration of the historical trend in the allocation of the labor force in favor of the service sector (possibly caused by the rapid growth of government services), a change in comparative advantages in relation to the newly industrialized countries, and special factors such as North Sea oil in the United Kingdom. Despite the reduction in the growth rate of L, we find that at the cyclical peak in 1979–80 there were still sizable gaps between L and L in Japan, France, the Federal Republic of Germany, and Italy. In contrast, the gaps were fairly small in the United States, Canada, and the United Kingdom because in these countries the residual unemployment, even though very high, seemed to correspond to frictional and mismatch unemployment. By 1982 the gaps were large in all seven countries.

Table 2.

Actual (L) and High-Employment (L) Labor Input in Manufacturing, Latest Peak Year and 1982¹

¹Notation: U, unemployment rate in the whole economy; U, unemployment rate corresponding to high employment in the whole economy; L, actual labor input in manufacturing in man-hours worked (for the United States, man-hours paid); L, high-employment labor input corresponding to the definition of the L series. The unemployment rate is in percent of the civilian labor force (except for Japan, France, and the United Kingdom, where it is in percent of the total labor force).

²Parentheses enclose standard errors. The estimates are obtained by using least-squares methods and annual observations of the period 1955–82.

³The latest peak year for L is 1979 for the United States, Canada, France, the Federal Republic of Germany, and the United Kingdom and 1980 for Japan and Italy.

⁴Average rate of growth during 1962–69.

Table 2.

Actual (L) and High-Employment (L) Labor Input in Manufacturing, Latest Peak Year and 1982¹

		Unemployment in Latest Peak Year for L (In percent)3			Unemployment in 1982 (In percent)		Average Rate of Growth of L (In percent)
Country	â42	U	U	(L − L)/L	U	(L − L)/L	1956–69	1970–73	1974–82
United States	4.9 (0.3)	5.8	5.7	0.5	9.7	14.4	0.8	0.6	0.6
Canada	4.0 (0.4)	7.5	7.2	1.2	11.0	11.2	1.1	0.7	0.3
Japan	14.7 (2.6)	2.0	1.6	5.7	2.4	8.1	4.0	1.0	0.2
France	4.0 (1.1)	5.9	3.6	9.2	8.0	15.0	0.8	0.3	−0.9
Germany, Fed. Rep.	5.1 (0.8)	3.8	2.2	8.2	7.5	13.6	−0.14	−1.0	−1.5
Italy	4.2 (1.5)	7.6	6.5	5.0	9.1	11.7	1.6	−0.8	0.0
United Kingdom	4.1 (0.7)	5.1	4.6	2.1	11.7	19.7	−0.3	−2.6	−2.4

¹Notation: U, unemployment rate in the whole economy; U, unemployment rate corresponding to high employment in the whole economy; L, actual labor input in manufacturing in man-hours worked (for the United States, man-hours paid); L, high-employment labor input corresponding to the definition of the L series. The unemployment rate is in percent of the civilian labor force (except for Japan, France, and the United Kingdom, where it is in percent of the total labor force).

²Parentheses enclose standard errors. The estimates are obtained by using least-squares methods and annual observations of the period 1955–82.

³The latest peak year for L is 1979 for the United States, Canada, France, the Federal Republic of Germany, and the United Kingdom and 1980 for Japan and Italy.

⁴Average rate of growth during 1962–69.

View Table

Table 2.

Actual (L) and High-Employment (L) Labor Input in Manufacturing, Latest Peak Year and 1982¹

		Unemployment in Latest Peak Year for L (In percent)3			Unemployment in 1982 (In percent)		Average Rate of Growth of L (In percent)
Country	â42	U	U	(L − L)/L	U	(L − L)/L	1956–69	1970–73	1974–82
United States	4.9 (0.3)	5.8	5.7	0.5	9.7	14.4	0.8	0.6	0.6
Canada	4.0 (0.4)	7.5	7.2	1.2	11.0	11.2	1.1	0.7	0.3
Japan	14.7 (2.6)	2.0	1.6	5.7	2.4	8.1	4.0	1.0	0.2
France	4.0 (1.1)	5.9	3.6	9.2	8.0	15.0	0.8	0.3	−0.9
Germany, Fed. Rep.	5.1 (0.8)	3.8	2.2	8.2	7.5	13.6	−0.14	−1.0	−1.5
Italy	4.2 (1.5)	7.6	6.5	5.0	9.1	11.7	1.6	−0.8	0.0
United Kingdom	4.1 (0.7)	5.1	4.6	2.1	11.7	19.7	−0.3	−2.6	−2.4

¹Notation: U, unemployment rate in the whole economy; U, unemployment rate corresponding to high employment in the whole economy; L, actual labor input in manufacturing in man-hours worked (for the United States, man-hours paid); L, high-employment labor input corresponding to the definition of the L series. The unemployment rate is in percent of the civilian labor force (except for Japan, France, and the United Kingdom, where it is in percent of the total labor force).

²Parentheses enclose standard errors. The estimates are obtained by using least-squares methods and annual observations of the period 1955–82.

³The latest peak year for L is 1979 for the United States, Canada, France, the Federal Republic of Germany, and the United Kingdom and 1980 for Japan and Italy.

⁴Average rate of growth during 1962–69.

These estimates of L are, of course, subject to a large margin of error. In particular, they assume that: (1) at each cyclical peak, the relation between aggregate unemployment (in excess of U) and man-hours worked in manufacturing is the same; (2) the change in the distribution of employment among sectors between two cyclical peaks is due to long-run changes in comparative advantage and in the pattern of demand, rather than to real wage problems in manufacturing; (3) the rate of change of U and the rate of change in the distribution of employment among sectors between the last two cyclical peaks, 1973–74 and 1979–80, can be extrapolated to the early 1980s. On balance it is much more likely that these assumptions lead to an underestimation, rather than an overestimation, of L during the late 1970s and early 1980s, at least in Japan and in European countries. One cannot but wonder whether the marked shift in the distribution of employment toward the service sector, especially government services, that took place in Japan and in European countries between the years 1973–74 and 1979–80 was really warranted by long-run growth considerations. In part this shift may itself be the result of an excessive real wage rate in manufacturing. Some of the persons that could not get a job in this sector (because labor contracts prevented entrepreneurs from offering them a wage corresponding to their marginal product) were probably recruited in the government sector to limit the rise in unemployment, or they may have been absorbed by the private service sector in occupations involving very low marginal product and very low real wages. In addition, it may be unduly pessimistic to extrapolate into the early 1980s the rapid rise in U observed between the years 1973–74 and 1979–80. Indeed, the evidence does not suggest a further shift of the Beveridge curve in the early 1980s, except possibly in France and the United Kingdom.

Model A

The parameter estimates for model A are presented in Table 3. We find that the rate of disembodied productivity growth increased during the 1960s and then fell back during the 1970s and early 1980s. The only exception is the Federal Republic of Germany, where the rate of disembodied productivity growth was constant throughout the period 1961–82. An examination of the regression residuals for 1981 and 1982 suggests that productivity growth may have picked up again in the United Kingdom in recent years, but it is still too early to say. These results corroborate the findings of Denison (1982), Bosworth (1982), and others, that a significant part of the decline in the growth of labor productivity during the last decade is not accounted for by the decline in the rate of capital accumulation. This is true even when, as in the present study, the rate of capital accumulation is adjusted downward to take account of the rise in pollution-abatement investment and premature obsolescence.¹⁸

Table 3.

Estimates of Parameters, Model A¹

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.78 (0.09); Canada, 0.50 (0.15); France, 0.60 (0.19); and Italy, 0.82 (0.06).

³The trend rates of growth of disembodied productivity are in percent per year.

⁴The estimate of 4.87 (0.15) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 9.39 (0.42).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.45 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.08 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.

Table 3.

Estimates of Parameters, Model A¹

Parameter	United States2	Canada2	Japan	France2	Germany, Fed. Rep.	Italy2	United Kingdom
In λ	−0.023	−0.003	−0.019	−0.034	0.036	0.016	0.011
	(0.003)	(0.002)	(0.006)	(0.007)	(0.004)	(0.004)	(0.003)
λt3	1955–60	1955–60	1955–66	1955–64		1955–60	1955–60
	1.44	1.81	4.874	4.02		3.11	1.08
	(0.24)	(0.20)	(0.15)	(0.09)		(0.26)	(0.30)
	1961–73	1961–73	1967–70	1965–70	1961–82	1961–73	1961–73
	2.74	3.63	6.73	5.33	3.05	4.69	3.05
	(0.06)	(0.05)	(0.29)	(0.22)	(0.08)	(0.07)	(0.07)
	1974–82	1974–82	1971–82	1971–82		1974–82	1974–82
	0.89	1.15	3.68	2.29		2.85	1.04
	(0.14)	(0.13)	(0.21)	(0.17)		(0.15)	(0.21)
δ05	26.2	33.1	54.8	39.5	31.8	37.7	27.9
	(0.3)	(0.2)	(0.4)	(0.2)	(0.4)	(0.2)	(0.2)
δ15	0.56	0.14	0.456	0.66	0.54	−0.14	0.62
	(0.11)	(0.04)	(0.03)	(0.09)	(0.04)	(0.05)	(0.26)
β	1.014	0.189	0.225	0.569	0.406	0.306	0.789
	(0.163)	(0.049)	(0.012)	(0.095)	(0.020)	(0.036)	(0.246)
ρ	0.426	0.469	0.704	0.524	0.328	0.487	0.373
	(0.028)	(0.037)	(0.026)	(0.081)	(0.069)	(0.050)	(0.072)
		Goodness-of-fit statistics
Production function SE7	1.69	1.49	1.81	1.92	3.99	2.54	2.33
Share function SE7	0.59	0.17	0.23	0.26	0.12	0.17	0.28

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.78 (0.09); Canada, 0.50 (0.15); France, 0.60 (0.19); and Italy, 0.82 (0.06).

³The trend rates of growth of disembodied productivity are in percent per year.

⁴The estimate of 4.87 (0.15) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 9.39 (0.42).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.45 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.08 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.

View Table

Table 3.

Estimates of Parameters, Model A¹

Parameter	United States2	Canada2	Japan	France2	Germany, Fed. Rep.	Italy2	United Kingdom
In λ	−0.023	−0.003	−0.019	−0.034	0.036	0.016	0.011
	(0.003)	(0.002)	(0.006)	(0.007)	(0.004)	(0.004)	(0.003)
λt3	1955–60	1955–60	1955–66	1955–64		1955–60	1955–60
	1.44	1.81	4.874	4.02		3.11	1.08
	(0.24)	(0.20)	(0.15)	(0.09)		(0.26)	(0.30)
	1961–73	1961–73	1967–70	1965–70	1961–82	1961–73	1961–73
	2.74	3.63	6.73	5.33	3.05	4.69	3.05
	(0.06)	(0.05)	(0.29)	(0.22)	(0.08)	(0.07)	(0.07)
	1974–82	1974–82	1971–82	1971–82		1974–82	1974–82
	0.89	1.15	3.68	2.29		2.85	1.04
	(0.14)	(0.13)	(0.21)	(0.17)		(0.15)	(0.21)
δ05	26.2	33.1	54.8	39.5	31.8	37.7	27.9
	(0.3)	(0.2)	(0.4)	(0.2)	(0.4)	(0.2)	(0.2)
δ15	0.56	0.14	0.456	0.66	0.54	−0.14	0.62
	(0.11)	(0.04)	(0.03)	(0.09)	(0.04)	(0.05)	(0.26)
β	1.014	0.189	0.225	0.569	0.406	0.306	0.789
	(0.163)	(0.049)	(0.012)	(0.095)	(0.020)	(0.036)	(0.246)
ρ	0.426	0.469	0.704	0.524	0.328	0.487	0.373
	(0.028)	(0.037)	(0.026)	(0.081)	(0.069)	(0.050)	(0.072)
		Goodness-of-fit statistics
Production function SE7	1.69	1.49	1.81	1.92	3.99	2.54	2.33
Share function SE7	0.59	0.17	0.23	0.26	0.12	0.17	0.28

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.78 (0.09); Canada, 0.50 (0.15); France, 0.60 (0.19); and Italy, 0.82 (0.06).

³The trend rates of growth of disembodied productivity are in percent per year.

⁴The estimate of 4.87 (0.15) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 9.39 (0.42).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.45 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.08 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.

Aside from the evolution of λ_t over time, it is striking how much λ_t, varies among countries and how stable the cross-country differences are. For the past three decades, λ_t has tended to be about 3 percentage points higher in Japan and 2 percentage points higher in France and Italy than in the United States, Canada, and the United Kingdom. The Federal Republic of Germany, by avoiding a decline in λ_t, has moved from the low group during the 1960s and early 1970s to the middle group during the last ten years. These large and persistent differences have obvious implications, not only for the growth of the real wage rate but also for employment and capital formation. During the last ten years, manufacturing production had to grow by more than 4 percent a year in Japan and by more than 2 to 3 percent a year in France, the Federal Republic of Germany, and Italy to lead to an increase in the demand for labor and capital services in manufacturing. In the United States, Canada, and the United Kingdom, the same result could be achieved with an increase in production of only slightly more than 1 percent.

We find that in most countries the weight on the capital stock, δ, is increasing over time; that is, δ₁ is positive. Again, this is not a surprising result; the tendency for a gradual increase in the capital intensity of production techniques has been in evidence for quite a long time. For Japan, however, the data suggest that the tendency for an increase in δ was interrupted during 1971–82. A possible reason for this development is that Japan experienced a marked change in its structure of production during this period as a result of a deliberate policy to move away from industries involving high levels of raw material and energy imports. Many of these industries, such as the steel industry, were also capital intensive. In addition, the value of δ was already quite high during the second half of the 1950s and during the 1960s—much higher than in other industrial countries.

The parameter β is positive for all countries; the corresponding elasticity of substitution between labor and capital (η, with η = 1/(1 + β)) is thus lower than 1. For most countries, η is between 0.5 and 0.8, a result that matches the finding of other studies, such as those by Griffin and Gregory (1976) and Pindyck (1979). The important implication of this result is that a rise in the capital-labor ratio tends to increase the equilibrium labor share of incomes (see equation (12)). In periods when the capital-labor ratio rises rapidly, this effect may dominate the effect of the gradual rise in the weight on the capital stock, and the equilibrium labor share may increase. In other periods the weight effect may dominate, and the equilibrium labor share may decrease.

The estimates of the parameter p are between 0.4 to 0.5 for most countries. This means that a sudden decline in the rate of growth of output of 10 percentage points is normally accompanied by a decline in the intensity of use of labor and capital resources within firms corresponding to a decline in total factor productivity of 4 to 5 percent. Then, the intensity of use of labor and capital is gradually brought back to normal as firms reduce their work force and their capital stock (see equation (16)). In the second year, assuming no further shock, the apparent total factor productivity is only 1½ to 2½ percent below what it would have been without the output decline. In Japan the adjustment is significantly slower. Factor productivity is cut by 7 percent the first year, and it takes four years for the reduction in productivity to become less than 2 percent.

Practically all the parameter estimates have low asymptotic standard errors. This result, however, should not be viewed as an indication that the estimates are highly precise and reliable. First, from a statistical standpoint we are working with fairly small samples, so that the asymptotic standard errors have limited relevance. Unfortunately, the small-sample properties of estimates for parameters in these nonlinear models are unknown. Second, the estimates for δ₀, δ₁, and β are, for all practical purposes, determined by the share function. When both equations are estimated separately, the estimates of δ₀, δ₁, and β derived from the share function¹⁹ are practically identical to the estimates presented in Table 3, whereas the estimates derived from the production function have often implausible values and high standard errors. Therefore, the estimates presented in Table 3 must be viewed as highly dependent on the assumption that, during the estimation period, $S_L^{\prime }$ was in fact the equilibrium labor share corresponding to the cyclically adjusted level of employment.

The results above are based on the assumptions that during 1974–76, and then again during 1980–82, 10 percent of the existing capital stock was prematurely retired. When the assumption was changed from 10 to 15 percent (and then from 10 to 5 percent), the estimated values of λ_t for the 1970s and early 1980s were raised (and then lowered), but only by 0.1 to 0.2. All the other estimates remained roughly unchanged.

Model B

The first step in the estimation of model B is the estimation of β_EK and δ_E, the parameters that are needed to calculate the flow of services corresponding to the K* input (see equation (15)). In this study, the parameter δ_E is assumed to be equal to the share of energy cost in the total of energy and capital costs in 1972, the last year before the first wave of oil price increases. To obtain an estimate of δ_EK, we have used the following equation:

which is the equivalent of equation (6) for the competitive allocation of income between capital and energy. Focusing on the 1972–82 period of doubling real energy prices, we have solved equation (19) for the value of β_EK corresponding to the observed changes in S_E and in ln(K/E).²⁰ The results, in Table 4, indicate that the value of β_EK is in the order of 2 to 3, which corresponds to a relatively small elasticity of substitution (η_EK) of 0.25 to 0.35.²¹ For purposes of the present study, we have taken a value of β_EK of 2.5 for all seven countries. Because this estimate is subject to a large margin of error, experiments with estimates ranging from 1.5 to 3.5 were carried out, with little effect on the estimates of the other parameters of model B or on the estimates of the equilibrium labor shares and real wage rates.

Table 4.

Estimates of β_EK, Model A¹

$({\beta }_{\mathrm{EK}}={\mathrm{\Delta }}_{72-82}{S}_E/{\delta }_E(1-{\delta }_E){\mathrm{\Delta }}_{72-82}\ln (K/E))$

¹In this table, S_E refers to the share of energy cost in the total of energy and capital costs, while δ_E refers to the weight of energy in the constant elasticity of substitution (CES) function defining the composite E-K input. The value of δ_E is approximated by the value of S_E in 1972. The symbol Δ₇₂₋₈₂S_E refers to the change in S_E from 1972 to 1982.

²For Canada, the estimate refers to the change in In (K/E) from 1972 to 1982, net of the change from 1962 to 1972. The adjustment was made to take account of what appeared to be a long-run trend in In (K/E).

Table 4.

Estimates of β_EK, Model A¹

$({\beta }_{\mathrm{EK}}={\mathrm{\Delta }}_{72-82}{S}_E/{\delta }_E(1-{\delta }_E){\mathrm{\Delta }}_{72-82}\ln (K/E))$

Country	Δ72−82SE	Δ72−82ln(K/E)	δE	β	ηEK = 1/(1 + βEK)
United States	0.242	0.500	0.217	2.86	0.26
Canada	0.183	0.4832	0.137	3.21	0.24
Japan	0.118	0.514	0.108	2.39	0.29
France	0.114	0.387	0.135	2.52	0.28
Germany, Fed. Rep.	0.125	0.368	0.155	2.59	0.28
Italy	0.105	0.175	0.219	3.51	0.22
United Kingdom	0.147	0.476	0.224	1.77	0.36

¹In this table, S_E refers to the share of energy cost in the total of energy and capital costs, while δ_E refers to the weight of energy in the constant elasticity of substitution (CES) function defining the composite E-K input. The value of δ_E is approximated by the value of S_E in 1972. The symbol Δ₇₂₋₈₂S_E refers to the change in S_E from 1972 to 1982.

²For Canada, the estimate refers to the change in In (K/E) from 1972 to 1982, net of the change from 1962 to 1972. The adjustment was made to take account of what appeared to be a long-run trend in In (K/E).

View Table

Table 4.

Estimates of β_EK, Model A¹

$({\beta }_{\mathrm{EK}}={\mathrm{\Delta }}_{72-82}{S}_E/{\delta }_E(1-{\delta }_E){\mathrm{\Delta }}_{72-82}\ln (K/E))$

Country	Δ72−82SE	Δ72−82ln(K/E)	δE	β	ηEK = 1/(1 + βEK)
United States	0.242	0.500	0.217	2.86	0.26
Canada	0.183	0.4832	0.137	3.21	0.24
Japan	0.118	0.514	0.108	2.39	0.29
France	0.114	0.387	0.135	2.52	0.28
Germany, Fed. Rep.	0.125	0.368	0.155	2.59	0.28
Italy	0.105	0.175	0.219	3.51	0.22
United Kingdom	0.147	0.476	0.224	1.77	0.36

¹In this table, S_E refers to the share of energy cost in the total of energy and capital costs, while δ_E refers to the weight of energy in the constant elasticity of substitution (CES) function defining the composite E-K input. The value of δ_E is approximated by the value of S_E in 1972. The symbol Δ₇₂₋₈₂S_E refers to the change in S_E from 1972 to 1982.

²For Canada, the estimate refers to the change in In (K/E) from 1972 to 1982, net of the change from 1962 to 1972. The adjustment was made to take account of what appeared to be a long-run trend in In (K/E).

The estimated values of the other parameters of model B are presented in Table 5. The results are similar to those obtained for model A, with two exceptions. First, as could be expected, the estimated value of δ₀ is now significantly larger. In all cases, it is approximately equal to the share of capital and energy cost in the total value added corresponding to capital, energy, and labor in the mid-1950s. Second, the reduction in the value of λ_t from the 1960s to the 1970s and early 1980s is now smaller, but in general not by much. Thus, even when the reduction in energy use achieved during the past ten years is explicitly taken into account, as it is in model B, there is still a sizable unexplained reduction in the rate of growth of disembodied productivity.

Table 5.

Estimates of Parameters, Model B¹

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.81 (0.08); Canada, 0.81 (0.09), France, 0.60 (0.21); and Italy, 0.83 (0.05).

³The trend rates of growth of total disembodied productivity are indicated in percent per year.

⁴The estimate of 4.22 (0.19) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 7.76 (0.52).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.50 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.12 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.

Table 5.

Estimates of Parameters, Model B¹

Parameter	United States2	Canada2	Japan	France2	Germany, Fed. Rep.	Italy2	United Kingdom
In γ	-0.022	-0.026	-0.006	-0.029	0.038	0.016	0.013
	(0.003)	(0.002)	(0.008)	(0.006)	(0.004)	(0.003)	(0.003)
λt3	1955–60	1955–60	1955–66	1955–64		1955–60	1955–60
	1.45	1.89	4.224	3.78		2.39	0.49
	(0.23)	(0.18)	(0.19)	(0.08)		(0.27)	(0.31)
	1961–73	1961–73	1967–70	1965–70	1961–82	1961–73	1961–73
	2.57	3.36	5.45	4.99	2.80	4.37	2.80
	(0.06)	(0.05)	(0.37)	(0.21)	(0.08)	(0.07)	(0.07)
	1974–82	1974–82	1971–82	1971–82		1974–82	1974–82
	0.90	1.05	4.41	2.27		2.57	1.46
	(0.14)	(0.12)	(0.26)	(0.17)		(0.16)	(0.22)
δ05	29.2	37.0	59.2	42.9	36.2	42.9	32.6
	(0.4)	(0.3)	(0.4)	(0.8)	(0.3)	(0.2)	(0.2)
δ15	0.53	0.15	0.506	0.71	0.47	-0.14	0.22
	(0.14)	(0.05)	(0.03)	(0.13)	(0.03)	(0.05)	(0.29)
β	0.813	0.182	0.254	0.587	0.372	0.258	0.460
	(0.172)	(0.050)	(0.014)	(0.118)	(0.016)	(0.037)	(0.310)
ρ	0.412	0.447	0.772	0.458	0.349	0.492	0.325
	(0.028)	(0.034)	(0.033)	(0.073)	(0.072)	(0.052)	(0.077)
		Goodness-of-fit statistics
Production function SE7	1.64	1.32	2.44	1.73	4.00	2.56	2.43
Share function SE7	0.72	0.13	0.24	0.26	0.11	0.33	0.31

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.81 (0.08); Canada, 0.81 (0.09), France, 0.60 (0.21); and Italy, 0.83 (0.05).

³The trend rates of growth of total disembodied productivity are indicated in percent per year.

⁴The estimate of 4.22 (0.19) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 7.76 (0.52).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.50 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.12 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.

View Table

Table 5.

Estimates of Parameters, Model B¹

Parameter	United States2	Canada2	Japan	France2	Germany, Fed. Rep.	Italy2	United Kingdom
In γ	-0.022	-0.026	-0.006	-0.029	0.038	0.016	0.013
	(0.003)	(0.002)	(0.008)	(0.006)	(0.004)	(0.003)	(0.003)
λt3	1955–60	1955–60	1955–66	1955–64		1955–60	1955–60
	1.45	1.89	4.224	3.78		2.39	0.49
	(0.23)	(0.18)	(0.19)	(0.08)		(0.27)	(0.31)
	1961–73	1961–73	1967–70	1965–70	1961–82	1961–73	1961–73
	2.57	3.36	5.45	4.99	2.80	4.37	2.80
	(0.06)	(0.05)	(0.37)	(0.21)	(0.08)	(0.07)	(0.07)
	1974–82	1974–82	1971–82	1971–82		1974–82	1974–82
	0.90	1.05	4.41	2.27		2.57	1.46
	(0.14)	(0.12)	(0.26)	(0.17)		(0.16)	(0.22)
δ05	29.2	37.0	59.2	42.9	36.2	42.9	32.6
	(0.4)	(0.3)	(0.4)	(0.8)	(0.3)	(0.2)	(0.2)
δ15	0.53	0.15	0.506	0.71	0.47	-0.14	0.22
	(0.14)	(0.05)	(0.03)	(0.13)	(0.03)	(0.05)	(0.29)
β	0.813	0.182	0.254	0.587	0.372	0.258	0.460
	(0.172)	(0.050)	(0.014)	(0.118)	(0.016)	(0.037)	(0.310)
ρ	0.412	0.447	0.772	0.458	0.349	0.492	0.325
	(0.028)	(0.034)	(0.033)	(0.073)	(0.072)	(0.052)	(0.077)
		Goodness-of-fit statistics
Production function SE7	1.64	1.32	2.44	1.73	4.00	2.56	2.43
Share function SE7	0.72	0.13	0.24	0.26	0.11	0.33	0.31

¹Parentheses enclose asymptotic standard errors.

²For these countries, the share function had to be adjusted for first-order autocorrelation. The estimates of the coefficients of autocorrelation are: United States, 0.81 (0.08); Canada, 0.81 (0.09), France, 0.60 (0.21); and Italy, 0.83 (0.05).

³The trend rates of growth of total disembodied productivity are indicated in percent per year.

⁴The estimate of 4.22 (0.19) is for the 1955–58 and 1962–66 periods. For 1959–61, the estimate is 7.76 (0.52).

⁵The weights of capital services are indicated in percent. The trend rates of change in these weights are in percentage points per year.

⁶The estimate of 0.50 (0.03) is for the 1955–70 period. For 1971–82, the estimate is −0.12 (0.07).

⁷SE denotes the standard error of estimate of the estimated equation. The standard error is in percent for the production function and in percentage points for the share function. The R² is not given because it conveys little information in the present case.