To design a quota allocation scheme that fits the development reality of the industrial sector in Henan Province, we conducted a quota allocation study for its subsectors. First, we use the Stochastic Impacts by Regression on Population, Affluence and Technology (STIRPAT) model to determine a carbon peak scenario and peak value that best meet development requirements. Then, we construct a multi-index allocation system and use the TOPSIS model to make initial quota allocations between subsectors. We also use the ZSG-DEA model to optimize the efficiency of the allocation scheme. Finally, we use the environmental Gini coefficient and NDDF model for quantitative evaluation to select the optimal quota allocation scheme. The research framework of this paper is shown in Fig. 1.
2.1.1 Total Quota Prediction Model
(1) STIRPAT model
The STIRPAT model is an extension of the classic IPAT model, and it can simultaneously consider various factors that influence carbon emissions, such as energy, economy and environment (Pan et al. 2023). Currently, the STIRPAT model is widely used in analyzing the factors influencing carbon emissions and predicting their trends (Chai et al. 2022, Noorpoor &Kudahi 2015). The basic form of the model is:
$$\text{l}\text{n}\text{I}=\text{a}{\text{P}}^{\text{b}}{\text{A}}^{\text{c}}{\text{T}}^{\text{d}}\text{e}$$
1
where I represents carbon emissions; P represents demographic factors; A represents Per capita GDP; and T represents technical level (represented by energy intensity). a is the model coefficient; b, c, and d are the elasticity coefficients of each factor respectively; and e is the error term.
Based on the modeling ideas and development characteristics of the industrial sector in Henan Province, the STIRPAT model for predicting carbon emissions is constructed by introducing total industrial output (S), industrial structure (IS) and carbon intensity (CI) indicators (Liu et al. 2020). To facilitate calculation and data analysis, the model is transformed into a linear equation, and the expansion formula is:
$$\text{l}\text{n}\text{I}=\text{l}\text{n}\text{a}+\text{b}\text{l}\text{n}\text{P}+\text{c}\text{l}\text{n}\text{A}+\text{d}\text{l}\text{n}\text{T}+\text{e}\text{l}\text{n}\text{S}+\text{f}\text{l}\text{n}\text{I}\text{S}+\text{g}\text{l}\text{n}\text{C}\text{I}+\text{l}\text{n}\text{e}$$
2
The model may face multicollinearity issue due to the interaction among input variables. To ensure the validity of the model estimation results, ridge regression is adopted for fitting (Zhong et al. 2019). Based on the relevant data of industrial sector in Henan Province from 2010 to 2020, ridge regression modeling is conducted using SPSS. With a unit length of 0.02 and a ridge regression coefficient k between 0 and 1, equations, ridge traces and R2 corresponding to different values of k are obtained. The smaller the value of the ridge regression parameter k, the less information loss in the sample data and hence higher accuracy of the model.
(2) STIRPAT parameter setting
The scenario analysis is designed based on the development plans and historical data (Zhong et al. 2019). With the STIRPAT model as a basis, eight development scenarios aimed at the socio-economic level of the industrial sector in Henan Province are established to predict carbon emissions. We divide the future trends of each factor into three different rates (low, medium and high), considering the differences in social, economic and technological levels and the relevant development goals (Liu et al. 2019). The change rates for each factor are shown in Appendix Table A1.
Based on the variation of factors, eight scenarios are designed for predicting carbon emissions trends from 2021 to 2060. In simple terms, all factors are divided into two groups. The first group comprises population, Per capita GDP and total industrial output, which mainly maintain an upward trend in the future and have a positive impact on carbon emissions. The second group includes technology level, industrial structure and carbon intensity, which are mainly on a downward trend in the future and will help to stop the increase of carbon emissions. Table 1 provides specific description of the different scenarios.
Table 1
Parameter designs of scenarios.
Scenario
|
P
|
A
|
S
|
T
|
IS
|
CI
|
Low
|
L
|
L
|
L
|
L
|
L
|
L
|
Medium
|
M
|
M
|
M
|
M
|
M
|
M
|
High
|
H
|
H
|
H
|
H
|
H
|
H
|
High-Medium
|
H
|
H
|
H
|
M
|
M
|
M
|
High- Low
|
H
|
H
|
H
|
L
|
L
|
L
|
Medium - High
|
M
|
M
|
M
|
H
|
H
|
H
|
Medium- Low
|
M
|
M
|
M
|
L
|
L
|
L
|
Low - Medium
|
L
|
L
|
L
|
M
|
M
|
M
|
According to the “Double Carbon” target and the development plans of industrial sector in Henan Province(Anonymous 2022b), this paper divides the period from 2021 to 2060 into eight stages, with every five years as a stage. Based on the design parameters, the predicted values of each variable during the study period are obtained. These predicted values are logarithmically processed and incorporated into Eq. (2) to obtain the predicted carbon emission values.
2.1.2 Quota Allocation Model
(1) Quota allocation index system
The formulation of a quota allocation scheme is influenced by numerous factors (Shojaei &Mokhtar 2022, Zhan 2022). Therefore, this paper chose the multi-index method for the initial allocation of quotas. Referring to previous studies, the fairness principle can be divided into four dimensions: emission reduction responsibility, emission reduction potential, emission reduction capacity and social welfare. Allocation indicators include GDP, carbon emissions, number of employees and so on. Based on the principle of fairness, we select eight indicators from the above dimensions to construct a quota allocation index system (Li 2020, Wu &Zhang 2019, Zhang 2019). The details of these indicators are shown in Table 2.
Table 2
The dimensions, indexes, and interpretation of the principle of fairness.
Dimension
|
Indicator
|
Interpretation
|
Emissions reduction responsibility (Z1)
|
Carbon emissions (Z11)
|
Major energy carbon emissions
|
Carbon intensity (Z12)
|
Carbon emissions per GDP
|
Emissions reduction potential (Z2)
|
Energy mix (Z21)
|
Coal consumption proportion
|
Energy intensity (Z22)
|
Energy consumption per GDP
|
Emissions reduction capacity (Z3)
|
Total industrial output (Z31)
|
Main revenue
|
Profitability (Z32)
|
Profit/Net assets
|
Social welfare (Z4)
|
Number of employees (Z41)
|
Number of employees
|
Per capita output (Z42)
|
Total industrial output /Employees
|
(2) The TOPSIS-based initial allocation model |
To ensure fairness, the TOPSIS method is selected for the initial allocation of quotas (Zhu et al. 2020a, Zhuang &Liu 2015). This method has been widely used in multi-criteria decision-making problems, but less applied in quota allocation studies (Fan et al. 2023, Feizi et al. 2017). First, the Analytic Hierarchy Process (AHP) and entropy weight method are used to assign weights to the indicators respectively, and then the combined subjective and objective weights of the indicators can be calculated (see Appendix Table A2). Finally, incorporating indicator comprehensive weights into the TOPSIS model can lead to an ideal initial allocation scheme.
①Analytic Hierarchy Process (AHP)
AHP was proposed by Saaty in the 1970s, which is a simple method to determine the weight of various indicators at each level (Saaty 1988). This paper uses a single layer model to weight the indicators.
Step1: Build the comparison matrix. Referring to relevant research, we rank the importance of eight indicators in Table 2. We also invite six experts from the Development and Reform Commission of Henan Provincial Government and the Bureau of Geology and Mineral Resources to score the indicators. By integrating expert opinions, we obtain the comparison matrix \({\left({\text{a}}_{\text{i}\text{j}}\right)}_{n\times n}\):
$$\begin{array}{cccccccc}{\text{A}}_{1}& {\text{A}}_{2} & {\text{A}}_{3}& {\text{A}}_{4}& {\text{A}}_{5}& {\text{A}}_{6}& {\text{A}}_{7}& {\text{A}}_{8}\end{array}$$
$$\left[\begin{array}{ccccccccc}{\text{A}}_{1}& 1& 1/2& 2& 3& 3& 5& 6& 7\\ {\text{A}}_{2}& 2& 1& 2& 3& 3& 4& 5& 6\\ {\text{A}}_{3}& 1/2& 1/2& 1& 1/3& 2& 2& 3& 4\\ {\text{A}}_{4}& 1/3& 1/3& 3& 1& 2& 3& 3& 4\\ {\text{A}}_{5}& 1/3& 1/3& 1/2& 1/2& 1& 2& 3& 4\\ {\text{A}}_{6}& 1/5& 1/4& 1/2& 1/3& 1/2& 1& 2& 3\\ {\text{A}}_{7}& 1/6& 1/5& 1/3& 1/3& 1/3& 1/2& 1& 2\\ {\text{A}}_{8}& 1/7& 1/6& 1/4& 1/4& 1/4& 1/3& 1/2& 1\end{array}\right]$$
3
where \({A}_{i}\) denotes the importance of indicator \(i=\text{1,2},\dots ,8\) respectively. \({a}_{ij}\) denotes the importance of\({A}_{i}\) compared with \({A}_{j}\).
Step2: Matrix consistency test. Calculate the eigenvalues and eigenvectors of the comparison matrix. We denote \({\lambda }_{max}\) as the largest eigenvalue and calculate the consistency ratio (CR):
$$\text{C}\text{R}=\text{C}\text{I}/\text{R}\text{I}=\left(\frac{{{\lambda }}_{\text{m}\text{a}\text{x}}-\text{n}}{\text{n}-1}\right)/\text{R}\text{I}$$
4
where CI is consistency indicator, which is used to estimate the consistency of the comparison matrix. RI is the average random consistency index, which takes a value that increases with increasing order n (Zong et al. 2022). When CR ≤ 0.1, the consistency of this matrix is acceptable.
Step3: Determine indicator weights. The unit positive eigenvector corresponding to \({\lambda }_{max}\) is the weight of each indicator.
②Entropy weight method
The entropy weight method is applied to calculate the entropy \({\text{E}}_{j}\) of the indicator by the information of the indicator. If \({\text{E}}_{j}\) is smaller, it means that the greater dispersion of the indicator data and the more information provided, the greater its weight (Chen et al. 2021b). The steps are as follows:
Step1: Normalize the matrix. Let \({x}_{ij}\) be the original data value of indicator j of subsector i, where \(i=1,2,···,m\) and \(j=1,2···,n\). We normalize the matrix \(\text{X}={\left({x}_{ij}\right)}_{m\times n}\) according to the properties of each indicator:
$${\tilde{\text{x}}}_{\text{i}\text{j}}=\left\{\begin{array}{c}\frac{{\text{x}}_{\text{i}\text{j}}-\text{m}\text{in}\left({\text{x}}_{\text{i}}\right)}{\text{max}\left({\text{x}}_{\text{i}}\right)-\text{min}\left({\text{x}}_{\text{i}}\right)}, when {\text{x}}_{\text{i}\text{j}} is positive indicator.\\ \frac{\text{m}\text{a}\text{x}\left({\text{x}}_{\text{i}}\right)-{\text{x}}_{\text{i}\text{j}}}{\text{max}\left({\text{x}}_{\text{i}}\right)-\text{min}\left({\text{x}}_{\text{i}}\right)},when {\text{x}}_{\text{i}\text{j}} is negative indicator. \end{array}\right.$$
5
Step2: Calculate the ratio for each indicator. The ratio of indicator j of subsector i refers to the change in size of the indicator.
\({\text{P}}_{\text{i}\text{j}}=\frac{{\tilde{\text{x}}}_{\text{i}\text{j}}}{{\sum }_{\text{i}=1}^{\text{m}}{\tilde{\text{x}}}_{\text{i}\text{j}}}\) , i = 1, ···, m, j = 1, ···, n (6)
Step3: Calculate the entropy and weights of the indicators. If \({\text{P}}_{ij}>\)0, the entropy of the data is defined as \({\text{E}}_{j}=-\frac{1}{\text{ln}\text{m}}{\sum }_{i=1}^{\text{m}}{\text{P}}_{ij}\text{ln}\left({\text{P}}_{ij}\right)\), and \({\text{E}}_{j}\)=0 otherwise. The calculation formula for indicator weight is as follows:
\({{\omega }}_{\text{j}}=\frac{1-{\text{E}}_{\text{j}}}{{\sum }_{\text{j}=1}^{\text{n}}\left(1-{\text{E}}_{\text{j}}\right)}\) j = 1, ···, n. (7)
③The weight of indicator j calculated by AHP and entropy weight method are \({\omega }_{1j}\) and \({\omega }_{2j}\), respectively. The comprehensive weight \({\omega }_{j}\) of indicator j can be calculated using Eq. (8):
\({{\omega }}_{\text{j}}=\frac{{{\omega }}_{1\text{j}}\bullet {{\omega }}_{2\text{j}}}{\sum _{\text{j}=1}^{\text{n}}{{\omega }}_{\text{i}\text{j}}\bullet {{\omega }}_{2\text{j}}}\) , j = 1,2,⋯,n (8)
④The TOPSIS quota allocation model
The basic idea of TOPSIS is that the optimal solution should be closest to the positive-ideal solution and farthest from the negative-ideal solution (Zhu et al. 2020b). Based on the above indicator comprehensive weight \({\omega }_{j}\), this paper calculates the initial quotas for each subsector using the TOPSIS method.
Step1: Normalize the matrix. We normalize the original data matrix \(\text{X}={\left({x}_{ij}\right)}_{m\times n}\) and write the normalized matrix of X as Z.
Step2: Calculate the positive-ideal solution and negative-ideal solution. The optimal values of all indicators in the evaluation system constitute the positive-ideal solution, denoted as Z+. The definition of the negative-ideal solution Z− is exactly opposite.
Step3: Calculate the distance from the positive-ideal solution and negative-ideal solution. The weight of each indicator is defined as \({\omega }_{j} (\text{j}=\text{1,2},\cdots ,\text{n})\). Let \({{D}_{i}}^{+}\) be the distance between the indicator vector of subsector i (i=1, ···, m) and the positive-ideal solution. The definition of \({{D}_{i}}^{-}\)is exactly opposite.
$${{\text{D}}_{\text{i}}}^{+}=\sqrt{\sum _{\text{j}=1}^{\text{n}}{{{\omega }}_{\text{j}}({{\text{Z}}_{\text{j}}}^{+}-{\text{Z}}_{\text{i}\text{j}})}^{2}}, {{\text{D}}_{\text{i}}}^{-}=\sqrt{\sum _{\text{j}=1}^{\text{n}}{{{\omega }}_{\text{j}}({{\text{Z}}_{\text{j}}}^{-}-{\text{Z}}_{\text{i}\text{j}})}^{2}}$$
9
Step4: Determine the allocation ratio. The score for subsector i is recorded as \({S}_{i}=\frac{{{D}_{i}}^{-}}{{{D}_{i}}^{-}+{{D}_{i}}^{+}}\). The score \({S}_{i}\) (i = 1, ..., m) is then normalized to obtain the initial allocation ratio for each subsector.
$${{\text{S}}_{\text{i}}}^{\text{*}}=\frac{{\text{S}}_{\text{i}}}{\sum _{\text{i}=1}^{\text{m}}{\text{S}}_{\text{i}}}$$
10
Step5: Calculate the initial quotas for subsectors. Multiply the allocation ratio \({{\text{S}}_{\text{i}}}^{\text{*}}\) with the total quotas I to obtain the quotas for subsector i:
$${\text{I}}_{\text{i}}=\text{I}\bullet {{\text{S}}_{\text{i}}}^{\text{*}}$$
11
(3) The ZSG-DEA-based reallocation model
Scholars generally believe that efficiency is one of the main criteria for quota allocation (Zhan 2022). Therefore, it is necessary to optimize the efficiency of the initial allocation scheme obtained from the TOPSIS model. Considering the zero-sum game situation among subsectors, this paper uses the ZSG-DEA model for efficiency optimization so as to complete the quota reallocation (Lins et al. 2003). Since carbon emissions are weakly disposable, they are usually considered as undesirable outputs in energy-economic-environmental systems and are commonly used as input factors in DEA models. Therefore, this paper considers the initial quotas as an input factor and the total output, number of employees and energy consumption of each subsector as output factors. According to the idea of proportional abatement strategy, inefficient decision-making units (DMUs) must reduce their inputs to improve efficiency. Considering the total quotas limit, other DMUs must increase their inputs (Ma et al. 2018, Zheng 2012). The input-oriented model of ZSG-DEA is written as Eq. (12):
$$\begin{array}{c}\text{min}{{\phi }}_{\text{o}} \\ \text{s}.\text{t}.\sum _{\text{i}=1}^{\text{n}}{{\lambda }}_{\text{i}}{\text{x}}_{\text{j}\text{i}}\left[1+\frac{{\text{x}}_{\text{j}0}(1-{{\phi }}_{0})}{\sum _{\text{i}\ne 0}{\text{x}}_{\text{j}\text{i}}}\right]\le {{\phi }}_{0}{\text{x}}_{\text{j}0}, \text{j}=1\\ \sum _{\text{i}=1}^{\text{n}}{{\lambda }}_{\text{i}}{\text{y}}_{\text{i}\text{k}}\ge {\text{y}}_{\text{o}\text{k}}, \text{k}=\text{1,2},3 \\ \sum _{\text{i}=1}^{\text{n}}{{\lambda }}_{\text{i}}=1, \\ {{\lambda }}_{\text{i}}\ge 0, \text{i}=\text{1,2},\cdots ,\text{n} \end{array}$$
12
where \({\text{φ}}_{\text{o}}\) is the allocation efficiency under a fixed quota; \({\lambda }_{i}\) is the contribution of the ith subsector to the overall efficiency; \({X}_{ji}\)(i = 1,2,\(\cdots\),n, j = 1) is the jth input factor of the ith DMU (i.e., subsector i), which is the initial quotas; \({y}_{ik}(\text{k}=\text{1,2},3)\) is the kth output factor of subsector i; \({x}_{j0}\) and \({y}_{ok}\) are the corresponding input and output factors for each subsector.`
2.1.3 Quota Allocation Scheme Evaluation Model
(1) Environmental Gini coefficient
Fairness is a fundamental condition for a scheme to be accepted by different audiences (Yang et al. 2020). The environmental Gini coefficient has been widely used to measure the fairness of resources allocation, which is developed from the Gini coefficient (Tan et al. 2020b). The Gini coefficient was first proposed by Corrado Gini and widely used to measure income unfairness in economics. The Gini coefficient is calculated based on the Lorenz curve, and the curvature of the curve can reflect the unfairness of income allocation. The Gini coefficient ranges from 0 to 1, with a larger coefficient indicating greater unfairness in income allocation. If the income is changed to carbon emission quotas, the Gini coefficient becomes the environmental Gini coefficient, which can reflect the unfairness of quota allocation.
Referring to the literatures (Chen et al. 2021a, Yang &Yang 2020, Yao et al. 2020), carbon emissions have a significant positive correlation with economic level. Therefore, in this study, the horizontal and vertical coordinates of the Lorenz curve are total output and carbon emission quotas, respectively. The environmental Gini coefficient based on total output can be computed as follows:
$$\text{G}=1-{\sum }_{\text{i}=1}^{\text{n}}\left({\text{x}}_{\text{i}}-{\text{x}}_{\text{i}-1}\right)\left({\text{y}}_{\text{i}}+{\text{y}}_{\text{i}-1}\right) \text{i}=\text{1,2},\cdots ,\text{n}$$
13
where G denotes the environmental Gini coefficient. When the subsectors are ranked from low to high according to their share of total output,\({x}_{i}\) and \({y}_{i}\) denote the cumulative share of total output and quotas up to subsector i, respectively, with \({x}_{0}=0\), \({y}_{0}=0\). According to the classification definition of the Gini coefficient, G < 0.2 denotes absolute fairness of the scheme; 0.2 < G < 0.4 denotes relative fairness; G > 0.4 denotes that the scheme is unfair; and 0.4 is the threshold for the scheme to changes from fair to unfair.
(2) The non-radial direction distance function (NDDF)
To ensure the effectiveness of the scheme, the quantitative evaluation of its economics is necessary. This paper selects marginal abatement cost (MAC) and total abatement cost (TRC) as evaluation indicators, and constructs NDDF model to obtain them. Referring to (Lin &Zhou 2021, Long et al. 2015), we select total assets (K), number of employees (L) and energy consumption (E) as input variables, and total output (Y) and carbon emission quotas (Q) as desired and undesired outputs, respectively. Following Kuosmanen (Kuosmanen &Podinovski 2009) and Zhou (Zhou et al. 2012) et al., we apply NDDF model based on Kuosmanen technology to calculate the value of specific decision-making unit (\({\text{D}\text{M}\text{U}}_{\text{o}}\)), which is denoted as\(\overrightarrow{\text{N}\text{D}}\left(\text{K},\text{L},\text{E},\text{Y},\text{Q};\text{g}\right)\):
$$\begin{array}{c}\overrightarrow{\text{N}\text{D}}\left(\text{K},\text{L},\text{E},\text{Y},\text{Q};\text{g}\right)=\text{m}\text{a}\text{x}({{\omega }}_{\text{K}}{{\beta }}_{\text{K}}+{{\omega }}_{\text{L}}{{\beta }}_{\text{L}}+{{\omega }}_{\text{E}}{{\beta }}_{\text{E}}+{{\omega }}_{\text{Y}}{{\beta }}_{\text{Y}}+{{\omega }}_{\text{Q}}{{\beta }}_{\text{Q}})\\ \text{s}.\text{t}.{\sum }_{\text{i}=1}^{\text{N}}{\text{Z}}_{\text{i}}{\text{K}}_{\text{i}}\le {\text{K}}_{0}+{{\beta }}_{\text{K}}{\text{g}}_{\text{K}}\\ {\sum }_{\text{i}=1}^{\text{N}}{\text{Z}}_{\text{i}}{\text{L}}_{\text{i}}\le {\text{L}}_{0}+{{\beta }}_{\text{L}}{\text{g}}_{\text{L}}\\ {\sum }_{\text{i}=1}^{\text{N}}{\text{Z}}_{\text{i}}{\text{E}}_{\text{i}}\le {\text{E}}_{0}+{{\beta }}_{\text{E}}{\text{g}}_{\text{E}}\\ {\sum }_{\text{i}=1}^{\text{N}}{{{\theta }}_{\text{i}}\text{Z}}_{\text{i}}{\text{Y}}_{\text{i}}\ge {\text{Y}}_{0}+{{\beta }}_{\text{Y}}{\text{g}}_{\text{Y}}\\ {\sum }_{\text{i}=1}^{\text{N}}{{{\theta }}_{\text{i}}\text{Z}}_{\text{i}}{\text{Q}}_{\text{i}}={\text{Q}}_{0}+{{\beta }}_{\text{Q}}{\text{g}}_{\text{Q}}\\ {\sum }_{\text{i}=1}^{\text{N}}{\text{Z}}_{\text{i}}=1\\ {\text{Z}}_{\text{i}}\ge 0,\text{i}=1,\cdots ,\text{N}\\ 0\le {{\theta }}_{\text{i}}\le 1,\text{i}=1,\cdots ,\text{N}\end{array}$$
14
where \({{{\omega }}^{\text{T}}=({\omega }}_{\text{K}},{{\omega }}_{\text{L}},{{\omega }}_{\text{E}},{{\omega }}_{\text{Y}},{{\omega }}_{\text{Q}})\) is the normalized weight vector of the inputs and outputs; \({Z}_{i}\)is the intensity variables used to extend or reduce individual inputs and outputs; the multiplier\({\theta }_{i}\) is the individual abatement factor; \(\text{g}=({\text{g}}_{\text{K}},{\text{g}}_{\text{L}},{\text{g}}_{\text{E}},{\text{g}}_{\text{Y}},{\text{g}}_{\text{Q}})\) is the directional vector. Referring to Zhou et al. (Zhou et al. 2012), we set the weight vector \({{\omega }}^{\text{T}}=(\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3},\frac{1}{3})\) and the direction vector \(\text{g}=(-\text{K},-\text{L},-\text{E},\text{Y},-\text{B})\).
Let \({\lambda }_{i}={\theta }_{i}{Z}_{i}\), \({\mu }_{i}=\left({1-\theta }_{i}\right){Z}_{i}\), the Eq. (14) can be transformed as follows:
$$\begin{array}{c}\overrightarrow{\text{N}\text{D}}\left(\text{K},\text{L},\text{E},\text{Y},\text{Q};\text{g}\right)=\text{m}\text{a}\text{x}({{\omega }}_{\text{K}}{{\beta }}_{\text{K}}+{{\omega }}_{\text{L}}{{\beta }}_{\text{L}}+{{\omega }}_{\text{E}}{{\beta }}_{\text{E}}+{{\omega }}_{\text{Y}}{{\beta }}_{\text{Y}}+{{\omega }}_{\text{Q}}{{\beta }}_{\text{Q}})\\ \text{s}.\text{t}.{\sum }_{\text{i}=1}^{\text{N}}({\lambda }_{i}+{\mu }_{i}){\text{K}}_{\text{i}}\le {\text{K}}_{0}+{{\beta }}_{\text{K}}{\text{g}}_{\text{K}}\\ {\sum }_{\text{i}=1}^{\text{N}}({\lambda }_{i}+{\mu }_{i}){\text{L}}_{\text{i}}\le {\text{L}}_{0}+{{\beta }}_{\text{L}}{\text{g}}_{\text{L}}\\ {\sum }_{\text{i}=1}^{\text{N}}({\lambda }_{i}+{\mu }_{i}){\text{E}}_{\text{i}}\le {\text{E}}_{0}+{{\beta }}_{\text{E}}{\text{g}}_{\text{E}}\\ {\sum }_{\text{i}=1}^{\text{N}}{\lambda }_{i}{\text{Y}}_{\text{i}}\ge {\text{Y}}_{0}+{{\beta }}_{\text{Y}}{\text{g}}_{\text{Y}}\\ {\sum }_{\text{i}=1}^{\text{N}}{\lambda }_{i}{\text{Q}}_{\text{i}}={\text{Q}}_{0}+{{\beta }}_{\text{Q}}{\text{g}}_{\text{Q}}\\ {\sum }_{\text{i}=1}^{\text{N}}({\lambda }_{i}+{\mu }_{i})=1\\ {\lambda }_{i},{\mu }_{i}\ge 0,\text{i}=1,\cdots ,\text{N}\\ 0\le {{\theta }}_{\text{i}}\le 1,\text{i}=1,\cdots ,\text{N}\end{array}$$
15
As the marginal abatement cost equals the ratio of carbon emission shadow price to economic output, we construct a dual model to obtain this indicator. The dual model is as follows:
$$\begin{array}{c}\overrightarrow{\text{N}\text{D}}\left(\text{K},\text{L},\text{E},\text{Y},\text{Q};\text{g}\right)=\text{m}\text{i}\text{n}({{\mu }}_{\text{k}\text{i}}{\text{K}}_{\text{i}}+{{\mu }}_{\text{L}\text{i}}{\text{L}}_{\text{i}}+{{\mu }}_{\text{E}\text{i}}{\text{E}}_{\text{i}}+{{\mu }}_{\text{Y}\text{i}}{\text{Y}}_{\text{i}}+{{\mu }}_{\text{Q}\text{i}}{\text{Q}}_{\text{i}}+{\varphi }_{i})\\ \text{s}.\text{t}.{\sum }_{\text{i}=1}^{\text{N}}({{\mu }}_{\text{k}\text{i}}{\text{K}}_{\text{i}}+{{\mu }}_{\text{L}\text{i}}{\text{L}}_{\text{i}}+{{\mu }}_{\text{E}\text{i}}{\text{E}}_{\text{i}}+{{\mu }}_{\text{Y}\text{i}}{\text{Y}}_{\text{i}}+{{\mu }}_{\text{Q}\text{i}}{\text{Q}}_{\text{i}}+{\varphi }_{i})\ge 0\\ {\sum }_{\text{i}=1}^{\text{N}}{({\mu }}_{\text{k}\text{i}}{\text{K}}_{\text{i}}+{{\mu }}_{\text{L}\text{i}}{\text{L}}_{\text{i}}+{{\mu }}_{\text{E}\text{i}}{\text{E}}_{\text{i}}+{\varphi }_{i})\ge 0\\ {{\mu }}_{\text{k}\text{i}}{\text{K}}_{\text{i}}\ge 1;{{\mu }}_{\text{L}\text{i}}{\text{L}}_{\text{i}}\ge 1;{{\mu }}_{\text{E}\text{i}}{\text{E}}_{\text{i}}\ge 1;\\ {{\mu }}_{\text{Y}\text{i}}{\text{Y}}_{\text{i}}\ge 1;{{\mu }}_{\text{Q}\text{i}}{\text{Q}}_{\text{i}}\ge 1\\ {{\mu }}_{\text{k}\text{i}}\ge 0;{{\mu }}_{\text{L}\text{i}}\ge 0;{{\mu }}_{\text{E}\text{i}}\ge 0;{{\mu }}_{\text{Y}\text{i}}\ge 0;\\ {{\mu }}_{\text{Q}\text{i}} and {\varphi }_{i} free\end{array}$$
16
where \({\varphi }_{i},{{\mu }}_{\text{k}\text{i}},{{\mu }}_{\text{L}\text{i}},{{\mu }}_{\text{E}\text{i}},{{\mu }}_{\text{Y}\text{i}}\) and \({{\mu }}_{\text{Q}\text{i}}\)are all dual variables. \({\varphi }_{i}\) represents the scale payoff of \({\text{D}\text{M}\text{U}}_{\text{i}}\). By solving model (16), the marginal abatement cost can be expressed by Eq. (17):
$${\text{M}\text{A}\text{C}}_{\text{i}}={{\rho }}_{\text{Y}\text{i}}\bullet \frac{{{\mu }}_{\text{Q}\text{i}}}{{{\mu }}_{\text{Y}\text{i}}}$$
17
where \({\rho }_{Yi}\) represents the absolute shadow price of the total output, which is set to 1 yuan/t according to the definition (Wu et al. 2019b). \({MAC}_{i}\) represents the marginal abatement cost of carbon emission for subsector i.
However, MAC only reflects the local marginal cost-effectiveness of emission reduction work. Accordingly, we choose TRC as the indicator of the overall effectiveness of the scheme. The significance of the TRC is to maintain the current carbon emission share of each subsector as a baseline in the year when carbon peak is reached, and compare it with the designed quota allocation scheme in order to quantify the implementation cost of the scheme. Combined with MAC, TRC can be quantified by Eq. (18):
$$\text{T}\text{R}\text{C}={\sum }_{\text{i}=1}^{\text{n}}({\text{I}}_{\text{i}}-{\text{D}}_{\text{i}})\bullet {\text{M}\text{A}\text{C}}_{\text{i}}$$
18
where \({I}_{i}\) denotes the carbon emissions of subsector i at the year of carbon peak, assuming that the carbon emission proportion from 2010 to 2020 is maintained. \({D}_{i}\) denotes the quotas for subsector i in the designed schemes. If TRC > 0, it means that additional costs are needed compared to current carbon emissions. If TRC < 0, it means that the scheme is cost-effective compared to current carbon emissions. Obviously, a smaller TAC of the scheme is more likely to be favored by policy makers.