Elasticity in CES Production Functions: New Estimates for Europe and Different Nesting Structures

Abstract

This study addresses the scarcity of empirical estimates on the elasticity of production factor substitutions, a crucial parameter in general equilibrium models for policy analysis. Employing the non-linear least squares estimation method, we determine the elasticity of substitution between capital, labor, energy, and materials in the Constant Elasticity of Substitution model across Europe at both aggregate and sectoral levels. Through rigorous analysis, we identify the optimal nesting structure of the production function for our data, rejecting widely used CES functional forms such as Cobb-Douglas and Leontief. Our findings reveal changes in elasticity of substitution over time, with Eastern Europe exhibiting greater ease of substitution compared to Western Europe, particularly between capital and labor. While elasticities in tertiary sectors diverge over time, those in energy-intensive sectors converge, though they remain statistically different, underscoring the necessity of region-specific elasticity sets in CGE models.

JEL Classification: C51, Model Construction and Estimation; D24, Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity

1. Introduction

In modern applied economics and especially in the field of environmental and climate policy, Computable General Equilibrium (CGE) models have become one of the leading tools for evaluating policy measures and scenarios (Böhringer et al. 2003; Dwyer 2015; Fossati and Wiegard 2002). Besides CGE models, macro-econometric, input-output, or linear programming models also use different types of nested production function with Constant Elasticity of Substitution (CES) to describe the production of an economy (Kemfert 1998). As noted by Jacoby et al. (2006) in a sensitivity analysis of the structural parameters of the MIT-EPPA CGE model, elasticities of substitution (EoS) are among the main drivers of the model results that determine the behavioral response of economic agents, such as producers, since they measure the ease or difficulty of substitution between inputs in economic production. Similarly, Antimiani et al. (2015) confirms the importance of the magnitude of substitution elasticities using a dynamic CGE model based on the GTAP framework with sector-specific values for capital-energy and inter-fuel elasticities. According to their results, a change in the elasticity values generates a different sectoral and regional distribution of impacts, and a lower flexibility of the energy substitution possibilities induces more expensive abatement efforts.

The criticism of EoS (denoted by $\sigma$ ) estimates for CGE models has several dimensions. First, however, as van der Werf (2008) points out, many models use different values of EoS, even when the models use the same nesting structure. Second, many research papers use elasticity estimates taken from the literature, but empirical validations for the nesting structures and values chosen are missing (van der Werf 2008). Furthermore, Sorrell (2014) adds other difficulties in using empirical studies to infer values of EoS specifically for CGE models originating from the fact that CGE models typically: (1) differ from empirical studies in the manner in which individual inputs are aggregated and in the level of sectoral aggregation; (2) require estimates of the elasticity of substitution between nests of inputs, while the parameters estimated by most empirical studies relate to individual pairs of inputs; and (3) define production function by means of Hicks/Direct Elasticity of substitution (HES), while the most empirical studies estimate other types of elasticity of substitution.

An applied general equilibrium model SAGE (Marten et al. 2019) of the U.S. economy is calibrated using the sector-specific elasticity estimates by Koesler and Schymura (2015) for a pool of forty countries using the World Input-Output Database (WIOD). The sectors in the SAGE model adhere to the sector structure used by Koesler and Schymura (2015) only partly. EoS between capital, labor, and energy in the CGE model ICES (Intertemporal Computable Equilibrium System; Parrado and De Cian 2014) are calibrated based on Carraro and De Cian (2012) who use a non-nested CES production function with an elasticity of substitution estimated as 0.38. Parrado and De Cian (2014) use this same elasticity value for eight world regions.

Another example of regional data incoherence is a CGE model for Korea by Oh et al. (2020) with a three-level CES nesting structure ((KL)E)M. ¹ The Korean model is calibrated based on Okagawa and Ban (2008) who estimate substitution elasticities for a pool of fourteen major world economies. The well-known Emissions Prediction and Policy Analysis (EPPA) recursive-dynamic CGE model for the world economy (Paltsev et al. 2005) uses for its elasticity calibration values proposed by Cossa (2004) who conducted a literature review and expert elicitation. The EPPA model uses the same elasticity values across sectors and countries.

The I3E model (de Bruin and Yakut 2020) is a country-specific intertemporal CGE model focused on assessing the economic and environmental impacts of climate policies specifically for Ireland. The elasticities of the CES production function in the I3E model are based on expert judgment without any further specification. The same complication applies for the calibration of the JRC-GEM-E3 model (Vandyck et al. 2016), which uses expert-based substitution elasticity values in a two-level (KL)(EM) CES production function. Kiuila et al. (2019) propose for the substitution between capital and labor and capital and electricity a value of 0.2 without further specifying the source for calibration.

An elasticity for capital-energy substitution of 0.5 is assumed across sectors in the GTAP-E model (Burniaux and Truong 2002) based on a literature review. The substitution between the capital-labor-energy composite and materials is assumed to be in Leontief form without empirical verification of its suitability for the GTAP-E model. Similarly, the authors assume a Leontief specification in the energy-materials nest and a Cobb-Douglas specification in the capital-labor nest in the case of a Japanese CGE model (Huang and Kim 2019). Montalbano et al. (2021) explore the relationship between energy efficiency and productivity using firm-level data using the Cobb-Douglas production function. An empirically not validated Cobb-Douglas structure for capital-labor substitution is also used by the World Induced Technical Change Hybrid (WITCH) integrated assessment model (Emmerling et al. 2016). Other elasticities used to calibrate the WITCH model are based on a review of the literature from the 1990s. Moreover, the model uses the same elasticities across sectors and countries. Analogously, the CGE model NEWAGE (National European World Applied General Equilibrium) developed within the REEM Pathways project (REEEM Project 2019) assumes a Cobb-Douglas EoS between capital and labor and a Leontief EoS between KLE composite and materials, the (KL)E EoS is calibrated at 0.5. The values are calibrated based on Beestermöller (2016). The HyBGEM model calibrated for Portugal assumes the same nesting structure as NEWAGE and uses the empirical elasticity estimates provided by Okagawa and Ban (2008).

Regarding the estimation methods for the parameters of the CES production function, Gechert et al. (2022) collected 121 studies that estimated substitution elasticities. The vast majority of estimates come from single-level production functions with capital and labor as inputs. Almost 70 percent of the considered studies estimated EoS using single first-order conditions (FOCs) for capital, labor, or their systems. Another large part of the studies used the linear Kmenta (1967) approximation of the production function. Henningsen and Henningsen (2011) cites main complications of this approach as a method reliable only for the Cobb-Douglas production function, for example, when $\sigma \to 1$ , and the Kmenta approximation is by itself a truncated Taylor series with the remainder term being an omitted variable.

Non-linear estimation techniques have been applied by only a limited number of research papers (Gechert et al. 2022). Kemfert (1998) estimated the substitution elasticity for three nested two-level CES production functions for the entire German industry and individual industrial sectors. Henningsen and Henningsen (2011) re-estimated this study using the same data and a non-linear least squares estimation method. They applied several estimation approaches that yield robust results significantly different from those provided in the original paper. Koesler and Schymura (2015) used the World-Input-Output Database (WIOD) to estimate EoS for a three-level four-input nested CES ((KL)E)M production function using non-linear least squares estimation method developed by Henningsen and Henningsen (2011). They estimated the substitution elasticity for thirty-six sectors pooled across the forty countries included in WIOD over a period of twelve years (1995–2006). This dataset has the advantage of having a larger number of observations, but since the WIOD covers not only European countries but also thirteen other major world countries, Koesler and Schymura (2015) loses geographic consistency. Moreover, they assumed one particular nesting of CES production function without empirically validating it.

The different conclusions of the mentioned studies serve as an example of limited regional transferability of substitution elasticity estimates. The vast majority of them focus on well-developed western countries, but to our knowledge, there is no study focused primarily on Central and Eastern European (CEE) countries so far. As a consequence, the lack of adequate estimates of the elasticity of substitution specific for the CEE countries is greater than in other regions. In this paper, we seek to fill this gap by providing consistent sectoral estimates of elasticity of production factors’ substitution for the all the European countries included in the WIOD (further denoted as EU) and two regions—the CEE countries and the rest constituting the old EU member states plus Norway and Switzerland (WEST). We report the accompanying substitution elasticity for the three- and four-input CES production functions in six different nesting structures. This is particularly useful for country-specific research such as Miess et al. (2022) where a hybrid CGE model ² is applied for a specific case of Austria representing the WEST region and an older work by this author team focusing on applying the hybrid CGE model on Poland as a representative of the CEE region.

The remainder of the paper is organized as follows. First, we specify our models and describe the data and our estimation approach. Next, we present our results and verify whether the often used CES production functions in the Cobb-Douglas and Leontief form fit our data. We test our estimated EoS for the regional and time differences. The last section concludes.

2. Methodology

2.1. Nesting Structure Specifications

The CES production function as a general form of the Cobb-Douglas (CD) one was introduced by Solow (1956) and later popularized by Arrow et al. (1961). In contrast to CD, CES allows for non-unity elasticities of substitution between production factors. As mentioned in Zha and Zhou (2014), the nesting of production factors allows for different elasticities, since factors on the same level are substituted with the same elasticity.

In our analysis, we benefit from the flexibility of the CES production function (Böhringer et al. 2003) and employ three different ways of specification with a total of six nesting structures that are chosen based on the usual functional CES forms found in the existing literature in the field. The list of reviewed papers with the CES forms used in them is provided in Table B1 in the Appendix.

In a two-level CES production function, the three inputs of capital (K), labor (L), and energy (E) can be combined as follows:

$$y_t=\gamma e^{t\lambda }{[\alpha {({\alpha }_1{K}_t^{-{\rho }_1}+(1-{\alpha }_1)E_t^{-{\rho }_1})}^{\frac{\rho }{{\rho }_1}}+(1-\alpha )L_t^{-\rho }]}^{\frac{-\nu }{\rho }},$$

(1)

$$y_t=\gamma e^{t\lambda }{[\alpha {({\alpha }_1{L}_t^{-{\rho }_1}+(1-{\alpha }_1)E_t^{-{\rho }_1})}^{\frac{\rho }{{\rho }_1}}+(1-\alpha )K_t^{-\rho }]}^{\frac{-\nu }{\rho }},$$

(2)

$$y_t=\gamma e^{t\lambda }{[\alpha {({\alpha }_1{K}_t^{-{\rho }_1}+(1-{\alpha }_1)L_t^{-{\rho }_1})}^{\frac{\rho }{{\rho }_1}}+(1-\alpha )E_t^{-\rho }]}^{\frac{-\nu }{\rho }},$$

(3)

where $y$ is the output, $\gamma \in (0,\infty )$ is an efficiency parameter, $\lambda \ge 0$ is the rate of technological change, t is a time variable, $\alpha$ and ${\alpha }_1\in (0,\infty )$ set the optimal distribution of inputs, $\rho$ and ${\rho }_1\in (-1,0)\cup (0,\infty )$ determine the (constant) elasticity of substitution, and $\nu \in (0,\infty )$ is equal to one in the case of constant returns to scale.

Including intermediate inputs (M) as the fourth production factor proved to be valid, as shown by Okagawa and Ban (2008) who estimated a nested CES function using an OECD dataset with nineteen sectors. Thus, we include M in our estimation and add the four-input nested CES production function in three different forms.

We estimate a three-level CES nesting structure ((KL)E)M as in Koesler and Schymura (2015) based on Sato (1967) and Henningsen and Henningsen (2011). The production function has the following form:

$$y_t=\gamma e^{t\lambda }{[{\alpha }_2{M}_t^{-{\rho }_2}+(1-{\alpha }_2){({\alpha }_1{E}_t^{-{\rho }_1}+(1-{\alpha }_1){(\alpha K_t^{-\rho }+(1-\alpha )L_t^{-\rho })}^{-\frac{{\rho }_1}{\rho }})}^{\frac{{\rho }_2}{{\rho }_1}}]}^{\frac{-\nu }{{\rho }_2}}$$

(4)

Based on Koesler and Schymura (2015), the equation (4) can be separated into a system as follows:

$$y_t=\gamma e^{t\lambda }{[{\alpha }_2{M}_t^{-{\rho }_2}+(1-{\alpha }_2){\left({({\alpha }_1{E}_t^{-{\rho }_1}+(1-{\alpha }_1){\mathrm{VA}}_t^{-{\rho }_1})}^{\frac{1}{-{\rho }_1}}\right)}^{-{\rho }_2}]}^{\frac{-\nu }{{\rho }_2}}$$

(5)

with

$$V{A}_t={(\alpha K_t^{-\rho }+(1-\alpha )L_t^{-\rho })}^{\frac{1}{-\rho }}$$

(6)

where VA is a compound of value added (K and L). The separability implied by the CES framework allows us to divide the three-level nesting structure into two equations (5) and (6). This overcomes the limitation of the micEconCES package software Henningsen and Henningsen (2011) used for our estimation that allows only two-level nesting structures. ³

A (KL)(EM) specification introduced by Sato (1967) follows, used as a functional form, that is, the CGE model AMOS-ENVI (ENVIronmental impact version of A Macro-micro model Of Scotland; Lecca et al. 2011):

$$\begin{array}{c}{y}_t=\gamma e^{t\lambda }{[\alpha {({\alpha }_1{K}_t^{-{\rho }_1}+(1-{\alpha }_1)L_t^{-{\rho }_1})}^{\frac{\rho }{{\rho }_1}}+(1-\alpha ){({\alpha }_2{E}_t^{-{\rho }_2}+(1-{\alpha }_2)M_t^{-{\rho }_2})}^{\frac{\rho }{{\rho }_2}}]}^{\frac{-\nu }{\rho }}\hfill \end{array}$$

(7)

A functional form of ((KE)L)M nesting completes our list of estimated equations. This nesting structure is present in well-known models such as GTAP-E (Burniaux and Truong 2002), ICES (Parrado and De Cian 2014), or GDYN-E (Antimiani et al. 2015). The specification is as follows:

$$y_t=\gamma e^{t\lambda }{[{\alpha }_2{M}_t^{-{\rho }_2}+(1-{\alpha }_2){({\alpha }_1{L}_t^{-{\rho }_1}+(1-{\alpha }_1){(\alpha K_t^{-\rho }+(1-\alpha )E_t^{-\rho })}^{\frac{{\rho }_1}{\rho }})}^{\frac{{\rho }_2}{{\rho }_1}}]}^{\frac{-\nu }{{\rho }_2}}$$

(8)

However, as stated above, this three-level nesting structure cannot be estimated by the micEconCES package nor can it be separated into a system of two equations as in the case of the equation (4), since there are no data available for the composite (KE). Thus, we need to estimate the equation (8) using the R package minpack.lm (Elzhov et al. 2023) which is intended to solve non-linear least-squares problems using a variation of the Levenberg Marquardt algorithm. More details on the estimation process using minpack.lm are provided in Section 2.3.

For 1 to 6, the elasticities of substitution $\sigma$ , ${\sigma }_1$ and ${\sigma }_2$ are defined as:

$$\sigma =\frac{1}{1+\rho },$$

(9)

$${\sigma }_1=\frac{1}{1+{\rho }_1},$$

(10)

and

$${\sigma }_2=\frac{1}{1+{\rho }_2}.$$

(11)

We estimate the Hicks-McFadden (direct) elasticity of substitution (HES) between the inputs in the lower nest and the Allen-Uzawa (partial) elasticity of substitution (AES) between the nests. HES elasticity of substitution describes the input substitutability of two inputs $i$ and $j$ along an isoquant given that all other inputs are constant. The AES describes the input substitutability of two inputs when all other input quantities are allowed to be adjusted. ⁴ Two inputs within an individual nest are necessary HES substitutes, they may at the same time be AES complements (Sorrell 2014).

2.2. Data and Estimation Procedure

2.2.1. Dataset Composition

For our analysis, we use WIOD (Timmer et al. 2012) as a consistent data source. Given the limited transferability of substitution elasticity, we focus on western European (WEST) and Central and Eastern European (CEE) countries as a subregion. Our dataset covers thirty-four industries over a fifteen-year period (2000–2014) across thirty European countries, including the eleven CEE countries listed in Table B2 in the Appendix.

We integrate Gross Output (Y), Intermediate Inputs (M), Gross Value Added (VA), Labor Compensation (L), and Nominal Capital Stock (K) from the Socio-Economic Accounts with Gross Energy Use (E) from the Environmental Accounts. The WIOD database provides gross output and value added at current basic prices in millions of national currencies. Intermediate inputs, labor compensations, and nominal capital stocks are also stated in millions of national currencies at current purchasers’ prices.

Gross energy use is reported in physical units (terajoules, TJ), which we convert into tons of oil equivalent (toe). Since all other variables are expressed in monetary units, E expressed in toe is converted using energy prices per toe unit from the IEA dataset on Energy Prices (International Energy Agency [IEA] 2021). These prices, provided in national currency per net calorific value per toe (NCV), are specific to each energy source. ⁵ By multiplying the physical units of E by the prices specific to the energy source, we obtain the final monetary E composite.

To standardize Y, VA, M, K, L, and E values across countries, we use exchange rates provided by WIOD, expressed in USD per unit of local currency, for all thirty countries (WIOD database 2016). Subsequently, the USD values are converted to EUR at the 2010 constant prices to account for inflation. The price indices for VA composite (K and L), M, Y, and E by sector are available in the WIOD Socio-Economic Accounts.

The thirty-four WIOD sectors are aggregated according to Baccianti (2013) in two ways: first, by the classical division into primary, secondary, and tertiary sectors, and second, by categorizing industries as energy-intensive or energy-efficient. A sector is deemed energy-intensive if its average energy costs exceed 5 percent of its total costs. Such sectors include agriculture, mining, energy generation, transport, and specific manufacturing industries. A comprehensive overview of sector aggregation is provided in Table B3 in the Appendix.

After excluding observations with missing or zero values, as well as the first and last 4 percentiles of outlier values for the capital/output ratio and extreme values of the energy/output ratio, we obtain 20,640 observations for Europe, 7,888 for CEE, and 12,752 for the WEST countries.

2.2.2. Data Summary

Summary statistics for the three regions and the aggregated sectors are presented in Table 1.

Table 1.

Summary Statistics of Our Data Sample Across Regions and Aggregated Sectors.

Region/Agg. sector	Variable	Obs.	Mean	Std	Min	Max
EU	Y	20,640	15,009	33,195	1.1	381,017
	VA	20,640	7,149	17,462	0.3	177,558
	K	20,640	13,429	43,129	0.8	709,530
	L	20,640	4,868	12,980	0.27	141,041
	E	20,640	580	1,981	0.01	50,649
	M	20,640	7,889	17,842	0.32	271,011
	Y	7,888	3,037	5,518	2.47	125,406
	VA	7,888	1,311	2,462	0.4	25,131
CEE	K	7,888	2,723	4,761	1.33	62,530
	L	7,888	748	1,522	0.743	18,683
	E	7,888	188	576	0.013	10,387
	M	7,888	1,735	3,555	1.45	109,648
	Y	12,752	21,354	39,416	1.1	381,017
	VA	12,752	10,243	20,873	0.3	177,558
WEST	K	12,752	19,103	52,356	0.8	709,530
	L	12,752	7,052	15,581	0.27	141,041
	E	12,752	787	2,388	0.03	50,649
	M	12,752	11,152	21,205	0.32	271,011
	Y	1,626	6,381	14,918	2.47	122,667
	VA	1,626	3,346	9,579	0.67	86,326
Primary	K	1,626	11,017	27,655	1.88	221,476
	L	1,626	1,490	2,967	0.39	22,412
	E	1,626	405	823	0.01	5,240
	M	1,626	3,066	6,721	1.33	47,699
	Y	8,775	13,232	32,065	1.05	381,017
	VA	8,775	4,289	11,053	0.3	131,205
Secondary	K	8,775	8,261	27,522	0.84	455,751
	L	8,775	2,781	7,961	0.27	111,280
	E	8,775	720	2,738	0.03	50,649
	M	8,775	8,964	21,547	0.32	271,010
	Y	10,239	17,902	35,805	6.92	286,092
	VA	10,239	10,204	21,835	0.40	177,558
Tertiary	K	10,239	18,240	54,151	9.78	709,530
	L	10,239	7,193	16,522	3.14	141,041
	E	10,239	488	1,159	0.08	11,733
	M	10,239	7,735	15,231	1.17	147,240
	Y	6,536	9,186	17,583	1.78	128,414
	VA	6,536	3,390	7,456	0.40	86,326
Energy intensive	K	6,536	10,834	26,078	1.88	229,849
	L	6,536	1,898	3,984	0.27	43,049
	E	6,536	1,194	3,307	0.01	50,649
	M	6,536	5,820	11,285	0.32	109,648
	Y	14,104	17,707	38,031	1.05	381,017
	VA	14,104	8,891	20,270	0.3	177,558
Energy non-intensive	K	14,104	14,631	49,015	0.84	709,530
	L	14,104	6,244	15,271	0.46	141,041
	E	14,104	295	646	0.03	7,409
	M	14,104	8,848	20,099	0.73	271,010

Figure A1 in the Appendix shows the average K/Y ratio in the CEE region, which is on average 11 percentage points (pp) higher than in the WEST countries during the whole period 2000 to 2014. This indicates a higher average capital efficiency in the WEST countries. There was a drop after 2003 that stemmed from a decrease in the K/Y ratio in the CEE region, but after the financial crisis, the ratio increased again and remained fluctuating around 1.1. In the WEST region, the ratio gradually converged to one over the observed period.

The L/Y ratio remains approximately the same throughout the period for both regions. On average, the ratio rests about four pp lower in the CEE countries than in the WEST countries. This reflects the lower average wage in the CEE countries compared to Western Europe (Goraus-Tanska and Lewandowski 2016). The labor/capital ratio shows a slightly decreasing trend in both regions presumably arising from productivity improvements obtained through capital investments and automatizing of production procedures.

The biggest difference between the CEE region and the rest of Europe is in the energy intensity (Figure A2 in the Appendix). In the period 2000 to 2008 the difference between the energy intensity in production in the CEE countries and the rest of Europe exceeded 50 percent in the initial years. In the period 2000 to 2008 the difference between the energy intensity in production in the CEE countries and the rest of Europe exceeded 50 percent in the initial years.

After 2009, the energy intensity in the CEE countries has been slowly converging to the level of the WEST countries. This could be a consequence of a change in the economic structure of the CEE countries, that is, reorienting toward less energy-intensive industries such as services. The energy/capital ratios for both regions are steadily decreasing over the observed period and the gap between them is slowly closing and ending with a difference of 21 percent in 2014. Two possible reasons lie behind this development: First, it could indicate the increase in the energy efficiency of machinery and equipment and overall technological progress as well as the modernization of equipment in the CEE countries, and second, a relatively declining energy demand not directly related to technology or equipment but rather to secondary reasons, such as better infrastructure and logistics.

2.3. Econometric Approach

2.3.1. Estimation with the micEconCES Package

A non-linear least squares (NLS) model is a common choice for estimating functions with non-linear parameters, such as the CES framework. However, NLS performance on real data can be problematic, as the model can estimate parameters outside of reasonable ranges, fail to converge, or converge to a local minimum (Henningsen and Henningsen 2011).

To avoid these potential complications, we use the micEconCES package in R developed by Henningsen and Henningsen (2011) to estimate the substitution elasticity. The micEconCES package provides a robust estimation tool for the elasticity of substitution (Koesler and Schymura 2015).

One of the features of this package is the choice for the estimation method determined by the argument provided to the function cesEst designed for the estimation of the CES function. When cesEst set to “Kmenta,” the CES function is estimated using ordinary least squares via the Kmenta approximation. Otherwise, it is estimated using non-linear least squares with various optimizers, namely the Levenberg-Marquardt (LM; Marquardt 1963) and the Conjugate Gradients (CG) method (Fletcher and Reeves 1964), among others listed below. The Kmenta approximation was the initial approach for estimating non-linear CES function by log-linearizing the model. Henningsen and Henningsen (2011) critique this methodology and further note that the LM algorithm tends to bias elasticity estimates toward zero. Additionally, the CG method, which is designed for well-behaved, approximately quadratic objective functions and large, sparse Hessian matrices, is less suitable for CES estimation. For these reasons, we excluded the Kmenta approximation, CG, and LM optimization algorithms from our analysis.

The other optimization algorithms that are part of the micEconCES package, and that we are employing for our estimation of the CES function parameters, are the Newton algorithm (Dennis and Schnabel 1983), Nelder-Mead routines (NM; Nelder and Mead 1965), the Simulated Annealing algorithm (SANN; Kirkpatrick et al. 1983), both the Broyden-Fletcher-Goldfarb-Shanno (BFGS) and the restricted BFGS (L-BFGS-B; Broyden 1970; Fletcher 1970; Goldfarb 1970; Shanno 1970) routines, Differential Evolution (DE; Storn and Price 1997), PORT routines (Gay 1990) and a two-dimensional grid search for the starting values of $\rho$ , ${\rho }_1$ , and eventually ${\rho }_2$ using PORT algorithm (Henningsen and Henningsen 2011).

Analytical gradients are utilized in several methods, including LM, BFGS, CG, L-BFGS-B, Newton, and PORT. Starting values, if not provided, are set to pre-defined defaults. The lower and upper bounds for the parameters are automatically determined based on the estimation method, but can be manually specified. Furthermore, grid searches for $\rho$ parameters can be performed if multiple values are provided, optimizing the CES by minimizing the sum of squared residuals. The function preserves the state of the random number generator to ensure consistency and reproducibility in methods that require random numbers, such as SANN and DE.

Henningsen and Henningsen (2011) further describes SANN as a “robust global optimizer,” and its advantage is the flexibility and ability to find global minima in large, complex solution spaces, making it suitable for applications in combinatorial and continuous optimization problems. ⁶

The SANN algorithm yields results with the best fit to our data in all nesting structures, most significant estimates, and the least residual sum of squares in most cases. ⁷

Despite the undeniable performance advantages of the micEconCES package, a notable drawback is its inability to estimate three-level nesting structures, as mentioned in Section 2.1. Thus, for the estimation of the ((KE)L)M nesting given by equation (8), we employ the direct NLS model and interpret its results in light of the critique outlined in the beginning of this section 2.3, particularly the risk of obtaining a local minimum solution.

2.3.2. Non-linear System of Equations

For the estimation of this three-level nesting structure, we use the minpack.lm R package.

Besides the estimation of the ((KE)L)M specification, the minpack.lm package allows for an implementation of a non-linear system of equations. Such a framework is useful since a non-linear system facilitates the implementation of cross-equation restrictions, ensuring coherent parameter estimation across multiple equations. In this context, the parameters $\gamma$ (scale) and $\lambda$ (growth rate) are restricted to be identical in all equations (1) to (4), (7), and (8), reflecting the uniformity of the production scale and technological progress across different output equations. The cross-equation restrictions enhance the model’s coherence and reduce the number of free parameters ensuring that the model remains parsimonious while accurately capturing the shared dynamics among the production function equations. The aim of this exercise is to verify the fit to data of various nesting structures. However, the drawbacks of this approach stated in the beginning of this section 2.3.1 still apply.

The methodology involves defining a residual function for each equations (1) to (4), (7), and (8), calculating the discrepancy between predicted values $y_{\mathrm{pre}{d}_i}$ and actual output values $y$ . These residuals are then minimized using non-linear least squares optimization to estimate the model’s parameters.

These residuals are minimized to identify the parameter values that best fit the data. The nls.lm function from the minpack.lm package applies a modification of the Levenberg-Marquardt algorithm to solve the non-linear least squares (NLS) problem. ⁸

Unlike the micEconCES package, the fitting process involves explicitly defining initial parameter guesses, with the $\gamma =1$ , $\lambda =0.001$ , and all $\rho$ and $\alpha$ parameters set to 0.5. Lower bounds are set to ensure the validity of the parameters: $\gamma >0$ , $\lambda \ge 0$ , each $\alpha$ parameter $>0$ , and each $\rho$ parameter > $-1$ . The parameters are then iteratively optimized to minimize the sum of squared residuals.

3. Results

3.1. Choice of Nesting Structure

Estimates of substitution elasticity for the two-level three-input nesting structures of the CES function (KE)L, (KL)E, and (LE)K given by equations (1) to (3), respectively, are displayed in Tables 2 to 4, respectively, together with their standard errors ⁹ in parentheses. Estimates are provided separately for the aggregated sectors and for the CEE and WEST regions, as well as the whole EU. Table B4 provides the goodness of fit $R^2$ for each estimation.

Table 2.

Estimates of Elasticities of Substitution for (KE)L Specification.

	EU		CEE		WEST
Region sector	KE	(KE)L	KE	(KE)L	KE	(KE)L
Whole economy	0.67*** (0.04)	1.08*** (0.02)	0.72*** (0.05)	0.54*** (0.03)	0.93*** (0.18)	1.07*** (0.03)
Primary	1.49*** (0.26)	n.a.	1.89 (1.42)	0.88* (0.45)	1.41 (0.35)	n.a.
Secondary	1.01*** (0.07)	0.74*** (0.02)	1.13*** (0.21)	0.83*** (0.06)	1.01*** (0.09)	0.74*** (0.02)
Tertiary	0.66*** (0.23)	1.54*** (0.08)	0.67 (0.66)	0.49*** (0.02)	0.66** (0.28)	1.54*** (0.1)
Energy intensive	0.64 (0.45)	5.56*** (0.97)	42.34 (436)	1.24*** (0.36)	0.57 (1.04)	28.23 (33.25)
Energy non-intensive	0.51*** (0.03)	300.44 (7,200)	0.52*** (0.1)	1.10*** (0.17)	0.59*** (0.05)	2.97*** (0.89)

Note. Standard errors are displayed in parentheses; *, **, *** indicates that the coefficient differs from zero at the 10%, 5%, 1% level of significance; R² is displayed in Table B4.

Table 3.

Estimates of Elasticities of Substitution for (LE)K Specification.

	EU		CEE		WEST
Region sector	LE	(LE)K	LE	(LE)K	LE	(LE)K
Whole economy	0.93*** (0.02)	0.45*** (0.04)	0.66*** (0.03)	2.30*** (0.87)	0.85*** (0.02)	0.68*** (0.08)
Primary	0.54 (4.61)	1.15*** (0.15)	0.8 (0.77)	1.34** (0.64)	0.59 (1.5)	1.08*** (0.25)
Secondary	0.74*** (0.02)	0.54*** (0.03)	0.69*** (0.05)	0.67*** (0.13)	0.79*** (0.02)	0.57*** (0.03)
Tertiary	1.11*** (0.05)	0.74*** (0.26)	0.55*** (0.02)	0.53*** (0.12)	1.39*** (0.1)	0.80 (0.98)
Energy intensive	3.27*** (0.31)	2.01 (3.97)	6.09 (9.47)	1.50*** (0.47)	2.72*** (0.26)	0.86 (1.68)
Energy non-intensive	0.97*** (0.06)	0.67*** (0.11)	0.40*** (0.04)	0.66*** (0.09)	0.85*** (0.06)	1.40 (1.33)

Note. Standard errors are displayed in parentheses; *, **, *** indicates that the coefficient differs from zero at the 10%, 5%, 1% level of significance; R² is displayed in Table B4.

Table 4.

Estimates of Elasticities of Substitution for (KL)E Specification.

	EU		CEE		WEST
Region sector	KL	(KL)E	KL	(KL)E	KL	(KL)E
Whole economy	0.36*** (0.08)	0.98*** (0.01)	2.51*** (0.53)	0.42*** (0.02)	0.49** (0.2)	1.10*** (0.02)
Primary	2.08 (1.51)	0.55*** (0.04)	1.23** (0.56)	0.67 (0.46)	4.2 (14.37)	0.46*** (0.04)
Secondary	0.53*** (0.02)	0.45*** (0.01)	0.54*** (0.05)	0.87*** (0.05)	0.51*** (0.02)	0.38*** (0.01)
Tertiary	0.48*** (0.05)	1.03*** (0.05)	0.59*** (0.19)	0.51*** (0.02)	0.46*** (0.08)	1.07*** (0.07)
Energy intensive	0.94*** (0.11)	1.31*** (0.04)	1.14*** (0.15)	2.13*** (0.19)	1.00*** (0.11)	1.16*** (0.04)
Energy non-intensive	0.56*** (0.19)	1.05*** (0.06)	0.90*** (0.29)	0.82*** (0.05)	0.65 (0.75)	1.81*** (0.2)

Note. Standard errors are displayed in parentheses; *, **, *** indicates that the coefficient differs from zero at the 10%, 5%, 1% level of significance; R² is displayed in Table B4.

Kemfert (1998) and van der Werf (2008) suggest the goodness of fit $R^2$ as a criterion to identify the nesting structure most suited. However, Feng and Zhang (2018) point out the compliance of estimated CES parameters to economic meaning and convergence to assumed elasticities as important points in the decision process.

Table B4 shows that all three nesting specifications provide very similar goodness of fit to the data across regions and sectors. Only the average $R^2$ for the (KL)E specification is about two pp higher than the other two nesting structures. The residual sum of squares (RSS) appears to be almost equivalent for the (KE)L and (KL)E nests and slightly higher for the (LE)K nest. ¹⁰

(KE)L Specification

For the sigma parameters in equation (1), Feng and Zhang (2018) propose ${\sigma }_{K,E}=0.5$ and ${\sigma }_{\mathrm{KE},L}=1$ as initial points taken from the GTAP-E model (Burniaux and Truong 2002). Our estimates of ${\sigma }_{K,E}$ are higher than these values on the level of the entire economy but still below one. However, on sectoral level, estimates differ and range between 0.51 and 1.49 for the EU, 0.52 and 42.34 for CEE (the former estimate being, however, statistically insignificant), and 0.57 and 1.41 for the WEST countries. The vastest inconsistency in estimates for the EU regions is present in energy-intensive industries. Five estimates ${\sigma }_{K,E}$ are insignificant at the 10% level. Estimates ${\sigma }_{\mathrm{KE},L}$ exhibit even greater regional inconsistency, particularly for energy-intensive sectors (with a difference of nearly 27) and energy non-intensive sectors (with a difference of almost 300). The average ${\sigma }_{\mathrm{KE},L}$ for the CEE region is 0.85 with a standard error of 0.18 while it is 6.91 (se = 6.86) for the WEST countries. Some of the parameters’ estimates do not comply with their economic meaning. Specifically, Feng and Zhang (2018) propose to verify that $\alpha$ , ${\alpha }_1$ , $\nu$ , and $\lambda$ all lie in acceptable ranges. In eight cases, $\lambda$ is negative and in two cases ${\alpha }_1$ is also negative. Two of ${\sigma }_{\mathrm{KE},L}$ estimates are insignificant, and two estimates of $\rho$ are not within acceptable range—hence the n.a. values, see Table 2.

(LE)K Specification

According to the review by Lagomarsino (2020), the (LE)K specification given in equation (2) has been chosen for CGE models only very sporadically. Few examples include CES elasticity estimates by Turner et al. (2012) for the UK, Su et al. (2012) and Shen and Whalley (2013) for China, and Dissou et al. (2015) for Canada. While in our case, the elasticity estimates $\sigma$ and ${\sigma }_1$ are more or less consistent across regions, with the exception of the energy-intensive sectors, estimates of the $\lambda$ parameter are negative in eight cases. The average ${\sigma }_{L,E}$ for CEE region is 1.53 (se = 1.73) while ${\sigma }_{L,E}$  = 1.2 (se = 0.33) for the WEST countries. On the upper nest, the average ${\sigma }_{\mathrm{LE},K}$ is 1.17 (se = 0.39) for CEE and 0.9 (se = 0.72) for WEST region. On both the upper and lower nest, four estimates of ${\sigma }_{L,E}$ and four estimates of ${\sigma }_{\mathrm{LE},K}$ are statistically insignificant. The RSS for the (LE)K specification is the highest and $R^2$ the lowest among the three-input forms.

(KL)E Specification

For the last three-input specification given by equation (3), the initial points for ${\sigma }_{K,L}$ and ${\sigma }_{\mathrm{KL},E}$ are 0.59 and 0.94, respectively (Burniaux and Truong 2002; Feng and Zhang 2018;). Three of our estimates ${\sigma }_{K,L}$ are insignificant. With the exception of one, all estimates of σ_KL,E are significant at least at the 1% level. The estimated average ${\sigma }_{K,L}$ for the CEE region is 1.15 (se = 0.29) and 1.22 (se = 2.59) for the WEST countries. For the upper nest, the average ${\sigma }_{\mathrm{KL},E}$ is 0.90 (se = 0.13) for CEE and 1.0 (se = 0.06) for the WEST region, which is close to the values considered by the GTAP-E model. The differences between the estimates ${\sigma }_{K,L}$ and ${\sigma }_{\mathrm{KL},E}$ for the three regions are less than for the (KE)L structure. They appear also to be more consistent across sectors except for the lower nest in the primary sector and the level of the economy. Only two estimates of the $\lambda$ parameter are negative.

The (LE)K nesting specification appears less suitable due to the high RSS and lower $R^2$ and a high number of negative $\lambda$ parameter estimates. The (KL)E and (KE)L nesting specifications provide a similar fit to our data, but the (KL)E structure shows lower elasticity inconsistencies across regions. Moreover, a high number of elasticity estimates in the (KE)L case are statistically insignificant or out of acceptable range. Kemfert (1998) came to a similar conclusion that the specification with capital and energy in one nest (KE)L fits best for the aggregated analysis of the entire German industry, but a nest of capital and labor (KL)E might be closer to reality for several industrial sectors. The appropriateness of the (KL)E nesting structure was also confirmed for industrial level data from twelve developed OECD countries for the 1978 to 1996 time horizon by van der Werf (2008). Our results support those findings.

((KL)E)M Specification

In the three-level four-input ((KL)E)M CES nesting structure, ¹¹ four estimates at the bottom and one at the upper nest are not significant or out of the acceptable range, see Table 5. The residual sum of squares is two orders lower than the three-input specification. The average $R^2$ of the ((KL)E)M specification is 0.90. However, in eight cases in total, $\lambda$ is negative, which violates the assumption of a nonnegative rate of technological change. The average estimate of ${\sigma }_{K,L}$ is 4.0 (se = 10.22) for the CEE countries (or 1.13 (se = 0.22 if we do not take insignificant estimates into account), and 0.64 (se = 0.05) for the WEST region. The average regional estimates of ${\sigma }_{\mathrm{KL},E}$ are similar, 0.58 (se = 0.07) and 0.61 (se = 0.04) for CEE and WEST, respectively. At the upper level, the average estimate of ${\sigma }_{\mathrm{KLE},M}$ is equal to 29.64 (se = 15.42) for CEE and 6.73 (se = 0.28) for the WEST region. The regional estimates of the elasticities are rather inconsistent across the three regions in almost all nests.

Table 5.

Estimates of Elasticities of Substitution for ((KL)E)M Specification.

	EU			CEE			WEST
Region sector	KL	(KL)E	((KL)E)M	KL	(KL)E	((KL)E)M	KL	(KL)E	((KL)E)M
Whole economy	0.65*** (0.03)	0.53*** (0.01)	6.39*** (0.12)	n.a.	0.65*** (0.22)	2.55*** (0.04)	0.65*** (0.04)	0.51*** (0.01)	17.19*** (1.01)
Primary	0.74*** (0.02)	0.82*** (0.05)	n.a.	1.37 (0.93)	0.56*** (0.06)	0.93*** (0.06)	0.77*** (0.03)	0.63*** (0.06)	3.27*** (0.12)
Secondary	0.57*** (0.03)	0.48*** (0.02)	0.82*** (0.01)	1.06*** (0.06)	0.52*** (0.03)	2.61*** (0.06)	0.57*** (0.03)	0.78*** (0.03)	1.29*** (0.02)
Tertiary	1.01*** (0.32)	0.38*** (0.01)	1.05*** (0.01)	15.22 (49.5)	0.67*** (0.02)	12.27*** (0.98)	0.64*** (0.1)	0.33*** (0.01)	3.68*** (0.09)
Energy intensive	n.a.	0.60*** (0.02)	11.81*** (0.28)	0.84*** (0.05)	0.48*** (0.1)	1.04*** (0.03)	n.a.	0.60*** (0.02)	11.81*** (0.38)
Energy non-intensive	0.75*** (0.09)	0.79*** (0.06)	3.17*** (0.04)	1.5*** (0.56)	0.59*** (0.02)	158.45* (91.33)	0.56*** (0.04)	0.79*** (0.09)	3.17*** (0.05)

Note. Standard errors are displayed in parentheses; *, **, *** indicates that the coefficient differs from zero at the 10%, 5%, 1% level of significance; R² is displayed in Table B5.

(KL)(EM) Specification

For this estimated nesting structure given in equation (7), the average goodness of fit $R^2$ is 0.98 and the residual sum of squares of the (KL)(EM) model is similar to the ((KL)E)M CES function, see Table B6. In the bottom nest, two estimates of ${\sigma }_{K,L}$ for the CEE region and two estimates of ${\sigma }_{E,M}$ are not significant at the level 10%. In the upper nest, all the estimated ${\sigma }_{\mathrm{KL},\mathrm{EM}}$ are significant at least at the level 5%. The estimates on the lower nest are consistent across regions with the exception of the tertiary sector in the energy-materials nest. The average estimates of ${\sigma }_{K,L}$ are 0.62 (se = 0.37) and 0.67 (se = 0.08), ${\sigma }_{E,M}$ are 0.62 ¹² (se = 0.13) and 0.83 (se = 0.33), and ${\sigma }_{\mathrm{KL},\mathrm{EM}}$ are 1.01 (se = 0.06) and 1.42 (se = 0.06) for CEE and WEST, respectively. Consequently, the (KL)(EM) model appears to be the best fit for our data out of all the nesting structures analyzed. The detailed analysis on this nesting is provided in the next section.

3.2. Elasticity of Factor Substitution: Central Estimates

Table 6 reports the summary of the substitution elasticity estimates recommended as the best fit for the European countries. Gechert et al. (2022) prove the irrelevancy of the Cobb-Douglas production function and find that the most representative substitution elasticity between capital and labor found in the literature is 0.5 when considering aggregated economy-level data, and 0.3 under the restriction of industry-level country-level data.

Table 6.

Central Estimates of Elasticities of Substitution.

	EU			CEE			WEST
Region sector	KL	EM	(KL)(EM)	KL	EM	(KL)(EM)	KL	EM	(KL)(EM)
Whole	0.55*** (0.02)	0.84*** (0.06)	2.89*** (0.07)	0.48** (0.21)	0.5*** (0.02)	1.88*** (0.04)	0.64*** (0.03)	1.3 (1.18)	1.93*** (0.04)
Primary	0.84*** (0.13)	0.78* (0.36)	0.66*** (0.07)	0.61 (0.45)	0.52*** (0.2)	0.48** (0.19)	0.74*** (0.15)	0.72*** (0.12)	0.55*** (0.04)
Secondary	0.64*** (0.11)	0.69* (0.24)	1.4*** (0.05)	0.89*** (0.09)	0.72*** (0.06)	0.65*** (0.02)	0.64*** (0.14)	0.69* (0.29)	1.4*** (0.07)
Tertiary	0.85*** (0.15)	0.74*** (0.12)	1.05*** (0.02)	0.37*** (0.04)	8.3 (223.62)	0.67*** (0.02)	0.64*** (0.03)	0.81*** (0.07)	1.27*** (0.03)
Energy intensive	0.84*** (0.06)	0.53*** (0.05)	2.73*** (0.34)	0.75** (0.24)	0.52* (0.29)	0.8*** (0.03)	0.78*** (0.08)	0.69** (0.19)	1.48*** (0.14)
Energy non-intensive	0.58*** (0.05)	0.79*** (0.11)	1.92*** (0.03)	0.63 (1.2)	0.83*** (0.06)	1.57*** (0.04)	0.58*** (0.07)	0.79*** (0.13)	1.92*** (0.04)

Note. Standard errors are displayed in parentheses; *, **, *** indicates that the coefficient differs from zero at 10%, 5%, 1% levels of significance; R² is displayed in Table B6.

When closely examining the results of the (KL)(EM) specification given in Table B6, we find that the substitution elasticity estimates on the aggregate level between capital and labor for the CEE region ( ${\sigma }_{K,L}=0.48$ ) and the EU ( ${\sigma }_{K,L}=0.55$ ) approach the value 0.3 found by Gechert et al. (2022). The estimate for WEST is slightly higher with ${\sigma }_{K,L}=0.64$ . Our estimates of ${\sigma }_{K,L}$ are 0.48 for CEE, 0.64 for WEST, and 0.55 for the EU. As Kahoulia and Pautrel (2023) point out, the amount of time an economy needs to transition to a new steady state increases with a higher substitution elasticity.

However, the substitution elasticities differ on the level of sectors. The divisions into three sectors (I., II., and III.) and according to the energy intensity show that industries with a necessity of machine use report an easier substitution between capital and labor.

For energy-intensive industries, ${\sigma }_{K,L}$ rounds 0.75 for the CEE and 0.78 for the WEST regions, and for the primary sector, the average elasticity reaches 0.61 for CEE and 0.74 for WEST. In contrast, substitution is more complicated in the tertiary sector, where human capital is important and often cannot be replaced by automatization. Thus, ${\sigma }_{K,L}$ is equal to 0.37 and 0.64, for the CEE and the WEST countries, respectively. Since the energy non-intensive industries overlap with the secondary and tertiary sector, the estimates are consequently also lower than in the energy-intensive case, 0.63 for CEE and 0.58 for the WEST countries.

While energy and materials substitution is more difficult in energy-intensive industries and the primary sector, as well as on the economy-wide level in comparison to capital-labor substitution, the opposite is true for energy non-intensive, secondary and tertiary sectors. This result is expected since we assume that energy-demanding sectors can replace interchange machines and workers relatively easily, but it may be more complicated to replace intermediate inputs with energy. The opposite appears to be true for the sectors focused on human capital.

The estimation of ${\sigma }_{\mathrm{KL},\mathrm{EM}}$ across regions and sectors indicates that the substitution between the two composites is relatively easy in the EU and WEST region, with an exception in the primary sector. For the CEE region, the results are rather inconclusive regarding the relative difficulty of substitution. The average ${\sigma }_{\mathrm{KL},\mathrm{EM}}$ being 1.01 (se = 0.06) and 1.42 (se = 0.06) in the WEST and CEE region, respectively. At the EU level, the compounds can also be substituted straightforwardly, with the average ${\sigma }_{\mathrm{KL},\mathrm{EM}}$ equal to 1.77 (se = 0.10). According to Kahoulia and Pautrel (2023), the impact of a higher substitution elasticity is opposite on the transition path toward a new steady state than in the case of capital—labor elasticity and decreases the time needed for the economy to adjust since labor supply is a transmission channel.

3.3. Cobb-Douglas and Leontief Production Function

The Cobb-Douglas and Leontief forms, often found in the literature, are both a special case of the CES production function. The former occurs when $\rho \to 0$ and consequently $\sigma \to 1$ while for the latter $\rho \to \infty$ and $\sigma \to 0$ .

Using the two-sided Wald test, we test the hypotheses of both Cobb-Douglas and Leontief specifications’ suitability for all the nests within the (KL)(EM) nesting structure.

The assumption of Leontief function ( ${\sigma }_{\mathrm{KL}}=0,{\sigma }_{E,M}=0$ or ${\sigma }_{\mathrm{KL},\mathrm{EM}}=0$ ) can be rejected on a 5% significance level for all the estimates with the exception of the substitution elasticity between energy and materials in the CEE’s tertiary sector. However, this particular estimate is insignificant.

The same result is true for the Cobb-Douglas simplification hypothesis of the production function that assumes ${\sigma }_{\mathrm{KL}}=1,{\sigma }_{E,M}=1$ or ${\sigma }_{\mathrm{KL},\mathrm{EM}}=1$ . The hypothesis is rejected for all elasticity estimates on all levels within the (KL)(EM) nesting structure with the same exception as mentioned above. Our results support the findings of van der Werf (2008). Gechert et al. (2022) also reject the Cobb-Douglas specification, as a special case of the CES production function based on an extensive meta-analysis. In line with these authors and Koesler and Schymura (2015), we all conclude that a production function specified in a Leontief or Cobb-Douglas form may lead to inaccurate conclusions of an analysis.

3.4. Spatial and Time Differences in Substitution Elasticities

Both the summary statistics in Table 1 and Figures A1 and A2 suggest that there exists a difference in the substitution elasticities of production factors across regions. To assess the statistical significance of the difference between the CEE and the WEST countries, as well as the EU, we employ the double-sided Welch’s t-test.

Table B7 presents the results of the Welch’s t-test with the null hypothesis H0 of unequal elasticity estimates. The ones in Table B7 suggest that H0 can be rejected at a level of 10% significance (p-value > .1). The inequivalence of the elasticity is tested for all nests within the nesting structure (KL)(EM).

Substitution elasticities are different between WEST and CEE and also between EU and CEE in all nests for all sectors. When comparing the WEST countries and EU, the null hypothesis can be rejected for both bottom and upper nests within tertiary and energy non-intensive sectors. The hypothesis of elasticity inequivalence in the WEST countries and the EU cannot be rejected for all nests in the remaining sectors.

Hence, the difference between the CEE results and the WEST and EU results supports the need for specific estimates for the CEE region.

Since the 1990s, many structural and institutional changes affected economic development of sectors in the CEE countries which transformed from centrally planned communistic regime to market-based democratic system, see for example, Ščasný et al. (2021). In WEST, economic structures can change over time due to technological progress or macroeconomic trends. To analyze possible changes in economic production, we further conduct an analysis of change in input substitutability over time in two periods divided by the financial crisis: 2000 to 2008 and 2009 to 2014. Equation (7) is reestimated for the two time-restricted subsamples, and the accompanying elasticity estimates are compared using the double-sided Welch’s test. Table B8 in the Appendix summarizes the results, similar to Table B7, with one that indicates that the null hypothesis of unequal substitution elasticities in both time periods can be rejected at the 10% level. Some of the time-restricted elasticity estimates are not significant at the 10% level, particularly for the primary sector due to the low number of observations, less than 250 in the CEE region. The tests performed on such estimates are reported in Table B8 with an asterisk and are to be interpreted with caution.

On the aggregated level as well as in the secondary, tertiary and energy non-intensive sectors, the hypothesis H0 of unequal substitution elasticities over time cannot be rejected for all nests and for both regions. The null hypothesis is rejected for the bottom nest in the primary sector in the CEE region, for the E-M nest in the WEST region, and for the upper nest in CEE in the energy-intensive sector. This indicates that the elasticities concerned were evaluated as time-invariant. However, most estimates in the primary sector were statistically insignificant at the level 10%, as mentioned above.

An analysis of elasticity estimates over time reveals regional convergence in energy-intensive sectors and divergence in energy non-intensive sectors. Specifically, the differences between Central and Eastern Europe (CEE) and Western Europe (WEST) are decreasing in the industrial sectors but increasing in the service sectors. Several factors contribute to these contrasting trends.

• Technological and Innovation Diffusion: Energy-intensive sectors, such as manufacturing and heavy industries, benefit from standardized and easily transferable technologies, leading to convergence in production practices between CEE and WEST (Grela et al. 2017). Conversely, the tertiary sector is more heterogeneous and innovation-driven, relying on localized, non-standardized technologies and human capital, resulting in divergence due to varying levels of technological adoption and innovation capacity.

• Regulatory Environment: EU-wide regulations, such as emission standards and energy efficiency directives, are stringently enforced in energy-intensive sectors, promoting uniformity in production practices across regions (Ibekwe et al. 2024; Kitzing et al. 2012). However, policies in the tertiary sector are more fragmented and influenced by national differences in education, infrastructure, and cultural factors, leading to divergent development paths (van der Marel et al. 2016).

• Foreign Direct Investment (FDI): FDI in the CEE countries has mainly targeted manufacturing and heavy industries, introducing modern technologies and practices from Western Europe, thus driving convergence (Gál and Lux 2022). In contrast, FDI in the tertiary sector is uneven, with significant investment in high-tech services in some areas and lag investment in others, causing divergence.

• Economic Structure and Development: Both CEE and WEST have experienced structural changes, with CEE transitioning from agriculture to industry and services (Burnete 2014; Havlik 2005). The convergence in industrial sectors is driven by innovation, trade, and similar institutional environments (Dobrinsky and Havlik 2014). However, the development of the tertiary sector depends on various factors such as consumer preferences, urbanization, and availability of digital infrastructure, leading to regional divergence (Gál and Lux 2022).

• Global Market Forces: Global competition requires the adoption of similar technologies in energy-intensive industries in CEE and WEST to remain competitive. In contrast, the service sector is less exposed to global competition and more influenced by local market conditions, fostering unique regional service offerings and divergence.

Over time, the substitution elasticities between capital and labor, energy and material, and value-added composite and energy-material composite for the CEE and WEST region tend to increase. The only exception is the K-L substitution elasticity in the WEST region. Knoblach and Stöckl (2019) identify several possible reasons for this development related to the institutional framework. First, in centrally planned economies where prices of goods and services tend to be distorted by state intervention, and rigid economic policies predominate, conditions for an efficient substitution between capital and labor are not created. Thus, after the CEE countries had started to transform from centrally planned to free-market economies; the use of production factors was not centrally planned anymore, and hence the input substitution has reflected market signals; the elasticity of substitution naturally increased as a result. Second, a stronger financial and monetary system and greater liquidity in free-market economies open the door to greater possibilities to finance costly mechanization to replace human labor (Klump and Preissler 2000), which again contributes to higher substitution elasticity. Third, stronger trade unions particularly present in Scandinavia, Belgium, Ireland, Austria, Italy, and Germany may provide regulatory limits to capital-labor substitution inducing a lower elasticity value, as debated by de La Grandville (2016). Lastly, the open economies in the WEST region have generally well-developed international trade, but the rate of growth has not even been across the countries. As Saam (2008) points out using a Heckscher-Ohlin-Samuelson model, if two countries with the same level of technological development but different rates of capital accumulation trade together, the substitution elasticity is decreased in the country with the lower capital growth rate.

In general, our analysis suggests that the elasticity of substitution of production factors has changed over time. Future research on the elasticity of substitution of production factors should therefore take into account the time aspect and investigate it more rigorously.

3.5. Direct Non-linear Least Squares Estimation

The estimates from a non-linear system of equations, ¹³ as well as a discussion on them, are reported in the Appendix, Tables C1 to C6.

In conclusion, on the level of the entire economy, generally KE-L and KL-E show significant substitution across regions, indicating flexibility in substituting energy and labor for capital. The primary and tertiary sectors show more consistent substitution patterns, while the secondary sector shows high variability. Energy-intensive sectors generally have higher elasticities, indicating more substitution capability than non-intensive sectors. CEE often shows higher and more variable elasticities compared to the EU and the WEST, indicating different substitution dynamics in these regions. The high standard errors and significant variability in some sectors highlight the need for cautious interpretation of these elasticities, and in general, the direct NLS system estimation proved to be not very reliable.

4. Conclusion

We apply non-linear estimation techniques to estimate substitution elasticity directly from the CES production function using the WIOD database as a data source. We focus on the CEE countries and their differences from western European countries due to a general lack of specific empirical evidence for this region.

We estimate five different nesting specifications of a CES production function with and without materials as the fourth input besides capital, labor, and energy with the help of the micEconCES R package. Estimation is done on the economy-wide level, as well as on the level of five aggregated sectors—primary, secondary, tertiary, energy-intensive, and energy non-intensive. In the nesting of two levels without material with three inputs, we confirmed the findings by van der Werf (2008) of the substitution of value-added composite with energy as the best fit to our data. However, based on the performance of the models, adding materials as a fourth production input seems to be a reasonable choice. Of the two four-input nesting structures, (KL)(EM) performs significantly better and is thus the preferred nesting structure of the CES production function in our case. Based on our estimation, we conclude that, while the values of substitution elasticities on the economy-wide level conform to the literature, they differ on the level of sectors.

With higher elasticity of substitution comes a quicker change in the input mix in response to a potential change in relative production factors’ prices. For example, human labor intensive sectors (tertiary, energy non-intensive, and partly secondary) show a more difficult substitution between capital and labor than capital intensive industries (primary, energy-intensive). This implies that even if the relative price of human labor increased, the input mix would change considerably less in the tertiary sector than in the primary sector since it is still very difficult to replace people by automation in sectors such as education, health, and social work, etc. The substitution between energy and materials is lower in energy-intensive industries and in the primary sector compared to the substitution of capital-labor. This means that if the relative price of energy increases, replacement by intermediate inputs in the input mix would be more difficult. The opposite is true for the energy efficient, secondary and tertiary sectors, as well as on the economic-wide level. With a few exceptions, the KL and EM composites can be substituted more easily than the bottom nests. Thus, should the relative price of the value-added increase, producers can replace it more easily by increasing the energy-materials composite in the input mix to preserve the volume of output produced.

In line with Gechert et al. (2022), van der Werf (2008), and Koesler and Schymura (2015), using the two-sided Wald test, we reject the suitability of the production function specified in a Leontief or Cobb-Douglas form often found in the literature and possibly thus leading to inaccurate conclusions. The hypothesis is rejected across sectors for both regions as well as for the EU as a whole.

Based on significant differences in capital efficiency, energy intensity, energy/capital, and capital/labor ratios between the CEE and western European countries, we test the hypothesis that the elasticity of substitution between production factors is different in these two regions. The substitution elasticities of the production factors are found to be statistically different in all nests for almost all sectors between the two regions. The difference between the EoS for the CEE and WEST and the EU supports the need for region-specific elasticity estimates in economic modeling.

In addition, we find that the elasticity of substitution of production factors changes over time. The elasticities for KL, EM, and (KL)(EM) are increasing over time in the CEE countries, indicating easier substitution, while the opposite is true for the KL nest in WEST. Over time, the elasticities of the primary, secondary, and energy-intensive sectors exhibit convergence between the two regions. In contrast, the elasticities in the tertiary, labor-intensive, sectors diverge. Hence, CGE and other macrostructural models should take into consideration not only the regional but also the temporal aspect and choose values for its calibration carefully.

Lastly, a direct estimation of the ((KE)L)M) nesting structure indicates that at the economy-wide level, elasticity estimates on the lowest level are either insignificant or unreasonably high. Considerable variability, particularly in the CEE countries, and high standard errors indicate that caution is needed when interpreting these estimates due to the limitations of direct NLS system estimation.

The assumption of Hicks-neutral technological change translating into the same rate of technology progress for all included production factors may be considered unrealistic. This strict assumption might be relaxed in future research. Moreover, future analysis might also be extended by including the rest of the world in addition to Europe.