3.1. Power Decomposition Framework of Carbon Productivity
Carbon productivity is defined as the GDP output per unit of carbon equivalent emissions in a certain period. Since different types of energy consumption can estimate carbon equivalent emissions, and energy consumption per unit of emissions reflects the influence of energy mix, we further isolate the energy mix based on the defining equation, which can be obtained as:
In Equation (1),
CP is carbon productivity,
Y is GDP,
C is carbon equivalent emissions, and
E is fossil energy consumption.
ES is the energy consumption per unit emission, reflecting the influence of energy structure on carbon productivity;
YE is the GDP per unit energy consumption output, which is the energy productivity. According to the production theory, energy productivity can form an output relationship with the unit energy factor input. Thus, the effects of factor accumulation and total factor productivity can be further separated. The global DEA model is used in this paper to ensure the continuity and stability of the decomposition of technological progress.
KE and
LE denote the capital input and labor input per unit of energy consumption, respectively, and
is the distance function of output direction under the constant payoff of scale (CRS). Then, the global optimal energy productivity of the production unit in period t is from
. Further, the decomposition formula of drivers is obtained as follows:
In Equation (2),
YEC and
TFP denote GDP growth and total factor productivity change per energy consumption unit, respectively. Since the factor input per unit of energy consumption reflects the substitution relationship of capital and labor for energy input,
XEC is called the factor substitution effect here. According to the principle of DEA model solving,
is known that the inverse reflects the global technical
efficiency of a given production unit in the first period. Total factor productivity is thus defined as the ratio of global technical efficiency of two adjacent periods, while the factor substitution effect is defined as the optimal output. Further dynamizing Equation (1) and bringing Equation (2) gives the following decomposition:
CPC and ESC denote carbon productivity change and energy mix change, respectively. Equation (3) shows that three mechanisms mainly drive carbon productivity growth: first, energy structure optimization, i.e., the reduction in carbon emissions due to the substitution of low-carbon energy for high-carbon energy; second, total factor productivity growth, i.e., the increase in energy production due to the improvement of global technical efficiency in two adjacent periods; third, the factor substitution effect, i.e., the change in global optimal unit productivity due to the substitution of capital and labor for energy under the same technological conditions.
Combining single-period DEA with global DEA, TFP can be further decomposed into technological progress changes and technical efficiency changes. Let
be the CRS output distance function of the technical reference set for a single period, thus defining technological progress as the ratio of the relative distances of a given production unit to the single-period frontier and the global frontier, while technical efficiency is still defined as the ratio of the single-period technical efficiencies of the two adjacent periods, calculated as follows:
TPC and
TEC denote the change in technical progress and the change in technical efficiency, respectively,
. Since factor substitution includes the substitution of capital and labor for energy, it can still be further decomposed into the capital substitution effect (denoted as
KEC) and labor substitution effect (denoted as
LEC) in the global DEA framework. To avoid the uncertainty of the fixed factor reference, the geometric mean under the two-period consideration was chosen to respectively measure the capital substitution effect and labor substitution effect in this paper, and we have:
According to the nature of the metafrontier DEA model, production units with the same combination of factor input have the same output projection in the frontier, i.e.,
and
. The product of the two equals the total effect of factor substitution (
XEC) and is easily verifiable. Carbon productivity growth is mainly driven by the factor substitution effect and the total-factor productivity. The former includes low-carbon energy substitution, capital energy substitution, and labor energy substitution, and the latter contains technical progress and efficiency improvement. Therefore, by plugging the decomposition equations of total-factor productivity and factor substitution effect into Formula (3), the decomposition equation of the quintuple growth drivers of carbon productivity can be obtained as below:
It is easy to see that the change in carbon productivity is mainly driven by the energy structure effect, technological progress change, technical efficiency improvement, capital substitution effect, and labor substitution effect. Based on the above decomposition framework, growth accounting for carbon productivity changes in a country or region can be performed to assess the pattern of temporal changes dominated by the contributions of different factors.
3.2. β-Convergence Model of Carbon Productivity
Convergence models derived from economic growth theory can examine whether economic variables in different regions have normal and other steady states, i.e., whether lagging areas can catch up with the trends in developed areas. Common steady-state convergence is often referred to as absolute β-convergence. In contrast, different steady-state convergence is conditional β-convergence when considering the impact of differences in resource endowments and technological conditions between regions on steady-state equilibrium. We use the Mankiw–Romer–Weil (MRW) convergence model [
64] as the basis, and first construct the following carbon productivity convergence model:
where
denotes the
growth rate of regional carbon productivity in two periods, and there is a log-transformation
relationship with that in the decomposition model above.
denotes the logarithm
of the initial carbon productivity level of the region, then
is the logarithm of the effective depreciation rate of capital in the corresponding region.
is a constant term, and the random
, error assignment, follows a normal distribution. When
, the coefficient, is less than 0, it indicates that the carbon productivity in less-developed regions grows faster than that in developed areas, and there is a convergence trend during the study period. When the differences in regional endowment factors are
not considered, i.e., Equation (9) examines
the absolute convergence, otherwise,
is the conditional convergence.
Based on the proof of the MRW model,
it is clear that the parameters are defined by an exponential decay function, i.e.,
In Equation (10), is the rate of convergence and is the period interval examined. If the parameter takes the values falling in the gap, this implies that carbon productivity can converge directly to a steady state without oscillation. Thus, according to the speed of convergence implied of by the parameter, we can easily calculate the regression results for .
The traditional literature ignores the spatial dependence of carbon productivity, leading to bias in estimation convergence models. The resource endowments and market conditions of neighboring provinces are generally more similar, factor flows and technological cooperation between regions will be adequate, and carbon productivity exhibits significant spatial clustering characteristics. Therefore, it is necessary to adopt a spatial panel model to revise the traditional convergence model, and we start from a spatial Durbin model (SDM) of panel data.
For the convenience of writing, the subscripts of the variables in Equation (11) are omitted to represent them in matrix and vector form. Among these, the
matrix is used to describe the dependence between regions, and the commonly used spatial adjacency weight matrix (Hainan is adjacent to Guangdong) was chosen here.
, the product with the relative variables, represents the spatial lag term, and the other variables are defined the same as in Equation (9). It is easy to see that the SDM simultaneously examines the spatial association and mutual influence of carbon productivity and effective investment rate between regions. When the spatial lag parameter
sum
is zero, the SDM model degenerates to a spatial lag model (SLM), which means that the growth of carbon productivity comes only from the spatial spillover effect of the dependent variable. In addition, the unidentifiable spatial effects can be further degraded to a spatial error model (SEM) by attributing them to random error shocks if all spatial lag terms are insignificant, i.e.,
In Equation (12), is the random error term for the presence of random spatial effects, and is the same, subject to the normal distribution.
Further, the spatial convergence mechanism of carbon productivity can be further tested by combining the power decomposition Equation (8) and the spatial convergence Equation (11). Substituting
,
,
,
, and
, these denote the energy structure effect, capital substitution effect, labor substitution effect, technical progress effect, and technical efficiency improvement, respectively. The corresponding growth rate forms are obtained by taking the logarithm according to the decomposition equation. Replacing in Equation (9),
as the spatial convergence equation with the five major driving factors as the explanatory variables is further set. As a result, the set of joint cubic equations based on the decomposition model extension constitutes a more rigorous fully mediated effects model, and the following relationship exists between the explanatory variables of the different convergence equations
, as well as the regression coefficients, i.e.,
Thus, by examining the magnitude and direction of the parameters of each driver around the
, the initial level of carbon productivity can be used to assess the specific mechanisms by which carbon productivity convergence is achieved. In order to provide the accuracy of parameter estimation, quasi-maximum likelihood (QML) estimation is used for the spatial convergence model [
65].
3.3. Variable Selection and Data Description
According to the guidelines developed by the United Nations Intergovernmental Panel on Climate Change (IPCC), the reference method accounts for fossil energy production sites. Still, many primary energy sources do not enter final consumption, resulting in severe risk of overestimation [
66], so this paper chooses those fossil energy carbon emissions accounted for by the sectoral method. Based on data availability and consistency, 30 provinces, autonomous regions, and cities in China, with the exceptions of Tibet and Hong Kong, Macao, and Taiwan, were selected for the study during 1997–2017, with output as GDP, energy as total consumption, and labor as total social employees. Capital was physical capital stock and was estimated using the perpetual inventory method, which was extended to 2017 following the classical literature [
67]. The underlying data are obtained from the China Statistical Yearbook, China Energy Statistical Yearbook, provincial statistical yearbooks, and the China Carbon Accounting Database (CEADs) for previous years. Considering that fossil energy is not fully converted into CO
2, the direct use of CO
2 emission coefficients is likely to cause estimation bias, so we used uniform carbon equivalents and reconstructed them according to provincial CO
2 emission inventories; GDP and physical capital stock were adjusted to compare prices in 2000, and individual missing data were completed by linear interpolation.
Regarding economic growth theory, the logarithm of the ratio of capital investment rate to effective depreciation rate (
) was chosen as the control variable in this paper, and
and
were logarithms of the effective investment rate of physical capital and the effective investment rate of human capital, respectively. The physical capital investment rate was measured as the share of GDP’s total fixed capital formation. The human capital investment rate used the share of the population with a high school education or above in the 15–19 age group as a proxy variable [
64],
as the GDP growth rate, and
still used the results measured in the classical literature [
67]. Corresponding data were obtained from the China Statistical Yearbook and the China Human Capital Index Report database (CHLR) for previous years. The main variables and their measures are summarized in
Table 1.