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Relative Income and Investment Comparisons among OECD Nations
Relative Income and Investment Comparisons among OECD Nations GERALD W. SCULLY and FRANK M. BASS
ABSTRACT Cross-section and time series studies of convergence generally have led to opposite conclusions about the convergence hypothesis. The methodologies employed in these studies suffer certain conceptual or statistical weaknesses. In this study, an entirely different approach is taken. The analytics of convergence is modeled using differential equations and the necessary and sufficient condition for absolute (steady state) convergence is derived. While our results reject absolute convergence for the OECD nations in a differential equation-time only model and in a differential equation model with capital accumulation and time as arguments, we find evidence of relative convergence with a weaker differential equation model. 1998 Elsevier Science Inc.
Introduction For roughly half of the post-war period, many Organization for Economic Cooperation and Development (OECD) nations, particularly Germany and Japan, grew rapidly, had low unemployment, and had a modicum of social tranquility. Many of the European Union nations had growth rates that were twice that of the United States, and Japan’s growth was extraordinary. This growth differential was due to a much higher rate of investment and to a rapid closing of the technology gap. Unemployment rates in many of the European nations were a third of that in the United States. German unemployment was less than 1% in the 1960s. French unemployment was around 2%. One spoke of the German and the Japanese miracles. But, other OECD countries did well in this period, also. There was every prospect that many of them would eventually achieve a living standard similar to that in the United States. The second half of the period witnessed a sharp reversal of fortune for a number of OECD nations. Growth rates in Europe began to lag in the 1970s and 1980s. In recent years (1992–1994), European growth rates have been one-third of that of the United States, and economic growth virtually stopped in Japan. This decline in growth has in part been due to a relative decline in investment and saving. Many of the OECD nations have unemployment rates that are twice or more of that in the United States. GERALD W. SCULLY is Professor of Economics at the School of Management, University of Texas at Dallas. FRANK M. BASS is the Eugene C. McDermott University of Texas System Professor of Management at the School of Management, University of Texas at Dallas. Address reprint requests to: G. W. Scully, School of Management, University of Texas at Dallas, Box 830688, Richardson, TX 75083.
Technological Forecasting and Social Change 59, 167–182 (1998) 1998 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010
0040-1625/98/$–see front matter PII S0040-1625(97)00162-5
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The prospect of social discord looms in a number of these countries. Social peace has been bought by huge income transfers, but this deficit-financed fiscal policy is not sustainable in the long run. Now, one talks in terms of sclerosis in the economies of many OECD countries. Clearly, an implication of this reversal is that the prospects for convergence of living standards has been mitigated. What is the evidence on convergence among OECD nations? There have been a number of studies that have investigated convergence in different contexts. In crosssection studies, the standard technique for measuring convergence is to regress the interperiod change in per capita income against the initially observed level. An inverse relationship is taken to mean that nations with low initial per capita incomes are growing faster than nations with high initial levels, and, hence, are catching up. Generally, crosssection empirical studies reject the no convergence null hypothesis (for a representative sample, see [1–7]). However, cross-section models have been extensively criticized. They are not able to distinguish between nations which are converging or not or between absolute and relative convergence [8]. Furthermore, because of the specification and the period of observation, cross-section models cannot establish an upper bound on per capita income (i.e., the steady state) and test whether nations will converge to that at steady state at some future time T.1 Moreover, the cross-section approach has been criticized as suffering from the fallacy of regression toward the mean [9, 10]. We discuss this issue later in the article. In time series analyses, convergence is interpreted to mean that differences in per capita income are not sustainable in the long run. Per capita income differences between nations cannot contain unit roots or time trends, and per capita income levels may be cointegrated. Generally, time series tests have accepted the no convergence null [11–13]. (For contrasting results on convergence and the speed of convergence with respect to exchange rates, prices, and interest rates in the case of the UK economy, see Johansen and Juselius [14] and Pesaran and Shin [15].) Criticisms of the time series methodology are that the power of unit root tests are known to be very low against the alternative of trend stationarity and that tests for cointegranting ranks are very sensitive to the nuisance parameter in finite samples [16, 17]. Hence, a certain caution needs to be exercised about conclusions on convergence based on tests the powers of which are known to be weak. Leaving aside these methodological issues, Bernard and Durlauf [8] point out that cross-section and time series tests applied to the same data necessarily may yield opposite conclusions, since they require different implications (i.e., first difference of crossnational per capita income differences have a non-zero mean, while a time series of the differences have a zero mean). Having different conclusions about convergence based on different testing frameworks is not satisfying. The contribution of this article to the literature on convergence is to approach the question from an entirely different time series framework, one that does not rely on the standard time series tests, and yet one that still can shed light on the question of convergence (both absolute and relative) among a cross-section of nations. We model the analytics of convergence using differential equations and derive the necessary condition for absolute (steady state) convergence. Although we reject absolute convergence, we demonstrate empirically, for the first time to our knowledge, that the time paths of 1 Time series test can determine if any pair of economies have converged to the same per capita income over some finite data length, but cannot distinguish this per capita income from the neoclassical steady state per capita income.
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per capita income do narrow over time across nations, and we provide a theoretical explanation for this observed phenomenon. The article will unfold as follows. First, the meaning of convergence within the context of the neoclassical growth model is briefly discussed. This leads us to first specify, estimate, and test for absolute convergence a simple differential equation-time only model of per capita income. Absolute convergence in this model is rejected. Next, we specify, estimate, and test a differential equation model of absolute convergence with capital accumulation and time as arguments. Absolute convergence in this model is rejected, also. Finally, we specify, estimate, and test a weaker differential equation model of converging incomes. This differential equation model works very well empirically. The coefficient of variation of the predicted per capita income from the parameter estimates of this model are found to mimic the coefficient of variation of the actual data.
Convergence in the Neoclassical Growth Model The neoclassical growth model predicts that low per capita income countries will have higher rates of economic growth than those with high per capita incomes. Hence, an important welfare implication of the model is that, under certain conditions, there will be eventual intercountry convergence of per capita incomes. The conditions under which absolute convergence of per capita income across geographical units will occur are stringent in the Solow [18] growth model. The production function (assumed well behaved, exhibiting nonincreasing returns to scale, and obeying the Inada restrictions) in intensive form is y ⫽ f(k), where y ⫽ Y/L and k ⫽ K/L. Concavity is guaranteed with f⬘⬘ (k) ⬍ 0. Solow [18] shows that k (y) reaches an upper bound at (dk/dt ⫽ dy/dt ⫽ 0) for a value of k (denoted as k⬚) that satisfies the differential equation sf(k) ⫺ nk ⫽ 0. In the equation s is a constant savings rate and n is an exogenous population (labor force) growth rate. Let there be two economies with identical preferences, production functions, and a fixed level of technology. Country i is relatively rich; nation j is relatively poor. Hence, yi ⬎ yj and ki ⬎ kj. Given the concavity of the production function, the rate of return on capital will be higher in economy j. As such, dki/dt (dyi/dt) ⬍ dkj/dt (dyj/dt). For si ⫽ sj and ni ⫽ nj, both economies will converge at some time equals T to the same absolute (steady state) per capita income. Because kj is further from k⬚ than is ki, economy j will reach the upper bound at a later date. Thus, over the time interval of t to t ⫹ n ⬍ T (assuming identical probability distributions of exogenous shocks), economy j will be observed as ‘‘catching up’’ to economy i. At time equals T, economy j will have converged to economy i. These definitions of convergence as relative and absolute are consistent with the literature [8]. Differences in national savings rates and population growth rates need not obviate absolute convergence. If si ⬎ sj, but the risk premium on capital is internationally invariant, international capital flows will make up the saving deficiency. If ni ⬍ nj, the inverse relationship between population and income growth will eventually cause convergence in population growth rates. However, if neutral technical change is introduced into the neoclassical growth model, convergence in the sense of an upper bound or steady state per capita income no longer holds. If the rate of technical change is internationally invariant, economies converge to the same per capita income level, but continue to grow at a rate equal to that of the technological shift parameter. If rates of neutral technical change differ, then economies neither reach an upper bound per capita income nor do they converge to the same per capita income.
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Thus, convergence of per capita income has several meanings in the neoclassical growth model. When technical change is ignored, convergence means a terminal absolute per capita income. Among nations, if savings and population growth rates are the same, then per capita income will converge to the same absolute value. If they differ and these differences are sustainable, then nations will converge to different absolute per capita income levels. When technical change is introduced, there is no upper bound per capita income. Naturally, if rates of technical change differ among the nations, there will be no convergence.
The Necessary Condition for Absolute Convergence and a Time Only Model of Convergence (Equations 1–9) Consider the evolution of yt. yt ⫽ ␣ ⫹ yt⫺1.
(1)
Then, yt ⫽ t[y0 ⫺ ␣/(1 ⫺ )] ⫹ ␣/(1 ⫺ ),
(1a)
where y0 is some initial value of per capita income. The increment in yt over time is: dy/dt ⫽ tln() [y0 ⫺ ␣/(1 ⫺ )],
(2)
which is positive Iff 0 ⬍  ⬍ 1 and y0 ⬍ ␣/(1 ⫺ ). The interperiod change is negatively correlated with yt⫺1: ⌬y ⫽ yt ⫺ yt⫺1 ⫽ t⫺1( ⫺ 1)[y0 ⫺ ␣/(1 ⫺ )] ⫽ ␣ ⫺ (1 ⫺ )yt⫺1.
(3)
The term ⫺(1 ⫺ )yt⫺1 is the problem of regression toward the mean that has been criticized in the literature. The implication of Galton’s fallacy is that the negative correlation between ⌬y and yt⫺1 guarantees convergence. This implication is false and also is a fallacy. While the negative correlation is a necessary condition for convergence, it is not sufficient. Observe that the ␣-term is positive in equation 3. If |j| ⬍ 1 and |i| ⬍ 1, the necessary condition for the convergence of yj to yi is: ␣j/(1 ⫺ j) ⫽ ␣i/(1 ⫺ i).
(4)
We can illustrate this condition for the two economies. Let the initial values be $2,000 in economy j and $6,000 in economy i. Also, let ␣j ⫽ $800, ␣i ⫽ $1,000, j ⫽ .96, and i ⫽ .95. Then, as is shown in the simulation in Figure 1, per capita incomes converge absolutely at time ⫽ T, where $800/(1 ⫺ .96) ⫽ $1,000/(1 ⫺ .95) ⫽ $20,000. At T, dy/dt ⫽ 0. A more flexible approach to the analytics of convergence than that of the difference equation is one using a differential equation. Definitionally, [(dy/dt)/yt⫺1] ⫽ ␣ ⫺ yt⫺1 or, [dy/yt⫺1(␣ ⫺ yt⫺1)] ⫽ dt.
(5) (5a)
Integrating both sides of the differential equation 5a yields the following. 1/␣ ln[y/(␣ ⫺ y)] ⫽ t ⫹ C, where C is a constant of integration. Then,
(6)
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Fig. 1. Convergence in difference equation model.
y/(␣ ⫺ y) ⫽ e␣(t⫹C).
(6a)
Simple algebra leads to y ⫽ ␣e␣(t⫹C)/(1 ⫹ e␣(t⫹C)).
(6b)
Multiplying numerator and denominator of the r.h.s. by e⫺␣(t⫹C) yields yt ⫽ ␣/( ⫹ ␦e⫺␣t),
(6c)
where ␦ ⫽ e⫺␣C. yt has the following limits: lim yt ⫽ ␣/( ⫹ ␦); t→0
(7)
lim yt ⫽ ␣/. t→∞
(8)
That yt has an upper bound at t ⫽ ∞ is an important property of the model. It means that for a concave function ( ⬎ 0), no matter what the observed per capita income at time t, the asymptotic (steady state) per capita income at some future time T will be ␣/. Hence, the function has a property consistent with the steady state in neoclassical growth theory. Moreover, from an empirical perspective, estimation of the parameters of the differential equation permit calculation of the steady state per capita income and comparison across economies. This is an advance in the empirical testing of the very strong predictions made by the neoclassical growth model about the time paths of economies.
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Fig. 2. Convergence in differential equation model.
The model is illustrated in Figure 2. The coefficient values are: ␣j ⫽ ␣i ⫽ .05, j ⫽ i ⫽ .25E⫺05, ␦j ⫽ .225E⫺04, and ␦i ⫽ .583E⫺05. Absolute convergence of per capita incomes occurs at time T, where ␣j/j ⫽ ␣i/i ⫽ $20,000. For empirical purposes, we chose to examine the 24 member states of the OECD, because many of these countries are homogeneous, because the data length is fairly long (1950–1990), and the data are more reliable. We use the Summers and Heston [19] RGDP (in 1985 dollars) data. Figure 3 is a plot of the coefficient of variation of real gross domestic product per capita (also, real investment per head) for the OECD nations. Clearly, per capita incomes have not converged, but there is a strong downward trend in the coefficient of variation until the late 1970s. The parameters of equation 6c are estimated by iterative nonlinear least squares with correction for autocorrelation. The parameter estimates are highly significant and the fits very good, but it is clear that absolute convergence over all countries is inconsistent with the data. A strong test of absolute convergence requires ␣i ⫽ ␣ and i ⫽ , but ␦i may vary. Using an asymptotically valid F-test based on Fisher [20], absolute convergence is rejected. A less stringent, but valid, test requires ␣i/i ⫽ ␣/. This test is implemented by regressing ␣i on i and testing for a zero intercept. The intercept is statistically significant and absolute convergence is rejected.2 2 Nevertheless, subgroups of nations have converged in a statistical sense to the same absolute per capita income. The largest group contained 11 nations. The intercept had a t-value of 1.62, prob. ⫽ .139. These nations were: New Zealand, the Netherlands, Austria, France, Italy, Germany, Ireland, Denmark, Japan, Sweden, and Belgium. Smaller groups of nations also converged to each other at different values of ␣/. For the difference equation model, the condition for convergence is ␣i/(1 ⫺ i) ⫽ ␣/(1 ⫺ ) for all i. ␣i was regressed on 1 ⫺ i and the intercept was statistically significant. Hence, absolute convergence is rejected.
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Fig. 3. Coefficients of variation for RGDP and investment.
Absolute Convergence Model with Capital Accumulation (Equations 9–10) The next step is to broaden the analysis by introducing capital accumulation into the absolute convergence model. We specify a differential equation in which there are diminishing returns of percentage income increases to income levels. [(dy/dt)/y] ⫽ (1 ⫹ ␥[(dk/dt)/k])(␣⫺y).
(9)
In this model percentage increases in income are the product of two factors. The first, the capital accumulation factor, will increase percentage in per capita income with percentage changes in the capital–labor ratio. This is a natural feature of the neoclassical growth model. The second, the productivity factor, will have a constant component and one that reflects diminishing returns to the income level. The ␣ coefficient might be interpreted as a natural productivity coefficient that reflects a nation’s stock of human capital and level of technology. The parameter ␥ is a nation’s productivity multiplier that reflects the influence of percentage changes in the capital–labor ratio. The product of the coefficients, ␣␥, may be interpreted as the efficiency of the capital–labor ratio and (1 ⫺ ) as retention efficiency. Integrating equation 9 over the interval 0 to t, the solution to the differential equation is: yt ⫽ ␣/( ⫹ ␦e⫺␣(t⫹␥ln(k(t)/k(0))).
(10)
As t → ∞, yt converges to ␣/. Thus, the time only model in equation 6⬘⬘⬘ is nested in equation 10, since equation 10 reduces to equation 6⬘⬘⬘ when ␥ ⫽ 0. Data on the capital stock for the full period are not available in the Summers and Heston [19] data. The gross investment share is provided, which when multiplied by RGDP gives real investment per capita (It). We use It as a proxy for kt.
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Nonlinear estimates of the parameters of equation 10, using ln(It/I0), were highly significant and plausible, but ␣i/i ⬆ ␣/ across countries. Moreover, when we estimated the differential equation model in equation 9 as expanded below [(dy/dt)/y] ⫽ ␣ ⫺ y ⫹ ␣␥[(dI/dt)/I] ⫺ ␥[(dI/dt)/I]y we found that ⫺␥ was not significant and/or was implausible. This leads us to reject equation 9 and the logic that a nation’s per capital income would converge in time to a steady state value. As long as there are productivity gains that are independent of capital accumulation (that is, ␣ in equation 9) real per capita income will continue to increase regardless of capital accumulation. In any case, the absolute convergence model is convincingly rejected by the estimation of the differential equation version of the model.
Model of Converging Incomes (Equations 11–14) While the evidence rejects absolute convergence of income, there is reason to believe that real per capita investment will converge to a limit. This convergence need not be to the same level in each nation, since factor price differences imply different factor mixes. There are several reasons for thinking that convergence in investment is a plausible assumption. Partly, it is a feature of the Cobb-Douglas function in which (1 ⫺ ␣) is a constant [18]. Partly, it is a feature of the concavity of the production function. As capital accumulates diminishing returns arise and the rate of return falls. Partly, it is due to falling savings rates as per capita incomes rise, and social insurance and national health insurance reduce the incentive to save. Finally, holding risk constant, as investment levels rise a country’s attractiveness relative to others with lower investment levels declines. Hence, international capital flows adjust to differential rewards. Thus, we have the following differential equation model for investment: [(dI/dt)/I] ⫽ a ⫺ bI, or [dI/I(a ⫺ bI)] ⫽ dt.
(11) (11a)
Integrating both sides and manipulating results in: It ⫽ a/(b ⫹ ce⫺at).
(12)
The limit of It as t → ∞ is a/b. As is revealed in Table 1, the interative NLLS estimates of equation 12 produced good fits and mostly statistically significant parameter estimates of the right sign. The sign exceptions are Canada and Luxembourg. Weak statistical significance applies to the results for the United States, Spain, Switzerland, United Kingdom, and New Zealand, and is mainly due to the correction for autocorrelation. Global convergence was obtained for the parameter estimates for 20 of the 24 nations (i.e., the equations were re-estimated with different starting values and the parameter estimates remained unchanged). For four of the nations (Japan, Belgium, Luxembourg, and Switzerland) global convergence could not be obtained with the correction for autocorrelation. The estimates without the correction, which are globally stable, appear in Table 1. For the per capita income model, our investigation leads to a model in which percentage increases in income increase with: 1) a productivity factor that is independent of time and investment and 2) with percentage increases in investment and that decreases with the level of investment due to ‘‘diminishing returns to the investment level.’’ The differential equation is:
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TABLE 1 NLLS Estimates of Investment Model, 1950–90 Country
a
b*105
c*105
Canada
0.018 (1.52) 0.032 (1.78) 0.134 (6.72) 0.119 (6.04) 0.111 (5.57) 0.229 (3.86) 0.084 (5.30) 0.156 (4.00) 0.114 (4.44) 0.277 (7.15) 0.101 (6.41) 0.141 (4.68) 0.105 (4.87) ⫺0.232 (3.19) 0.194 (4.00) 0.095 (3.84) 0.173 (5.10) 0.140 (1.40) 0.155 (3.12) 0.060 (2.30) 0.110 (4.24) 0.065 (1.09) 0.080 (4.22) 0.033 (0.55)
⫺0.432 (1.18) 0.436 (0.78) 2.560 (4.47) 3.411 (4.44) 3.770 (4.48) 7.230 (3.56) 1.660 (3.57) 4.181 (3.44) 3.138 (3.64) 18.874 (5.86) 2.620 (4.75) 6.472 (3.73) 3.281 (3.82) ⫺7.183 (93.34) 6.959 (3.69) 2.091 (2.89) 13.340 (4.30) 5.572 (1.08) 4.845 (2.79) 0.869 (1.36) 12.225 (3.10) 2.188 (0.69) 1.806 (3.27) 0.682 (0.30)
1.619 (3.07) 1.194 (2.41) 45.146 (2.21) 23.061 (2.44) 10.766 (2.78) 58.429 (1.12) 5.882 (3.06) 30.596 (1.29) 7.822 (2.30) 879.410 (2.15) 10.198 (3.14) 59.411 (1.54) 10.071 (2.49) 0.0002 (0.38) 28.802 (1.37) 7.010 (1.87) 178.290 (1.44) 44.990 (0.53) 13.863 (1.32) 2.416 (1.93) 78.026 (1.70) 6.233 (0.78) 3.177 (2.75) 1.241 (0.75)
United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
1
2
0.425 (2.89) 0.608 (4.11) —
— ⫺0.304 (2.08) —
.779 (4.81) —
⫺.281 (1.67) —
0.608 (5.23) 0.859 (6.37) 0.748 (6.68) 0.937 (6.28) 0.743 (6.96) —
— ⫺0.600 (3.82)
⫺0.502 (2.97) — —
0.783 (4.99) 0.465 (3.20) —
⫺0.277 (1.81)
0.875 (5.80) 0.740 (4.54) 0.995 (7.21) 1.502 (11.83) 0.988 (6.85) —
⫺0.267 (1.79) ⫺0.192 (1.14) ⫺0.481 (3.52) ⫺0.663 (4.46) ⫺0.404 (2.48) —
0.840 (5.36) 0.927 (5.90) 0.204 (1.36) 0.489 (3.66)
⫺0.281 (1.78) ⫺0.233 (1.24) —
—
—
Log L R2 193.6 0.96 193.4 0.90 167.4 0.94 207.7 0.97 182.2 0.83 187.1 0.91 186.3 0.95 205.4 0.97 202.6 0.96 214.4 0.94 173.7 0.87 207.2 0.94 207.9 0.96 161.5 0.51 201.1 0.92 186.3 0.94 227.3 0.95 222.2 0.98 197.7 0.93 154.0 0.80 250.8 0.95 212.0 0.94 184.3 0.88 177.7 0.68
DW 1.80 1.98 0.26 1.75 0.48 1.85 1.91 1.88 1.76 1.98 1.69 1.95 1.86 0.82 1.87 1.95 1.94 2.17 1.79 0.40 2.01 1.81 2.10 1.84
Note: Asymptotic t-values in parentheses. Abbreviations: Log L ⫽ Log Likelihood; R2 ⫽ between the observed and predicted values; DW ⫽ DurbinWatson statistic.
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[(dy/dt)/y] ⫽ ␣ ⫹ ␣␥[(dI/dt)/I] ⫺ I.
(13)
Using the solution to the investment differential equation in equation 12, the solution to equation 13 is: yt ⫽ ␦e(␣ ⫺ (a/b))t ⫹ (␣␥ ⫹ (/b))ln(I(t)).
(14)
The iterative NLLS estimates of the parameters of equation 14 appear in Table 2. For every country the parameters are highly significant and the fits are extremely good. Global convergence was obtained for all of the nations. The individual parameter values in the exponent of equation 14 are not identified, but a and b can be obtained from equation 12 and ␣, ␣␥, and  from the OLS estimation of the differential equation in equation 13. These parameter estimates appear in Table 3. The estimates of the coefficient of time and of ln(It) obtained in equation 14 indicate that these coefficients are positive and highly significant. Thus, even with investment going to an asymptote, per capita income will continue to rise. A particularly nice feature of the model is that diminishing returns is allowed but not required. Where  ⬎ 0 in Table 3, it is always statistically insignificant. Thus, treating statistically insignificant cases as  ⫽ 0, (a/b) and (/b) drop from the exponent in equation 14. The computed values of ␣ ⫺ (a/b) and ␣␥ ⫹ (/b) appear in Table 4 and may be compared with the NLLS parameter estimates in Table 2.
The Meaning of Convergence How well does the differential equation model given in equations 12 and 14 track the pattern of convergence? We use equation 12 to project the path of real investment per head. The projected values of investment are used in equation 14 to obtain predicted values of real income per capita. We calculate year by year the mean and standard deviation of predicted investment and income per head for the OECD nations and compute the coefficient of variation. The paths of the coefficient of variation are shown in Figure 4. Clearly, the differential equation model works well, since the coefficients of variation of the predicted values mimic those of the actual values in Figure 3. Note that equation 14 may also be written as: yt ⫽ ␦e(␣ ⫺ (a/b))tIt(␣␥ ⫹ (/b)),
(15)
and, as such, income elasticity with respect to investment is a constant at (␣␥ ⫹ (/b)). Income change with respect to investment change will depend on the country’s ‘‘investment efficiency’’ and upon decay rates,  and b. Further, note that euqation 13 may be written as: [(dy/dt)/y] ⫽ (␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[dI/dt)/I].
(16)
The proportion of a given change in income of a country, then, may be partitioned into two components: the first term, call it A, is productivity growth that reflects a nation’s stock of human capital and level of technology; the second, B, is due to percentage changes in investment. Thus, we have: A ⫽ (␣ ⫺ (a/b))/(␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I] ⫽ 1 ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I] and
(17)
B ⫽ (␣␥ ⫹ (/b))[(dI/dt)/I]/(␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I].
(18)
During the period of rapid capital accumulation, the dominant component of income
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TABLE 2 NLLS Estimates of Income Model, 1950–90 Country
␦*102
Canada
0.199 (10.51) 0.239 (14.27) 0.138 (6.30) 0.111 (10.11) 0.143 (7.89) 0.138 (18.38) 0.124 (10.39) 0.152 (9.19) 0.195 (11.17) 0.065 (7.21) 0.079 (8.56) 0.062 (6.60) 0.146 (9.13) 0.146 (10.92) 0.127 (5.37) 0.055 (8.70) 0.049 (4.82) 0.145 (6.49) 0.142 (5.06) 0.230 (12.80) 0.070 (3.70) 0.094 (7.33) 0.171 (13.34) 0.142 (5.06)
United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
␣⫺(a/b)
␣␥⫹(/b)
0.018 (9.68) 0.014 (19.82) 0.024 (10.79) 0.019 (9.33) 0.018 (6.45) 0.023 (18.86) 0.023 (13.17) 0.016 (5.80) 0.014 (5.74) 0.022 (6.05) 0.027 (12.22) 0.026 (15.49) 0.020 (5.29) 0.020 (29.75) 0.014 (2.88) 0.031 (29.54) 0.030 (9.74) 0.016 (6.08) 0.012 (3.05) 0.010 (5.62) 0.016 (8.72) 0.019 (24.41) 0.016 (10.99) 0.012 (3.04)
0.282 (13.89) 0.271 (15.50) 0.321 (8.15) 0.190 (7.55) 0.233 (8.75) 0.224 (15.44) 0.268 (12.00) 0.231 (9.92) 0.258 (9.36) 0.199 (5.78) 0.149 (6.38) 0.192 (6.80) 0.276 (8.83) 0.242 (9.62) 0.150 (4.58) 0.066 (2.39) 0.215 (5.78) 0.308 (8.73) 0.132 (4.46) 0.251 (10.70) 0.277 (6.22) 0.133 (4.49) 0.247 (12.52) 0.132 (4.48)
1 1.338 (9.02) 1.071 (7.04) 0.959 (28.36) 0.945 (32.80) 0.970 (26.48) 0.868 (8.79) 0.882 (9.10) 1.377 (9.45) 0.960 (41.16) 0.958 (27.10) 1.293 (8.72) 0.736 (5.99) 1.464 (10.45) 0.822 (5.41) 1.277 (8.22) 1.363 (9.10) 1.204 (7.86) 1.312 (8.48) 0.973 (32.01) 0.942 (25.08) 0.563 (4.03) 0.941 (6.73) 0.900 (12.01) 0.973 (32.16)
2 ⫺0.427 (2.79) ⫺0.257 (1.71) — — — — — ⫺0.409 (2.78) — — ⫺0.529 (3.28) — ⫺0.486 (3.53) ⫺0.226 (1.54) ⫺0.306 (1.98) ⫺0.702 (4.06) ⫺0.317 (2.11) ⫺0.349 (2.31) — — — ⫺0.482 (3.05) — —
Log L R2 215.6 0.99 219.0 0.99 224.2 0.99 232.0 0.99 223.9 0.99 231.3 0.99 219.3 0.99 238.3 0.99 238.8 0.99 233.0 0.99 189.1 0.99 221.5 0.99 241.8 0.99 199.9 0.99 221.9 0.99 209.5 0.99 227.4 0.99 238.3 0.99 220.7 0.99 210.5 0.99 237.8 0.99 213.8 0.99 213.8 0.99 220.7 0.99
DW 1.97 2.16 1.76 2.10 1.63 2.08 1.69 2.04 1.84 1.86 1.90 2.00 2.17 1.88 2.11 2.07 2.08 2.09 1.65 1.62 1.77 1.82 1.74 1.65
Note: Asymptotic t-values in parentheses. Abbreviations: Log L ⫽ Log Likelihood; R2 ⫽ between the observed and predicted values; DW ⫽ DurbinWatson statistic.
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TABLE 3 OLS Estimates of Percentage Change in Income, 1951–1990 Country a
Canada
United States Japan Austria Belgium Denmark Finland Francea Germany Greece Iceland Ireland Italya Luxembourg Netherlandsa Norwaya Portugal Spain Sweden Switzerland Turkeya United Kingdom Australia New Zealand
␣*102 2.517 (2.86) 1.003 (1.11) 6.124 (7.81) 4.677 (9.26) 1.551 (2.03) 1.556 (2.67) 2.726 (3.28) 3.730 (5.02) 6.389 (10.08) 3.890 (4.04) 3.131 (1.77) 1.525 (1.63) 3.410 (3.35) 1.729 (0.82) 1.550 (1.28) 3.090 (2.63) 4.076 (3.30) 3.382 (3.29) 3.387 (4.15) 1.492 (2.16) 1.250 (0.80) 0.816 (0.69) 1.282 (1.15) 3.970 (2.10)
␣␥ 0.272 (12.60) 0.277 (12.91) 0.188 (6.28) 0.185 (11.53) 0.227 (8.48) 0.221 (16.07) 0.278 (11.53) 0.212 (8.71) 0.284 (13.09) 0.173 (6.06) 0.179 (5.41) 0.162 (6.20) 0.292 (15.76) 0.282 (10.82) 0.210 (10.25) 0.069 (2.10) 0.179 (4.07) 0.349 (8.58) 0.130 (5.35) 0.247 (13.56) 0.327 (4.93) 0.151 (4.40) 0.255 (11.90) 0.215 (7.52)
⫺*105 ⫺0.286 (1.01) 0.143 (0.50) ⫺0.736 (3.17) ⫺0.897 (4.15) 0.229 (0.67) 0.099 (0.46) ⫺0.149 (0.59) ⫺0.566 (2.14) ⫺1.426 (6.59) ⫺0.721 (0.91) ⫺0.345 (0.52) 0.681 (1.09) ⫺0.358 (0.83) ⫺0.022 (0.04) 0.221 (0.43) ⫺0.057 (0.16) ⫺0.762 (0.60) ⫺0.828 (1.45) ⫺0.580 (1.93) ⫺0.782 (0.46) 0.468 (0.19) 0.562 (0.85) 0.024 (0.07) ⫺1.201 (1.63)
R2 0.800 0.811 0.645 0.822 0.643 0.875 0.791 0.747 0.891 0.533 0.455 0.483 0.878 0.765 0.766 0.196 0.320 0.700 0.508 0.838 0.353 0.310 0.798 0.639
a Serial correlation is present on the OLS equations for these countries. The regressions presented are corrected for first-order autocorrelation. The -value, Durbin-Watson statistic, and Durbin-Watson h statistic are as follows: Canada, ⫽ 0.349, DW ⫽ 2.01, and DW h ⫽ ⫺0.255; France, ⫽ 0.388, DW ⫽ 1.86, and DW h ⫽ 0.195; Italy, ⫽ 0.543, DW ⫽ 2.12, and DW h ⫽ 0.785; Netherlands, ⫽ 0.311, DW ⫽ 2.05, and DW h ⫽ ⫺0.926; Norway, ⫽ 0.277, DW ⫽ 1.93, and DW h ⫽ 0.357; and, Turkey, ⫽ ⫺0.366, DW ⫽ 1.50, and DW h ⫽ 1.10.
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TABLE 4 Computed Values of ␣ ⫺ (a/b) and ␣␥ ⫹ (/b) Country Canada United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
␣ or ␣ ⫺ (a/b)
␣␥ or ␣␥ ⫹ (/b)
0.025 0.010 0.023 0.016 0.016 0.016 0.027 0.016 0.012 0.039 0.031 0.015 0.034 0.017 0.016 0.031 0.041 0.034 0.015 0.015 0.013 0.008 0.013 0.040
0.272 0.277 0.475 0.448 0.227 0.221 0.278 0.347 0.738 0.173 0.179 0.162 0.292 0.282 0.210 0.069 0.179 0.349 0.250 0.247 0.327 0.151 0.255 0.215
change arises from the second term, B. As diminishing returns to capital set in the weight of B in income change diminishes. As investment per head reaches its steady state value (i.e., as t → ∞, [(dI/dt)/I] or B → 0), further change in per capita income is due solely to the A term, which is a constant. Thus, in this differential equation model of convergence, the growth path of nations is influenced by two factors: capital accumulation and a productivity factor that is independent of capital accumulation (human capital and technical change). Nations will reach their steady-state values of capital accumulation. As investment per head goes to its asymptotic value, no further contribution to economic growth comes from capital accumulation. But, growth continues through human capital accumulation and technical change, and these contributions need not be internationally invariant. Nations differ in a variety of ways. They have different levels of skill, rates of technical change, and efficiencies [21].3 These differences prevent convergence of per capita incomes to the same level. However, income differences will narrow as capital formation grows more rapidly in countries with lower incomes. The coefficient of variation of per capita income of the OECD nations shows a steady decline from 1950 to 3 Nations differ as well by variables wholly outside of the neoclassical growth model, such as policies, industrial structure, and so on. A recent study [21] finds that one-digit sectoral decomposition improves the understanding of the convergence phenomenon. They find evidence of convergence (measured as labor and multifactor productivity) in the non-manufacturing sectors but little evidence of convergence in manufacturing. Yet, their conclusions point to a danger of sectoral disaggregation in the analysis of convergence. In the nontradeable sectors, such as services and construction, technology tends to diffuse over time. In the tradeablegoods sectors comparative advantage implies specialization, with no a priori reason for technological diffusion over time.
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Fig. 4. Coefficients of variation for predicted RGDP and investment.
the late 1970s, but appears to have reached a lower limit. Much of this narrowing is due to rapid capital accumulation of those nations that were defeated or occupied during World War II relative to the victorious nations. The coefficient of variation of investment per head rapidly declined, until the early 1970s. It rises, thereafter, mainly due to more rapid capital accumulation of the lagging OECD nations (for example, Greece, Portugal, Spain, and Turkey). As investment levels reach an upper bound in all of these nations, income growth will be influenced by the first term in the exponent of equation 14 that reflects learning productivity and retention efficiency. To the extent that these rates differ among OECD nations one might well expect a divergence of per capita incomes.
Conclusions The large literature on convergence leaves in doubt whether the very strong predictions of the neoclassical growth model hold or do not. Generally, cross-section studies reject the no convergence null, while time series studies accept it. The different conclusions partly rest on different testing frameworks. Moreoever, neither methodology distinguishes between absolute convergence to the neoclassical steady state, fixed, upper bound per capita income and some finite (data length constrained) per capita income that will continue to grow. This article contributes to the understanding of convergence within the neoclassical growth model by specifying a differential equation, the limit of which is the neoclassical steady state per capita income. We tested two versions of the model: a time only differential equation and a differential equation with capital accumulation and time as arguments. Our empirical analysis, which is not constrained by the data length, rejects the absolute convergence null at the limit (i.e., ␣i/i ⫽ ␣/ for all i). We then specified
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a weaker model of converging per capita incomes in which percentage changes in income are due to capital accumulation, the limit of which is a steady state capital–labor ratio, which may vary internationally, and to a productivity factor that is independent of time and capital accumulation. This pair of differential equations yields plausible results in the sense that the coefficient of variation of the predicted values mimics that of the coefficient of variation of the actual data. We conclude that for the whole period examined per capita incomes across the OECD nations have been converging. But, convergence slowed and then stopped by the late 1970s. In fact, there has been a slight divergence in per capita incomes since then. The pattern in income convergence is related to the pattern in investment convergence. When per capita incomes were strongly converging, per capita investment was strongly converging. In the late 1970s per capita investment began diverging, and per capita incomes also show slight divergence. We conclude that even if all of the variation in per capita incomes due to variation in capital accumulation per head is eliminated (i.e., all nations reach their steady state capital–labor ratios), there is no guarantee that per capita incomes will further converge toward one another. This will depend crucially on the time path of the productivity factor that is independent of time and capital accumulation (e.g., technical change and human capital). A narrowing (widening) of differences in this source of per capita income growth will determine whether further convergence (divergence) occurs.
Reference 1. Mankiw, M., Romer, D., and Weil, D. N.: A Contribution to the Empirics of Economic Growth, Quarterly Journal of Economics 107(2), 407–437 (1992). 2. Barro, R. J.: Economic Growth in a Cross-section of Countries, Quarterly Journal of Economics 106(2), 407–443 (1991). 3. Barro, R. J., and Sala-i-Martin, X.: Convergence Across States and Regions, Brookings Papers on Economic Activity 1, 107–182 (1991). 4. Barro, R. J., and Sala-i-Martin, X.: Convergence, Journal of Political Economy 100(2), 223–251 (1992). 5. Baumol, W. J.: Productivity Growth, Convergence, and Welfare, American Economic Review 76(5), 1072–1085 (1986). 6. DeLong, J. B.: Productivity Growth, Convergence, and Welfare: A Comment, American Economic Review 78(5), 1138–1155 (1988). 7. Dowrick, S., and Nguyen, D.: OECD Comparative Economic Growth 1950–85: Catch-up and Convergence, American Economic Review 79(5), 1110–1130 (1989). 8. Bernard, A. B., and Durlauf, S. N.: Interpreting Tests of the Convergence Hypothesis, Journal of Econometrics 71(1–2), 161–173 (1996). 9. Friedman, M.: Do Old Fallacies Ever Die? Journal of Economic Literature 30(4), 2129–2132 (1992). 10. Quah, D.: Galton’s Fallacy and the Tests of the Convergence Hypothesis, Scandinavian Journal of Economics 95(4), 427–443 (1993). 11. Quah, D.: International Patterns of Growth: I. Persistence in Cross-country Disparities. Working Paper. London School of Economics, London, 1992. 12. Bernard, A. B.: Empirical Implications of the Convergence Hypothesis. Working Paper. MIT, Cambridge, MA, 1992. 13. Evans, P., and Karras, G.: Do Economies Converge? Evidence from a Panel of U.S. States, Review of Economics and Statistics 78, 384–388 (1996). 14. Johansen, S., and Juselius, K.: Testing Structural Hypotheses in a Multivariate Cointegration Analysis of the PPP and UIP for UK, Journal of Econometrics 53, 211–244 (1992). 15. Pesaran, M. H., and Shin, Y.: Cointegration and Speed of Convergence to Equilibrium, Journal of Econometrics 71, 117–143 (1996). 16. Reimers, H. E.: Comparisons of Tests for Multivariate Cointegration, Statistical Papers 33, 335–359 (1992). 17. Toda, H. Y.: Finite Sample Performance of Likelihood Ratio Tests for Cointegrating Ranks in Vector Autoregressions, Econometric Theory 11 (1995). 18. Solow, R. M.: A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics 70, 65–94 (1956).
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19. Summers, R., and Heston, A.: The Penn World Table (Mark 5): An Expanded Set of International Comparisons 1950–1988, Quarterly Journal of Economics 106(2), 327–368 (1991). 20. Fisher, F. M.: Tests of Equality Between Sets of Coefficients in Two Linear Regressions: An Expository Note, Econometrica 38(3), 361–366 (1970). 21. Bernard, A. B., and Jones, C. I.: Comparing Apples and Oranges: Productivity Convergence and Measurement Across Industries and Countries, American Economic Review 86(5), 1216–1238 (1996).
Received 1 April 1997; accepted 14 October 1997
ABSTRACT Cross-section and time series studies of convergence generally have led to opposite conclusions about the convergence hypothesis. The methodologies employed in these studies suffer certain conceptual or statistical weaknesses. In this study, an entirely different approach is taken. The analytics of convergence is modeled using differential equations and the necessary and sufficient condition for absolute (steady state) convergence is derived. While our results reject absolute convergence for the OECD nations in a differential equation-time only model and in a differential equation model with capital accumulation and time as arguments, we find evidence of relative convergence with a weaker differential equation model. 1998 Elsevier Science Inc.
Introduction For roughly half of the post-war period, many Organization for Economic Cooperation and Development (OECD) nations, particularly Germany and Japan, grew rapidly, had low unemployment, and had a modicum of social tranquility. Many of the European Union nations had growth rates that were twice that of the United States, and Japan’s growth was extraordinary. This growth differential was due to a much higher rate of investment and to a rapid closing of the technology gap. Unemployment rates in many of the European nations were a third of that in the United States. German unemployment was less than 1% in the 1960s. French unemployment was around 2%. One spoke of the German and the Japanese miracles. But, other OECD countries did well in this period, also. There was every prospect that many of them would eventually achieve a living standard similar to that in the United States. The second half of the period witnessed a sharp reversal of fortune for a number of OECD nations. Growth rates in Europe began to lag in the 1970s and 1980s. In recent years (1992–1994), European growth rates have been one-third of that of the United States, and economic growth virtually stopped in Japan. This decline in growth has in part been due to a relative decline in investment and saving. Many of the OECD nations have unemployment rates that are twice or more of that in the United States. GERALD W. SCULLY is Professor of Economics at the School of Management, University of Texas at Dallas. FRANK M. BASS is the Eugene C. McDermott University of Texas System Professor of Management at the School of Management, University of Texas at Dallas. Address reprint requests to: G. W. Scully, School of Management, University of Texas at Dallas, Box 830688, Richardson, TX 75083.
Technological Forecasting and Social Change 59, 167–182 (1998) 1998 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010
0040-1625/98/$–see front matter PII S0040-1625(97)00162-5
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The prospect of social discord looms in a number of these countries. Social peace has been bought by huge income transfers, but this deficit-financed fiscal policy is not sustainable in the long run. Now, one talks in terms of sclerosis in the economies of many OECD countries. Clearly, an implication of this reversal is that the prospects for convergence of living standards has been mitigated. What is the evidence on convergence among OECD nations? There have been a number of studies that have investigated convergence in different contexts. In crosssection studies, the standard technique for measuring convergence is to regress the interperiod change in per capita income against the initially observed level. An inverse relationship is taken to mean that nations with low initial per capita incomes are growing faster than nations with high initial levels, and, hence, are catching up. Generally, crosssection empirical studies reject the no convergence null hypothesis (for a representative sample, see [1–7]). However, cross-section models have been extensively criticized. They are not able to distinguish between nations which are converging or not or between absolute and relative convergence [8]. Furthermore, because of the specification and the period of observation, cross-section models cannot establish an upper bound on per capita income (i.e., the steady state) and test whether nations will converge to that at steady state at some future time T.1 Moreover, the cross-section approach has been criticized as suffering from the fallacy of regression toward the mean [9, 10]. We discuss this issue later in the article. In time series analyses, convergence is interpreted to mean that differences in per capita income are not sustainable in the long run. Per capita income differences between nations cannot contain unit roots or time trends, and per capita income levels may be cointegrated. Generally, time series tests have accepted the no convergence null [11–13]. (For contrasting results on convergence and the speed of convergence with respect to exchange rates, prices, and interest rates in the case of the UK economy, see Johansen and Juselius [14] and Pesaran and Shin [15].) Criticisms of the time series methodology are that the power of unit root tests are known to be very low against the alternative of trend stationarity and that tests for cointegranting ranks are very sensitive to the nuisance parameter in finite samples [16, 17]. Hence, a certain caution needs to be exercised about conclusions on convergence based on tests the powers of which are known to be weak. Leaving aside these methodological issues, Bernard and Durlauf [8] point out that cross-section and time series tests applied to the same data necessarily may yield opposite conclusions, since they require different implications (i.e., first difference of crossnational per capita income differences have a non-zero mean, while a time series of the differences have a zero mean). Having different conclusions about convergence based on different testing frameworks is not satisfying. The contribution of this article to the literature on convergence is to approach the question from an entirely different time series framework, one that does not rely on the standard time series tests, and yet one that still can shed light on the question of convergence (both absolute and relative) among a cross-section of nations. We model the analytics of convergence using differential equations and derive the necessary condition for absolute (steady state) convergence. Although we reject absolute convergence, we demonstrate empirically, for the first time to our knowledge, that the time paths of 1 Time series test can determine if any pair of economies have converged to the same per capita income over some finite data length, but cannot distinguish this per capita income from the neoclassical steady state per capita income.
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per capita income do narrow over time across nations, and we provide a theoretical explanation for this observed phenomenon. The article will unfold as follows. First, the meaning of convergence within the context of the neoclassical growth model is briefly discussed. This leads us to first specify, estimate, and test for absolute convergence a simple differential equation-time only model of per capita income. Absolute convergence in this model is rejected. Next, we specify, estimate, and test a differential equation model of absolute convergence with capital accumulation and time as arguments. Absolute convergence in this model is rejected, also. Finally, we specify, estimate, and test a weaker differential equation model of converging incomes. This differential equation model works very well empirically. The coefficient of variation of the predicted per capita income from the parameter estimates of this model are found to mimic the coefficient of variation of the actual data.
Convergence in the Neoclassical Growth Model The neoclassical growth model predicts that low per capita income countries will have higher rates of economic growth than those with high per capita incomes. Hence, an important welfare implication of the model is that, under certain conditions, there will be eventual intercountry convergence of per capita incomes. The conditions under which absolute convergence of per capita income across geographical units will occur are stringent in the Solow [18] growth model. The production function (assumed well behaved, exhibiting nonincreasing returns to scale, and obeying the Inada restrictions) in intensive form is y ⫽ f(k), where y ⫽ Y/L and k ⫽ K/L. Concavity is guaranteed with f⬘⬘ (k) ⬍ 0. Solow [18] shows that k (y) reaches an upper bound at (dk/dt ⫽ dy/dt ⫽ 0) for a value of k (denoted as k⬚) that satisfies the differential equation sf(k) ⫺ nk ⫽ 0. In the equation s is a constant savings rate and n is an exogenous population (labor force) growth rate. Let there be two economies with identical preferences, production functions, and a fixed level of technology. Country i is relatively rich; nation j is relatively poor. Hence, yi ⬎ yj and ki ⬎ kj. Given the concavity of the production function, the rate of return on capital will be higher in economy j. As such, dki/dt (dyi/dt) ⬍ dkj/dt (dyj/dt). For si ⫽ sj and ni ⫽ nj, both economies will converge at some time equals T to the same absolute (steady state) per capita income. Because kj is further from k⬚ than is ki, economy j will reach the upper bound at a later date. Thus, over the time interval of t to t ⫹ n ⬍ T (assuming identical probability distributions of exogenous shocks), economy j will be observed as ‘‘catching up’’ to economy i. At time equals T, economy j will have converged to economy i. These definitions of convergence as relative and absolute are consistent with the literature [8]. Differences in national savings rates and population growth rates need not obviate absolute convergence. If si ⬎ sj, but the risk premium on capital is internationally invariant, international capital flows will make up the saving deficiency. If ni ⬍ nj, the inverse relationship between population and income growth will eventually cause convergence in population growth rates. However, if neutral technical change is introduced into the neoclassical growth model, convergence in the sense of an upper bound or steady state per capita income no longer holds. If the rate of technical change is internationally invariant, economies converge to the same per capita income level, but continue to grow at a rate equal to that of the technological shift parameter. If rates of neutral technical change differ, then economies neither reach an upper bound per capita income nor do they converge to the same per capita income.
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Thus, convergence of per capita income has several meanings in the neoclassical growth model. When technical change is ignored, convergence means a terminal absolute per capita income. Among nations, if savings and population growth rates are the same, then per capita income will converge to the same absolute value. If they differ and these differences are sustainable, then nations will converge to different absolute per capita income levels. When technical change is introduced, there is no upper bound per capita income. Naturally, if rates of technical change differ among the nations, there will be no convergence.
The Necessary Condition for Absolute Convergence and a Time Only Model of Convergence (Equations 1–9) Consider the evolution of yt. yt ⫽ ␣ ⫹ yt⫺1.
(1)
Then, yt ⫽ t[y0 ⫺ ␣/(1 ⫺ )] ⫹ ␣/(1 ⫺ ),
(1a)
where y0 is some initial value of per capita income. The increment in yt over time is: dy/dt ⫽ tln() [y0 ⫺ ␣/(1 ⫺ )],
(2)
which is positive Iff 0 ⬍  ⬍ 1 and y0 ⬍ ␣/(1 ⫺ ). The interperiod change is negatively correlated with yt⫺1: ⌬y ⫽ yt ⫺ yt⫺1 ⫽ t⫺1( ⫺ 1)[y0 ⫺ ␣/(1 ⫺ )] ⫽ ␣ ⫺ (1 ⫺ )yt⫺1.
(3)
The term ⫺(1 ⫺ )yt⫺1 is the problem of regression toward the mean that has been criticized in the literature. The implication of Galton’s fallacy is that the negative correlation between ⌬y and yt⫺1 guarantees convergence. This implication is false and also is a fallacy. While the negative correlation is a necessary condition for convergence, it is not sufficient. Observe that the ␣-term is positive in equation 3. If |j| ⬍ 1 and |i| ⬍ 1, the necessary condition for the convergence of yj to yi is: ␣j/(1 ⫺ j) ⫽ ␣i/(1 ⫺ i).
(4)
We can illustrate this condition for the two economies. Let the initial values be $2,000 in economy j and $6,000 in economy i. Also, let ␣j ⫽ $800, ␣i ⫽ $1,000, j ⫽ .96, and i ⫽ .95. Then, as is shown in the simulation in Figure 1, per capita incomes converge absolutely at time ⫽ T, where $800/(1 ⫺ .96) ⫽ $1,000/(1 ⫺ .95) ⫽ $20,000. At T, dy/dt ⫽ 0. A more flexible approach to the analytics of convergence than that of the difference equation is one using a differential equation. Definitionally, [(dy/dt)/yt⫺1] ⫽ ␣ ⫺ yt⫺1 or, [dy/yt⫺1(␣ ⫺ yt⫺1)] ⫽ dt.
(5) (5a)
Integrating both sides of the differential equation 5a yields the following. 1/␣ ln[y/(␣ ⫺ y)] ⫽ t ⫹ C, where C is a constant of integration. Then,
(6)
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Fig. 1. Convergence in difference equation model.
y/(␣ ⫺ y) ⫽ e␣(t⫹C).
(6a)
Simple algebra leads to y ⫽ ␣e␣(t⫹C)/(1 ⫹ e␣(t⫹C)).
(6b)
Multiplying numerator and denominator of the r.h.s. by e⫺␣(t⫹C) yields yt ⫽ ␣/( ⫹ ␦e⫺␣t),
(6c)
where ␦ ⫽ e⫺␣C. yt has the following limits: lim yt ⫽ ␣/( ⫹ ␦); t→0
(7)
lim yt ⫽ ␣/. t→∞
(8)
That yt has an upper bound at t ⫽ ∞ is an important property of the model. It means that for a concave function ( ⬎ 0), no matter what the observed per capita income at time t, the asymptotic (steady state) per capita income at some future time T will be ␣/. Hence, the function has a property consistent with the steady state in neoclassical growth theory. Moreover, from an empirical perspective, estimation of the parameters of the differential equation permit calculation of the steady state per capita income and comparison across economies. This is an advance in the empirical testing of the very strong predictions made by the neoclassical growth model about the time paths of economies.
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Fig. 2. Convergence in differential equation model.
The model is illustrated in Figure 2. The coefficient values are: ␣j ⫽ ␣i ⫽ .05, j ⫽ i ⫽ .25E⫺05, ␦j ⫽ .225E⫺04, and ␦i ⫽ .583E⫺05. Absolute convergence of per capita incomes occurs at time T, where ␣j/j ⫽ ␣i/i ⫽ $20,000. For empirical purposes, we chose to examine the 24 member states of the OECD, because many of these countries are homogeneous, because the data length is fairly long (1950–1990), and the data are more reliable. We use the Summers and Heston [19] RGDP (in 1985 dollars) data. Figure 3 is a plot of the coefficient of variation of real gross domestic product per capita (also, real investment per head) for the OECD nations. Clearly, per capita incomes have not converged, but there is a strong downward trend in the coefficient of variation until the late 1970s. The parameters of equation 6c are estimated by iterative nonlinear least squares with correction for autocorrelation. The parameter estimates are highly significant and the fits very good, but it is clear that absolute convergence over all countries is inconsistent with the data. A strong test of absolute convergence requires ␣i ⫽ ␣ and i ⫽ , but ␦i may vary. Using an asymptotically valid F-test based on Fisher [20], absolute convergence is rejected. A less stringent, but valid, test requires ␣i/i ⫽ ␣/. This test is implemented by regressing ␣i on i and testing for a zero intercept. The intercept is statistically significant and absolute convergence is rejected.2 2 Nevertheless, subgroups of nations have converged in a statistical sense to the same absolute per capita income. The largest group contained 11 nations. The intercept had a t-value of 1.62, prob. ⫽ .139. These nations were: New Zealand, the Netherlands, Austria, France, Italy, Germany, Ireland, Denmark, Japan, Sweden, and Belgium. Smaller groups of nations also converged to each other at different values of ␣/. For the difference equation model, the condition for convergence is ␣i/(1 ⫺ i) ⫽ ␣/(1 ⫺ ) for all i. ␣i was regressed on 1 ⫺ i and the intercept was statistically significant. Hence, absolute convergence is rejected.
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Fig. 3. Coefficients of variation for RGDP and investment.
Absolute Convergence Model with Capital Accumulation (Equations 9–10) The next step is to broaden the analysis by introducing capital accumulation into the absolute convergence model. We specify a differential equation in which there are diminishing returns of percentage income increases to income levels. [(dy/dt)/y] ⫽ (1 ⫹ ␥[(dk/dt)/k])(␣⫺y).
(9)
In this model percentage increases in income are the product of two factors. The first, the capital accumulation factor, will increase percentage in per capita income with percentage changes in the capital–labor ratio. This is a natural feature of the neoclassical growth model. The second, the productivity factor, will have a constant component and one that reflects diminishing returns to the income level. The ␣ coefficient might be interpreted as a natural productivity coefficient that reflects a nation’s stock of human capital and level of technology. The parameter ␥ is a nation’s productivity multiplier that reflects the influence of percentage changes in the capital–labor ratio. The product of the coefficients, ␣␥, may be interpreted as the efficiency of the capital–labor ratio and (1 ⫺ ) as retention efficiency. Integrating equation 9 over the interval 0 to t, the solution to the differential equation is: yt ⫽ ␣/( ⫹ ␦e⫺␣(t⫹␥ln(k(t)/k(0))).
(10)
As t → ∞, yt converges to ␣/. Thus, the time only model in equation 6⬘⬘⬘ is nested in equation 10, since equation 10 reduces to equation 6⬘⬘⬘ when ␥ ⫽ 0. Data on the capital stock for the full period are not available in the Summers and Heston [19] data. The gross investment share is provided, which when multiplied by RGDP gives real investment per capita (It). We use It as a proxy for kt.
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Nonlinear estimates of the parameters of equation 10, using ln(It/I0), were highly significant and plausible, but ␣i/i ⬆ ␣/ across countries. Moreover, when we estimated the differential equation model in equation 9 as expanded below [(dy/dt)/y] ⫽ ␣ ⫺ y ⫹ ␣␥[(dI/dt)/I] ⫺ ␥[(dI/dt)/I]y we found that ⫺␥ was not significant and/or was implausible. This leads us to reject equation 9 and the logic that a nation’s per capital income would converge in time to a steady state value. As long as there are productivity gains that are independent of capital accumulation (that is, ␣ in equation 9) real per capita income will continue to increase regardless of capital accumulation. In any case, the absolute convergence model is convincingly rejected by the estimation of the differential equation version of the model.
Model of Converging Incomes (Equations 11–14) While the evidence rejects absolute convergence of income, there is reason to believe that real per capita investment will converge to a limit. This convergence need not be to the same level in each nation, since factor price differences imply different factor mixes. There are several reasons for thinking that convergence in investment is a plausible assumption. Partly, it is a feature of the Cobb-Douglas function in which (1 ⫺ ␣) is a constant [18]. Partly, it is a feature of the concavity of the production function. As capital accumulates diminishing returns arise and the rate of return falls. Partly, it is due to falling savings rates as per capita incomes rise, and social insurance and national health insurance reduce the incentive to save. Finally, holding risk constant, as investment levels rise a country’s attractiveness relative to others with lower investment levels declines. Hence, international capital flows adjust to differential rewards. Thus, we have the following differential equation model for investment: [(dI/dt)/I] ⫽ a ⫺ bI, or [dI/I(a ⫺ bI)] ⫽ dt.
(11) (11a)
Integrating both sides and manipulating results in: It ⫽ a/(b ⫹ ce⫺at).
(12)
The limit of It as t → ∞ is a/b. As is revealed in Table 1, the interative NLLS estimates of equation 12 produced good fits and mostly statistically significant parameter estimates of the right sign. The sign exceptions are Canada and Luxembourg. Weak statistical significance applies to the results for the United States, Spain, Switzerland, United Kingdom, and New Zealand, and is mainly due to the correction for autocorrelation. Global convergence was obtained for the parameter estimates for 20 of the 24 nations (i.e., the equations were re-estimated with different starting values and the parameter estimates remained unchanged). For four of the nations (Japan, Belgium, Luxembourg, and Switzerland) global convergence could not be obtained with the correction for autocorrelation. The estimates without the correction, which are globally stable, appear in Table 1. For the per capita income model, our investigation leads to a model in which percentage increases in income increase with: 1) a productivity factor that is independent of time and investment and 2) with percentage increases in investment and that decreases with the level of investment due to ‘‘diminishing returns to the investment level.’’ The differential equation is:
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TABLE 1 NLLS Estimates of Investment Model, 1950–90 Country
a
b*105
c*105
Canada
0.018 (1.52) 0.032 (1.78) 0.134 (6.72) 0.119 (6.04) 0.111 (5.57) 0.229 (3.86) 0.084 (5.30) 0.156 (4.00) 0.114 (4.44) 0.277 (7.15) 0.101 (6.41) 0.141 (4.68) 0.105 (4.87) ⫺0.232 (3.19) 0.194 (4.00) 0.095 (3.84) 0.173 (5.10) 0.140 (1.40) 0.155 (3.12) 0.060 (2.30) 0.110 (4.24) 0.065 (1.09) 0.080 (4.22) 0.033 (0.55)
⫺0.432 (1.18) 0.436 (0.78) 2.560 (4.47) 3.411 (4.44) 3.770 (4.48) 7.230 (3.56) 1.660 (3.57) 4.181 (3.44) 3.138 (3.64) 18.874 (5.86) 2.620 (4.75) 6.472 (3.73) 3.281 (3.82) ⫺7.183 (93.34) 6.959 (3.69) 2.091 (2.89) 13.340 (4.30) 5.572 (1.08) 4.845 (2.79) 0.869 (1.36) 12.225 (3.10) 2.188 (0.69) 1.806 (3.27) 0.682 (0.30)
1.619 (3.07) 1.194 (2.41) 45.146 (2.21) 23.061 (2.44) 10.766 (2.78) 58.429 (1.12) 5.882 (3.06) 30.596 (1.29) 7.822 (2.30) 879.410 (2.15) 10.198 (3.14) 59.411 (1.54) 10.071 (2.49) 0.0002 (0.38) 28.802 (1.37) 7.010 (1.87) 178.290 (1.44) 44.990 (0.53) 13.863 (1.32) 2.416 (1.93) 78.026 (1.70) 6.233 (0.78) 3.177 (2.75) 1.241 (0.75)
United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
1
2
0.425 (2.89) 0.608 (4.11) —
— ⫺0.304 (2.08) —
.779 (4.81) —
⫺.281 (1.67) —
0.608 (5.23) 0.859 (6.37) 0.748 (6.68) 0.937 (6.28) 0.743 (6.96) —
— ⫺0.600 (3.82)
⫺0.502 (2.97) — —
0.783 (4.99) 0.465 (3.20) —
⫺0.277 (1.81)
0.875 (5.80) 0.740 (4.54) 0.995 (7.21) 1.502 (11.83) 0.988 (6.85) —
⫺0.267 (1.79) ⫺0.192 (1.14) ⫺0.481 (3.52) ⫺0.663 (4.46) ⫺0.404 (2.48) —
0.840 (5.36) 0.927 (5.90) 0.204 (1.36) 0.489 (3.66)
⫺0.281 (1.78) ⫺0.233 (1.24) —
—
—
Log L R2 193.6 0.96 193.4 0.90 167.4 0.94 207.7 0.97 182.2 0.83 187.1 0.91 186.3 0.95 205.4 0.97 202.6 0.96 214.4 0.94 173.7 0.87 207.2 0.94 207.9 0.96 161.5 0.51 201.1 0.92 186.3 0.94 227.3 0.95 222.2 0.98 197.7 0.93 154.0 0.80 250.8 0.95 212.0 0.94 184.3 0.88 177.7 0.68
DW 1.80 1.98 0.26 1.75 0.48 1.85 1.91 1.88 1.76 1.98 1.69 1.95 1.86 0.82 1.87 1.95 1.94 2.17 1.79 0.40 2.01 1.81 2.10 1.84
Note: Asymptotic t-values in parentheses. Abbreviations: Log L ⫽ Log Likelihood; R2 ⫽ between the observed and predicted values; DW ⫽ DurbinWatson statistic.
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[(dy/dt)/y] ⫽ ␣ ⫹ ␣␥[(dI/dt)/I] ⫺ I.
(13)
Using the solution to the investment differential equation in equation 12, the solution to equation 13 is: yt ⫽ ␦e(␣ ⫺ (a/b))t ⫹ (␣␥ ⫹ (/b))ln(I(t)).
(14)
The iterative NLLS estimates of the parameters of equation 14 appear in Table 2. For every country the parameters are highly significant and the fits are extremely good. Global convergence was obtained for all of the nations. The individual parameter values in the exponent of equation 14 are not identified, but a and b can be obtained from equation 12 and ␣, ␣␥, and  from the OLS estimation of the differential equation in equation 13. These parameter estimates appear in Table 3. The estimates of the coefficient of time and of ln(It) obtained in equation 14 indicate that these coefficients are positive and highly significant. Thus, even with investment going to an asymptote, per capita income will continue to rise. A particularly nice feature of the model is that diminishing returns is allowed but not required. Where  ⬎ 0 in Table 3, it is always statistically insignificant. Thus, treating statistically insignificant cases as  ⫽ 0, (a/b) and (/b) drop from the exponent in equation 14. The computed values of ␣ ⫺ (a/b) and ␣␥ ⫹ (/b) appear in Table 4 and may be compared with the NLLS parameter estimates in Table 2.
The Meaning of Convergence How well does the differential equation model given in equations 12 and 14 track the pattern of convergence? We use equation 12 to project the path of real investment per head. The projected values of investment are used in equation 14 to obtain predicted values of real income per capita. We calculate year by year the mean and standard deviation of predicted investment and income per head for the OECD nations and compute the coefficient of variation. The paths of the coefficient of variation are shown in Figure 4. Clearly, the differential equation model works well, since the coefficients of variation of the predicted values mimic those of the actual values in Figure 3. Note that equation 14 may also be written as: yt ⫽ ␦e(␣ ⫺ (a/b))tIt(␣␥ ⫹ (/b)),
(15)
and, as such, income elasticity with respect to investment is a constant at (␣␥ ⫹ (/b)). Income change with respect to investment change will depend on the country’s ‘‘investment efficiency’’ and upon decay rates,  and b. Further, note that euqation 13 may be written as: [(dy/dt)/y] ⫽ (␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[dI/dt)/I].
(16)
The proportion of a given change in income of a country, then, may be partitioned into two components: the first term, call it A, is productivity growth that reflects a nation’s stock of human capital and level of technology; the second, B, is due to percentage changes in investment. Thus, we have: A ⫽ (␣ ⫺ (a/b))/(␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I] ⫽ 1 ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I] and
(17)
B ⫽ (␣␥ ⫹ (/b))[(dI/dt)/I]/(␣ ⫺ (a/b)) ⫹ (␣␥ ⫹ (/b))[(dI/dt)/I].
(18)
During the period of rapid capital accumulation, the dominant component of income
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TABLE 2 NLLS Estimates of Income Model, 1950–90 Country
␦*102
Canada
0.199 (10.51) 0.239 (14.27) 0.138 (6.30) 0.111 (10.11) 0.143 (7.89) 0.138 (18.38) 0.124 (10.39) 0.152 (9.19) 0.195 (11.17) 0.065 (7.21) 0.079 (8.56) 0.062 (6.60) 0.146 (9.13) 0.146 (10.92) 0.127 (5.37) 0.055 (8.70) 0.049 (4.82) 0.145 (6.49) 0.142 (5.06) 0.230 (12.80) 0.070 (3.70) 0.094 (7.33) 0.171 (13.34) 0.142 (5.06)
United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
␣⫺(a/b)
␣␥⫹(/b)
0.018 (9.68) 0.014 (19.82) 0.024 (10.79) 0.019 (9.33) 0.018 (6.45) 0.023 (18.86) 0.023 (13.17) 0.016 (5.80) 0.014 (5.74) 0.022 (6.05) 0.027 (12.22) 0.026 (15.49) 0.020 (5.29) 0.020 (29.75) 0.014 (2.88) 0.031 (29.54) 0.030 (9.74) 0.016 (6.08) 0.012 (3.05) 0.010 (5.62) 0.016 (8.72) 0.019 (24.41) 0.016 (10.99) 0.012 (3.04)
0.282 (13.89) 0.271 (15.50) 0.321 (8.15) 0.190 (7.55) 0.233 (8.75) 0.224 (15.44) 0.268 (12.00) 0.231 (9.92) 0.258 (9.36) 0.199 (5.78) 0.149 (6.38) 0.192 (6.80) 0.276 (8.83) 0.242 (9.62) 0.150 (4.58) 0.066 (2.39) 0.215 (5.78) 0.308 (8.73) 0.132 (4.46) 0.251 (10.70) 0.277 (6.22) 0.133 (4.49) 0.247 (12.52) 0.132 (4.48)
1 1.338 (9.02) 1.071 (7.04) 0.959 (28.36) 0.945 (32.80) 0.970 (26.48) 0.868 (8.79) 0.882 (9.10) 1.377 (9.45) 0.960 (41.16) 0.958 (27.10) 1.293 (8.72) 0.736 (5.99) 1.464 (10.45) 0.822 (5.41) 1.277 (8.22) 1.363 (9.10) 1.204 (7.86) 1.312 (8.48) 0.973 (32.01) 0.942 (25.08) 0.563 (4.03) 0.941 (6.73) 0.900 (12.01) 0.973 (32.16)
2 ⫺0.427 (2.79) ⫺0.257 (1.71) — — — — — ⫺0.409 (2.78) — — ⫺0.529 (3.28) — ⫺0.486 (3.53) ⫺0.226 (1.54) ⫺0.306 (1.98) ⫺0.702 (4.06) ⫺0.317 (2.11) ⫺0.349 (2.31) — — — ⫺0.482 (3.05) — —
Log L R2 215.6 0.99 219.0 0.99 224.2 0.99 232.0 0.99 223.9 0.99 231.3 0.99 219.3 0.99 238.3 0.99 238.8 0.99 233.0 0.99 189.1 0.99 221.5 0.99 241.8 0.99 199.9 0.99 221.9 0.99 209.5 0.99 227.4 0.99 238.3 0.99 220.7 0.99 210.5 0.99 237.8 0.99 213.8 0.99 213.8 0.99 220.7 0.99
DW 1.97 2.16 1.76 2.10 1.63 2.08 1.69 2.04 1.84 1.86 1.90 2.00 2.17 1.88 2.11 2.07 2.08 2.09 1.65 1.62 1.77 1.82 1.74 1.65
Note: Asymptotic t-values in parentheses. Abbreviations: Log L ⫽ Log Likelihood; R2 ⫽ between the observed and predicted values; DW ⫽ DurbinWatson statistic.
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TABLE 3 OLS Estimates of Percentage Change in Income, 1951–1990 Country a
Canada
United States Japan Austria Belgium Denmark Finland Francea Germany Greece Iceland Ireland Italya Luxembourg Netherlandsa Norwaya Portugal Spain Sweden Switzerland Turkeya United Kingdom Australia New Zealand
␣*102 2.517 (2.86) 1.003 (1.11) 6.124 (7.81) 4.677 (9.26) 1.551 (2.03) 1.556 (2.67) 2.726 (3.28) 3.730 (5.02) 6.389 (10.08) 3.890 (4.04) 3.131 (1.77) 1.525 (1.63) 3.410 (3.35) 1.729 (0.82) 1.550 (1.28) 3.090 (2.63) 4.076 (3.30) 3.382 (3.29) 3.387 (4.15) 1.492 (2.16) 1.250 (0.80) 0.816 (0.69) 1.282 (1.15) 3.970 (2.10)
␣␥ 0.272 (12.60) 0.277 (12.91) 0.188 (6.28) 0.185 (11.53) 0.227 (8.48) 0.221 (16.07) 0.278 (11.53) 0.212 (8.71) 0.284 (13.09) 0.173 (6.06) 0.179 (5.41) 0.162 (6.20) 0.292 (15.76) 0.282 (10.82) 0.210 (10.25) 0.069 (2.10) 0.179 (4.07) 0.349 (8.58) 0.130 (5.35) 0.247 (13.56) 0.327 (4.93) 0.151 (4.40) 0.255 (11.90) 0.215 (7.52)
⫺*105 ⫺0.286 (1.01) 0.143 (0.50) ⫺0.736 (3.17) ⫺0.897 (4.15) 0.229 (0.67) 0.099 (0.46) ⫺0.149 (0.59) ⫺0.566 (2.14) ⫺1.426 (6.59) ⫺0.721 (0.91) ⫺0.345 (0.52) 0.681 (1.09) ⫺0.358 (0.83) ⫺0.022 (0.04) 0.221 (0.43) ⫺0.057 (0.16) ⫺0.762 (0.60) ⫺0.828 (1.45) ⫺0.580 (1.93) ⫺0.782 (0.46) 0.468 (0.19) 0.562 (0.85) 0.024 (0.07) ⫺1.201 (1.63)
R2 0.800 0.811 0.645 0.822 0.643 0.875 0.791 0.747 0.891 0.533 0.455 0.483 0.878 0.765 0.766 0.196 0.320 0.700 0.508 0.838 0.353 0.310 0.798 0.639
a Serial correlation is present on the OLS equations for these countries. The regressions presented are corrected for first-order autocorrelation. The -value, Durbin-Watson statistic, and Durbin-Watson h statistic are as follows: Canada, ⫽ 0.349, DW ⫽ 2.01, and DW h ⫽ ⫺0.255; France, ⫽ 0.388, DW ⫽ 1.86, and DW h ⫽ 0.195; Italy, ⫽ 0.543, DW ⫽ 2.12, and DW h ⫽ 0.785; Netherlands, ⫽ 0.311, DW ⫽ 2.05, and DW h ⫽ ⫺0.926; Norway, ⫽ 0.277, DW ⫽ 1.93, and DW h ⫽ 0.357; and, Turkey, ⫽ ⫺0.366, DW ⫽ 1.50, and DW h ⫽ 1.10.
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TABLE 4 Computed Values of ␣ ⫺ (a/b) and ␣␥ ⫹ (/b) Country Canada United States Japan Austria Belgium Denmark Finland France Germany Greece Iceland Ireland Italy Luxembourg Netherlands Norway Portugal Spain Sweden Switzerland Turkey United Kingdom Australia New Zealand
␣ or ␣ ⫺ (a/b)
␣␥ or ␣␥ ⫹ (/b)
0.025 0.010 0.023 0.016 0.016 0.016 0.027 0.016 0.012 0.039 0.031 0.015 0.034 0.017 0.016 0.031 0.041 0.034 0.015 0.015 0.013 0.008 0.013 0.040
0.272 0.277 0.475 0.448 0.227 0.221 0.278 0.347 0.738 0.173 0.179 0.162 0.292 0.282 0.210 0.069 0.179 0.349 0.250 0.247 0.327 0.151 0.255 0.215
change arises from the second term, B. As diminishing returns to capital set in the weight of B in income change diminishes. As investment per head reaches its steady state value (i.e., as t → ∞, [(dI/dt)/I] or B → 0), further change in per capita income is due solely to the A term, which is a constant. Thus, in this differential equation model of convergence, the growth path of nations is influenced by two factors: capital accumulation and a productivity factor that is independent of capital accumulation (human capital and technical change). Nations will reach their steady-state values of capital accumulation. As investment per head goes to its asymptotic value, no further contribution to economic growth comes from capital accumulation. But, growth continues through human capital accumulation and technical change, and these contributions need not be internationally invariant. Nations differ in a variety of ways. They have different levels of skill, rates of technical change, and efficiencies [21].3 These differences prevent convergence of per capita incomes to the same level. However, income differences will narrow as capital formation grows more rapidly in countries with lower incomes. The coefficient of variation of per capita income of the OECD nations shows a steady decline from 1950 to 3 Nations differ as well by variables wholly outside of the neoclassical growth model, such as policies, industrial structure, and so on. A recent study [21] finds that one-digit sectoral decomposition improves the understanding of the convergence phenomenon. They find evidence of convergence (measured as labor and multifactor productivity) in the non-manufacturing sectors but little evidence of convergence in manufacturing. Yet, their conclusions point to a danger of sectoral disaggregation in the analysis of convergence. In the nontradeable sectors, such as services and construction, technology tends to diffuse over time. In the tradeablegoods sectors comparative advantage implies specialization, with no a priori reason for technological diffusion over time.
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Fig. 4. Coefficients of variation for predicted RGDP and investment.
the late 1970s, but appears to have reached a lower limit. Much of this narrowing is due to rapid capital accumulation of those nations that were defeated or occupied during World War II relative to the victorious nations. The coefficient of variation of investment per head rapidly declined, until the early 1970s. It rises, thereafter, mainly due to more rapid capital accumulation of the lagging OECD nations (for example, Greece, Portugal, Spain, and Turkey). As investment levels reach an upper bound in all of these nations, income growth will be influenced by the first term in the exponent of equation 14 that reflects learning productivity and retention efficiency. To the extent that these rates differ among OECD nations one might well expect a divergence of per capita incomes.
Conclusions The large literature on convergence leaves in doubt whether the very strong predictions of the neoclassical growth model hold or do not. Generally, cross-section studies reject the no convergence null, while time series studies accept it. The different conclusions partly rest on different testing frameworks. Moreoever, neither methodology distinguishes between absolute convergence to the neoclassical steady state, fixed, upper bound per capita income and some finite (data length constrained) per capita income that will continue to grow. This article contributes to the understanding of convergence within the neoclassical growth model by specifying a differential equation, the limit of which is the neoclassical steady state per capita income. We tested two versions of the model: a time only differential equation and a differential equation with capital accumulation and time as arguments. Our empirical analysis, which is not constrained by the data length, rejects the absolute convergence null at the limit (i.e., ␣i/i ⫽ ␣/ for all i). We then specified
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a weaker model of converging per capita incomes in which percentage changes in income are due to capital accumulation, the limit of which is a steady state capital–labor ratio, which may vary internationally, and to a productivity factor that is independent of time and capital accumulation. This pair of differential equations yields plausible results in the sense that the coefficient of variation of the predicted values mimics that of the coefficient of variation of the actual data. We conclude that for the whole period examined per capita incomes across the OECD nations have been converging. But, convergence slowed and then stopped by the late 1970s. In fact, there has been a slight divergence in per capita incomes since then. The pattern in income convergence is related to the pattern in investment convergence. When per capita incomes were strongly converging, per capita investment was strongly converging. In the late 1970s per capita investment began diverging, and per capita incomes also show slight divergence. We conclude that even if all of the variation in per capita incomes due to variation in capital accumulation per head is eliminated (i.e., all nations reach their steady state capital–labor ratios), there is no guarantee that per capita incomes will further converge toward one another. This will depend crucially on the time path of the productivity factor that is independent of time and capital accumulation (e.g., technical change and human capital). A narrowing (widening) of differences in this source of per capita income growth will determine whether further convergence (divergence) occurs.
Reference 1. Mankiw, M., Romer, D., and Weil, D. N.: A Contribution to the Empirics of Economic Growth, Quarterly Journal of Economics 107(2), 407–437 (1992). 2. Barro, R. J.: Economic Growth in a Cross-section of Countries, Quarterly Journal of Economics 106(2), 407–443 (1991). 3. Barro, R. J., and Sala-i-Martin, X.: Convergence Across States and Regions, Brookings Papers on Economic Activity 1, 107–182 (1991). 4. Barro, R. J., and Sala-i-Martin, X.: Convergence, Journal of Political Economy 100(2), 223–251 (1992). 5. Baumol, W. J.: Productivity Growth, Convergence, and Welfare, American Economic Review 76(5), 1072–1085 (1986). 6. DeLong, J. B.: Productivity Growth, Convergence, and Welfare: A Comment, American Economic Review 78(5), 1138–1155 (1988). 7. Dowrick, S., and Nguyen, D.: OECD Comparative Economic Growth 1950–85: Catch-up and Convergence, American Economic Review 79(5), 1110–1130 (1989). 8. Bernard, A. B., and Durlauf, S. N.: Interpreting Tests of the Convergence Hypothesis, Journal of Econometrics 71(1–2), 161–173 (1996). 9. Friedman, M.: Do Old Fallacies Ever Die? Journal of Economic Literature 30(4), 2129–2132 (1992). 10. Quah, D.: Galton’s Fallacy and the Tests of the Convergence Hypothesis, Scandinavian Journal of Economics 95(4), 427–443 (1993). 11. Quah, D.: International Patterns of Growth: I. Persistence in Cross-country Disparities. Working Paper. London School of Economics, London, 1992. 12. Bernard, A. B.: Empirical Implications of the Convergence Hypothesis. Working Paper. MIT, Cambridge, MA, 1992. 13. Evans, P., and Karras, G.: Do Economies Converge? Evidence from a Panel of U.S. States, Review of Economics and Statistics 78, 384–388 (1996). 14. Johansen, S., and Juselius, K.: Testing Structural Hypotheses in a Multivariate Cointegration Analysis of the PPP and UIP for UK, Journal of Econometrics 53, 211–244 (1992). 15. Pesaran, M. H., and Shin, Y.: Cointegration and Speed of Convergence to Equilibrium, Journal of Econometrics 71, 117–143 (1996). 16. Reimers, H. E.: Comparisons of Tests for Multivariate Cointegration, Statistical Papers 33, 335–359 (1992). 17. Toda, H. Y.: Finite Sample Performance of Likelihood Ratio Tests for Cointegrating Ranks in Vector Autoregressions, Econometric Theory 11 (1995). 18. Solow, R. M.: A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics 70, 65–94 (1956).
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19. Summers, R., and Heston, A.: The Penn World Table (Mark 5): An Expanded Set of International Comparisons 1950–1988, Quarterly Journal of Economics 106(2), 327–368 (1991). 20. Fisher, F. M.: Tests of Equality Between Sets of Coefficients in Two Linear Regressions: An Expository Note, Econometrica 38(3), 361–366 (1970). 21. Bernard, A. B., and Jones, C. I.: Comparing Apples and Oranges: Productivity Convergence and Measurement Across Industries and Countries, American Economic Review 86(5), 1216–1238 (1996).
Received 1 April 1997; accepted 14 October 1997