Productivity Development in Selected Central European Countries Measured by the Sato Production Function

Roubalová, Lenka; Viskotová, Lenka Article Productivity Development in Selected Central European Countries Measured by the Sato Production Function Review of Economic Perspectives Provided in Cooperation with: Masaryk University, Faculty of Economics and Administration Suggested Citation: Roubalová, Lenka; Viskotová, Lenka (2018) : Productivity Development in Selected Central European Countries Measured by the Sato Production Function, Review of Economic Perspectives, ISSN 1804-1663, De Gruyter, Warsaw, Vol. 18, Iss. 4, pp. 353-370, https://doi.org/10.2478/revecp-2018-0018 This Version is available at: https://hdl.handle.net/10419/194200 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ Review of Economic Perspectives – Národohospodářský obzor Vol. 18, Issue 4, 2018, pp. 353–370, DOI: 10.2478/revecp-2018-0018 © 2018 by the authors; licensee Review of Economic Perspectives / Národohospodářský obzor, Masaryk University, Faculty of Economics and Administration, Brno, Czech Republic. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 3.0 license, Attribution – Non Commercial – No Derivatives. Productivity Development in Selected Central European Countries Measured by the Sato Production Function Lenka Roubalová, 1 Lenka Viskotová 2 Abstract: In this paper, we investigate the relationship between economic output, labour and capital in the Visegrád Four, Austria and Germany. The main objective is to determine the type of technological progress in these countries over time, specifically in the period 1995–2015. The Sato production functions (a special case of the linearly homogeneous production function) for all the aforementioned countries are estimated using linear and nonlinear techniques. In addition to the original Sato production function, we propose modifying it in using a time variable, which allows us to analyse the development of productivity over time. Based on the NLS estimates of this modification, we create isoquant maps and calculate the value of the marginal rate of technical substitution of labour for capital to identify the nature of technological progress typical for each country. We also compare the properties of both the OLS and NLS estimates. The results are quite specific to individual countries, but there is some room for generalization. Key words: production function, Sato production function, isoquants, nonlinear least squares method, technological progress JEL Classification: C51, D24, O47 Received: 18 April 2018 / Accepted: 18 September 2018 / Sent for Publication: 28 November 2018 Introduction The idea that production is a function of factor inputs comes from classical economists, mainly Malthus and Ricardo (Blaug, 1985). In fact, it was Turgot that recognized and described the law of diminishing returns in agriculture, which is generally understood to 1 Mendel University in Brno, Faculty of Business and Economics, Department of Statistics and Operation Analysis, Zemědělská 1, 613 00 Brno, Czech Republic, e-mail: [email protected] 2 Mendel University in Brno, Faculty of Business and Economics, Department of Statistics and Operation Analysis, Zemědělská 1, 613 00 Brno, Czech Republic, e-mail: [email protected] Review of Economic Perspectives 354 be the basis of the production function theory. As Humphrey (1997) describes, Malthus later presented his logarithmic production function, Ricardo introduced his quadratic production function and Thünen described the exponential production function based on the statement that marginal product declines geometrically. He also discovered that output per labour unit can be defined as a function of capital per labour unit. We should also emphasize Marshall, who based a prototypal neoclassical growth model on the aggregate production function, and economists like Wicksteed, Walras and Wicksell, who used the production function to show how the total product is exhausted by the sum of factor payments distributed according to marginal productivity. The currently wellknown and frequently used Cobb-Douglas production function was originally described by Wicksell (Humphrey, 1997). In our field, standard production function specifications have been commonly used to analyse the relationship between inputs, typically labour (L), capital (K) and production (Q), at both the microeconomic and macroeconomic level. Inter alia, the Cobb-Douglas function, the Constant Elasticity of Substitution (CES) function in the form described by Kmenta, the linear function and the Leontief production function are all widely used as tools of economic analysis in various areas of the neoclassical tradition. Aggregate production functions have been used to investigate the productivity of capital and labour, the distribution of income at a national level, business cycles and also economic growth. The production function used for economic growth analysis pertains to an approach mainly related to the Keynesians. The concept of technological progress in the long run does not treat the production function as fixed, but it introduces changes in the production process. Specifically, technological progress decreases the quantity of inputs needed to achieve a given output. Economic theory classifies technological progress into three types, based on Hicks (1932). In the case of neutral technological progress, the ratio between the marginal product of labour and the marginal product of capital is not affected. Labour-saving technological progress is typical when technical advances in capital are experienced and they increase the marginal product of capital relative to the marginal product of labour. When educational or skill levels rise, marginal productivity of labour increases relative to the marginal product of capital and capital-saving technological progress can be observed (Besanko and Braeutigam, 2007). With suitable parametrization, the aforementioned production functions satisfy the assumption of linear homogeneity that corresponds to the economic assumption of constant returns to scale. These functions are characterized by some unique properties: both the average product of labour and capital and the marginal product of labour and capital can be expressed as a function of the capital-labour ratio, k = K/L, only. Thus, the average products and the marginal products will remain the same as long as the capitallabour ratio remains constant. Therefore, these products are homogeneous of degree zero in the variables K and L. Euler’s theorem 𝐾 𝜕𝑄 𝜕𝐾 + 𝐿 𝜕𝑄 𝜕𝐿 =𝑄 implies that the total quantity of output Q is exhausted by the distributive shares of all the inputs if each input factor is paid the amount of its marginal product and the pure economic profit is zero. Volume 18, Issue 4, 2018 355 The elasticity of substitution in production measures how easy or difficult it is to substitute one input for another, the typical case being the substitution of capital for labour. The elasticity of substitution is very often assumed to be constant. Production functions incorporating this property form the class of so-called constant elasticity of substitution (CES) functions. Among these functions, an important role is played by the CES function published by Kmenta (1967). Many authors have analysed the class of CES functions and modified it to overcome some interrelated difficulties such as the problem of constancy of the elasticity of substitution between labour and capital and the fact that obstacles are encountered when defining elasticity in the case of more than two inputs. For more detailed information related to the various CES function modifications, see Mishra (2007). The assumption of constant elasticity of substitution sometimes seems to be very confining, which makes the CES function a bit disadvantageous. Moreover, Henningsen and Henningsen (2012) state that convergence problems and instability are the main problems of the non-linear CES function estimation. These problems often arise due to a non-smooth objective function with large flat areas, or due to the discontinuity of the CES function where the elasticity of substitution is one. In other words, it is quite difficult to estimate the CES function parameters. To avoid these limitations, we draw our attention to the non-CES production function defined by Sato (1964), see also Sato (1975). Sato based his work on the modification of the CES function and proved that this function can be extended to the n-input case or also to the case of the variable elasticity of substitution, where the elasticity depends on the output level. For the Sato function, the elasticity of substitution might vary across the isoquant as the output varies. Therefore, the Sato production function can rather be characterized as a non-CES production function. The Sato production function is linearly homogeneous, so it exhibits constant returns to scale. Makin and Strong (2013) emphasize some advantageous properties of this function; especially, they were able to prove that this function can be applied not only to microeconomic data, but also at the aggregate level. Despite the fact that there is no assumption of constant elasticity of substitution, we also find it useful that the Sato function contains less parameters to be estimated than the CES production function, it can be linearized by inverse transformation instead of the logarithmic transformation that is needed in the case of the CES function and finally, this function is not frequently used, which creates an opportunity for new insight into using production functions to be gained. The main aim of this paper is to determine the nature of technological progress of the Visegrád Four countries, Austria and Germany. For this purpose, we use the original Sato production function and its modification employing a time variable, which allows us to analyse the development of productivity over time. In this paper, the Visegrád group is extended by two more countries, Austria and Germany. According to Heczková (2014), Slovakia, Poland, Austria and mainly Germany are the most important trading partners of the Czech Republic. The share of these countries in total turnover was about 46.5 % in 2013, which indicates that these countries affect the import and export dynamics of the Czech Republic substantially. Germany has been Hungary’s most important trading partner for years and in 2015, Austria was the third most important (European Commission, 2016). The geographical location of these countries also supports Review of Economic Perspectives 356 the idea of grouping these countries. As this paper analyses productivity at the country level, we consider these trading relationships to be the main reason for grouping all of these six countries. Material and Methods To estimate the parameters of the input factors of the Sato production function, selected methods of non-linear least squares (NLS) and ordinary least squares (OLS) are used. The nonlinear model estimation is computed in iterations using numerical methods; see Levenberg (1944) and Marquardt (1963). For more details related to the NLS method, see Greene (2012). The Sato production function is specified by 𝑌=𝐿2𝐾2/(𝑎𝐿3+𝑏𝐾3), (1) where Y represents output, K, L are capital and labour inputs, respectively, and a, b are parameters, where a > 0, b > 0 (Sato, 1975). The original Sato function assumes parameters a and b to be constant. Due to the purpose of the further analysis, which is focused on production function changes over time, we modify the original Sato production function as follows. Parameters a, b, are intended to be a function of time. This means both parameters can vary over time, which implies that the production function and the marginal product of the individual inputs vary as well. Using a linear function, parameters a and b can be expressed as 𝑎𝑡=𝑎0+𝑎1𝑡, (2) 𝑏𝑡=𝑏0+𝑏1𝑡, (3) where t is time. Then, the time-augmented version of the Sato production function (1) can be expressed as 𝑌=𝐿2𝐾2/(𝑎0𝐿3+𝑎1𝑡𝐿3+𝑏0𝐾3+𝑏1𝑡𝐾3), (4) where a0, a1, b0, b1 are the parameters of the time-augmented version of the original Sato function. Both types of the function, the original (1) and the augmented (4), are nonlinear in parameters. These functions can be linearized and then all parameters can be estimated using the OLS procedure. Using inverse transformation, the augmented Sato production function is described by the equation 1 𝑌=𝑎0 𝐿/𝐾2+𝑎1 𝑡𝐿/𝐾2+𝑏0 𝐾/𝐿2+𝑏1 𝑡𝐾/𝐿2. (5) If the time variable is omitted, we obtain a simplified model representing the linearized version of the original Sato production function 1 𝑌=𝑎𝐿/𝐾2+𝑏𝐾/𝐿2. (6) The quality of the model is evaluated by the coefficient of determination R2. Theil’s U (Theil, 1966) is used to compare the models to each other from the point of view of the overall accuracy of the forecast. Theil’s U has a minimum of 0, the naive models yield U = 1, the lower the value, the better the accuracy of the forecast. We also use information criteria AIC, BIC, HQC to compare models of the original and the time augmented function (with a different number of parameters). Volume 18, Issue 4, 2018 357 Considering the map of isoquants of the estimated Sato production function in the LKplane, we can find the economically efficient region of production, where the isoquants are downward-sloping and production costs are minimized. The suitable segment of each isoquant stretches between the two points where the slope of the tangent line is either zero (the tangent line is horizontal) or equal to ∞ (the tangent line is vertical). These points satisfy the assumption 𝑀𝑃𝐿=0, (7a) 𝑀𝑃𝐾=0, (7b) where MPL is the marginal product of labour and MPK is the marginal product of capital. The marginal products are defined as the partial derivatives of the production function. In the case of the Sato production function, the borders of the economically efficient region are defined by the formulas 𝐾= √2𝑎 𝑏 3𝐿, (8a) 𝐾= √𝑎 2𝑏 3𝐿. (8b) The plots of these equations are the straight lines passing through the origin of the isoquant map. The region where backward-bending or upward-sloping isoquants are situated corresponds to the uneconomic region of production. With respect to the objective of the technological progress classification, the marginal rate of technical substitution (MRTS) of capital for labour is given by the ratio of the marginal product of labour MPL and the marginal product of capital MPK. When calculating MRTS for the given production function, a straight line from the origin can be constructed, which represents a ray identifying the tangent point of the isoquant and its tangent. The MRTS can then be found for individual isoquants during the period under observation. If the marginal rate of technical substitution is decreasing during this period, the absolute value of the isoquant slope is decreasing, and the isoquants become flatter. This indicates that MPK increases more rapidly than MPL and labour-saving technological progress is identified. Likewise, if the marginal rate of technical substitution is increasing during the period being observed, the absolute value of the isoquant slope is increasing, and the isoquants become steeper. MPK therefore increases at a slower pace than MPL and capital-saving technological progress can be identified. In the case of neutral technological progress, lower amounts of labour and capital are needed; therefore, the isoquant corresponding to the given level of output shifts inward, but in contrast to the two previous situations, in this case, the marginal rate of technical substitution is unchanged along the ray from the origin. Neutral technological progress keeps the isoquant shape unchanged. For a more detailed microeconomic explanation, see Besanko and Braeutigam (2007). The methods outlined above are applied to data sourced from the EU KLEMS database. Gross output at current basic prices is used for economic output Y, capital K is represented by the nominal gross fixed capital formation, both in millions of national curren- Review of Economic Perspectives 358 cies, and labour L is represented by the total number of employees in thousands. The data are in the form of annual time series for the period 1995–2015, except for Poland, where the period 2003–2015 is used due to limited data availability. In our case, MRTS is determined based on the isoquant corresponding to the average output and the axis of the intersection of the economically efficient regions in all given years. There is a problem due to the empty intersection for Germany, which is why there are no values of MRTS for this country for 1995–1999. All calculations were run in the MATLAB R2017b computational system. The level of significance was set to 0.05, as usual. Results To avoid a spurious regression, we tested for cointegration of the time series used for this analysis (ADF and KPSS tests applied on the cointegration regression residuals) and we can conclude that the time series are cointegrated. Even though heteroskedasticity occurs rarely in time series regressions, it was tested using the Breusch-Pagan test. Homoskedasticity was confirmed with the exception of three cases (the OLS model of the original function (6) for the Czech Republic and for Slovakia and the OLS model of the augmented function (5) for Hungary), where p-values were slightly lower than the significance level – this can be the consequence of the relatively short time series. Graphical analysis of the residuals did not indicate the problem of heteroskedasticity. The Lilliefors test proved the normality of residuals in all cases. Autocorrelation of the residuals was investigated for up to a maximum lag of 10; there are no significant autocorrelations for any residuals of all models. As the form of the functions is determined by economic theory, we do not focus on specification tests. The most important results of t-tests and F-tests are presented and discussed below in the text regarding individual models. The Sato production function parameters obtained via NLS and OLS for the original production function, given by equations (1) and (6), are very similar, as well as the values of Theil’s U. The parameters meet the condition of a > 0 and b > 0. The coefficients of determination explained more than 90 % of the variability of output. The results summarized in Tab. 1 and 3 show that the original Sato production function parameters estimated via NLS and OLS do not differ dramatically and the models are very similar from the point of view of its quality. Despite the fact that not all parameters are statistically significant, the p-values of the F-test are under 0.001 for all countries, all these OLS models can thus be considered statistically significant. These results prove that the Sato production function is suitable for describing the output and input relationship. In Tab. 2 and Tab. 4, the p-values of the t-test for the original Sato production function parameters estimated via NLS and OLS are presented. Tab. 2 shows that parameters a, b estimated via the NLS method are statistically significant for the original Sato function models of all six countries. Tab. 4 shows that, in the case of the OLS estimation, the parameters a and b are statistically significant for Poland, the Czech Republic, Slovakia and Hungary; in the case of Austria and Germany, the estimated parameter a is not significant. Volume 18, Issue 4, 2018 359 Table 1 Original Sato production function (1) parameters estimated via NLS, coefficient of determination, Theil’s U and information criteria. PL CZ SK HU AT DE a 1.383 17.197 0.587 108.521 2.161 1.375 b 7.107×10-5 7.504×10-7 4.684×10-4 1.525×10-8 3.347×10-5 9.206×10-5 R2 88.53 % 97.02 % 90.59 % 94.11 % 96.67 % 78.04 % U 1.025 0.661 1.356 1.526 1.023 2.135 AIC 331 604 464 704 480 597 HQC 330 605 464 704 480 598 BIC 332 606 466 706 482 600 Source: own processing Table 2 P-values of the t-test for the original Sato production function (1) parameters estimated via NLS. PL CZ SK HU AT DE a 4.26×10-6 1.54×10-15 5.81×10-10 1.22×10-10 2.34×10-15 3.89×10-9 b 5.23×10-5 6.35×10-8 7.69×10-5 7.23×10-7 0.0290 0.0407 Source: own processing Table 3 Original Sato production function (6) parameters estimated via OLS, coefficient of determination, Theil’s U and information criteria. PL CZ SK HU AT DE a 1.168 16.130 0.396 49.410 2.291 1.384 b 8.772×10-5 9.299×10-7 1.035×10-3 3.645×10-8 1.324×10-5 1.029×10-4 R2 92.22 % 97.32 % 94.41 % 84.05 % 97.09 % 67.95 % U 1.029 0.759 2.375 2.205 1.118 2.105 AIC -379 -713 -473 -719 -612 -674 HQC -379 -712 -473 -719 -612 -674 BIC -378 -711 -471 -717 -610 -672 Source: own processing Table 4 P-values of the t-test for the original Sato production function (6) parameters estimated via OLS. PL CZ SK HU AT DE a 1.98×10-8 7.70×10-20 6.63×10-10 6.46×10-12 0.3978 0.1453 b 7.52×10-7 5.42×10-9 5.90×10-5 1.00×10-6 2.95×10-17 2.92×10-7 Source: own processing Review of Economic Perspectives 360 Our results also prove that both estimating techniques yield similar values of estimated parameters, so from this point of view, linearization could be considered an alternative approach that is equivalent to nonlinear estimation. The information criteria AIC, BIC, HQC are used to compare models of the original Sato production function (1) and (6) with the models of the time augmented function (4) and (5), see Tab. 5 and Tab. 7. It is also necessary to highlight that the values of the estimated parameters a and b differ significantly between countries. This is caused by the fact that the variable K is expressed in thousands of national currencies. Table 5 The augmented Sato production function (4) parameters estimated via NLS, coefficient of determination, Theil’s U and information criteria. PL CZ SK HU AT DE a0 0.860 16.201 6.060×10-1 91.922 2.117 1.251 a1 0.041 -0.202 -1.445×10-2 -1.817 6.861×10-2 -0.047 b0 1.507×10-4 1.601×10-6 1.188×10-3 4.041×10-8 1.239×10-4 0.003×10-2 b1 -7.603×10-6 -2.400×10-8 -2.223×10-5 -9.018×10-10 4.098×10-6 4.733×10-6 R2 97.56 % 99.32 % 97.12 % 97.36 % 99.21 % 97.86 % U 0.427 0.368 0.852 1.08 0.457 0.620 AIC 316 577 443 691 453 553 HQC 315 578 444 692 454 553 BIC 318 581 447 695 458 557 Source: own processing Table 6 P-values of the t-test for the augmented Sato production function (4) parameters estimated via NLS. PL CZ SK HU AT DE a0 1.01×10-4 1.08×10-14 2.14×10-8 2.15×10-7 2.46×10-10 3.38×10-6 a1 0.3852 0.0262 0.0376 0.2182 1.49×10-6 0.0005 b0 3.09×10-5 1.50×10-9 5.90×10-6 1.33×10-6 0.0088 0.0022 b1 0.0038 0.0134 0.0317 0.0031 0.0202 0.0269 Source: own processing Based on a comparison of the original and the augmented versions, we can state that the value of Theil’s U is higher for the original Sato function models. This indicates that the model with the time-augmented function is better. The value of Theil’s U is considerably higher than 1 in most cases of the original function; meanwhile, in the case of the time-augmented version, Theil’s U is 0 < U < 1 in most cases; in the case of Hungary, it is slightly higher than 1. The coefficient of determination proves that the nonlinear models explain a higher percentage of the variability of the dependent variable. For NLS (see Tab. 5), results were comparable with the results obtained via OLS models (see Tab. 7); in the majority of cases, the coefficients of determination and the Theil’s U values coming from OLS are very close to the ones obtained from the nonlinear models. Volume 18, Issue 4, 2018 367 dence of low capital productivity and limited capital stock (Biea, 2015), and also serious problems with a high unemployment rate connected in part to the large gap in unemployment rate between Roma and non-Roma workers. The developments in this area both in Germany and Slovakia show reasons to substantially improve labour force qualification. In spite of the fact that the Sato version of the production function is not widely used by authors dealing with the issue of productivity, there is an example in the research of Makin and Strong (2013), who have also used the Sato production function. They also use the time variable as an input, but they differ in rather focusing on the elasticity of substitution and factor productivity development and its change as a consequence of labour, product and capital market reforms. Their results confirm that the Sato production function is suitable for analysis at the aggregate level. Based on this analysis proving that the Sato function is suitable for the output and input relationship investigation at the aggregate level, the objective of further research could be to use this specification for economic output and production gap estimation. There are several papers using the Cobb-Douglas production function for this purpose. Conclusion In conclusion, this paper shows how the combination of labour and capital inputs has changed during the period of 1995–2015 in the Visegrád Four, Germany and Austria. The results show that the Sato production function is applicable for analysis at the aggregate level for the conditions of selected Central European countries. The OLS and NLS procedures provide very similar models and parameters compared to the original version of the Sato production function. However, the relationship between economic output, labour and capital is not fixed during the period under observation. Therefore, the time-augmented Sato production function is proposed in order to capture the development of the relationship in each year of the period. In the case of the time-augmented Sato function, the quality of NLS models is higher in respect to the original version of the Sato function. This means that the time variable is an important input. Additionally, it is the only way to estimate macroeconomic production functions for selected periods continuously year on year. Such a model is more appropriate because it provides information about technological progress. However, linearization was not successful in the case of the augmented Sato function. The development of the isoquant tangent line slope detects labour-saving technological progress in the Czech Republic, Hungary, Poland and Austria and capital-saving technological progress in Slovakia and Germany. In the Czech Republic, Hungary, Poland and Austria, where the results show that technological progress was achieved through capital equipment improvement, national policies should concentrate on this way of improvement and find ways and resources to support capital and technological enhancement to contribute to faster productivity growth. Similarly, in Slovakia and Germany, where the results show that technological progress is achieved through the increasing of skills and qualifications of the workforce, national policy should primarily focus on activities leading to the education of the workforce. It should also be supported by an adequate social benefits policy to avoid the lack of motivation to work. These recommendations stemming from the results are very general. To provide sufficient Review of Economic Perspectives 368 recommendations for national policies, it would be necessary to monitor the situation and repeat this kind of analysis periodically to verify whether the way technological progress occurs changes or not. More complex analysis could help to reveal whether there is still room for improvement in capital or labour, or whether it is exhausted and policies should rather concentrate on another input, labour in the case of the first group of countries and capital in the case of Slovakia and Germany. Additionally, analysis at the sector level would also be necessary to make policy targeting more efficient. The results we found can be verified by other authors, but there is also space for further research. The implementation of the time-augmented production function could be extended and applied for widely used functions, e. g. the Cobb-Douglas and the CES production function. This methodology could be used for all European countries to get an overview of the type of technological progress found in each of the countries of the EU. It can be a base for further research focusing on the sector level using selected countries - this could provide valuable information for economic entities operating within those sectors. We can presume that productivity growth of tradable sectors whose outputs are traded internationally will be faster than in the case of non-tradable sectors. 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