Abstract
The model developed here postulates that learning through experience raises labor productivity with three major consequences. First, the steady-state growth rate of output becomes endogenous and is influenced by government policies. Second, the speed of adjustment to steady-state growth increases, and enhanced learning further reduces adjustment time. Third, both steady state growth and the optimal net rate of return to capital are higher than the sum of the exogenous rates of technical change and population growth. Simulation results confirm the model’s faster speed of adjustment, and regression analysis finds that a large part of the divergent growth patterns across countries is related to the extent of economic openness, the depth of human development, and the quality of fiscal policies.
The basic neoclassical growth model developed by Solow (1956) and Swan (1956) has been the workhorse of growth theory for the past three and a half decades. Its simple structure—consisting of a well-behaved neoclassical production function, investment-saving relation, and a labor growth function—is an elegant solution to the “knife-edge” problem posed by Harrod (1939) and Domar (1946). By allowing smooth factor substitution and wage-price flexibility, the capital-output ratio is made a monotonic function of the capital-labor ratio. The growth rate of the capital stock (the warranted rate) adjusts to the exogenously given growth rate of the labor force (the natural rate) to maintain full-employment real output.
The Solow-Swan model, however, has certain equilibrium properties that bother many growth theorists: an increase in the saving rate, while raising the level of per capita real income, has no effect on the growth rate of output. This surprising result on growth neutrality has a simple explanation: although a higher saving rate raises the growth rate of output by increasing the investment rate, the increase in economic growth occurs only during the transition toward the next equilibrium; sooner or later, the labor input becomes a bottleneck, restricting further output expansion. The growth rate of output would eventually fall back to the constant natural rate of growth.
The time it takes the economy to reach this balanced growth path is of considerable interest—particularly to policymakers. In the context of the Solow-Swan model, if the objective of economic policy is to raise the equilibrium level of per capita real income (for example, by raising the government saving rate), a fast adjustment would be desirable.
Using a Cobb-Douglas production function with constant returns to scale and Harrod-neutral technical progress, Sato (1963) has shown that the time required for the Solow-Swan model to reach equilibrium is about a hundred years!1 Moreover, the lower the rate of depreciation or the higher the share of capital, the slower the adjustment. An intuitive explanation for these results is that a slower rate of depreciation or a larger share of capital would enable firms to substitute capital for labor and thus postpone for a longer period the bottleneck posed by a fixed rate of labor growth.
The Solow-Swan model’s prediction that the rates of saving, depreciation, and population growth as well as government policies cannot affect the equilibrium growth rate of per capita real income, which is fixed by an exogenously determined rate of labor-augmenting technological progress, appears counterfactual. It seems reasonable to conjecture that, over the long haul, countries that promote saving and investment, reduce the depreciation of the capital stock, and create more open trading systems tend to grow faster and that those with rapid population growth, sluggish expansion in expenditures on human development and basic needs, and high ratios of government deficits to GDP tend to grow slower.
The relatively slow adjustment of the Solow-Swan model toward its steady state is partly due to the (assumed) inability of the natural rate to adjust to changes in capital intensity as the economy moves from one equilibrium to another in response to an exogenous shock. It seems plausible to consider that a partly endogenous natural rate, via learning through experience, would contribute to a faster speed of adjustment. If so, the limiting behavior of the Solow-Swan model would assume much more relevance to policymakers.
This study is both theoretical and empirical and belongs to the class of new “endogenous growth” models.2 It is a variant of Conlisk’s (1967) endogenous-technical change model and Arrow’s (1962) “learning by doing” model, wherein experience (measured in terms of either cumulative past investment or output) plays a critical role in raising labor productivity over time. The presence of learning through experience has three major theoretical consequences. First, equilibrium growth becomes endogenous and is influenced by government policies.3 Second, the speed of adjustment to growth equilibrium is faster, and enhanced learning further reduces adjustment time. Third, both equilibrium economic growth and the optimal net rate of return to capital are higher than the sum of the exogenous rates of technical change and population growth.
The endogenous growth model’s equilibrium behavior is found to be consistent with the substantial diversity in per capita growth patterns observed among countries. Such diverse growth experiences, which are predicted by the model, can be explained by differences in saving rates, ratios of government deficits to GDP, population growth rates, and certain parameters that influence the learning coefficient, such as changes in openness to world trade and growth in government outlays on education and health.
I. Endogenous Growth
The model is summarized by the following relationships:
where the variables are defined as
Y = real GDP,
K = capital stock,
N = labor (manhours in efficiency units),
L = population (manhours),
T = labor productivity or technical change multiplier (index number),
k = ratio of K to N,
s = ratio of real saving to Y,
δ = depreciation of capital,
α = learning coefficient;
and the parameters are defined as
n = population growth rate,
X = change in ratio of foreign trade (sum of exports and imports) to GDP,
ξ = growth rate of real government expenditures on education and health,
ω = growth rate of real government expenditures on social security and housing,
μ = growth rate of real government expenditures on operations and maintenance,
θ = ratio of government deficits to GDP,
λ = rate of exogenous labor-augmenting technical change,
d(.)/dt = time derivative.
Equation (1) is a standard neoclassical production function satisfying the Inada (1963) conditions.4 Equation (2) is the expression for capital accumulation: the increment in the capital stock is equal to gross domestic saving less depreciation. The proportion s of GDP saved and invested is assumed to be sensitive to government policies, in particular to θ, the ratio of the fiscal deficit to GDP. High values of θ directly lower s, as the public sector dissaves. There are indirect effects as well. High levels of θ indicate large government borrowings from financial markets. Either through higher interest rates or lower credit availability, private sector capital accumulation is adversely affected. Thus, it is assumed that s’(θ) < 0.
There are other (unspecified) factors affecting s. For example, interest rate liberalization may increase the private saving rate, which would tend to pull up aggregate s, but may also entail increases in the rate of government dissaving in the presence of a large stock of public debt, which would drag down total s both directly and indirectly (via negative effects on the private saving-investment rate). It is also assumed that δ’(μ) < 0—the rate of depreciation, δ, is a negative function of the real growth of expenditures on operations and maintenance, μ. In other words, a higher μ lowers the rate of depreciation of existing capital stock K. The population grows at an exogenously constant rate n in equation (3).
The key relationship in the model is equation (4). It postulates that technical change dT/dt improves with the capital stock per capita Y/L. Output per capita K/L. can be used instead. For example, manhours in the production of an airframe (the structural frame of an airplane) during the 1930s tended to decline with the number of airframes produced. A more current example is the introduction of both high-speed and personal computers, which has improved the productivity of engineers and scientists (including economists). Since (dT/dt)/T is a function of Y/TL = Y/N = f(k), using K/L is equivalent to using Y/L as the forcing variable behind improvements in labor productivity. The parameter α is the learning coefficient. If α = 0, T grows exogenously at a constant rate X and the endogenous growth model collapses into the Solow-Swan model. (The restrictions α ≥ 0 are assumed and empirically tested in a later section. Since the assumption that α > 0 is crucial to the arguments in this paper, the extended discussion of its rationale is useful.)
The Solow-Swan model’s characterizing assumption α = 0 may be true in a world devoid of technical change, since labor supply may be measured by the size of population. In this case, it may be plausible to assume that labor has no endogenous growth component, since population in many countries appears to grow independently of the economic system. But the real world is one of continuous technical change. While a portion of this change may be exogenous, some technical change is clearly endogenous and partly labor augmenting. Workers learn through experience, and their productivity is likely to be enhanced by the arrival of new and advanced capital goods. That is, the endogenous growth model’s assumption that α > 0 seems more plausible than the Solow-Swan model’s assumption that α = 0.5
In the restriction α > 0, the learning coefficient a is allowed to vary positively with changes in the ratio of foreign trade to GDP, x; real growth of outlays on education and health, ξ; social security, housing, and recreation, ω and other unspecified factors. The role of rapid growth of foreign trade in stimulating a higher learning coefficient is twofold.6 First, the import-export sector serves as a vehicle for technology transfer through the importation of advanced capital goods, as elucidated by Bardhan and Lewis (1970), Chen (1979), and Khang (1987), and as a channel for positive intersectoral externalities through the development of efficient and internationally competitive management, the training of skilled workers, and the spillover consequences of scale expansion (Keesing (1967), and Feder (1983)). Second, rising exports relieve the foreign exchange constraint. The importation of technologically superior capital goods is enhanced by growing export receipts and higher flows of foreign credits and direct investment, which take into account the country’s ability to repay out of export earnings.7
It is also reasonable to posit that accelerated growth of real outlays on education and health would be associated with a higher learning potential of labor, as would growth in real expenditures on social security, housing, and recreation. Finally, equations (5) and (6) are standard definitional relations involving N and k.
Reduced Model
The growth rate of the capital stock is derived by dividing equation (2) by K, using equations (1) and (6):
The growth rate of efficient labor is derived by differentiating equation (5) with respect to time, using equations (1) and (3)–(6):
Differentiating equation (6) with respect to time and substituting equations (7) and (8), the growth rate of the capital-labor ratio k is thus equal to
The reduced model, equation (9), is a single differential equation involving the variables (dk/dt)/k and k alone.
Per capita income, Y/L, grows according to
which is also a single-valued function of k. Here, π is the share of income going to capital; in general, it is a function of k.8 The equilibrium capital intensity k* is the root of equation (9) equated to zero:
And the equilibrium growth rate of per capita income is given by
Stability
Given the Inada conditions on the production function, equations (7)–(9) are depicted according to Figure 1. The upper panel graphs equation (9), and the lower panel graphs equations (7) and (8). The downward slopes of the curves representing equations (7) and (9) and the upward slope of the curve representing equation (8) follow from the assumption of a positive but diminishing marginal product of capital. The reasons why the (dk/dt)/k curve lies partly in the first quadrant and partly in the fourth quadrant in Figure 1 are given by the other Inada conditions—that is, for some initial values of the capital-labor ratio, it is possible for capital to grow either faster or slower than labor. It is obvious by inspection that, at any point on the (dk/dt)/ke curve, the economy would move in the direction indicated by the arrows. Thus, k tends to settle at an equilibrium value
Endogenous and Solow-Swan Growth Models
Citation: IMF Staff Papers 1994, 001; 10.5089/9781451957037.024.A001
Equilibrium Capital Intensity and Growth
The Solow-Swan and endogenous growth models are graphically portrayed in Figure 1. In the lower panel, the natural rate schedule, Ne, is upward sloping in the endogenous growth model, owing to the presence of learning-by-doing and the assumption of a positive marginal product of capital. The natural rate schedule in the Solow-Swan model is shown as the horizontal line Ns with vertical height equal to a constant growth rate
In the upper panel, reflecting the different natural rate schedules, the capital accumulation schedules assume the shape and intersection with the k-axis indicated by the two curves, with (dk/dt)/ks flatter than and to the right of (dk/dt)/ke. The equilibrium positions of the two types of models are indicated by the points A and B, respectively, in the lower panel. The growth rate of output is higher in the endogenous growth model by the magnitude
Comparative Dynamics
Table 1 summarizes the qualitative effects of changes in the structural parameters on the equilibrium capital intensity, k*, and on the equilibrium per capita growth rate of income, g* – n. Algebraically, the partial derivatives of k* and g* – n with respect to any structural parameter may be obtained by the differentiation of equations (11) and (12a)–(12b).
Comparative Effects of Structural Parameters on Equilibrium Values of Capital Intensity (k*) and the Per Capita Growth Rate (g* — n)
Comparative Effects of Structural Parameters on Equilibrium Values of Capital Intensity (k*) and the Per Capita Growth Rate (g* — n)
| An increase in | Endogenous growth |
Solow-Swan | ||
|---|---|---|---|---|
| k* | g* – n | k* | g* – n | |
| Saving rate (s) | + | + | + | 0 |
| Ratio of foreign trade to GDP (χ) | + | + | na | na |
| Growth in real spending on education and health (ξ) | – | + | na | na |
| Growth in real spending on social security (ω) | – | + | na | na |
| Growth in real spending on operations and maintenance (μ) | + | + | na | na |
| Ratio of fiscal deficit to GDP (θ) | – | – | na | na |
| Population growth (n) | – | – | – | 0 |
| Exogenous technical change (λ) | – | + | – | + |
Comparative Effects of Structural Parameters on Equilibrium Values of Capital Intensity (k*) and the Per Capita Growth Rate (g* — n)
| An increase in | Endogenous growth |
Solow-Swan | ||
|---|---|---|---|---|
| k* | g* – n | k* | g* – n | |
| Saving rate (s) | + | + | + | 0 |
| Ratio of foreign trade to GDP (χ) | + | + | na | na |
| Growth in real spending on education and health (ξ) | – | + | na | na |
| Growth in real spending on social security (ω) | – | + | na | na |
| Growth in real spending on operations and maintenance (μ) | + | + | na | na |
| Ratio of fiscal deficit to GDP (θ) | – | – | na | na |
| Population growth (n) | – | – | – | 0 |
| Exogenous technical change (λ) | – | + | – | + |
Effects of a Higher Saving Rate10
The effects of an increase in the saving rate s on the transitional and equilibrium growth rates of output can be analyzed using Figure 2, in which the initial equilibrium positions in the endogenous growth and Solow-Swan models are indicated by points A and B, respectively. An increase in the saving rate shifts the warranted-rate curve to K’ in either model. The new equilibrium positions are indicated by points D in the endogenous growth model and C in the Solow-Swan model. In both models, the capital-labor ratio rises, although the new ratio remains lower in the endogenous growth model than in the Solow-Swan model, owing to positive learning-by-doing. However, the new equilibrium growth rate increases in the endogenous growth model but remains unchanged in the other. The discussion below traces the adjustment dynamics to the new growth equilibrium in the two models in the wake of an increase in the saving rate.
Effects of an Increase in the Saving Rate
Citation: IMF Staff Papers 1994, 001; 10.5089/9781451957037.024.A001
During the transition between equilibrium points B and C, the rate of output growth in the Solow-Swan model is momentarily higher—by EB—than the natural rate
In the endogenous growth model, an increase in the saving rate shifts the equilibrium from A to D. At the starting position A, capital would grow faster than labor (by FA), and the capital-labor ratio would rise (from
Effects of Openness, Human Development Spending, and Technical Change
The effects of these factors can be analyzed with the help of Figure 3. Since many of these parameters are absent from the Solow-Swan model,13 the illustrations refer only to the endogenous growth model. Changes in the ratio of foreign trade (sum of exports and imports) to GDP and growth in real outlays on education, health, social security, housing, and recreation are reflected in changes in the learning coefficient a, while changes in the exogenous rate of technical change X enter the natural rate schedule directly.
Effects of Increased Openness and Expenditures on Human Development in the Endogenous Growth Model
Citation: IMF Staff Papers 1994, 001; 10.5089/9781451957037.024.A001
An increase in any of these parameters shifts the capital accumulation schedule to the lower left (upper panel) and the natural rate schedule to the upper left (lower panel). With reference to Figure 3, the adjustment dynamics are the following. After the parametric increase, the rate of change in k is negative at the old equilibrium value
Effects of Fiscal Deficits and Population Growth
Finally, Figure 4 illustrates the effects of increases in the ratio of the fiscal deficit to GDP and in the rate of population growth on equilibrium capital intensity and on the growth rate of per capita output in the endogenous growth model. An increase in population growth or in the rate of government dissaving14 (by lowering the saving rate) shifts the capital accumulation schedule to the lower left in both panels. At the old equilibrium capital intensity, the rate of change in k turns negative (upper panel), implying that the warranted rate falls short of the natural rate (lower panel). As k falls, income per unit of capital increases, raising saving and investment, and hence the warranted rate. At the same time, the natural rate decreases, because a lower k induces a lower rate of learning. This process continues until the economy settles at a new equilibrium position characterized by a convergence of the warranted and natural rates, a lower level of capital intensity, and a slower growth rate of per capita income.
Effects of Increases in Ratio of Fiscal Deficits to GDP and in Population Growth
Citation: IMF Staff Papers 1994, 001; 10.5089/9781451957037.024.A001
II. Optimal Long-Run Growth
In the long run, output per unit of effective labor is y* = f(k*). If y* is considered a measure of the standard of living, and since f’(k*) > 0, it is possible to raise living standards by increasing k*. This can be done by adjusting the saving rate—for example, by lowering the ratio of the fiscal deficit to GDP. If consumption per unit of effective labor (or any monotonically increasing function of it) is taken as a measure of the social welfare of the society, the saving rate that maximizes social welfare by maximizing long-run consumption can be determined. Phelps (1966) refers to this path as the “Golden Rule of Accumulation.”
Consumption C per unit of effective labor is c = C/N = Y/N – S/N, where S is saving. Y/N is f(k) and S = I = dk/dt + δ(μ)K, where I is investment. Thus,
On the balanced growth path,
where α(.) = α(X,ξ, w,.) Thus,
Maximizing c* with respect to s,
Since ∂k*/∂s > 0, the Golden Rule condition is
where g*(k*) = α(.)k* + λ + n is the equilibrium growth rate of output. The second-order condition for a maximum is satisfied, since
Equation (15) says that for social welfare to be maximized the saving rate should be raised to a point where the net rate of return to capital (which is equal to capital’s marginal product less depreciation) equals the long-run growth rate of output plus the product of the learning coefficient and the equilibrium capital intensity. The second term is nothing more than the endogenous component of labor-augmenting technical change— the component of (dT/dt)/T induced by any learning that occurs at a higher level of capital intensity, which, in turn, is caused by a higher saving rate. If there is no learning (α = 0), equation (15) reduces to f’(k*) –δ = λ + n, which is the familiar Golden Rule result from standard neoclassical growth theory. It is evident that the optimal net rate of return to capital should be higher than λ + n when α > 0—when there is learning-by-doing—because of two factors. First, when the saving rate s is raised, the equilibrium growth g* will be higher than λ + n, by the amount α(.)∂k*/∂s. Second, capital should be compensated for the effect on equilibrium output growth through the induced learning term α(.)k*.
An alternative interpretation of the above Golden Rule can be given. A standard neoclassical result is that the optimal saving rate s should be set equal to the income share of capital π. With endogenous learning-by-doing, the optimal saving rate should be set at a fraction of π, the fraction being equal to (g* + δ)/[g* + δ + α(.)k*].15 Here, g* + δ + α(.)k* = f’(k*)—given by equation (15)—is the (gross) social marginal product of capital, inclusive of the positive externalities arising from the learning associated with capital accumulation in the endogenous growth model. Equivalently put, income going to capital as a share of total output should be a multiple of the amount saved and invested in order to compensate capital for the additional output generated by endogenous growth and induced learning. A value of π equal to s, implicit in the standard model, would undercompensate capital and thus be suboptimal from a societal point of view.
III. Speed of Adjustment Toward Equilibrium
The equilibrium results derived in the preceding section would not be relevant to the real world if the time period for the model to reach its equilibrium were unduly long. There are three approaches to the analysis of adjustment dynamics in the speed-of-approach literature:
—analytical approach, which garners less explicit results but does not resort to a full-scale numerical simulation;
—simulation, such as the work of Sato (1963), which uses a specific functional form for the production function and representative values of the structural parameters and which calculates adjustment paths from hypothetical disequilibria to obtain estimates of the time (in years) needed to reach equilibrium; and
—empirical approach, which examines whether the model’s equilibrium predictions accord with observed growth patterns of real economies over reasonably long periods.
Analytical Approach
The negative slope of the (dk/dt)/ke curve (see Figure 1) at the equilibrium capital intensity
The key feature of the endogenous growth model that distinguishes it from the Solow-Swan model is the assumed presence of learning-by-doing, represented by a positive learning coefficient α. In the absence of learning (α = 0), equation (17) reduces to the Solow-Swan expression. It is obvious from equation (17) that with α > 0 the slope of the (dk/dt)/ke curve is steeper than the slope of the (dk/dt)/ks curve (when α = 0). Thus, the endogenous growth model takes relatively less time to reach equilibrium. Moreover, it can be shown that enhanced learning—represented by an increase in α—would further reduce the adjustment time, provided that the elasticity of substitution is not less than one, such as when the production function is CES. This can be seen by differentiating equation (17) with respect to α, which yields
which is positive if the production function is CES (in which case,
Estimated Period of Adjustment in Years as yt Approaches a Limit of y∞, Using Various αa
Analysis uses a = 0.4, δ = 0.04, λ = 0.005, n = 0.025, and s = 0.2.
When α = 0, k* = 5.75 and y∞ = 0.03. When α = 0.01, k* = 3.00 and yoc = 0.06. When α = 0.02, k* = 2.40 and y∞ = 0.078.
Estimated Period of Adjustment in Years as yt Approaches a Limit of y∞, Using Various αa
| Adjustment ratio (Pt) |
Adjustment from above (y0 – y∞ > 0) |
Adjustment from below (y0 – y∞ <0) |
|||
|---|---|---|---|---|---|
| α = 0b | y0 = 0.045 | y0 = 0.035 | y0 = 0.015 | y0 = 0.025 | |
| 0.25 | 33.8 | 9.3 | 3.9 | 5.4 | |
| 0.50 | 55.7 | 21.1 | 10.1 | 13.6 | |
| 0.75 | 80.3 | 39.6 | 22.6 | 28.5 | |
| 0.90 | 105.9 | 62.6 | 41.7 | 49.3 | |
| α = 0.01b | y0 = 0.08 | y0 = 0.07 | y0 = 0.01 | y0 = 0.05 | |
| 0.25 | 6.4 | 4.3 | 1.5 | 2.6 | |
| 0.50 | 13.5 | 9.8 | 3.9 | 6.5 | |
| 0.75 | 23.3 | 18.6 | 9.1 | 13.6 | |
| 0.90 | 34.7 | 29.4 | 17.5 | 23.5 | |
| α = 0.02b | |||||
| 0.25 | 3.8 | 3.0 | 1.3 | 2.1 | |
| 0.50 | 8.4 | 6.9 | 3.3 | 5.1 | |
| 0.75 | 15.3 | 13.2 | 7.6 | 10.5 | |
| 0.90 | 23.6 | 21.2 | 14.2 | 18.0 | |
Analysis uses a = 0.4, δ = 0.04, λ = 0.005, n = 0.025, and s = 0.2.
When α = 0, k* = 5.75 and y∞ = 0.03. When α = 0.01, k* = 3.00 and yoc = 0.06. When α = 0.02, k* = 2.40 and y∞ = 0.078.
Estimated Period of Adjustment in Years as yt Approaches a Limit of y∞, Using Various αa
| Adjustment ratio (Pt) |
Adjustment from above (y0 – y∞ > 0) |
Adjustment from below (y0 – y∞ <0) |
|||
|---|---|---|---|---|---|
| α = 0b | y0 = 0.045 | y0 = 0.035 | y0 = 0.015 | y0 = 0.025 | |
| 0.25 | 33.8 | 9.3 | 3.9 | 5.4 | |
| 0.50 | 55.7 | 21.1 | 10.1 | 13.6 | |
| 0.75 | 80.3 | 39.6 | 22.6 | 28.5 | |
| 0.90 | 105.9 | 62.6 | 41.7 | 49.3 | |
| α = 0.01b | y0 = 0.08 | y0 = 0.07 | y0 = 0.01 | y0 = 0.05 | |
| 0.25 | 6.4 | 4.3 | 1.5 | 2.6 | |
| 0.50 | 13.5 | 9.8 | 3.9 | 6.5 | |
| 0.75 | 23.3 | 18.6 | 9.1 | 13.6 | |
| 0.90 | 34.7 | 29.4 | 17.5 | 23.5 | |
| α = 0.02b | |||||
| 0.25 | 3.8 | 3.0 | 1.3 | 2.1 | |
| 0.50 | 8.4 | 6.9 | 3.3 | 5.1 | |
| 0.75 | 15.3 | 13.2 | 7.6 | 10.5 | |
| 0.90 | 23.6 | 21.2 | 14.2 | 18.0 | |
Analysis uses a = 0.4, δ = 0.04, λ = 0.005, n = 0.025, and s = 0.2.
When α = 0, k* = 5.75 and y∞ = 0.03. When α = 0.01, k* = 3.00 and yoc = 0.06. When α = 0.02, k* = 2.40 and y∞ = 0.078.
The above results can be given an intuitive interpretation. It has been shown that the equilibrium growth rate of output is [(dN/dt)/N]* = [(dK/dt)/K]*. Both the natural and warranted rates adjust endogenously to changes in capital intensity. With the brunt of adjustment toward equilibrium being shared by changes in the natural rate, the time needed to reach equilibrium is much less in the endogenous growth model. In sharp contrast, the time required to reach equilibrium is much longer in the Solow-Swan model because the adjustment burden is borne entirely by changes in the warranted rate.
Simulation
The reduced model, equation (9), is
Assuming a Cobb-Douglas form for f(k) = ka, where 0 < a < 1 is the exponent of the capital stock (in this particular case, also equal to capital’s share in income π, which is constant and independent of k), the reduced model becomes
The solution to this differential equation is complicated because it is a nonlinear function. However, a linear approximation is possible in the neighborhood of the steady-state constant value k*:16
since g(k*) = 0. Or,
where A = ask*a–1 – 2αk* – (n + λ + δ) < 0.17
Equation (19) is of a “variables separable” form, which can be separated as
Integrating both sides,
where C is a constant of integration.18 Substituting equation (21) into equation (19)—
Now, from equation (10), the growth rate of output is given by
where
Substituting equations (22) and (24) into equation (23)—
Setting yt = y0 and t= 0 in equation (25),
which can be solved for the constant C,
Substituting equation (27) into equation (25),
Next, define the adjustment ratio pt as
Substituting equation (28) into equation (29) solves for the time t (in years) required to get a fraction pt of the way from y0 to y∞, from which Table 2 is computed:
where In is the natural logarithm operator.
Table 2 reveals that the adjustment times in an endogenous growth model are generally a quarter to a third of those in an exogenous growth model, depending on the value of the learning coefficient α;.19 For example, whereas an exogenous growth model (α = 0) takes from 42 to 106 years to approach equilibrium growth, an endogenous growth model (α > 0) takes anywhere from 14 to 35 years to achieve 90 percent adjustment to the steady-state growth path, depending on the learning coefficient α. (Table 2 alternately uses values of 0, 0.01, and 0.02 for α.)
Table 2 also illustrates the effects of an increase in the learning coefficient from 0.01 to 0.02: the equilibrium capital intensity falls from 3.0 to 2.4 and equilibrium growth rises from 6.0 to 7.8 percent annually; moreover, adjustment times are reduced by 30–50 percent.20
Empirical Approach
The model’s predictions about the per capita output growth and capital stock, which are summarized in Table 1, are reproduced below, with the directional impact given by the sign above each argument inside the two functions:
Equations (31) and (32) are nonlinear functions, in general. Without the fiscal deficit variable θ, a linear approximation to these two equations can produce coefficient estimates of arbitrary magnitude and significance. For example, suppose that growth rates initially rise and then fall as government expenditure continuously grows, with the attendant heavy financing burdens reflected in rising values of θ. In this case, positive coefficients on government expenditure will be obtained for linear regressions using data with low θ, negative coefficients will be obtained for those that rely on high θ, and coefficients biased toward zero will be obtained for linear regressions using both low and high θ. The endogenous growth model developed earlier and the linear regression results reported below thus include θ, the ratio of government deficits to GDP.
No data for k* exist in developing countries, so that equation (32) cannot be estimated. However, since there are data on g* – n, equation (31) can be tested. In general, the average per capita growth rate, g* – n, is inversely related to the starting value of per capita real income, y0—the familiar convergence property of neoclassical growth models (including the present one).21 Thus, for empirical testing, the following linear specification can be considered:
Of the nine explanatory variables in equation (33), data on only the last two are unavailable. Recall that μ is the real growth of expenditures on the operation and maintenance of capital assets, while λ is the exogenous rate of labor-augmenting technological progress. The parameter λ can be interpreted as capturing all the unobserved country-specific factors that raise labor productivity—cultural, social, ethnic, political, and religious. Regional dummy variables are included below to reflect such factors. The unobserved series μ is assumed to enter the error term in a well-behaved manner. For present purposes, the following multiple regression can be estimated:
The endogenous growth model’s equilibrium predictions (where the learning coefficient α>0) are that a1, a2, a3, a4 > 0 and that a5, a6, a7 < 0. The Solow-Swan model (where α = 0) predicts that a1 = a2 = a3 = a5 = a6 = 0 and a7 < 0. The data set consists of annual averages of observations for the period 1975–86 for 36 developing countries from five regions (see the Appendix for details).
The regression results are reported below, where the insignificant coefficients on the regional dummies are suppressed (t-values are in parentheses):
R2 = 0.7952; standard error of estimate (s.e.e.) = 0.0144.
An R2 of almost 0.8 is relatively high for a cross-country regression,22 and all the regression coefficients have the expected signs. The coefficients on the saving rate, the ratio of foreign trade to GDP, the ratio of fiscal deficits to GDP, and the initial level of per capita income are statistically significant at the 5 percent level or better. The coefficients on the growth of real expenditures on education and health and on the rate of population growth are statistically significant at the 10 percent level or better. The coefficient on the growth of real expenditures on social security, housing, and recreation is marginally significant.
Since θ (government dissaving) is a part of total s, further discussion of the coefficients on s and 8 would be useful. The endogenous growth model divides the total long-run impact of changes in s on g* – n into two components: (i) the element arising from changes in the private saving rate induced by changes in its determinants other than changes in θ; and (ii) the composite factor stemming from changes in s directly as a result of changes in θ and indirectly via induced changes in the private saving rate. Component (i) is measured by the coefficient on s, while component (ii) is captured by the coefficient on θ. Since the estimates of these two coefficients are nearly identical (with opposite signs), the results suggest a symmetric response of g* – n, though in opposite direction, either to a change in the private saving rate or to a change in the rate of government dissaving.
The empirical results clearly show that the following factors promote per capita economic growth: steady increases in the saving-investment rate, the ratio of foreign trade to GDP, and the growth of real expenditures on education and health. On the other hand, rapid population growth and a high ratio of fiscal deficit to GDP are followed by slow average growth of per capita output. There is also empirical support for the convergence property of the endogenous growth and Solow-Swan models—the significant negative relationship between the initial level of per capita real income and subsequent average growth.
IV. Conclusions
This paper has presented a simple neoclassical growth model with endogenous technical change and contrasted its equilibrium properties with those of the more standard growth model. It has found that, contrary to the predictions of the Solow-Swan model, the equilibrium growth rate of per capita output is influenced in a systematic way by changes in the private rates of saving, depreciation, and population growth, as well as by changes in public policies regarding trade liberalization, fiscal deficits, spending on human resource development, and net investment.
In the absence of learning-by-doing, the model’s optimal net rate of return to capital is equal to the sum of the population growth n and the exogenous rate of labor-augmenting technical change λ; alternatively, the optimal saving rate should be set equal to the share of capital in aggregate output—the familiar Golden Rule theorems from standard optimal growth theory. With learning-by-doing, these standard Golden Rule results are revised: the optimal net rate of return to capital is higher than n + λ, or the optimal saving rate should be set at only a fraction of capital’s income share because of endogenous growth and the induced learning that is associated with increases in the capital stock.
The analytic and simulation results appear to favor the endogenous growth over the Solow-Swan model. Simulations show that the speed of adjustment toward equilibrium is substantially faster in a model of endogenous growth. Moreover, an increase in learning-by-doing further shortens the adjustment period. The empirical results also validate the endogenous growth model, particularly those relating to the positive growth effects of public policies aimed at promoting greater openness of the trading system, high saving rates, and rapid growth in expenditures on human development, and those relating to the negative growth effects of rapid population growth and high ratios of fiscal deficits to GDP. Finally, the convergence property of the endogenous growth model has been confirmed (as has the convergence of the Solow-Swan model). However, the result on the saving rate-growth relationship is tenuous, in view of the short time interval (12 years) of the sample. Since the realized growth dynamics in the Solow-Swan model over this relatively short period would also show a positive relationship, the empirical results would hardly invalidate the Solow-Swan approach, pending additional research. Efforts are currently under way to use the very long time series (1950–85) from Summers and Heston (1988) in testing the equilibrium relationships among the growth rates of per capita real income, saving rates, population growth rates, and the growth and size of government. The 36 years spanned in these data would meet the adjustment time estimates of 14 to 35 years for equilibrium growth to be reached (but not the estimates of 42 to 106 years in a model without endogenous learning).
The policy implications are straightforward. Public policies that raise the capital-labor ratio magnify the effects on the growth rate of per capita income, owing to induced learning-by-doing associated with a rising capital stock. Policies that enhance the learning process also accelerate the speed of adjustment toward the balanced growth path. Examples of such policies include measures to raise saving and investment, to permit the steady expansion of the tradable sector, and to accelerate the growth of real expenditures on education and health. On the other hand, there are clear limits to the size of government in relation to GDP, because of the increasingly heavy costs of burgeoning deficits.
APPENDIX Data Used in the Study
The data, except for foreign trade flows, are drawn from Orsmond (1990), which are based on the IMF’s Government Financial Statistics and International Financial Statistics. Foreign trade flows are taken from the World Economic Outlook data base. The sample consists of observations averaged over the period 1975–86 for 36 developing countries.
| PYG | = | Real per capita GDP growth rate (annual average); |
| KY | = | Gross investment divided by nominal GDP (annual average); |
| XC | = | Change in ratio of the sum of nominal exports and imports to nominal GDP between 1975 and 1986; |
| EG | = | Growth rate of government expenditures on education and health (annual average), deflated by GDP deflator for budget year; |
| SG | = | Growth rate of government expenditures on social security, housing, and recreation (annual average), deflated by GDP deflator for budget year; |
| DY | Nominal fiscal deficits divided by nominal GDP (annual average); | |
| PG | = | Population growth rate (annual average); |
| GDP15 | = | Per capita income level in 1975 U.S. dollars; |
| DUM(i) | = | Dummy variable that assumes the value of 1 for region i and the value of 0 for other regions; i = Africa, Asia, Middle East, Western Hemisphere. |
| PYG | = | Real per capita GDP growth rate (annual average); |
| KY | = | Gross investment divided by nominal GDP (annual average); |
| XC | = | Change in ratio of the sum of nominal exports and imports to nominal GDP between 1975 and 1986; |
| EG | = | Growth rate of government expenditures on education and health (annual average), deflated by GDP deflator for budget year; |
| SG | = | Growth rate of government expenditures on social security, housing, and recreation (annual average), deflated by GDP deflator for budget year; |
| DY | Nominal fiscal deficits divided by nominal GDP (annual average); | |
| PG | = | Population growth rate (annual average); |
| GDP15 | = | Per capita income level in 1975 U.S. dollars; |
| DUM(i) | = | Dummy variable that assumes the value of 1 for region i and the value of 0 for other regions; i = Africa, Asia, Middle East, Western Hemisphere. |
The countries in the sample are
Botswana
Burkina Faso
Cameroon
Chile
Costa Rica
Dominican Republic
Egypt
El Salvador
Ethiopia
Fiji
Guatemala
Indonesia
Islamic Republic of Iran
Kenya
Korea
Liberia
Mauritius
Mexico
Morocco
Myanmar
Nepal
Pakistan
Panama
Paraguay
Singapore
Sri Lanka
Tanzania
Thailand
Togo
Tunisia
Turkey
Uruguay
Venezuela
Yemen Arab Republic
Zambia
Zimbabwe
| Country | PYG | KY | XC | EG | SG | DY | PG | GDP15 |
|---|---|---|---|---|---|---|---|---|
| Botswana | 7.4 | 29.8 | 3.5 | 17.5 | 38.7 | –4.2 | 4.7 | 350.0 |
| Korea | 7.1 | 28.8 | 10.6 | 10.2 | 12.9 | 1.7 | 1.4 | 580.0 |
| Singapore | 5.7 | 40.3 | 35.6 | 12.8 | 12.8 | –1.7 | 1.2 | 2,540.0 |
| Yemen Arab Republic | 3.7 | 27.9 | –9.4 | 32.1 | 2.0 | 10.8 | 2.8 | 140.0 |
| Pakistan | 3.4 | 17.0 | 1.1 | 10.1 | 32.2 | 7.5 | 3.1 | 140.0 |
| Cameroon | 3.1 | 22.4 | –7.6 | 0.0 | 13.3 | 0.5 | 3.0 | 310.0 |
| Mali | 3.9 | 24.1 | 11.1 | 9.0 | 6.1 | 4.1 | 2.1 | 360.0 |
| Indonesia | 4.0 | 24.4 | –4.0 | 9.3 | 2.0 | 2.2 | 2.0 | 210.0 |
| Paraguay | 2.6 | 23.7 | 11.6 | 1.2 | 9.2 | 0.3 | 3.2 | 550.0 |
| Myanmar | 3.2 | 15.9 | –0.3 | 7.0 | 10.0 | –0.1 | 2.4 | 150.0 |
| Sri Lanka | 3.6 | 23.3 | 13.8 | 6.7 | 0.8 | 10.5 | 1.6 | 220.0 |
| Tunisia | 2.5 | 29.4 | 2.5 | 5.0 | 9.5 | 4.7 | 2.6 | 710.0 |
| Kenya | 0.8 | 20.9 | –15.8 | 5.8 | 5.1 | 5.7 | 4.2 | 230.0 |
| Panama | 1.9 | 23.4 | –26.7 | 4.1 | 9.1 | 7.6 | 2.6 | 1,030.0 |
| Mauritius | 3.1 | 24.4 | –3.1 | 7.7 | –0.5 | 8.5 | 1.3 | 300.0 |
| Burkina Faso | 2.7 | 22.9 | 4.6 | 6.7 | 18.7 | 0.3 | 1.6 | 100.0 |
| Egypt | 1.8 | 25.8 | 25.2 | 9.4 | 0.3 | 13.1 | 2.5 | 310.0 |
| Turkey | 2.1 | 20.2 | 15.4 | 0.7 | 2.7 | 4.3 | 2.0 | 830.0 |
| Chile | 2.2 | 14.7 | 8.0 | 3.4 | 7.3 | –0.3 | 1.7 | 860.0 |
| Morocco | 1.3 | 25.2 | –4.3 | 5.3 | 6.5 | 10.8 | 2.5 | 500.0 |
| Nepal | 0.9 | 17.7 | 10.7 | 11.3 | 14.3 | 4.6 | 2.8 | 110.0 |
| Mexico | 0.9 | 21.1 | 5.2 | 4.0 | 0.0 | 6.6 | 2.6 | 1,360.0 |
| Ethiopia | –1.2 | 10.3 | 8.0 | 6.3 | 10.8 | 6.0 | 4.6 | 90.0 |
| Dominican Republic | 0.4 | 21.2 | –8.9 | 2.0 | 1.6 | 2.1 | 2.8 | 670.0 |
| Costa Rica | 0.0 | 21.8 | –38.9 | 1.5 | 9.5 | 3.3 | 3.1 | 950.0 |
| Zimbabwe | –0.5 | 17.8 | –29.8 | 10.9 | 5.0 | 7.9 | 2.9 | 570.0 |
| Fiji | 0.2 | 20.6 | –14.3 | 4.8 | 13.0 | 3.8 | 2.0 | 1,030.0 |
| Guatemala | –0.4 | 15.3 | –16.5 | 13.9 | –1.2 | 2.7 | 2.5 | 570.0 |
| Togo | –0.9 | 29.1 | 0.4 | 5.9 | 6.6 | 11.3 | 2.9 | 260.0 |
| Tanzania | –1.8 | 18.5 | –16.0 | –5.6 | 3.8 | 8.8 | 3.6 | 160.0 |
| Venezuela | –1.4 | 26.4 | –17.4 | 2.7 | 2.6 | 0.7 | 3.1 | 2,380.0 |
| Uruguay | 0.8 | 13.3 | 5.8 | –1.8 | 3.8 | 2.6 | 0.6 | 1,370.0 |
| Islamic Republic of Iran | –2.6 | 22.7 | –58.9 | 3.5 | 10.1 | 5.8 | 3.8 | 1,449.7 |
| Zambia | –3.2 | 18.6 | 9.0 | –3.4 | –0.5 | 13.7 | 3.5 | 550.0 |
| Liberia | –3.3 | 22.4 | –29.2 | 6.1 | 13.7 | 7.8 | 3.3 | 410.0 |
| El Salvador | –2.1 | 15.9 | –21.1 | –3.4 | 0.3 | 2.8 | 1.9 | 440.0 |
| Country | PYG | KY | XC | EG | SG | DY | PG | GDP15 |
|---|---|---|---|---|---|---|---|---|
| Botswana | 7.4 | 29.8 | 3.5 | 17.5 | 38.7 | –4.2 | 4.7 | 350.0 |
| Korea | 7.1 | 28.8 | 10.6 | 10.2 | 12.9 | 1.7 | 1.4 | 580.0 |
| Singapore | 5.7 | 40.3 | 35.6 | 12.8 | 12.8 | –1.7 | 1.2 | 2,540.0 |
| Yemen Arab Republic | 3.7 | 27.9 | –9.4 | 32.1 | 2.0 | 10.8 | 2.8 | 140.0 |
| Pakistan | 3.4 | 17.0 | 1.1 | 10.1 | 32.2 | 7.5 | 3.1 | 140.0 |
| Cameroon | 3.1 | 22.4 | –7.6 | 0.0 | 13.3 | 0.5 | 3.0 | 310.0 |
| Mali | 3.9 | 24.1 | 11.1 | 9.0 | 6.1 | 4.1 | 2.1 | 360.0 |
| Indonesia | 4.0 | 24.4 | –4.0 | 9.3 | 2.0 | 2.2 | 2.0 | 210.0 |
| Paraguay | 2.6 | 23.7 | 11.6 | 1.2 | 9.2 | 0.3 | 3.2 | 550.0 |
| Myanmar | 3.2 | 15.9 | –0.3 | 7.0 | 10.0 | –0.1 | 2.4 | 150.0 |
| Sri Lanka | 3.6 | 23.3 | 13.8 | 6.7 | 0.8 | 10.5 | 1.6 | 220.0 |
| Tunisia | 2.5 | 29.4 | 2.5 | 5.0 | 9.5 | 4.7 | 2.6 | 710.0 |
| Kenya | 0.8 | 20.9 | –15.8 | 5.8 | 5.1 | 5.7 | 4.2 | 230.0 |
| Panama | 1.9 | 23.4 | –26.7 | 4.1 | 9.1 | 7.6 | 2.6 | 1,030.0 |
| Mauritius | 3.1 | 24.4 | –3.1 | 7.7 | –0.5 | 8.5 | 1.3 | 300.0 |
| Burkina Faso | 2.7 | 22.9 | 4.6 | 6.7 | 18.7 | 0.3 | 1.6 | 100.0 |
| Egypt | 1.8 | 25.8 | 25.2 | 9.4 | 0.3 | 13.1 | 2.5 | 310.0 |
| Turkey | 2.1 | 20.2 | 15.4 | 0.7 | 2.7 | 4.3 | 2.0 | 830.0 |
| Chile | 2.2 | 14.7 | 8.0 | 3.4 | 7.3 | –0.3 | 1.7 | 860.0 |
| Morocco | 1.3 | 25.2 | –4.3 | 5.3 | 6.5 | 10.8 | 2.5 | 500.0 |
| Nepal | 0.9 | 17.7 | 10.7 | 11.3 | 14.3 | 4.6 | 2.8 | 110.0 |
| Mexico | 0.9 | 21.1 | 5.2 | 4.0 | 0.0 | 6.6 | 2.6 | 1,360.0 |
| Ethiopia | –1.2 | 10.3 | 8.0 | 6.3 | 10.8 | 6.0 | 4.6 | 90.0 |
| Dominican Republic | 0.4 | 21.2 | –8.9 | 2.0 | 1.6 | 2.1 | 2.8 | 670.0 |
| Costa Rica | 0.0 | 21.8 | –38.9 | 1.5 | 9.5 | 3.3 | 3.1 | 950.0 |
| Zimbabwe | –0.5 | 17.8 | –29.8 | 10.9 | 5.0 | 7.9 | 2.9 | 570.0 |
| Fiji | 0.2 | 20.6 | –14.3 | 4.8 | 13.0 | 3.8 | 2.0 | 1,030.0 |
| Guatemala | –0.4 | 15.3 | –16.5 | 13.9 | –1.2 | 2.7 | 2.5 | 570.0 |
| Togo | –0.9 | 29.1 | 0.4 | 5.9 | 6.6 | 11.3 | 2.9 | 260.0 |
| Tanzania | –1.8 | 18.5 | –16.0 | –5.6 | 3.8 | 8.8 | 3.6 | 160.0 |
| Venezuela | –1.4 | 26.4 | –17.4 | 2.7 | 2.6 | 0.7 | 3.1 | 2,380.0 |
| Uruguay | 0.8 | 13.3 | 5.8 | –1.8 | 3.8 | 2.6 | 0.6 | 1,370.0 |
| Islamic Republic of Iran | –2.6 | 22.7 | –58.9 | 3.5 | 10.1 | 5.8 | 3.8 | 1,449.7 |
| Zambia | –3.2 | 18.6 | 9.0 | –3.4 | –0.5 | 13.7 | 3.5 | 550.0 |
| Liberia | –3.3 | 22.4 | –29.2 | 6.1 | 13.7 | 7.8 | 3.3 | 410.0 |
| El Salvador | –2.1 | 15.9 | –21.1 | –3.4 | 0.3 | 2.8 | 1.9 | 440.0 |
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Such a slow adjustment would render somewhat irrelevant the equilibrium behavior of the model because of the likelihood that the other parameters of the system would have changed in the interim.
See, among others, Romer (1986), Lucas (1988), Becker, Murphy, and Tamura (1990), Grossman and Helpman (1990), and Rivera-Batiz and Romer (1991).
Equilibrium growth in Arrow’s learning-by-doing model, although a function of the “learning coefficient/’ nevertheless remains independent of the saving rate and the depreciation rate. See footnote 5 for details.
Lim ∂F/∂K = ∞ as K→0; lim ∂F/∂K = 0 as K→∞ f(0) ≥ 0; f’(k) > 0; and f’’(k) < 0.
Arrow’s (1962) learning-by-doing model has a steady-state solution for the growth rate of output equal to (λ +n)/(1 – α), where the technical change function is (dT/dt)/T = α(dK/dt)/K + λ,0 < α < 1. Although steady-state growth is thus a multiple of X + n, growth remains independent of s and 8; esides, this model has the property that ∂(g* – n)/∂n = α/(1 – α) > 0. That is, an increase in population growth raises equilibrium rate of per capita growth. This proposition is rejected by the empirical finding reported later that an increase in the rate of population growth depresses the average growth rate of per capita output during 1975–86 in a sample of 36 developing countries from five regions.
See the discussion on the production linkage summarized in Khan and Villanueva (1991). Edwards (1992) and Knight, Loayza, and Villanueva (1993) present evidence on the relationship between trade openness and economic growth.
The transfer of efficient technologies and the availability of foreign exchange have featured prominently in recent experiences of rapid economic growth (Thirlwall (1979)).
For a degree-p homogeneous production function Y = F(K, N), π(k) = kf’(k)/βf(k). The sign of π’(k) follows the sign of ε(k) – 1, where ε(k) = f’(k)[βf(k) – kf’(k)]/k[(β – 1)f’(k)2 – βf(k)f’’(k)] is the elasticity of substitution. If Fis Cobb-Douglas, π(k) = ρ, where a is the constant exponent of K, and ε(k) = 1. If F has constant elasticity of substitution (CES), π(k) = 1/[1 + (1 – ρ)(1/ρ)k-σ] and ε(k) = 1/(1 – σ). Notice that if σ = 0, CES reduces to Cobb-Douglas. Also see footnote 8.
The opposite sequence of events is true for points to the right of
The effects of a reduction in the rate of depreciation—exogenously in the Solow-Swan model and endogenously in the endogenous growth model via a higher rate of real expenditures on operations and maintenance—are similar.
The transitional growth rate of output, (dY/dt)/Y, is equal to λ + n + π(k)
And thus it raises the equilibrium level of real income per efficient worker.
One parameter that is present is the exogenous rate of technical change λ, whose effects on capital intensity and per capita growth are similar in the two models.
As noted earlier, as the public sector dissaves, fewer resources will be available to accumulate capital. Moreover, the ensuing large government borrowings from financial markets tend to raise interest rates or lower available credit, adversely affecting private capital accumulation.
Equations (11) and (15) and the definition π = k*f’(k*)/f(k*) are used to derive this result. When a = 0, the proportionality factor assumes a value of unity, and the standard neoclassical result holds. In terms of the parametric values assumed in the simulations reported in Table 2 below, when the learning coefficient a is greater than zero, the optimal saving rate should be set at about three-quarters of the assumed income share of capital π, or at 0.3 when π = 0.4. The simulations also show that the higher is the learning coefficient, the lower is the optimal saving rate as a proportion of capital’s income share. According to the standard model, the optimal saving rate should always be set equal to π, which is at 0.4 in the numerical examples. The higher saving rate implied by the standard model owes to its neglect of endogenous growth and positive externalities through the learning-by-doing associated with saving and capital accumulation. By contrast, in the endogenous growth model the economy benefits from such endogenous growth and positive externalities, so that a smaller saving-investment rate is all that is required (relative to the rate required by the standard model).
The constant k* is the unique root of expression (18) equated to zero: sk*a – αk*2 – (n + λ + δ)k* = 0. Given s = 0.2, a = 0.4, α = 0.01, n= 0.025, λ = 0.005, and δ = 0.04, k* assumes the value of 3.00, and the balanced growth path is equal to an annual rate of 0.06. If α = 0, as in the Solow-Swan model, and if the other parameters are unchanged, k* solves to a higher level at 5.75 and balanced growth to a lower rate of 0.03 per annum.
In a follow-up on the preceding footnote, a particular value for A equal to –0.0886 is obtained for α = 0.01.
Note that as t goes to infinity, the second term on the right-hand side of equation (21) goes to zero (since A < 0), and thus k approaches k*.
Except for the learning coefficient a, the parameter values for the saving and population growth rates used in the simulation represent historical averages of the data in the sample of countries listed in the Appendix. The value of the income share of capital is within the range of available empirical estimates. The value of the rate of depreciation is a standard approximation used in the literature.
These simulation results are confirmed by the earlier qualitative analysis of the endogenous growth model, summarized in Table 1.
See footnote 11.
Ramanathan (1982) notes that typical values of R2 for equations estimating the growth performance of developing countries, using cross-country data, fall in the 0.3–0.4 range.
