6 Estimation of the CES-production function
6.2 Estimation of CES production function based
We have noticed from the previous estimations of the nonlinear CES-production function that the elasticity of substitution between capital and labour is not stable over time. In order to gain more understanding about the development of the substitute elasticity between capital and labour we estimate the substitution elasticity from the first-order conditions for profit maximizations of the CES-production function. Once more, we estimate the CES-production function in the Hicks-neutral and Harrod-neutral case. We use data from real compensation per employee in order to measure the real value of marginal labour product.
Under Hicks’ neutral technological development, technological efficiency does not have an effect on the ratio of marginal products at given capital-labour ratio. The first-order condition of the marginal labour product is of the form
,t production. σ1 is the estimate of the substitution elasticity between capital and labour. However, it is shown in the literature (Pol Antràs, 2004) that allowing for biased technological change leads to estimates of the elasticity of substitution that are in general significantly lower than one. Under biased technological change the above equation is of the form
,t 2 L 2 t
t 2
2 t
t/L ) log(W /P) (1 ) t
Q
log( =α +σ + −σ λ ⋅ +ε (6.7)
The estimation results of the equations (6.7) and (6.8) are presented in the Table 5a and 5b. In the Hicks’ neutral case (equation 6.6) the estimated elasticity of substitution (Table 5b) is comparable to the earlier results by Antràs (2004) and Berndt (1976). Antràs (2004) shows that there is problem of omitted variables in the estimated equation, making the estimated elasticity of substitution biased towards unity.
When the econometric specification is changed to take into account biased technical change, the estimated substitution elasticity becomes clearly lower (Table 5a). The substitution parameter is estimated to be 0.78 in the whole sample. Both linear and linear technological trends are significant, the non-linear term being somewhat more precisely estimated.
Table 5a. GLS estimation results for the first order condition (6.7)
1970–2004
σ .71 .78 (33.87) (36.86)
γ1 .01
(24.22)
γ2 .0002
(22.04) R2 0.740 0.932
t-values are inside parentheses. Estimates are derived from the panel data. γ1 (γ2) indicates the coefficient of a linear (quadratic) trend.
Table 5b. Alternative estimates for equations (6.6) and (6.7) Estimated
parameter
1 2 3 4 5 6 σ 1.13 .79 1.15 .78 1.12 .78
(30.22) (17.18) (53.61) (36.86) (212.5) (176.3) γ2 .0002 .0002 .0002
(11.13) (22.04) (88.72)
Estimator OLS OLS GLS GLS SUR SUR
Nature of
technical change Hicks Harrod Hicks Harrod Hicks Harrod R2 0.669 0.741 0.864 0.932 0.991 0.995 t-values are inside parentheses. γ2 indicates the coefficient of a quadratic trend.
Estimation period is 1970–2004.
In addition, the significance level of the quadratic trend increases clearly when we control for fixed country effects. The substitution elasticity is also higher when we control the model with fixed effects. Allowing for random country effects does not change this conclusion. We estimate the CES-production function also with GLS and SUR estimation methods. It is evident that with SUR and GLS estimation methods the results are more precise than with OLS-method.
The estimation results of the first-order conditions of the CES-production function show that the substitution elasticity between capital and labour has been below one on average during the last 35 years. However, we are unable to say precisely what the right level is, because the estimation results between nonlinear and linear model differ. One relatively robust finding however is especially with nonlinear model that the substitution elasticity has increased during the estimation period, and this increase seems to be associated with speeding up of non-neutral technical change. This can be seen from the following Figure 4. in which we drawn the time varying estimate of the elasticity of substitution from equation (6.8).17 We have controlled the labour augmented technical progress by countries.
In the Figure, the elasticity of substitution between capital and labour increases from 0.53 to 0.83 during the sample period.
17 Castro and Coen–Pirani (2005) and Depuy and Marey (2004) find that the US production function shifted in a non neutral way over the last decades due to technical and organizational changes. They suggest that the elasticity of substitution depends in non neutral way of the technical change.
Figure 4. Time varying elasticity of substitution between
capital and labour
0.4 0.5 0.6 0.7 0.8 0.9 1
1970 1975 1980 1985 1990 1995 2000 2005
Dotted lines are 95% confidence intervals.
7 Conclusions
This paper’s estimation results suggest that technical progress differ widely across countries, both in levels and in growth rates. In particular, in the ICT – using and producing countries, technical progress has been rapid and even accelerating, while in a number of European large countries technical progress has been rather slow. The results provide some evidence that the country differences in technical progress and R&D investments are related. Those countries that have fulfilled the Lisbon agenda and invested heavily on R&D, have also enjoyed rapid growth, generated primarily by the speeding up of technical progress.
The estimation results based on the CES-production function show that the substitution elasticity between capital and labour may have been on average below one during the last 35 years. However, we are unable to say precisely what the right level is because the estimation results in various specifications differ widely. Nevertheless, one robust finding is that the substitution elasticity has increased over time.
Hence, our results suggest that there is no growth paradox in Europe: R&D factor and the elasticity of substitution between capital and labour that have been known to be important factors of economies’ growth potential, do explain a significant part of the divergent growth performance of the European economies as well.
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Appendix
Figure A1. Productivity and capital intensity
.015
.01
.045 .050 .055 .060 .065 .070 .075 .080
.06 .08 .10 .12 .14 .16 .18 K/L
Q/L
UNITED STATES
Figure A2.1 R&D expenditure, total business enterprise,
% of GDP
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Finland Sweden Denmark Ireland United States
Source: OECD STAN
Figure A2.2
Table A1. Pool unit root test
Statistic Probability Observations Transformations Levin, Lin&Chu 6.65 1.00 1344 level
-5.42 0.00 1341 first difference Null hypothesis: unit root
Table A2. Wald test for parameter equality by individual countries
F-statistics Degrees of freedom Probability
K 16.47 12.00 0.00
L 14.42 12.400 0.00 Linear trend 29.03 12.411 0.00
Quadratic trend 36.96 12.411 0.00 Null hypothesis: parameters equal
Table A3. Estimation results of CES production function t-ratios are inside parentheses. δ is set at 0.30.
Table A4. Estimation results of CES production function United Kingdom .59
(4.30) t-ratios are inside parentheses. δ is set at 0.30.
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