1. Neoclassical growth model

In the neoclassical growth model aggregate output is produced according to a well-behaved production function ^1/ as given by:

Where Q_t denotes the flow of (value added) output at time t, K and L refer to the inputs of the services of capital and labor, and A denotes technology or multifactor productivity. The production function F(.) is assumed to exhibit constant returns to scale to capital and labor but diminishing returns to each factor individually. ^2/ Improvements in technology--increases in A--are viewed as shifting the production function outward, allowing the economy to produce higher levels of output with given inputs of capital and labor. In the basic neoclassical model, these improvements in technology are assumed to be disembodied. ^3/

Under the assumptions of (i) an exogenous growth of labor input at the rate n ^1/; (ii) exogenous (labor-augmenting) technical change at the rate g ^2/; (iii) a fixed savings (equal investment) rate s ^3/; (iv) perfectly competitive factor markets; and (v) a Cobb-Douglas production function F(.) where the exponent on capital equals its income share (a) and that on labor equals its income share (1 - a) ^4/, the key dynamic equation of the neoclassical model can be written as:

where k (lower case) refers to the capital-labor ratio with labor measured in efficiency units (i.e., efficiency augmented labor), a dot indicates a time derivative, α is capital’s exponent in the Cobb-Douglas production function, and d is the (assumed) constant depreciation rate on capital, ^5/ Given the assumption that capital’s contribution to growth (given by a) is below unity ^6/, this equation is stable implying that over time the rate of growth of the capital-labor ratio (k) approaches zero.

Interpretation of equation (2) is facilitated by noting that the first term on the right-hand side measures the rate of capital accumulation as given by the economy’s level of savings, and the second measures the rate of accumulation required to hold the capital-labor ratio constant. ^1/ The implication of equation (2) is that the capital-labor ratio is increasing when the rate of capital accumulation exceeds the rate required to stabilize this ratio, and conversely when it falls short.

Given the assumed exogeneity of the growth rates of labor and technical progress--as well as diminishing returns to capital--the neoclassical model reaches a number of conclusions about the determinants of long-run (or steady-state) growth. Along a long-run growth path, the ratio of capital to augmented labor is constant (${\dot{\mathrm{k}}}_{\mathrm{t}}\mathrm{=}\mathrm{0}$ in equation (2)), implying that the following hold: first, output grows at a constant rate determined by the (exogenous) growth rates of labor input and (labor-augmenting) technical change; second, labor productivity grows at a constant rate given by the rate of (labor-augmenting) technical change; third, capital productivity (the inverse of the capital-output ratio) and the rate of return on capital are both constant. ^2/

The neoclassical model thus implies that the steady-state growth rate--which occurs when the capital-output ratio is constant--is not affected by changes in the share of resources allocated to investment in physical capital; ^1/ it is influenced only by the exogenous growth rates of labor input and technical change. The reason why changes in the investment rate do not change the long-run growth rate is the existence of diminishing returns to capital as indicated by an exponent on capital in the production function that is below unity. Given diminishing returns, each successive increase in capital has a smaller and smaller effect on output (the rate of return on capital declines) and it is only continuing increases in labor input and (exogenous) improvements in technology that sustain a positive growth rate. Of course, if society were prepared to devote an ever increasing share of its resources to investment, a higher growth rate could be sustained for some time (until the ratio of gross investment to GNP becomes one) but at the expense of a declining rate of return on capital and reduced current consumption.

Even though the neoclassical model implies that (exogenous) changes in the investment rate have no implications for long-run growth, such changes will influence the growth rate in the transition from one steady-state to another, and the transition can take a long time to be completed. In addition, the neoclassical model implies that a change in investment will influence the level of output (and other variables) across different steady state growth paths. ^2/ These points can be seen from Figure 1 which graphs the dynamics of the neoclassical model implied by equation (2). In this figure, the curved line marked KK relates the rate of capital accumulation to the capital-labor ratio (the curve is concave due to diminishing returns to physical capital) while the straight line RR indicates the pace of capital accumulation required to keep the capital-labor ratio unchanged. An initial steady state is assumed at point E where the two lines intersect and the ratio of capital to augmented labor is constant.

Dynamics of the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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Dynamics of the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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Dynamics of the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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An exogenous increase in the investment rate shifts the KK line up in Figure 1 leading to a new steady state at point F. In the transition from the old steady state (at E) to the new steady state (at F), the rate of capital accumulation exceeds the rate required to hold constant the ratio of capital to augmented labor; this ratio accordingly is rising. As capital deepening occurs, the economy grows faster than its steady-state rate, but the rate of return on capital diminishes. Once the new steady state has been reached, growth returns to its original rate but the level of capital to augmented labor is higher, implying a lower rate of return to capital and a higher level of labor productivity. Provided the economy is not dynamically inefficient, the level of consumption per capita will also be higher in the new steady state. ^1/

Using the neoclassical model, some quantification of investment’s effects on the level and growth of output can be obtained. ^2/ As noted earlier, the exponent for capital in the production function corresponds to capital’s income share and also measures its contribution to growth. Capital’s income share has averaged around 0.3 in the nonfarm business sector of the U.S. economy. Given an investment rate in this sector over the last five years of 16 percent, a capital-output ratio of 2.5, and a depreciation rate of 4 percent, an (exogenous) permanent increase in the (gross) investment rate of one percentage point ^3/ (a 6 1/4 percent increase in the investment rate: the one percentage point increase in the investment rate divided by the share of output allocated to investment) on impact will raise the growth rate of capital by 0.4 percentage points (the 1 percentage point increase in the investment rate divided by the capital-output ratio: 1/2.5) and that of output by 0.12 percentage points (the increase in the growth of capital multiplied by its income share: 0.4^*0.3). Over time, however, as noted above, the increment to the growth of output will diminish and approach zero in the new steady state. With the assumed parameter values, the “half-life” of the transition to a new steady state is about 10 years, ^1/ implying over a decade an annual average increase in the growth of output a shade above 0.1 percentage points.

In the new steady state, when growth has returned to its original rate, the level of output (and labor productivity) will be 2.7 percent higher. Reflecting the capital deepening implied by a higher investment rate, the capital-output ratio will have risen by 6 1/4 percent, with an associated decline in the marginal product of capital and its rate of return by 6 1/4 percent, and the rise in the ratio of capital to augmented labor of almost 9 percent. Alternatively expressed, the (gross) rate of return to raising the investment rate is approximately 12 percent, (capital’s income share divided by the capital-output ratio) while net of depreciation the rate of return is 8 percent (gross return less the depreciation rate).

Given the apparently small effect of (physical) capital accumulation in the basic neoclassical model, there have been many attempts to determine if an alternative modeling of technology could imply a larger role for investment. ^2/ Phelps (1969), Solow (1959), and Nelson (1964) have relaxed the assumption that technical change is disembodied and allowed for the possibility that new capital--which incorporates more efficient technologies--may be more productive than old capital. In these models, even under the assumption of diminishing returns, capital’s growth contribution can be larger than in the basic neoclassical model. This possibility is reflected in an exponent for capital in the aggregate production function derived from these models that exceeds capital’s income share by an amount related to the (net) additional benefit implied by the introduction of new technologies.

Following Nelson (1964) and Branson (1979), the basic elements of the neoclassical model with capital-embodied technical progress can be illustrated as follows. The production function given by equation (1) is replaced by equation (3) where J denotes an aggregate measure of capital of different vintages. ^3/

As in the basic neoclassical model, constant returns to scale of capital and labor are assumed as are diminishing return to each factor individually. Aggregate capital input (J) is defined by equation (4) below where h denotes the assumed exogenous rate of capital-embodied (and capital-augmenting) technical progress ^1/ and K_m refers to capital of vintage m. By assumption, the only way for the economy to benefit from new technology is to invest in capital goods.

Using equation (4), the growth of output can be shown to depend on the growth of capital, the rate of capital-embodied technical progress, and the average age of the capital stock (a). This leads to equation (5), where, for simplicity, the production function is assumed to be Cobb-Douglas.

The steady-state growth rate in this model--achieved when the average age of the capital stock is constant--is independent of exogenous shifts in the investment rate. ^2/ Embodiment effects make a difference with respect to the growth contribution of capital and the impact of changes in the investment rate between steady states.

As can be seen from equation (5), capital’s growth contribution is higher than in the basic model by an amount that depends on the (exogenous) growth rate of capital-augmenting technical change (h) and the average age of the capital stock (a). Intuitively, capital’s growth contribution is higher because each new unit of capital means not only that there is more capital but also that there is new technology. However, the introduction of capital with new technology makes part of the existing capital stock obsolete faster. ^3/ Under these conditions, the benefits from the introduction of new and more efficient capital are offset to some extent by a faster rate of scrapping of existing capital, leading to a net contribution of capital given by the coefficient attached to ${\dot{\mathrm{K}}}_{\mathrm{t}}\mathrm{/}{\mathrm{K}}_{\mathrm{t}}$ in equation (5).

Using the representative parameter values assumed earlier, some quantification of the possible importance of embodiment effects can be obtained. The average age of the capital stock is around 25 years (the inverse of the 4 percent depreciation rate assumed above) ^1/ and the rate of capital-embodied technical progress (h) is presently about 2 percent a year. ^2/ With these parameters, a one percentage point increase in the investment rate on impact will raise the annual growth rate of capital by 0.4 percentage points. As a result of embodiment effects, however, the annual rate of growth of output immediately rises by 0.18 percentage points (compared with 0.12 percentage points when technical change is not embodied) and over a ten-year period on average rises by 0.15 percentage points a year. Other things remaining unchanged, the effects on growth are larger the older is the capital stock (likely to be the case if investment has been weak for some time) and the more rapid the (exogenous) rate of capital embodied technical progress (h).

Based on the representative parameters for the U.S. economy assumed earlier, it is apparent that an allowance for capital-embodiment effects does not significantly raise the growth contribution of capital. ^3/ In addition, embodiment effects do not alter the conclusion that the steady-state growth rate is not affected by changes in the investment rate. To a first approximation, therefore, the neoclassical model--with and without embodied technical change--implies that capital’s growth contribution is reasonably well approximated by its income share and that steady-state growth is not influenced by the investment rate. Under these conditions, consideration is given to models which imply that capital’s exponent in the aggregate production function could be larger than its income share as implied by the model with embodied technological change, but in which shifts in investment may permanently affect the growth of output.

2. Endogenous growth models

In the last several years growing attention has been paid to models in which increasing returns to scale, learning-by-doing, and spill-over effects play important roles. ^1/ (See Sala-i-martin (1990 a,b), Rebelo (1990) and Romer (1986, 1987 a,b, 1990)). These models imply that capital’s income share may underestimate significantly its growth contribution and that under certain conditions shifts in the investment rate can permanently change the growth rate. However, there are a number of questions concerning their applicability, since they do not appear able to generate the kinds of steady-state growth paths many researchers believe provide reasonable approximations to the long-run growth experience. ^2/

For illustrative purposes, a particular endogenous growth model associated with Romer (1986, 1987 a,b, 1990) is considered. ^3/ This model stresses the possibility that the returns from (physical) capital accumulation may be larger than in the neoclassical growth model and may not decrease in response to faster rates of capital accumulation. The basic insight is that even though individual firms may perceive diminishing returns to capital accumulation (as in the neoclassical growth model) the economy as a whole may be able to avoid diminishing returns if there are positive externalities associated with firms’ investment decisions. Such external effects, which arise if the acquisition of knowledge by one firm expands the knowledge frontier of others or if there are learning-by-doing effects associated with investment, imply that the social rate of return on investment--which will exceed the (perceived) private rate of return--need not necessarily decline as capital input is increased. To the extent that such external effects are important, capital’s income share will provide a misleading indicator of capital’s contribution to aggregate output growth although it will continue to provide a reasonably good approximation of the return to capital as perceived by an individual firm.

The presence of external effects can be captured through the specification of a firm’s (Cobb-Douglas) production function of the form given by equation (6)

where K_it and L_it refer to the individual firm’s inputs of capital and labor, A_it is firm-specific multifactor productivity, and K_t denotes the economy’s total stock of capital which is the sum of each individual firm’s capital stock. The firm’s production function has all the properties of the neoclassical production function--diminishing returns to the firm’s inputs of capital and labor, and constant returns to scale to simultaneous increases in these factors--but it is shifted by other firms’ investment decisions. Provided the individual firm is small, it is assumed not to take into account the impact of its investment decisions on the economy’s aggregate capital stock. ^1/ The assumption that factor markets are competitive is retained, so that α and (1-α) continue to represent the income shares of capital and income, respectively.

Aggregating individual firms’ production functions gives rise to a production function for the whole economy of the form:

which--in contrast to the neoclassical production function--has an exponent on capital equal to α plus β, implying that capital’s growth contribution is different from its income share, a.

The model’s implications for the relationship between investment and growth depend on the size of the exponent on the aggregate capital stock (α + β). Given the assumption of a positive externality (β > 0) ^2/ there are two possibilities: (i) α + β < 1, and (ii) α + β > 1, with alternative implications for the effect of an increase in the investment rate on the growth rate of the economy. When the coefficient on aggregate capital (α+β) is below unity, there are diminishing returns to capital and the growth rate of the economy can not be permanently raised by an increase in the investment rate, even though the social returns to investment in this case will exceed the (perceived) private returns. ^3/ Conversely, when α + β > 1, there are nondiminishing returns to capital, and an increase in the investment rate will permanently raise the growth rate.

Following Romer (1986), consideration is given to the case where the exponent on capital in the aggregate production function is unity (α+β=1), but capital’s income share applies to individual firm’s production functions (α<1). Labor’s exponent in the production function (contribution to growth) is assumed to equal its income share. This case describes a situation in which private firms perceive diminishing returns to their individual investments (α <1), but the social returns to investment are constant (α+β=1). In such a case, a once-and-for-all exogenous increase in the investment rate will permanently raise the growth rate because reductions in the (perceived) private rate of return on capital are exactly offset by positive externalities from firms’ investment decisions. For the representative parameter values assumed earlier, this model implies that a one percentage point increase in the investment rate on impact will raise the growth rate of both capital and output by 0.4 percentage points, and that output growth will remain at this permanently higher rate. In contrast to the neoclassical growth model, the economy jumps immediately to the new growth path and the social and perceived private rates of return on investment are the same as on the original path.

A difficulty with this growth model is its implication that the capital-output ratio will be constantly changing if labor input is not fixed, and hence that under general conditions a steady state will not be approached. This result, which follows directly from the specification of the production function in equation (7) ^1/, implies that the model cannot account for the approximate long-run stability of the capital-output ratio. ^2/ Nevertheless, given that the central focus of the paper is on the growth contribution of capital, it is still useful to try to determine whether the rather large role the model ascribes to capital is appropriate in light of historical experience.

1. Growth contribution of capital

The first question concerns the (proximate) growth contribution of capital which, in the neoclassical model, is reasonably well approximated by its income share. In the endogenous growth model, on the other hand, capital’s contribution is higher than its income share and possibly equal to unity. By examining the ability of different models to account for growth with different weights attached to capital, information is obtained on the possible applicability of the models.

Table 1 decomposes output growth in the nonfarm business sector of the U.S. economy over various sub-periods since 1889 according to two different measures of the growth contributions of capital (and labor input) and multifactor productivity. Both decompositions are derived from the total differential of a Cobb-Douglas production function relating the growth of output to the growth of factor inputs and multifactor productivity but differ according to the weights assigned to capital (g_k)) and labor (g_L)).

Table 1.

Contributions to Growth, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Source: See appendix for data sources.

Table 1.

Contributions to Growth, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Period	Output	Neoclassical Model			Endogenous Growth Model
		Labor	Capital	Multifactor Productivity	Labor	Capital	Multifactor Productivity
1889−1909	4.94	2.08	1.56	1.30	2.08	5.03	−2.16
1909−1929	3.42	1.12	0.96	1.34	1.12	3.09	−0.79
1929−1948	2.61	0.67	0.28	1.66	0.67	0.90	1.03
1948−1968	3.78	0.77	1.20	1.81	0.77	3.88	−0.86
1968−1973	3.31	1.12	1.42	0.77	1.12	4.57	−2.38
1973−1979	2.50	1.31	1.25	−0.05	1.31	4.03	−2.83
1979−1989	2.86	1.21	1.11	0.54	1.21	3.59	−1.94
1889−1989	3.53	1.18	1.06	1.30	1.18	3.40	−1.05

Source: See appendix for data sources.

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Table 1.

Contributions to Growth, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Period	Output	Neoclassical Model			Endogenous Growth Model
		Labor	Capital	Multifactor Productivity	Labor	Capital	Multifactor Productivity
1889−1909	4.94	2.08	1.56	1.30	2.08	5.03	−2.16
1909−1929	3.42	1.12	0.96	1.34	1.12	3.09	−0.79
1929−1948	2.61	0.67	0.28	1.66	0.67	0.90	1.03
1948−1968	3.78	0.77	1.20	1.81	0.77	3.88	−0.86
1968−1973	3.31	1.12	1.42	0.77	1.12	4.57	−2.38
1973−1979	2.50	1.31	1.25	−0.05	1.31	4.03	−2.83
1979−1989	2.86	1.21	1.11	0.54	1.21	3.59	−1.94
1889−1989	3.53	1.18	1.06	1.30	1.18	3.40	−1.05

Source: See appendix for data sources.

The first decomposition, based on the neoclassical model and standard growth accounting (Denison (1962)), uses the income shares of capital and labor to measure their growth contributions. In the case of the nonfarm business sector, these shares have not shown sustained change and an average value of 0.31 for capital and 0.69 for labor was used. ^1/ The second decomposition follows Romer’s (1986, 1987 a, b) endogenous growth model with external effects and has a weight of 1 on capital and a weight of 0.7 on labor. ^2/ In both decompositions, the growth of multifactor productivity is defined residually as that part of output growth not accounted for by changes in capital or labor inputs.

Several features of the decompositions presented in Table 1 are evident. ^1/ Most notable, the neoclassical model attributes only about 1/3 of output growth to measured changes in capital inputs. The remainder is explained by growth in labor input and multifactor productivity with the latter viewed as occurring as a result of (exogenous) improvements in technology and any unmeasured influences on growth. ^2/ Second, the endogenous growth model, while attributing a much larger share of growth to changes in factor inputs--particularly capital, which has an exponent of unity in the production function--over explains growth insofar as it tends in most periods to have a large negative residual. ^3/

Finally, neither model is very successful in explaining the post-1968 growth slowdown in terms of changes in factor inputs. Consistent with much of the growth accounting literature, both ascribe the bulk of this slowdown to slower multifactor growth and neither assigns an important role to capital, the growth of which picked up after 1968. ^4/

The decompositions presented in Table 1 suggest that the endogenous growth model may overstate capital’s growth contribution, since it is very difficult to explain negative multifactor productivity growth over long periods, except by appealing to technical regress. The correlations presented in Figure 2 are intended to shed further light on this issue by plotting for various sub-periods since 1889 the relationship between the growth of multifactor productivity (calculated according to the neoclassical model) and the growth of capital input. ^5/ If the endogenous growth model is correct--or if capital’s contribution to growth is seriously understated for some other reason--there should be a positive relationship between the neoclassical measure of multifactor productivity growth and the growth of capital. That is to say, if the neoclassical model understates capital’s growth contribution, sub-periods when capital is growing rapidly should also be periods when multifactor productivity is growing rapidly. Of course, given that the neoclassical model predicts a positive association between multifactor productivity growth and capital accumulation in the long run, ^1/ the finding of a positive correlation does not permit an unambiguous interpretation. Provided, however, that the economy is not always in a steady state, the neoclassical prediction that faster multifactor productivity growth will cause higher capital accumulation should be relatively weak over the periods considered.

Rates of Growth of Multifactor Productivity (MFP) and Capital Input, Nonfarm Business, Selected Periods, 1889–1989, United States.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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Rates of Growth of Multifactor Productivity (MFP) and Capital Input, Nonfarm Business, Selected Periods, 1889–1989, United States.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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Rates of Growth of Multifactor Productivity (MFP) and Capital Input, Nonfarm Business, Selected Periods, 1889–1989, United States.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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The data plotted in Figure 2 do not suggest that the neoclassical model underestimates capital’s growth contribution. Rather than implying a positive correlation between capital accumulation and multifactor productivity growth, the figure suggests that there is probably no relation between these variables. ^2/ On average, periods when capital accumulation is rapid are not periods of above average multifactor productivity growth as would be suggested by the endogenous growth model associated with Romer (1986, 1987 a,b).

Consideration has thus far been given to the possibility that the neoclassical model may understate capital’s growth contribution. Such a focus is appropriate, because the issue has been whether the data are consistent with the large effects of capital on growth implied by the endogenous growth model. Nevertheless, it is also necessary to consider whether the neoclassical model may itself overstate capital’s growth contribution, as suggested, for example, by Summers (1990). There are two main reasons why capital’s income share may overstate its contribution to growth. ^3/ In the first place, part of what is measured as profit in the national accounts may include returns to monopoly power, research and development expenditures, or rents to extraordinary managerial talent. These returns should not be included in measuring the (social) rate of return to physical capital. Second, the fact that investments are risky implies that profits include a reward for taking risk. In determining the contribution of capital to growth, one might not want to include this risk component, but would want to obtain the certainty equivalent rate of return. The difficulty, however, is that, if the (ex post) real rate of return on long-term Government debt is used for this purpose, this rate of return appears to have been very low over much of the post-war period, ^4/ implying that capital’s income share may overstate its contribution to growth. ^1/

There does not at this stage appear to be a satisfactory resolution to these problems. Based on evidence presented by Summers (1990)--notably that Tobin’s measure of the ratio of the market value of capital to its reproduction cost in the U.S. has averaged less than unity over the postwar period--there is the possibility that factors such as monopoly may not be very important in the U.S. economy; in short, capital income may not include large monopoly profit elements. As regards the treatment of risk, the situation is a little less clear but--regardless of the fact that capital income presumably includes a return on risk taking--capital’s income should still provide a measure of the average but risky rate of return on investment. The problem instead is that the value society places on this income in risk adjusted terms may be much lower than the rate of return on capital. ^2/

2. Time series relationship between investment and growth

Another approach to determining the relevance of alternative growth models is to study the time-series relationship between investment and growth (Tables 2a and 2b). Care must be exercised, however, since the neoclassical model--while implying that exogenous shifts, in investment have no long-run effects on growth--suggests that a relationship between these variables will arise in the transition between steady states. If these transitions take a long time, one would expect even at relatively low frequencies to observe a relationship between investment and growth rates. In addition, when the assumption of an exogenous savings (investment) rate is relaxed, a long-run relationship could arise between these variables if the savings (investment) rate is affected by the economy’s growth rate. Accordingly, if a relationship between growth and the investment rate is uncovered, it is necessary to determine whether it reflects the impact of investment on growth or the consequences of alternative growth rates for the savings (investment) rate. ^3/ Nevertheless, given that a major difference between the neoclassical model and the endogenous growth model concerns the impact of exogenous shifts in investment on growth, it is important to study the relationship between these variables. ^{4/ 5/}

Interpretation of the equations presented in Table 2--estimated with both period average and yearly observations in order to distinguish longer and shorter-run effects--is facilitated by considering the implications of regressing the growth of output on the investment rate, the growth of labor (in the case of some of the equations), and a constant term (equations (1) through (3)). Such regressions, of course, suffer from all sorts of simultaneity problems but can provide some information about underlying economic relationships. We emphasize that the regressions should not be viewed as rigorous tests of any model. Rather, they are a way of formally examining the correlations in the data.

According to the neoclassical model, the regressions should deliver a coefficient on the investment rate equal to zero over the long run (since investment has no effect on long-run growth). Conversely, the endogenous growth model considered in this paper imply that the long-run coefficient on the investment rate should be positive and--given a growth contribution of capital of unity--equal to the inverse of the (average) capital-output ratio (0.4: unity divided by 2 1/2). The constant terms that are obtained, on the other hand, should depend on whether the growth of labor input is included in the equations ^1/ and on whether the rate of technical change is closely associated with the investment rate. In the case of the endogenous growth model, for example, the constant terms in equations that include the growth of labor input would be expected to be close to zero since the rate of technical advance is regarded as being closely associated with the investment rate that is included in the equation. ^2/ Finally, the last two equations capture the relationship between the growth of output and that of factor inputs. These equations are included to provide information on the extent to which the growth contributions of factors are related to their income shares.

To the extent of course that one never actually observes the long run, the estimated coefficients in all cases will be influenced by several factors, including the transition dynamics of the neoclassical model and shorter-run cyclical fluctuations.

Table 2a.

Explaining Long-run Growth Using Period Averages, Nonfarm business sector, 1889–1989. ^1/

Table 2a.

Explaining Long-run Growth Using Period Averages, Nonfarm business sector, 1889–1989. ^1/

1. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.70}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.30}\mathrm{\right)}}{\mathrm{0.10}}\mathrm{i}\end{array}$	R2 = 0.25, D.W. = 0.71
2. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.74}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.009}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.001}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.11}\mathrm{\right)}}{\mathrm{0.84}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.43, D.W. = 1.42
3. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{2.51}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.39}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.02}}\mathrm{+}\underset{\mathrm{(}2/\mathrm{)}}{\mathrm{1.00}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.66, D.W. = 1.90
4. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{2.46}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.92}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.93}\mathrm{\right)}}{\mathrm{0.83}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.43, D.W. = 2.03
5. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}2.35\mathrm{\right)}}^{\mathrm{*}}}{2.03}+\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.62}\mathrm{\right)}}{\mathrm{0.37}}\frac{\dot{\mathrm{K}}}{\mathrm{K}}\end{array}$	R2 = 0.34, D.W. = 0.66

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Table 2a.

Explaining Long-run Growth Using Period Averages, Nonfarm business sector, 1889–1989. ^1/

1. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.70}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.30}\mathrm{\right)}}{\mathrm{0.10}}\mathrm{i}\end{array}$	R2 = 0.25, D.W. = 0.71
2. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.74}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.009}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.001}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.11}\mathrm{\right)}}{\mathrm{0.84}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.43, D.W. = 1.42
3. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{2.51}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.93}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.39}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.02}}\mathrm{+}\underset{\mathrm{(}2/\mathrm{)}}{\mathrm{1.00}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.66, D.W. = 1.90
4. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{2.46}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.92}}\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.93}\mathrm{\right)}}{\mathrm{0.83}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.43, D.W. = 2.03
5. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}2.35\mathrm{\right)}}^{\mathrm{*}}}{2.03}+\mathrm{+}\underset{\mathrm{\left(}\mathrm{1.62}\mathrm{\right)}}{\mathrm{0.37}}\frac{\dot{\mathrm{K}}}{\mathrm{K}}\end{array}$	R2 = 0.34, D.W. = 0.66

Table 2b.

Explaining Annual Output Growth, Nonfarm Business Sector, 1889–1989.

^1/Sample consists of average values for periods: 1889-1909, 1909-29, 1929-48, 1948-68, 1968-73, 1973-79, 1979-89. An asterisk denotes statistical significance at the 5 percent level.

^2/Constrained value. Constraint cannot be rejected at 5 percent level.

Table 2b.

Explaining Annual Output Growth, Nonfarm Business Sector, 1889–1989.

1.$\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.79}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.63}}\mathrm{+}\underset{{\mathrm{\left(}\mathrm{2.86}\mathrm{\right)}}^{*}}{\mathrm{0.14}}\mathrm{i}\end{array}$	R2 = 0.08, D.W. = 2.29
2. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{3.20}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.72}}\mathrm{+}\underset{{\mathrm{\left(}\mathrm{0.081}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.003}}+\underset{{\mathrm{\left(}\mathrm{13.60}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.04}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.68
3. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{3.21}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.72}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.270}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.008}}\mathrm{+}\underset{\mathrm{(}2/\mathrm{)}}{\mathrm{1.00}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.60
4. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{4.52}\mathrm{\right)}}^{\mathrm{*}}}{1.75}+\underset{{\mathrm{\left(}14.51\mathrm{\right)}}^{*}}{\mathrm{1.04}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.68
5. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.08}\mathrm{\right)}}^{}}{\mathrm{0.69}}+\mathrm{+}\underset{{\mathrm{\left(}\mathrm{7.48}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{0.83}}\frac{\dot{\mathrm{K}}}{\mathrm{K}}\end{array}$	R2 = 0.36, D.W. = 2.37

^1/Sample consists of average values for periods: 1889-1909, 1909-29, 1929-48, 1948-68, 1968-73, 1973-79, 1979-89. An asterisk denotes statistical significance at the 5 percent level.

^2/Constrained value. Constraint cannot be rejected at 5 percent level.

View Table

Table 2b.

Explaining Annual Output Growth, Nonfarm Business Sector, 1889–1989.

1.$\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.79}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.63}}\mathrm{+}\underset{{\mathrm{\left(}\mathrm{2.86}\mathrm{\right)}}^{*}}{\mathrm{0.14}}\mathrm{i}\end{array}$	R2 = 0.08, D.W. = 2.29
2. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{3.20}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.72}}\mathrm{+}\underset{{\mathrm{\left(}\mathrm{0.081}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.003}}+\underset{{\mathrm{\left(}\mathrm{13.60}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.04}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.68
3. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{3.21}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{1.72}}\mathrm{-}\underset{{\mathrm{\left(}\mathrm{0.270}\mathrm{\right)}}^{\mathrm{i}}}{\mathrm{0.008}}\mathrm{+}\underset{\mathrm{(}2/\mathrm{)}}{\mathrm{1.00}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.60
4. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{4.52}\mathrm{\right)}}^{\mathrm{*}}}{1.75}+\underset{{\mathrm{\left(}14.51\mathrm{\right)}}^{*}}{\mathrm{1.04}}\frac{\dot{\mathrm{L}}}{\mathrm{L}}\end{array}$	R2 = 0.68, D.W. = 2.68
5. $\begin{array}{c}\frac{\dot{\mathrm{Q}}}{\mathrm{Q}}\mathrm{=}\underset{{\mathrm{\left(}\mathrm{1.08}\mathrm{\right)}}^{}}{\mathrm{0.69}}+\mathrm{+}\underset{{\mathrm{\left(}\mathrm{7.48}\mathrm{\right)}}^{\mathrm{*}}}{\mathrm{0.83}}\frac{\dot{\mathrm{K}}}{\mathrm{K}}\end{array}$	R2 = 0.36, D.W. = 2.37

^1/Sample consists of average values for periods: 1889-1909, 1909-29, 1929-48, 1948-68, 1968-73, 1973-79, 1979-89. An asterisk denotes statistical significance at the 5 percent level.

^2/Constrained value. Constraint cannot be rejected at 5 percent level.

Several features of the results presented in Table 2a and 2b are of interest. In the first place, with both period-average and yearly observations, the relationship between investment and output growth is very weak, and in most cases statistically insignificant. ^1/ The exception is the annual regression of output growth on the investment rate which shows a small and statistically significant relationship between these two variables. This equation suggests that a one percentage point increase in the investment rate raises the growth of output by 0.14 percentage points a year--approximately the transition effect implied by the numerical estimate of the neoclassical model provided in Section II. Second, adding labor input growth to the equations--unconstrained or constrained to equal unity--weakens the link between investment and growth in both the period average data and the annual regressions. ^2/ Third, the constant terms in equations (1)-(3) do not change very much when the right-hand variables are varied; this arises because there is a sizable component of growth that is uncorrelated with investment and the growth of labor input. Many of the constant terms in these equations are around 1.7 - 1.9, approximately equal to the average growth of labor-augmenting technical change over the sample period. ^3/

Finally, when separate regressions are run of output growth against the rates of growth of factor inputs, the coefficient on labor tends to exceed (slightly) its income share while that on capital is approximately equal to its income share in period-average data, but considerably above it in the yearly data. At this stage, it is unclear how to interpret the apparently large coefficient on capital in the yearly data and the (related) finding that a statistically significant but weak relationship between investment and growth can be found in the annual regressions. However, because the relationship between growth and investment disappears over the longer runs of the period average data--and is not robust to including the growth of labor input in the regressions--it seems unlikely that it reflects the prediction of the endogenous growth model regarding large and permanent effects of investment on growth.

While one should not, given numerous problems, draw strong conclusions from the regression results it is nevertheless interesting that the data do not generally uncover a strong and lasting relationship between investment and growth as implied by the endogenous growth model. With this as background, some related evidence on the applicability of the alternative models is considered.

3. Behavior of the rate of return on capital

Another approach to assessing the usefulness of the neoclassical model is to examine the behavior of the rate of return on capital and whether there is evidence in favor of diminishing returns. As noted in Section II, a key implication of the neoclassical model is that the rate of return on capital should fall when the ratio of capital to augmented labor is rising. By contrast, the endogenous growth model predicts no necessary relationship between the (social and private) rate of return on capital and the degree of capital deepening. The validity of the neoclassical model therefore can be considered by studying the behavior of the rate of return to capital in periods when the ratio of capital to augmented labor was changing.

Table 3 and Figure 3 summarize how key variables in the neoclassical model have behaved since 1889. ^1/ In addition, the table shows a measure of the (gross) rate of return on capital, defined as the ratio of capital income (as given in the national accounts) to capital input. ^2/ The presentation differs from that in Table 1 because some of the variables have been expressed in labor efficiency units; in addition, the rate of technical progress is expressed in terms of the growth of labor efficiency, which is the growth of multifactor productivity (calculated according to the neoclassical model) divided by labor’s income share. Presentation in efficiency units facilitates the analysis of the data by providing a direct link between the key variables of the neoclassical model and the economy’s growth experience. ^3/

Table 3.

Key Variables in the Neoclassical Model, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Source: See appendix for data sources.

Table 3.

Key Variables in the Neoclassical Model, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Period	Output	Average Labor	Average Labor Productivity	Labor Augmenting Tech. Change	Capital to Augmented Ratio	Capital to Output Ratio	Gross Rate of Return (% per year)
1889−1909	4.94	4.90	1.93	1.89	0.13	0.09	11.87
1909−1929	3.42	3.57	1.80	1.95	−0.48	−0.33	10.47
1929−1948	2.61	3.38	1.63	2.40	−2.47	−1.71	12.90
1948−1968	3.78	3.74	2.67	2.62	0.14	0.10	16.05
1968−1973	3.31	2.75	1.69	1.12	1.82	1.26	13.85
1973−1979	2.50	1.81	0.61	−0.08	2.22	1.53	12.69
1979−1989	2.86	2.53	1.11	0.78	1.06	0.73	11.50
1889−1989	3.53	3.58	1.82	1.88	−0.18	−0.13	12.73

Source: See appendix for data sources.

View Table

Table 3.

Key Variables in the Neoclassical Model, Nonfarm Business Sector, 1889–1989.

(Average Annual Percentage Change)

Period	Output	Average Labor	Average Labor Productivity	Labor Augmenting Tech. Change	Capital to Augmented Ratio	Capital to Output Ratio	Gross Rate of Return (% per year)
1889−1909	4.94	4.90	1.93	1.89	0.13	0.09	11.87
1909−1929	3.42	3.57	1.80	1.95	−0.48	−0.33	10.47
1929−1948	2.61	3.38	1.63	2.40	−2.47	−1.71	12.90
1948−1968	3.78	3.74	2.67	2.62	0.14	0.10	16.05
1968−1973	3.31	2.75	1.69	1.12	1.82	1.26	13.85
1973−1979	2.50	1.81	0.61	−0.08	2.22	1.53	12.69
1979−1989	2.86	2.53	1.11	0.78	1.06	0.73	11.50
1889−1989	3.53	3.58	1.82	1.88	−0.18	−0.13	12.73

Source: See appendix for data sources.

Capital Intensity, Nonfarm Business Sector 1889-1989.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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Capital Intensity, Nonfarm Business Sector 1889-1989.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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Capital Intensity, Nonfarm Business Sector 1889-1989.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

Source: See appendix for data sources.

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Of key interest is the relationship between the rate of return on capital and the behavior of the ratio of capital to augmented labor. Abstracting from the Great depression and World War II, ^4/ three sub-periods are of interest. The first, from 1889 to 1929, is marked by large swings in the ratio of capital to augmented labor (and the capital-output ratio) but no sustained change in these variables; over this period the (gross) rate of return on capital fluctuated in the 10 to 12 percent range. Between 1948 and 1968 the ratio of capital to augmented labor (and capital to output) again showed no sustained change--notwithstanding substantial year-to-year fluctuations--and there was also no permanent change in the (gross) rate of return on capital which averaged about 16 percent over this period. It is only in the period since 1968 that there have been sustained changes in the capital to augmented labor ratio (and the capital-output ratio) and that there is a direct test of the neoclassical model’s predictions about the rate of return to capital.

The period since 1968 can be characterized broadly as one of a sustained slowdown in the (exogenous) rate of growth of labor augmenting technical progress. ^1/ Interpretation of developments over this period is facilitated using Figure 4 which shows--based on the neoclassical growth model--the implications of an exogenous slowing in the rate of labor-augmenting technical change. Beginning in an initial steady state at point E, the slowdown rotates the capital requirements line downward leading to a new steady state at point F. In the transition from point E to point F, there is an immediate slowdown in the rate of growth of output and labor productivity but the rate of capital accumulation only slows gradually. As a result, the ratio of capital to augmented labor (and of capital to output) begins to rise and is higher in the new steady state. As a result of the capital deepening, the rate of return on capital declines and is lower in the steady state at point F.

Effect of an Exogenous Slowdown in the Rate of Labor Augmenting Technical Change in the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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Effect of an Exogenous Slowdown in the Rate of Labor Augmenting Technical Change in the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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Effect of an Exogenous Slowdown in the Rate of Labor Augmenting Technical Change in the Neoclassical Model.

Citation: IMF Working Papers 1992, 035; 10.5089/9781451978407.001.A001

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Consistent with the predictions of the neoclassical model, the data in Table 3 and Figure 3 show that the rise in the ratio of capital to augmented labor since 1968 has been accompanied by a decline in the rate of return on capital. Between 1968 and 1989, the ratio of capital to augmented labor rose by approximately 33 1/3 percent which, according to the neoclassical model, should lead to a decline in the rate of return on capital of 23 percent (the share of labor (0.7) times the percentage rise in the ratio of capital to augmented labor). In fact, the rate of return to capital over this period declined from around 15 percent a year to 12 percent a year--a 20 percent decline--which is broadly consistent with the prediction of the neoclassical model.

In short, the experience of the U.S. economy as regards the rate of return on capital is broadly consistent with the predictions of the neoclassical model and suggestive of the existence of diminishing returns to capital accumulation.