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The specification of the aggregate production function in the presence of inefficiency
Economics Letters 81 (2003) 223 – 226 www.elsevier.com/locate/econbase
The specification of the aggregate production function in the presence of inefficiency Richard Kneller a, Philip Andrew Stevens b,* a
Leverhulme Centre for Research on Globalisation and Economic Policy, School of Economics, University of Nottingham, Nottingham NG7 2RD, UK b National Institute of Economic and Social Research, 2 Dean Trench Street, Smith Square, London SW1 3HE, UK Received 9 January 2003; received in revised form 23 April 2003; accepted 6 May 2003
Abstract In this paper we consider the impact of the specification of production technology on technical efficiency. We reject the Cobb-Douglas specification of aggregate production in favour of a more general translog form. However, the effect of functional form on the measured efficiency terms is actually quite small. Both of these results are robust to the adjustment of labour to account for years of schooling. D 2003 Elsevier B.V. All rights reserved. Keywords: Production function; Cobb-Douglas; Translog; Efficiency; Panel data JEL classification: O40; O47
1. Introduction The majority of studies of economic growth and development have assumed that output is produced according to a two-input Cobb-Douglas aggregate production function. This reliance was questioned by Duffy and Papageorgiou (2000), who found that they could reject the Cobb-Douglas specification using a panel of 82 countries over a 28 year period. The specification of the production function has ramifications for any measure that uses it as a baseline (e.g. McGrattan and Schmitz, 1998). Since methods such as stochastic frontier analysis (SFA) measure the efficiency of a country’s production relative to the estimated production frontier, it is important that the frontier be appropriately specified. In this paper we estimate a more general stochastic production frontier using the same panel as Duffy * Corresponding author. Tel.: +44-20-7654-1927; fax: +44-20-7654-1900. E-mail address: [email protected] (P. Andrew Stevens). 0165-1765/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1765(03)00173-3
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and Papageorgiou (2000) in order to assess the impact of the specification of technology on estimated inefficiency in production. We utilise the translog specification of frontier technology because it provides a good first-order approximation to a broad class of functions, including the CES, and includes the Cobb-Douglas as a special case. We also investigate the effect of how the labour input is measured, by adjusting it to allow for the average years of schooling.
2. Empirical model and results Our empirical model is a translog production function of the form yit ¼ b0 þ b1 kit þ b2 lit þ b3 kit2 þ b4 lit2 þ b5 kit lit þ b6 t þ eit þ git
ð1Þ
where y is the logarithm of output for country i at time t, k is the (log of) capital stock, l is (the log of) labour, e reflects the random character of the frontier, due to measurement error or other effects not captured by the model and g represents the output-orientated Farrell measure of technical efficiency (0 < exp[g] V 1), i.e. exp[g] measures the ratio of actual to potential output (Table 1). Following Battese and Coelli (1995), the inefficiency effect is obtained by a truncation of the normal distribution N(lit, r2). The mean level of inefficiency is defined by lit ¼ d0 þ d1 t.1 The data for our analysis is identical to that of Duffy and Papageorgiou (2000) and comes from the World Bank STARS dataset. The data is for 82 countries and covers the period 1960–1987. Our results are presented in Table 2. Model 1 refers to the translog stochastic production frontier described in Eq. (1) above and Model 2 refers to the Cobb-Douglas specification. It is clear that the hypothesis that b3 = b4 = b5 = 0, i.e. that production is better described by the Cobb-Douglas specification, cannot be accepted (the likelihood ratio test of 207.9 is significant at the 1% level). For the second two sets of results, we follow Duffy and Papageorgiou (2000) and include a measure of human capitaladjusted labour supply l* = ln(HL) where H is the mean years of schooling of the labour force. Again, a likelihood ratio test (v2 = 117.2) rejects the hypothesis that b3 = b4 = b5 = 0 at the 1% level. Like Duffy and Papageorgiou (2000), we find a significant, but small negative trend in the production frontier2. However, we also find that the estimated efficiency scores are also trended, but the negative value for d1 implies convergence, i.e. that the average distance from the frontier is negatively trended. Both of these results are consistent across specifications. The effect of specification of the frontier on the efficiency scores themselves is presented in Table 3. The first half of the table shows the descriptive statistics for the estimated efficiency scores. The effect of using the translog rather than the Cobb-Douglas specification in the model with unadjusted labour is to increase the mean and median levels of efficiency by around 1.5%. When labour is adjusted for human capital, these differences disappear. However, descriptive statistics only tell part of the story, in order to understand the effect of specification on efficiency scores, it is more appropriate to examine the correlations between the series. These are presented in the bottom half of Table 3. Here we see that the effect of changing the 1
For a more thorough description of a similar model, as well as a discussion of the effects of functional-form in an industrylevel context, see Kneller and Stevens (2002). 2 We also estimated models with a broken trend, but this had little effect on our results.
R. Kneller, P. Andrew Stevens / Economics Letters 81 (2003) 223–226
225
Table 1 Description of untransformed variables Variable a
Y K L H
GDP (billions of US$) Physical capital stock (billions of US$)a Labour force (population of working age) (millions) Mean years of education of the labour force aged 15 to 64
Mean
S.D.
119 315 24 4.67
403 1100 70 2.94
a
The capital stock and output data are expressed in 1987 constant local prices which are converted to US$ using the 1987 exchange rate data.
specification of the frontier on both the values of the efficiency scores and their orderings is relatively minor. The highest correlations (both Pearson and Spearman) are between models sharing the same measurement of the labour input, rather than those sharing functional form. McGrattan and Schmitz (1998) found that measures of TFP generated by a growth-accounting methodology were sensitive to both changes in factor shares and the measurement of human capital. Our results concur with the latter finding but suggest that the effect of the specification of the frontier on measures of productive efficiency is less important.
3. Conclusion If the production relationship is incorrectly specified, this will affect the output of methods such as stochastic frontier analysis, which measure the efficiency of a country’s production relative to the technically feasible maximum. Using a stochastic frontier analysis we reject the Cobb-Douglas Table 2 Stochastic frontier results Model 1 Translog Production b0 b1 b2 b3 b4 b5 b6
Model 2 CD
Model 3 Translog
frontier 5.347 0.072 1.031 0.032 0.003 0.040 0.021
(5.8) (1.0) (12.5) (11.5) (0.7) (6.7) (12.5)
0.936 0.863 0.138
(9.2) (189) (21.7)
0.023
Efficiency effects 0.422 d0 d1 0.102
(0.6) (2.2) (2.0) (17.9)
r2 c LL LR
0.576 0.895 711.8 145.8
Model 4 CD (11.5) (2.4) (6.0) (7.0) (2.6) (4.2) (8.4)
1.447 0.853 0.115
(11.3) (132) (14.0)
(11.9)
10.25 0.266 0.692 0.039 0.016 0.046 0.028
0.028
(7.1)
0.184 0.088
(0.3) (2.4)
0.449 0.057
(2.2) (3.2)
0.372 0.063
(1.1) (2.3)
0.525 0.877 815.8 155.31
(2.3) (17.7)
0.244 0.652 874.4 84.427
(3.4) (6.9)
0.289 0.696 933 88.771
(2.5) (6.3)
LL, Log likelihood; LR, Likelihood ratio test of the one-sided error; t-values in parenthesis.
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R. Kneller, P. Andrew Stevens / Economics Letters 81 (2003) 223–226
Table 3 Descriptive statistics and correlations between efficiency scores Min Max Mean Median Std. Dev. Pearson correlation Model 1 Model 2 Model 3 Model 4
Model 1
Model 2
Model 3
Model 4
0.224 0.957 0.798 0.841 0.125
0.213 0.952 0.783 0.825 0.130
0.286 0.947 0.789 0.828 0.120
0.268 0.944 0.789 0.829 0.120
1
0.973 1
0.912 0.913 1
0.916 0.944 0.987 1
0.962 1
0.910 0.904 1
0.912 0.942 0.985 1
Spearman’s rank correlation Model 1 Model 2 Model 3 Model 4 All correlations significant at the 0.01 level (2-tailed).
specification of aggregate production in favour of the more general translog. This result is robust to the adjustment of labour for years of schooling. However, the effect of functional form on the estimated efficiency terms is relatively minor and one which is smaller than that of the measurement of labour. Acknowledgements We would like to thank Mary O’Mahony, Nigel Pain, Marin Weale and participants at the ESRC-funded NIESR conference ‘The Role of Efficiency as an Explanation of International Income Differences’.
References Battese, G.E., Coelli, T.J., 1995. A Model for Technical Efficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics 20, 325 – 332. Duffy, J., Papageorgiou, C., 2000. A cross-Country Empirical Investigation of the Aggregate Production Function Specification. Journal of Economic Growth 5, 87 – 120. Kneller, R., Stevens, P.A., 2002. Absorptive Capacity and Frontier Technology: Evidence from OECD Manufacturing Industries. NIESR, Discussion Paper No. 202. McGrattan, and E.R., Schmitz, J.A., 1998. Explaining Cross-Country Income Differences, Federal Reserve Bank of Minneapolis Research Department, Staff Report 250.
The specification of the aggregate production function in the presence of inefficiency Richard Kneller a, Philip Andrew Stevens b,* a
Leverhulme Centre for Research on Globalisation and Economic Policy, School of Economics, University of Nottingham, Nottingham NG7 2RD, UK b National Institute of Economic and Social Research, 2 Dean Trench Street, Smith Square, London SW1 3HE, UK Received 9 January 2003; received in revised form 23 April 2003; accepted 6 May 2003
Abstract In this paper we consider the impact of the specification of production technology on technical efficiency. We reject the Cobb-Douglas specification of aggregate production in favour of a more general translog form. However, the effect of functional form on the measured efficiency terms is actually quite small. Both of these results are robust to the adjustment of labour to account for years of schooling. D 2003 Elsevier B.V. All rights reserved. Keywords: Production function; Cobb-Douglas; Translog; Efficiency; Panel data JEL classification: O40; O47
1. Introduction The majority of studies of economic growth and development have assumed that output is produced according to a two-input Cobb-Douglas aggregate production function. This reliance was questioned by Duffy and Papageorgiou (2000), who found that they could reject the Cobb-Douglas specification using a panel of 82 countries over a 28 year period. The specification of the production function has ramifications for any measure that uses it as a baseline (e.g. McGrattan and Schmitz, 1998). Since methods such as stochastic frontier analysis (SFA) measure the efficiency of a country’s production relative to the estimated production frontier, it is important that the frontier be appropriately specified. In this paper we estimate a more general stochastic production frontier using the same panel as Duffy * Corresponding author. Tel.: +44-20-7654-1927; fax: +44-20-7654-1900. E-mail address: [email protected] (P. Andrew Stevens). 0165-1765/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1765(03)00173-3
224
R. Kneller, P. Andrew Stevens / Economics Letters 81 (2003) 223–226
and Papageorgiou (2000) in order to assess the impact of the specification of technology on estimated inefficiency in production. We utilise the translog specification of frontier technology because it provides a good first-order approximation to a broad class of functions, including the CES, and includes the Cobb-Douglas as a special case. We also investigate the effect of how the labour input is measured, by adjusting it to allow for the average years of schooling.
2. Empirical model and results Our empirical model is a translog production function of the form yit ¼ b0 þ b1 kit þ b2 lit þ b3 kit2 þ b4 lit2 þ b5 kit lit þ b6 t þ eit þ git
ð1Þ
where y is the logarithm of output for country i at time t, k is the (log of) capital stock, l is (the log of) labour, e reflects the random character of the frontier, due to measurement error or other effects not captured by the model and g represents the output-orientated Farrell measure of technical efficiency (0 < exp[g] V 1), i.e. exp[g] measures the ratio of actual to potential output (Table 1). Following Battese and Coelli (1995), the inefficiency effect is obtained by a truncation of the normal distribution N(lit, r2). The mean level of inefficiency is defined by lit ¼ d0 þ d1 t.1 The data for our analysis is identical to that of Duffy and Papageorgiou (2000) and comes from the World Bank STARS dataset. The data is for 82 countries and covers the period 1960–1987. Our results are presented in Table 2. Model 1 refers to the translog stochastic production frontier described in Eq. (1) above and Model 2 refers to the Cobb-Douglas specification. It is clear that the hypothesis that b3 = b4 = b5 = 0, i.e. that production is better described by the Cobb-Douglas specification, cannot be accepted (the likelihood ratio test of 207.9 is significant at the 1% level). For the second two sets of results, we follow Duffy and Papageorgiou (2000) and include a measure of human capitaladjusted labour supply l* = ln(HL) where H is the mean years of schooling of the labour force. Again, a likelihood ratio test (v2 = 117.2) rejects the hypothesis that b3 = b4 = b5 = 0 at the 1% level. Like Duffy and Papageorgiou (2000), we find a significant, but small negative trend in the production frontier2. However, we also find that the estimated efficiency scores are also trended, but the negative value for d1 implies convergence, i.e. that the average distance from the frontier is negatively trended. Both of these results are consistent across specifications. The effect of specification of the frontier on the efficiency scores themselves is presented in Table 3. The first half of the table shows the descriptive statistics for the estimated efficiency scores. The effect of using the translog rather than the Cobb-Douglas specification in the model with unadjusted labour is to increase the mean and median levels of efficiency by around 1.5%. When labour is adjusted for human capital, these differences disappear. However, descriptive statistics only tell part of the story, in order to understand the effect of specification on efficiency scores, it is more appropriate to examine the correlations between the series. These are presented in the bottom half of Table 3. Here we see that the effect of changing the 1
For a more thorough description of a similar model, as well as a discussion of the effects of functional-form in an industrylevel context, see Kneller and Stevens (2002). 2 We also estimated models with a broken trend, but this had little effect on our results.
R. Kneller, P. Andrew Stevens / Economics Letters 81 (2003) 223–226
225
Table 1 Description of untransformed variables Variable a
Y K L H
GDP (billions of US$) Physical capital stock (billions of US$)a Labour force (population of working age) (millions) Mean years of education of the labour force aged 15 to 64
Mean
S.D.
119 315 24 4.67
403 1100 70 2.94
a
The capital stock and output data are expressed in 1987 constant local prices which are converted to US$ using the 1987 exchange rate data.
specification of the frontier on both the values of the efficiency scores and their orderings is relatively minor. The highest correlations (both Pearson and Spearman) are between models sharing the same measurement of the labour input, rather than those sharing functional form. McGrattan and Schmitz (1998) found that measures of TFP generated by a growth-accounting methodology were sensitive to both changes in factor shares and the measurement of human capital. Our results concur with the latter finding but suggest that the effect of the specification of the frontier on measures of productive efficiency is less important.
3. Conclusion If the production relationship is incorrectly specified, this will affect the output of methods such as stochastic frontier analysis, which measure the efficiency of a country’s production relative to the technically feasible maximum. Using a stochastic frontier analysis we reject the Cobb-Douglas Table 2 Stochastic frontier results Model 1 Translog Production b0 b1 b2 b3 b4 b5 b6
Model 2 CD
Model 3 Translog
frontier 5.347 0.072 1.031 0.032 0.003 0.040 0.021
(5.8) (1.0) (12.5) (11.5) (0.7) (6.7) (12.5)
0.936 0.863 0.138
(9.2) (189) (21.7)
0.023
Efficiency effects 0.422 d0 d1 0.102
(0.6) (2.2) (2.0) (17.9)
r2 c LL LR
0.576 0.895 711.8 145.8
Model 4 CD (11.5) (2.4) (6.0) (7.0) (2.6) (4.2) (8.4)
1.447 0.853 0.115
(11.3) (132) (14.0)
(11.9)
10.25 0.266 0.692 0.039 0.016 0.046 0.028
0.028
(7.1)
0.184 0.088
(0.3) (2.4)
0.449 0.057
(2.2) (3.2)
0.372 0.063
(1.1) (2.3)
0.525 0.877 815.8 155.31
(2.3) (17.7)
0.244 0.652 874.4 84.427
(3.4) (6.9)
0.289 0.696 933 88.771
(2.5) (6.3)
LL, Log likelihood; LR, Likelihood ratio test of the one-sided error; t-values in parenthesis.
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R. Kneller, P. Andrew Stevens / Economics Letters 81 (2003) 223–226
Table 3 Descriptive statistics and correlations between efficiency scores Min Max Mean Median Std. Dev. Pearson correlation Model 1 Model 2 Model 3 Model 4
Model 1
Model 2
Model 3
Model 4
0.224 0.957 0.798 0.841 0.125
0.213 0.952 0.783 0.825 0.130
0.286 0.947 0.789 0.828 0.120
0.268 0.944 0.789 0.829 0.120
1
0.973 1
0.912 0.913 1
0.916 0.944 0.987 1
0.962 1
0.910 0.904 1
0.912 0.942 0.985 1
Spearman’s rank correlation Model 1 Model 2 Model 3 Model 4 All correlations significant at the 0.01 level (2-tailed).
specification of aggregate production in favour of the more general translog. This result is robust to the adjustment of labour for years of schooling. However, the effect of functional form on the estimated efficiency terms is relatively minor and one which is smaller than that of the measurement of labour. Acknowledgements We would like to thank Mary O’Mahony, Nigel Pain, Marin Weale and participants at the ESRC-funded NIESR conference ‘The Role of Efficiency as an Explanation of International Income Differences’.
References Battese, G.E., Coelli, T.J., 1995. A Model for Technical Efficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics 20, 325 – 332. Duffy, J., Papageorgiou, C., 2000. A cross-Country Empirical Investigation of the Aggregate Production Function Specification. Journal of Economic Growth 5, 87 – 120. Kneller, R., Stevens, P.A., 2002. Absorptive Capacity and Frontier Technology: Evidence from OECD Manufacturing Industries. NIESR, Discussion Paper No. 202. McGrattan, and E.R., Schmitz, J.A., 1998. Explaining Cross-Country Income Differences, Federal Reserve Bank of Minneapolis Research Department, Staff Report 250.