In this article, we aim to solve two basic problems: first, we analyze whether the production activities in various regions are efficient. If there is inefficiency, then we study the dependent factors of the inefficiency. Second, we estimate the substitution elasticity and directed technological change in 30 regions so as to obtain the improvement pathway of energy transition. The fixed-effect SFA method meets the research purpose of this article, and it can effectively solve the above-mentioned problems and prevent heterogeneity among different provinces, while DEA cannot calculate the substitution elasticity and directed technological change. For the production function, there are many production functions including C-D and CES production functions that can calculate the technological progress bias of substitution elasticity in various regions. However, the translog function becomes our preferred model with an estimable and inclusive advantage. Therefore, according to Diamond (1965), the general form is as follows: y i t = α i + β x i t + v i t − u i t (1) where i and t represent the province and the time in years, respectively; y denotes the output; and α i stands for the individual fixed effect. x means the vector set of input elements, and β is the vector set of the estimated coefficients of the input factors. v is a random error term, which represents the impact of statistical errors and various random factors on frontier output; u indicates a technical inefficiency term, which represents the gap between the actual output and the technological frontier output. The article focuses on analyzing the optimization path of the internal and external electric transition. Therefore, we regard clean energy power generation, thermal power generation, and traditional fossil energy as three independent production factors to identify the biased technological change and the elasticity substitution of the internal and external transition.

In practice, the stochastic frontier production function is widely approximated by a translog production as follows: ln ⁡ Y i t = α 0 + β 1 t + β 2 1 2 t 2 + β 3 ⁡ ln ⁡ K i t + β 4 ⁡ ln ⁡ N i t + β 5 ⁡ ln ⁡ R i t + β 6 ⁡ ln ⁡ F i t + β 7 t × ln ⁡ K i t + β 8 t × ln ⁡ N i t + β 9 t × ln ⁡ R i t + β 10 t × ln ⁡ F i t + 1 2 β 11 ⁡ ln ⁡ K i t × ln ⁡ K i t + 1 2 β 12 ⁡ ln ⁡ N i t × ln ⁡ N i t + 1 2 β 13 ⁡ ln ⁡ R × ln ⁡ R i t + 1 2 β 14 ⁡ ln ⁡ F i t × ln ⁡ F i t + 1 2 β 15 ⁡ ln ⁡ K i t × ln ⁡ N i t + 1 2 β 16 ⁡ ln ⁡ K i t × ln ⁡ R i t + 1 2 β 17 ⁡ ln ⁡ K i t ⁡ ln ⁡ F i t + 1 2 β 18 ⁡ ln ⁡ N i t ⁡ ln ⁡ R i t + 1 2 β 19 ⁡ ln ⁡ N i t ⁡ ln ⁡ F i t + 1 2 β 20 ⁡ ln ⁡ R i t ⁡ ln ⁡ F i t + v − u (2) where Y represents the output of each province; K is the capital; N , R , and F denote the traditional fossil energy, clean energy power generation, and thermal power generation, respectively. Following Liu and Wang (2019), traditional fossil energy, clean energy power generation, and thermal power generation are three independent factors in the translog production function. The variables used in the translog production function are described as follows:

(1) Output ( Y ): We deflate all current price raw data to the constant 2000 prices and GDP of each province as the output measurement indicator.

Regarding the influential factors of technical inefficiency, we selected the following four factors: R&D intensity ( r d ) , learning by exporting ( e x p ) , foreign direct investment ( f d i ) , regional endowment structure ( k l ) , energy consumption structure ( s t r ) , and labor productivity ( l p ) u i t = δ 1 r d i t + δ 2 ⁡ exp i t + δ 3 f d i i t + δ 4 L . e s i t + δ 5 k l i t + δ 6 l p i t (3)

(1) R&D intensity ( r d ) : Tu and Leeke (2011) examined the impact of technology on environmental technology efficiency from three aspects: independent research and development, technology introduction, and technological transformation, and confirmed that technology has a significant effect on the environmental technology efficiency. Based on the availability of data, we used the ratio of the internal expenditure of research and experimental development funds in each region to the regional GDP to measure the R&D intensity.

According to Diamond (1965), we can further reveal the biased technical change for each pair of input factors with the following equation: B i a s n q = ∂ M P n / ∂ t M P n − ∂ M P q / ∂ t M P q (4) where n and q are two different production factors (including K , N , R , F ); M P n and M P q are the marginal productivities of n and q, respectively. B i a s n q represents the relative proportional change over time in pairwise input production elasticities. A positive (negative) sign on B i a s n q indicates that the directed technical change is based to use n ( q ) and save q ( n ) ; B i a s n q = 0 means the directed technical change in the production process is Hicks neutral. Thus, from Eq. 2 and Eq. 4, we can calculate the biased technical change between any two factors.

For the biased technical change between clean energy power generation and thermal power generation, the following relationship is defined: B i a s R F = ∂ M P R / ∂ t M P R − ∂ M P F / ∂ t M P F = β 9 η R − β 10 η F (5)

In the same vein, the biased technical change between clean energy power generation and traditional fossil energy is calculated by the following specification: B i a s R N = ∂ M P R / ∂ t M P R − ∂ M P N / ∂ t M P N = β 9 η R − β 8 η N (6)

Similarly, the biased technical change between thermal power generation and traditional fossil energy is specified as follows: B i a s F N = ∂ M P F / ∂ t M P F − ∂ M P N / ∂ t M P N = β 10 η F − β 8 η N (7) where η R , η F , and η N are the output elasticities of clean energy power generation, thermal power generation, and traditional fossil energy, respectively. The marginal productivity of clean energy power generation ( M P R ), thermal power generation ( M P F ), and traditional fossil energy ( M P N ) can be obtained as follows: M P R = ∂ Y ∂ R = Y R ∂ ln ⁡ Y ∂ ln ⁡ R = Y R η R = Y R ( β 5 + β 9 t + β 13 ⁡ ln ⁡ R i t + 1 2 β 16 ⁡ ln ⁡ K i t + 1 2 β 18 ⁡ ln ⁡ N i t + 1 2 β 20 ⁡ ln ⁡ F i t ) (8) M P F = ∂ Y ∂ F = Y F ∂ ln ⁡ Y ∂ ln ⁡ F = Y F η R = Y F ( β 6 + β 10 t + β 14 ⁡ ln ⁡ F i t + 1 2 β 17 ⁡ ln ⁡ K i t + 1 2 β 19 ⁡ ln ⁡ N i t + 1 2 β 20 ⁡ ln ⁡ R i t ) (9) M P N = ∂ Y ∂ N = Y N ∂ ln ⁡ Y ∂ ln ⁡ N = Y N η N = Y N ( β 4 + β 8 t + β 12 ⁡ ln ⁡ N i t + 1 2 β 15 ⁡ ln ⁡ K i t + 1 2 β 18 ⁡ ln ⁡ R i t + 1 2 β 19 ⁡ ln ⁡ F i t ) (10)

The elasticity substitution of input factor is the core indicator to measure the strength of the substitution relationship between factors. Its initial definition was given by Hicks in “Wage Theory”. The factor substitution elasticity (when a given output is constant) is the percentage change in the factor ratio caused by the change in the marginal substitution rate. The elasticity of substitution is as follows: S u b s R F = d ⁡ ln ( R / F ) d ⁡ ln ( M P R / M P F ) = d ⁡ ln ( F / R ) d ⁡ ln ( M P F / M P R ) = S u b s F R (11) M P R M P F = Y R ∂ ln ⁡ Y ∂ ln ⁡ F / Y F ∂ ln ⁡ Y ∂ ln ⁡ R = F R η R η F (12)

S u b s R F > 0 ( S u b s R F < 0 ) indicates the relationship between factors is substitution (complementary). The substitution relationship between factors indicates that an increase in the input of one factor will lead to a decrease in the input of another factor. According to Eq. 11 and Eq. 12, we can obtain the elasticity substitution of inputs factors in internal and external transition. The elasticities of substitution between R and F , R and N , and F and N are as follows, respectively, S u b s R F = [ 1 + 2 ( β 20 − η F η R β 13 − η R η F β 14 ) ( η R + η F ) − 1 ] − 1 (13) S u b s R N = [ 1 + 2 ( β 18 − η N η R β 13 − η R η N β 12 ) ( η R + η F ) − 1 ] − 1 (14) S u b s F N = [ 1 + 2 ( β 19 − η N η F β 14 − η F η N β 12 ) ( η F + η N ) − 1 ] − 1 (15)

Based on the available data, we selected panel data from 30 provinces in Mainland China from 2000 to 2017 as the research sample. The Tibet area is not included in the statistics due to incomplete data. In China, the “5-year plan” is an important part of China’s national economic plan. The data from 2000 to 2017 cover the end of the “9th Five-Year Plan” period, “10th Five-Year Plan” period, “11th Five-Year Plan” period, “12th Five-Year Plan” period, and the early stage of the “13th Five-Year Plan “. Therefore, the data we select are of wide statistical significance.

We obtain the data of traditional fossil energy including coal, oil, and natural gas from the district energy balance table in the China Energy Statistical Yearbook. The terminal consumption of coal, oil, and natural gas is adopted to prevent the impact of energy processing and conversion. The clean energy generation capacity is selected from hydro power, nuclear, wind, and solar technologies in the China Electric Power Yearbook. Restricted by the unavailability of data, hydro power and nuclear power data are from 2000 to 2017, and wind energy data are selected from 2006 to 2017. Solar energy data range from 2010 to 2017. The thermal power data come from the thermal power generation in the China Electric Power Yearbook. The units of all energy data are uniformly converted into tce according to the energy discount standard coal reference coefficient in the China Energy Statistical Yearbook. In order to eliminate the influence of inflation and other factors, the capital stock and GDP are deflated to the constant 2000 prices according to the price index and GDP index. Data such as capital stock, GDP, GDP index, and price index are from the China Statistical Yearbook. The descriptive statistics of the above-mentioned variables are shown in Table 1. The capital stock is calculated using the equation as follows: K t = K t − 1 ( 1 − δ ) + I t (16) where K is the capital stock, δ indicates the depreciation rate, and I t means the investment.

In order to test whether the model setting is correct, the following aspects should be tested successively. The results of specification tests of the production function are shown in Table 2.

(1) Whether the stochastic frontier model is applicable: H 0 : γ = 0 . If the null hypothesis is rejected, it indicates that there are inefficiencies in the model, and the stochastic frontier production model can be used for parameter estimation; otherwise, the stochastic frontier analysis is not needed.

The results in Table 2 show that the LR statistic of the above test (2) is greater than the critical value of the mixed Chi-square distribution, indicating that the null hypothesis should be rejected. Therefore, it is more reasonable to use the translog production function. The results of test (3) and test (4) indicate that there are technical changes in the model, and this change is non-neutral. As shown in Table 3, the regression results show that γ significantly passes the t test, which shows that the null hypothesis of the above test (1) is also rejected and the inefficiency term exists. After the above tests, it can be concluded that the stochastic frontier model is applicable, and the production function adopted by the model is the translog production function.

Considering that local governments can determine the share of fossil energy, e s may have obvious endogenous problems in the inefficiency equation. In this case, the estimation results of the stochastic frontier model may be inaccurate. Therefore, according to Yang et al. (Alataş, 2020), when estimating Eq. 3, we use the first-order lag e s ( L . e s ) to control the endogenous problem. The estimation results are shown in Table 3. Most of the coefficients in the translog production function (3) are statistically significant. The maximum likelihood function value of the model and the LR test result show that the stochastic frontier model has a strong explanatory power. Therefore, the model we establish can reasonably reflect the changes in the technical efficiency of the 30 provinces.

For the determinants of the inefficient equation, although the coefficient of r d is positive, it is not significant, indicating that the R&D intensity has little effect on promoting the technical efficiency. From the energy production mode of China, it can be found that the current innovation activities of enterprises aim at product upgrading rather than improving energy saving. This result is consistent with some of the existing studies. For example, Xuan and Zhou (Yazan et al., 2022) have no evidence of a significant positive relationship between original innovation activity and energy efficiency. In addition, Yang et al. (2018) believes that the original enterprise innovation is uncertain and cyclical, and an increased cost of innovation may make it difficult to get reports in the short term.

The e x p coefficient is negative, meaning that exports can help improve the technical efficiency. This shows that the expected “Learning by Exporting” effect appears in China. It means that enterprises can acquire new knowledge from competitors when exporting so that export behavior improves the technological efficiency.

The f d i coefficient is significantly negative, indicating that f d i can effectively improve the technical efficiency. There are two theories of “pollution paradise” and “pollution halo” on the impact of f d i on technical efficiency. Our empirical results support the latter. It indicates that f d i drives the promotion of more efficient technologies in multinational corporations.

The coefficient of L . e s is significantly positive, indicating that increased fossil energy consumption is detrimental to improving the technical efficiency. At present, China’s energy structure seriously depends on fossil energy, which also shows that China’s current energy structure limits the improvement of technical efficiency.

The coefficient of k l is significantly positive, indicating that the rising organic composition of capital will lead to a decrease in technical efficiency. This result is consistent with the views of Tu and Leeke (2011) and shao et al. (2016). It also confirms that “capital-intensive” industries tend to be heavy polluting industries, while labor-intensive industries tend to be light polluting industries.

The coefficient of l p is significantly negative, indicating that l p can promote the technical efficiency. On one hand, the improvement of labor productivity is conducive to enterprises to achieve better production with more and other resources. On the other hand, improving living standards can help enhance people’s willingness to improve the environment and the “green” technical efficiency.

To discuss the biased technical changes in internal and external transition, we show the mean value of 2000–2017 in Table 4. Bias-NR and Bias-NF refer to the biased technical change in the external transition, while Bias-FR means the biased technical change in the internal transition.

In the external transition, for the pair of R and N, only eight provinces prefer to use clean energy, and the remaining 22 regions prefer to use traditional fossil energy. For the pair of F and N, 22 regions prefer thermal power generation, and the other eight provinces prefer traditional power generation. In the internal transition, technological changes are biased toward R in nine of the 30 provinces, while the remaining 21 provinces are biased F. This suggests that the government should continue to encourage producers to value clean production.

In addition, Table 5 shows the directed technical change bias order of the three input factors. In 20 of the 30 regions, the technical changes are more biased to thermal power generation, which is the first factor of the biased order for the 20 regions. However, only five regions are biased to clean energy generation. The production mode of the above eight provinces is relatively green and sustainable. It can be explained that the larger the production scale of renewable energy, the more advanced the corresponding level of renewable energy production technology is. For example, Hubei ranks the third largest in the renewable energy scale in China, so the renewable energy production technology in Hubei is relatively advanced. Five regions have more preference to traditional fossil energy, which is the first factor of biased order for 14 regions. Therefore, technological changes in these provinces tend to use traditional fossil energy rather than clean power generation or thermal power generation. Therefore, as a whole, the directed technical change in China is more biased to thermal power generation and deviated from clean energy generation.

However, technological changes in 18 regions deviate from clean energy generation, which is the last factor of the biased order for the 18 provinces in Table 5. In addition, technological changes in seven regions are more likely to deviate from thermal power generation, and technological changes in five regions are more likely to deviate from traditional fossil energy, which is the last factor of the bias order for the five regions. The above results show that overall, China’s provincial scope prefers to use thermal power over clean energy and traditional fossil energy. They are less inclined to use clean energy rather than thermal power or traditional fossil energy. On one hand, these results confirm the fact that the thermal power generation is popular in China. On the other hand, the results also show that the government departments should encourage the emphasis on clean production.

Based on some studies (Hicks, 1932; Acemoglu et al., 2015; Fredriksson and Sauquet, 2017; Naqvi and Engelbert, 2017; Fried, 2018; Kha, 2019), the degree of factor bias of directed technology changes is determined by the price and scale effects. Adjusting the relative price will timely adjust the relative demand and actual input between factors in the production process and gradually reduce the difference in the marginal output growth rate of the two energy factors so as to change the degree of factor bias of directed technical changes between factors. Therefore, in the internal transition, the governments can adjust the technical change bias of the provinces by raising the price of thermal power generation or increasing the subsidies for clean energy power generation. In the external transition, the governments can adjust the energy policies of the province by increasing the carbon tax prices or increasing subsidies for clean energy power generation and low-coal thermal power generation. These changes will continue to alter the relative price between factors.

In Table 6, we list the substitution elasticities between factors in the 30 provinces. From the perspective of external transition, for the pair of F and N, there are only nine provinces with substitution relations, and the other 21 other regions have complementary relationships. The complementary relationship between F and N in most provinces can be explained by their need for more energy to meet production demand. In addition, there are complementary relationships between R and N in only nine regions, while the 21 remaining regions have substitution relationships. From the perspective of the internal electric transition, there is a complementary relationship between F and R in only two regions, and the other 28 regions all have substitution relations. It indicates that increased clean energy power generation can currently be used to reduce thermal power generation in these provinces. Although at the provincial level, different regions show an obvious difference in substitution elasticity, there is a substitution relationship between other energy factors except for traditional fossil generation and thermal power generation with a complementary relationship on a whole.

In the production process, the internal and external transition can be conducive to the green development and transformation. Therefore, we analyze three transition ways and study the improvement path of energy transition in different regions based on the degree of biased directed technical changes and the substitution elasticity between factors. The classification results of eight external transition pathways and three internal transition pathways are shown in Tables 7–9.

First, the classification results between factor N and R are shown in Table 7. There are four production patterns according to the factor N and R. Among them, the ideal production mode shows Bias-NR<0 and Subs-NR>0. In this kind of production mode, these regions (Beijing, Fujian, Hainan, Tianjin, and Chongqing) are more inclined to use the clean energy power generation rather than use the traditional fossil energy. Increasing the use of clean energy power generation will reduce the use of fossil energy, which will help to promote the transformation of clean energy to traditional fossil energy.

The mode-like Bias-NR>0 and Subs-NR>0 suggests that the region prefers to use traditional fossil energy over clean energy. In addition, an increase in traditional fossil energy use would lead to a decline in the use of clean energy generation. For these areas, the directed technological change needs to be adjusted; Bias-NR<0 and Subs-NR<0 suggest that the region prefers the use of clean energy generation, and increasing the use of clean energy generation leads to the increased use of traditional fossil energy. These regions need to adjust the alternative relationship between the two energy sources; for areas with Bias-NR>0 and Subs-NR<0, the government needs to encourage technological changes inclined to use clean energy power generation and change the complementary relationship between clean energy power generation and traditional fossil energy.

Second, the classification results with four production patterns between F and N are shown in Table 8. The ideal production mode is Bias-NF<0 and Subs-NF>0. Areas that are in line with this production mode (Gansu, Neimenggu, Ningxia, Tianjin, and Xinjiang) are more inclined to use thermal power generation. When expanding the scale of production, they will increase the thermal power generation and reduce traditional fossil energy so as to promote the external transformation of thermal power generation to traditional fossil energy.

Finally, the classification results between F and R are shown in Table 9. The mode with Bias-FR>0 and Subs-FR>0 indicates that the province prefers thermal power to clean energy power generation, and there is an alternative relationship in the production process between clean energy power and thermal power. Therefore, the areas with the above mode prefer to use thermal power generation rather than clean energy generation, and the increase of thermal power generation use will lead to the decline of clean energy power generation. For these areas, the technical change needs to be adjusted between the two factors. Areas with Bias-FR>0 and Subs-FR<0 prefer thermal power generation rather than clean energy power generation. For these areas, technological change that is biased to clean energy power generation and use need to be encouraged, and the alternative relationship between thermal and clean energy power generation needs to be adjusted. The ideal production model shows Bias-FR<0 and Subs-FR>0, where the provinces (Beijing, Fujian, Guangxi, Hainan, Hubei, Jilin, Jiangsu, Qinghai, and Zhejiang) prefer the use of clean energy power generation rather than the use of thermal power generation. In addition, increasing the scale of the use of clean energy power generation can reduce the use of thermal power generation in these provinces.

Hence, in order to improve the external and internal electric transition, the Chinese governments should promote the reform of the market-oriented energy pricing mechanism according to characteristic transition modes in different regions. For the provinces with production patterns which can automatically benefit the energy transition, we suggest a moderate policy, while for the other provinces, we suggest a policy of energy price and tax. Moreover, for the enterprises, their production patterns are not easy to change. They usually benefit by minimizing costs under the conditions of homogeneous products and unchanged price. Therefore, the change of the inter-fule price will have an effect on the production costs and the structure of production factors, which improves the energy transition. Finally, the results from the analysis of China show that it is also possible for other countries to treat their energy transition differently due to their characteristic production patterns.