A test of the Solow Groth Model. Willem Elbers Joop Adema Derck Stäbler. May 29, 2015

Transcription

1 A test of the Solow Groth Model Willem Elbers Joop Adema Derck Stäbler May 29, 2015 Abstract In this paper, we investigate the relationship between the savings rate and aggregate output per worker. Using data from the World Bank, we have tested both the classic Solow model and the augmented Solow model with human capital. In both models, we find a positive relation between saving and output per worker. Moreover, in the augmented model we find more realistic income shares for physical and human capital. Our results show that the effect of population growth is stronger for developing nations, whereas in Europe, we see a reversal of this effect due to migration flows from Eastern Europe to richer countries in the West. Overall, these results provide some renewed evidence for the Solow model, although empirical issues remain. 1

2 1 Introduction The neo-classical growth model developed by Solow (1956) and Swan (1956) is an important pillar of growth economics. This model explains growth of aggregate output in terms of population growth, capital accumulation and technological progress. It predicts that a country s income per worker converges towards a steady state value, determined by the propensity to save, depreciation, technological progress and population growth. In particular, a higher savings rate leads to a higher steady state level of output per worker, while higher population growth leads to a lower steady state level. In the Solow model, the savings rate explains cross-country differences in the amount of physical capital available to each worker and the resulting levels of output per worker. Every investment corresponds to an act of saving; if someone did not use their resources to consume but rather to invest in capital. The model thus assumes that the savings rate has the same effect on output per worker as the investment rate. Over the years, the effect of aggregate savings and population growth on the level of output per worker has been tested extensively (Mankiw et al., 1992; Levine and Renelt, 1992). However, empirical tests of the basic Solow model have never produced completely satisfactory results, in particular with regards to the implied income share of capital (Mankiw et al., 1992). In order to solve this issue, Mankiw et al. proposed an augmented Solow model, which incorporated human capital as a factor of production. Savings were split up into savings for human capital and savings for physical capital. Human capital is found to be vital to the process of growth. According to Kendrick (1976), over half of the U.S. capital stock in 1969 was human capital. Both the classic and the augmented Solow models have been tested in the past. Of course, it is always prudent to re-test models with the most recent data. However, two developments have made a renewed evaluation of the augmented Solow model particularly interesting. Over the past decade, new indicators for human capital have become available in the form of internationally comparable test scores for students and adults (PISA, The Programme of International Student Assessment, 2012; PIAAC, The Survey of Adult Skills, 2013). This opens up new opportunities to test the Solow model with potentially more reliable indicators for human capital. Furthermore, the growing importance of ICT and R&D-intensive capital in recent years has caused an increase in the depreciation rate (Oulton and Srinivasan, 2003). This may significantly impact the results of the Solow model, in which the effect of population growth is mixed up with that of the depreciation rate. In this paper, we focus in particular on the relation between savings and 2

3 output, as this has been historically one of the most robust results in growth economics (Levine and Renelt, 1992). We hypothesise that both versions of the model continue to show a positive relationship between the savings rate and output per worker. Additionally, we test for differences between developed and developing nations, and for regional differences in the magnitude of this relationship. In the next section, we discuss and test the basic Solow model. In section 3, we add a proxy for human capital and test the augmented Solow model. Finally, we offer some conclusions in section Theory 2 Basic Solow model The Solow-Swan model, as described above, equates total investment to aggregate savings, under the assumption of autarky. Considering additional capital per worker as the difference of new capital creation and depreciation and population dilution, a differential equation describing the level of capital per worker can be derived. Finally, assuming a Cobb-Douglas production function, one arrives at a steady state value of capital and output per worker. A detailed derivation is given in Appendix B. The production function is: Y = F (K, L) = AK α L 1 α = y = k α, (1) in which A denotes the productivity of the country, K the capital stock and L the labour force, which is defined as the number of people employed. Dividing Equation (1) by effective labour AL, one obtains a simplified production function y = k α in per worker units. The increase in capital stock per unit of effective labour is zero if a country has arrived at its steady state value of capital k. This level is reached when capital creation due to investment and capital depreciation equal each other, such that: 0 = k t = s(f(k, l)) (δ + n L + n z )k, where n L is the growth rate of labour, n Z is the growth rate of productivity, and s is the savings rate. Substituting the production function per worker f(k, l) allows us to derive the steady state level of capital per effective worker: ( ) 1 α k nl + n Z + δ =. s 3

4 Substituting these relations into the production function, we find a relation Y/L = f(s, n L, n Z, δ). Taking logarithms, we obtain ( ) Y ln = A(0) + α L 1 α ln(s) α 1 α ln(δ + n Z + n L ), (2) where A(0) is the initial productivity. Henceforth we refer to this equation as the basic Solow model. 2.2 Data We estimate (2) using data from the World Bank (World Bank, World Development Indicators, 2012). In order to smooth out year-to-year fluctuations, saving/investment rates and population growth values are averaged over the 25-year period This procedure would not be appropriate for GDP per employed person, as this is expected to be a non-stationary variable. We therefore use income per worker values from 2012, the most recent year for which almost all GDP per capita data is available. Out of the 248 countries and supra-national groupings in the World Bank index, a total of 138 observations were removed due to missing data, leaving a set of 113 countries and no groupings. In some cases, data were missing for some of the years of the 25-year period When there was at least one data point in the 5 years before 2012, the country was still included. In the theory of the Solow model, investment and saving are assumed to be equal. This is possible due to the assumptions of autarky. In practice, saving is likely to deviate from investment. For instance, due to current account deficits or government outlays, neither of which contribute to the capital stock. Due to these differences, total savings (aggregate income net of consumption) is not expected to contribute directly to output growth. A more appropriate measure is the rate of investment, which adequately describes contributions to the capital stock. In estimating (2), we therefore use the rate of investment rather than the savings rate, referring to both as s. In their paper, Mankiw et al. (1992) estimate the sum of the rates of the advancement of knowledge n Z and depreciation δ to be around 5% annually. However, depreciation rates in the US have been trending upwards since the 1980s. This has likely much to do with the increased importance of ICT applications, which depreciate on much shorter timescales than the traditional capital stock (Oulton and Srinivasan, 2003). Nadiri and Prucha (1996) estimate that in the US, plant equipment depreciates at a rate of 5.9%, whereas R&D capital depreciates at 12% annually. In countries where ICT and R&Dintensive capital is less important, a lower rate may be in order. Nevertheless, 4

5 we expect that the rates of the advancement of knowledge and depreciation have increased since Mankiw et al. published their seminal paper. Therefore, we use an estimate based on averages of n Z + δ = 7% per annum. 2.3 Econometric model We estimate the following equation, which directly follows from the theoretical model (2). ln(y i ) = α 1 + α 2 ln s i + α 3 ln(n L,i + n Z + δ) + ɛ i, (3) where ɛ i is a random error term. The expected coefficients are: α 1 = A(0) α 2 = α 1 α α 3 = α 1 α. In order to test whether the model has been correctly specified, we carry out a Ramsey RESET test of order 1. We strongly reject the null hypothesis that higher order terms of the explanatory variables do not improve the model. Potential causes for this result are an incorrect functional form or a violation of the OLS assumptions. Additionally, the results might indicate that a relevant variable was omitted, provided that the omitted variable is correlated to the included variables, since the omitted variable is then expected to influence the dependent variable through the higher order terms (Hill et al., 2011, p. 239). We can exclude the case of an OLS violation on the basis of the following observations. The results of a White test with cross terms show that the residuals are uncorrelated with the explanatory variables. Hence, the basic Solow model does not suffer from heteroskedasticity. Moreover, the negligible R 2 = for the regression between savings and population growth shows that collinearity is not a problem. Finally, we note that the values for output per worker are from 2012, whereas the values for savings and population growth are 25-year averages from the period Hence, there can be no question of reverse causation and we conclude that endogeneity is not a problem either. We also test for differences between developed and developing nations, and for regional differences between the four continents: Europe, Asia, Africa, and the Americas. In both cases, the only significant differences involve the coefficient for population growth. The coefficient for the investment rate is clearly unaffected. Moreover, neither the intercept coefficient for the OECD nor any of the coefficients for Asia or the Americas are significant. We thus proceed with two additional models: one with an interaction term between OECD-status and population growth, and one with intercept terms for Europe 5

6 and Africa and interaction terms between Europe/Asia and population growth. The misspecification properties of the two extended models are the same as for the basic model, following from a Ramsey RESET test. 2.4 Results Table 1: Estimation results for (3). Standard errors of the estimators are given in brackets. The dependent variable is log(y/l). OECD countries have D = 1, European countries have E = 1, and African countries have A = 1. Investment rate s and population growth n L are averaged over the 25-year period We assume that n Z + δ = 7%. Variable Basic model Basic model Basic model + OECD + regions Intercept (1.15) (1.02) (1.35) log(s) (0.18) (0.16) (0.16) log (n L + n Z + δ) (0.45) (0.41) (0.62) log (n L + n Z + δ) D 0.42 (0.068) E (2.30) log (n L + n Z + δ) E 3.91 (1.13) A (5.12) log (n L + n Z + δ) A (2.28) n R The estimation results for (3) are listed in table 1. The results for the basic model offer some evidence in support of the Solow model. First of all, we observe that the coefficient for the investment rate log(s) is positive and highly significant. Hence, there is strong evidence for the prediction that countries with higher saving rates have a higher output per worker. Similarly, the coefficient for log (n L + n Z + δ) is negative and significant, confirming the prediction 6

7 that a higher population growth depresses output per worker. Furthermore, the R 2 -value of 0.52 indicates that a sizeable portion of the cross-country variation in Y/L is explained by the model. These observations are promising. However, the results also suggest that the model is far from perfect. Considering equation (2), the coefficients for log(s) and log(n L + n Z + δ) are expected to be equal in magnitude. This hypothesis is strongly rejected (p < ). Moreover, the implied value of the income share of capital α = 0.65 is much higher than previously found (Solow, 1958). One possible explanation is the absence of a major production factor: human capital. This is consistent with the result of the Ramsey RESET test, which indicates a possible misspecification of the model. Incorporating human capital into the model might mitigate these issues. This will be addressed in the next section. First, we turn to the differences between developed and developing nations. As seen in table 1, the coefficient for the saving rate is approximately equal to that of the basic model. The coefficient for the interaction between population growth and the dummy indicating OECD-status is positive, showing that the effect of population growth is significantly smaller for OECD countries. A similar result is seen in the regional model. Even though the negative effect of population growth is much stronger for African countries, the effect turns out to be positive for European countries, contrary to Solow s prediction.this can be observed from the coefficient of the interacting dummy times the population growth plus the coefficient of the non region-specific population growth being positive. This difference is illustrated by figure 1, which shows that the correlation between output and population growth is negative for most countries, but positive within the sub-sample of European countries. On closer inspection of the data, it seems that this difference can be entirely attributed to the population decline in Eastern Europe. Many countries in Eastern Europe have seen a sizeable portion of the population migrate to Western Europe and Russia. In combination with falling birth rates, this has caused a net decline in their populations. At the same time, neighbouring countries with larger incomes have faced a net inflow of workers. This could explain the positive relationship within the region. When we exclude the countries with negative population growth rates, the effect of population growth is no longer significantly positive even within Europe. 7

8 Figure 1: The relation between output per worker log(y/l) and population growth (plus depreciation and technological progress) log(n L + n Z + δ). Even though there is a positive relationship for countries in Europe, the relationship is clearly negative for most of the rest of the world Income per worker vs. population growth Europe Non-Europe 4.0 log(y/l) log(n l +7) 3.1 Theory 3 Augmented Solow model As suggested by the results of the previous section, we add human capital as a production factor, which allows us to specify which share of national income is due to physical capital and which is due to human capital. The production function now becomes Y = f(k, H, L) = AK α H β L 1 α β = y = k α h β, where H is human capital. The change in capital stock per worker is zero if a 8

9 country has achieved steady state levels of physical k and human capital h : 0 = k t = s k(f(k, l, h)) (δ + n L + n z )k, 0 = h t = s h(f(k, l, h)) (δ + n L + n z )h, where s k is the saving rate for physical capital and s h the saving rate for human capital. Substituting the production function f(k, l, h) yields the steady state levels of both forms of capital: ( ) 1 α β ( ) 1 α β k n L + n Z + δ =, h nl + n Z + δ =. s 1 β k s β s α h k s1 α h Substituting these relations into the production function, we derive a relation Y/L = f(s k, s h, n L, n Z, δ). Taking logarithms, we finally obtain ( ) Y ln = A(0) + L α 1 α β ln (s k) + β 1 α β ln (s h) α + β 1 α β ln(δ + n Z + n L ), (4) where A(0) is the initial productivity. This equation will henceforth be referred to as the augmented Solow model. 3.2 Data In the augmented Solow model, we use the same data as in the basic model, but with the addition of human capital. Mankiw et al. (1992) consider an equation with the saving rate for human capital as in equation (4). In this case, we require data describing the accumulation rate of human capital. It is also possible to write (4) in terms of levels of human capital per worker h. In this research, we have tried both approaches. There are many possible proxies for the accumulation rate of human capital s h. Like Mankiw et al. (1992), we have chosen secondary school enrollment rates, because data on school enrollment is widely available for a large number of years and countries. We suspect that there are better indicators for human capital, particularly for the level of human capital, because all developed countries have high secondary school enrollment rates even though large differences in levels of human capital are likely to persist. We expect that adult test scores (PIAAC, The Survey of Adult Skills, 2013) and test scores 9

10 for 15-year olds (PISA, The Programme of International Student Assessment, 2012) are better indicators for the level of human capital. These indicators refer directly to the literacy and numeracy of a country s workforce, reflecting the quality of the schooling, whereas school enrollment rates only reflects the number of participants. The PIAAC scores are only for the 23 OECD and partner countries, whereas the PISA scores are for a set of 56 countries. We have run the model with both enrollment rates and PISA and PIAAC test scores. Unfortunately the datasets for the test scores are still too limited to produce significant results. Hence, we proceed now with the analysis using secondary school enrollment rates alone. 3.3 Econometric model Following the theoretical model (4), we estimate the following equation: ln(y i ) = α 1 + α 2 ln s k,i + α 3 ln s h,i + α 4 ln(n L,i + n Z + δ) + ɛ i, (5) where s k,i is investment in physical capital and school enrollment rates s h,i measure the accumulation of human capital. The expected coefficients are α 1 = A(0) α 4 = (α 2 + α 3 ) α β α 2 = α 3 = 1 α β 1 α β We test for any possible misspecification of the model using a Ramsey RESET test of order 1. As was the case with the basic model, we reject the null hypothesis that the inclusion of higher order terms does not improve the model, albeit less strongly (p = rather than p < ). We argued before that such a result could indicate an incorrect functional form, a violation of the OLS assumptions, or an omitted relevant variable (Hill et al., 2011). For the basic model, we could exclude heteroskedasticity as the cause. In order to detect whether heteroskedasticity may be an issue for the augmented model, we carry out a White test with cross terms. We find that the residuals are correlated with the explanatory variables. Hence, the augmented Solow model is indeed plagued by heteroskedasticity. We deal with this issue by using heteroskedastic-consistent White estimators. Since the addition of human capital, there is now a sizeable correlation between some of the explanatory variables. In particular, school enrollment and investment are strongly correlated. Because of this, we expect that collinearity might now be an issue. A regression of investment s k on the other two explanatory variables yields a relatively low R 2 of Hence, we conclude that collinearity is not a direct issue for the augmented Solow model. 10

11 Finally, we consider the case of endogeneity. Our explanatory data consists of 25-year averages from the period and the dependent variable is a one-year value for It is impossible that income in the final year influences savings, school enrollment and population growth in previous years. Hence, there can be no reversed causation or endogeneity. We also test for differences between OECD countries and the rest of the world. The only significant difference is in the coefficient for population growth. Ignoring the other interaction terms, we construct an extended version of the augmented model, incorporating an interaction term between OECDstatus and population growth. A Ramsey test indicates that the extended model is correctly specified, contrary to both the basic and the augmented models. A model incorporating regional differences between the four continents: Europe, Africa, Asia, and the Americas yields no significant results. 3.4 Results Table 2: Estimation results for (5). Heteroskedastic-consistent (White) standard errors of the estimators are given in brackets. The dependent variable is log(y/l). OECD countries have D = 1, non-oecd countries have D = 0. Investment rate s k, school enrollment rate s h, and population growth n L are averaged over the 25-year period We assume that n Z + δ = 7%. Variable Augmented Augmented + model OECD Intercept (1.20) (0.99) log(s k ) (0.18) (0.17) log(s h ) (0.12) (0.11) log (n L + n Z + δ) (0.44) (0.38) log (n L + n Z + δ) D 0.30 (0.042) n R We present the estimation results for (5) in table 2. At first sight, the results are similar to those of the basic Solow model (2). All coefficients have 11

12 the expected signs and are significant. However, the coefficients for log (s k ) and log (n L + n Z + δ) are much smaller in magnitude compared to the basic model. This can be explained by the fact that these variables are strongly correlated with school enrollment rates s h. Hence, part of the cross-country variation in Y/L that was explained by s k and n L + n Z + δ is now explained by s h. The estimate for R 2 has increased significantly from 0.52 to 0.72, suggesting that human capital contributes much to the share of cross-country variation that can be explained. Contrary to the basic model, we now find that the coefficients match the prediction of the augmented model (4) that the coefficients sum to 0. The hypothesis of a non-zero sum cannot be rejected (p=0.47). Moreover, the implied values for the income shares of capital α = 0.16 and human capital β = 0.45 are more realistic than the capital share α = 0.65 implied by the basic model. When considering differences between developed and developing nations, we find that the only significant difference is in the coefficient for population growth. Specifically, the effect of population growth is about 2 /3 smaller in magnitude for OECD countries. However, the total effect is still negative. This is consistent with our findings for the basic model. Finally, we note that the standard errors in table 2 are not estimated by BLUE estimators, but the use of robust heteroskedastic-consistent (White) estimators implies that the variances are not skewed towards incorrect values. Nevertheless, the estimators are still not best as they would have been in a GLS estimation. 4 Conclusion We have tested both the basic and the augmented Solow models. The test of the basic Solow model shows a highly significant positive relationship between the investment rate and output per worker. It also shows a negative relationship between population growth and output per worker. These findings match the predictions of the theoretical Solow model. The effect of population growth is larger for developing nations. In particular, we found that the negative relation between population growth and output per worker is 4.5 times stronger for African countries. Contrary to that, European countries exhibit a positive relation between population growth and output per worker, which does not follow from the Solow model. This can be explained by migration flows from Eastern European countries to Russia and richer countries in Western Europe. Overall, we find that a large share of the cross-country variation in output per worker can be explained by the model. Nonetheless, the results imply an unrealistically high value for the income share of capital and the magnitudes of 12

13 the coefficients do not match. To mitigate these problems, we turn to the augmented Solow model with human capital as an additional factor of production. The estimation results for this model are promising. The coefficients have not only their predicted signs, as was the case for the basic model, but also magnitudes closer to the expected values. The implied income shares of physical and human capital are more realistic than was the case for the basic model. Furthermore, the effect of population growth is still found to be stronger for developing nations. Some issues remain however, as the use of human capital introduced heteroskedasticity and collinearity into the model, obfuscating the results. Further improvements on our test of the Solow model could be made. First of all, the assumption that output levels always equal their steady state level could be relaxed, in which case a more elaborate estimate for the steady state level is necessary. Secondly, when test scores become available for a larger set of countries, it will be interesting to see whether these provide a more accurate proxy for human capital. Finally, the use of country-specific estimates for depreciation and technological progress are likely to offer new insights into the Solow model. In summary, the tests of both the basic and augmented Solow models provide evidence for the predicted positive relationship between aggregate savings and output per worker. This conclusion is in line with previous results (Mankiw et al., 1992; Levine and Renelt, 1992). Moreover, we confirm the result from Mankiw et al. (1992) that the augmented model better predicts the magnitudes of the effects. Although we were not able to utilise the promising new indicators for human capital, we were still able to verify the Solow model with renewed data and an updated estimate of the depreciation rate. All in all, the effect of savings on output per worker seems to hold up to this day. Acknowledgement The authors would like to thank the referee Dr. Gerard Kuper for his constructive comments. 13

14 References Heijdra, B. (2009). Foundations of Modern Macroeconomics. OUP Oxford, 2nd edition. Hill, R., Griffiths, W., and Lim, G. (2011). Principles of Econometrics. Wiley, 4th edition. Kendrick, J. W. (1976). The formation and stocks of total capital. NBER Books. Levine, R. and Renelt, D. (1992). A sensitivity analysis of cross-country growth regressions. The American economic review, 82(4): Mankiw, N. G., Romer, D., and Weil, D. N. (1992). A contribution to the empirics of economic growth. Quarterly Journal of Economics, 103(2): Nadiri, M. I. and Prucha, I. R. (1996). Estimation of the depreciation rate of physical and r&d capital in the us total manufacturing sector. Economic Inquiry, 34(1): Oulton, N. and Srinivasan, S. (2003). Capital stocks, capital services, and depreciation: an integrated framework. Technical report, Bank of England. PIAAC, The Survey of Adult Skills (2013). Indicator for adult numeracy and literacy. Retrieved from PISA, The Programme of International Student Assessment (2012). Indicator for teenage numeracy and literacy. Retrieved from pisa/. Solow, R. M. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70(1): Solow, R. M. (1958). A skeptical note on the constancy of relative shares. The American Economic Review, 48(4): Swan, T. W. (1956). Economic growth and capital accumulation. Economic record, 32(2): World Bank, World Development Indicators (2012). GDP per capita, Gross saving. Retrieved from 14

15 A Econometric tests To test whether both the Solow models are well specified, we conduct several tests. Tests for autocorrelation and stationarity are not included since the data is not time dependent. A.1 RESET Test To test for possible model misspecification, which means that important explanatory variables are not included and that irrelevant explanatory variables could have been included. Besides, a bad RESET-score could also cause other model problems that violate the OLS assumptions. The tests in the next sections (except for non-exact collinearity) could cause bad RESET-scores. The reset test is an F -test in which the unrestricted model is of the form of: y i = α 1 + α 2 x i γ i ŷ i 2 Where ŷ i is the predicted value of the dependent variable. One can perform this test with higher polynomials of ŷ i. If the value of the F -statistic is bigger than the value of F (1 α,j,n K), one rejects the null of no misspecification (Hill et al., 2011). A.2 Heteroskedasticity A model with heteroskedastic properties has a variance behavior that depends on an explanatory variable x i as such: var(e i ) = σ 2 i = σ 2 x i Hence, we see that this can be tested for the multiple explanatory variables in the model. A test to detect heteroskedasticity is the Lagrange multiplier test. The LM-test estimates the following regression, in which e 2 i is the array of error terms in the LS regression: ê 2 i = α 1 + α 2 z i α s z i s + v i Where the z i terms are variables that could explain the error term to be correlated with. Most often the explanatory variables, its squares and its cross-terms are used for this purpose. This is a so-called White test. The actual test statistic is based upon the explanatory power of the z i -terms on the e 2 i term: χ 2 = N R 2 15

16 The test statistic should be compared to the appropriate table value of χ 2 S 1. If the test statistic is the biggest of those two, one rejects the null of no heteroskedasticity(hill et al., 2011). A.3 Collinearity Collinearity can be detected by regressing the explanatory variables on eachother, such that: x 1 = α 1 x 2 + α 2 x α n 1 x n If this regression yields a high correlation coefficient and therefore a high R 2, there is collinearity. This causes larger variances and hence less significance of the estimators. If the value of R 2 approaches 0.8, there is problematic collinearity (Hill et al., 2011). A.4 Endogeneity Endogeneity of several of the multiple regression its explanatory variables can be detected by the Hausman-test. This test regresses the explanatory variable, x i, that is suspected to be endogenous on instrumental variables and the exogenous explanatory variables z i. This is done for all explanatory variables that could be endogenous. The residuals of this regression should be saved. After this, the final regression could be executed, which regresses the dependent variable on all the explanatory variables and on all the residuals found in the first part. The test statistic is obtained by restricting this model by setting the coefficients of the residual-terms in this regression to 0. An F-test can be performed. If the coefficients of the residuals are significantly nonzero, the residuals of the original regression are correlated with the explanatory variables of interest. B Derivation of Solow Model Let aggregate output Y in a closed economy be given by the production function 1 Y (t) = F (K(t), L(t), t), 1 The discussion in this section follows the treatment in Heijdra (2009, pp ). 16

17 where K(t) is capital and L(t) is labour. Assume constant returns to scale and a constant population growth rate L/L = n L. Assume furthermore a constant propensity to save s, such that total savings equal S(t) = sy (t). In a closed economy, Y (t) = C(t) + I(t). Hence, S(t) = I(t) = δk(t) + K(t), (6) where δ is the constant depreciation rate of capital. Finally, let technological progress be entirely disembodied (such that it affects all existing production factors as well as new ones) and Harrod neutral (labour augmenting). This implies that the production function can be written as Y (t) = F (K(t), L(t)), where L(t) Z(t)L(t) is the effective labour force augmented by technological progress occurring at a constant rate Ż/Z = n Z. Dividing both sides of (6) by L(t), sf (K(t), L(t)) L(t) = δk(t) + K(t) L(t), (7) where k(t) K(t)/L(t). Abusing constant returns to scale, we find F (K(t), L(t)) = L(t)F (K(t)/L(t), 1) L(t)f(k(t)). (8) Differentiating k(t) K(t)/L(t), we obtain k(t) = K(t)L(t) K(t) L(t) L(t) 2 = K(t) L(t) (n L + n Z )k(t). Combining this result with (7) and (8), we derive the fundamental differential equation k(t) = sf(k(t)) (δ + n L + n Z )k(t). In the steady state, k(t) = 0. Hence, we obtain the steady state level of output per unit L(t) y = δ + n L + n Z k. (9) s This equation implies that, without specifying a particular production function, steady state capital and output grow at the same rate n L + n Z. Now 17

18 assume a Cobb-Douglas production function f(k(t)) = k(t) α, where α is the income share of capital. In that case, (9) collapses to: ( δ + y nl + n Z = s ) α α 1. (10) As 0 < α < 1, (10) implies that steady state output strictly increases with the propensity to save s and decreases with the depreciation rate δ. 18