Provincial carbon emission performance analysis in China based on a Malmquist data envelopment analysis approach with fixed-sum undesirable outputs

Abstract

The environmental issue has attracted ever-increasing attention from governments and academics in recent years. Environmental performance analysis is widely considered a crucial and useful tool for effectively protecting the environment and developing a sustainable society. Many analytical techniques have been used to assess carbon emission performance, among which data envelopment analysis (DEA) is prominent. However, few previous DEA-related carbon emission performance studies recognize that the total amount of carbon dioxide emissions is limited to a specific level by authorities; ignoring this fixed-sum requirement may lead to distortions and deviations in empirical results. This paper proposes an alternative Malmquist DEA approach for evaluating the carbon emission performance while considering fixed-sum undesirable outputs. For this purpose, we develop a generalized equilibrium efficient frontier DEA model with fixed-sum undesirable outputs and combine the model with the Malmquist productivity index (MPI). The proposed approach is applied to assess the carbon emission performance of 30 provincial regions in China from 2009 to 2015. Then we provide analytical results and policy suggestions regarding the provincial carbon emission performance in China.

Appendix

Model (3) is a nonlinear model that needs to be transformed into a linear model. The first step is to convert model (3) into model (A1):

$$ \begin{array}{*{20}l} {Min\;\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{a = 1}^{l} {w_{a} \left| {\delta_{aj}^{u} } \right|} } } \hfill & {} \hfill \\ {s.t.\;\frac{{\sum\nolimits_{r = 1}^{s} {u_{r} y_{rj} + \mu_{0} } }}{{\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} + \sum\nolimits_{a = 1}^{l} {w_{a} (f_{aj}^{u} + \delta_{aj}^{u} )} } }} = 1} \hfill & {\forall j} \hfill \\ {\sum\nolimits_{j = 1}^{n} {\delta_{aj}^{u} = 0} } \hfill & {\forall a} \hfill \\ {f_{aj}^{u} + \delta_{aj}^{u} \ge 0} \hfill & {\forall a,j} \hfill \\ {u_{r} ,v_{i} ,w_{a} \ge 0} \hfill & {\forall r,i,a} \hfill \\ {{\kern 1pt} \delta_{aj}^{u} \;{\text{and}}\;\mu_{0} \;are\;free} \hfill & {} \hfill \\ \end{array} $$

(A1)

For model (A1), although $\left| {\delta_{aj}^{u} } \right|$ represents both positive and negative adjustments, the second constraint of model (3) shows that for $\forall a$, the sum of the positive adjustment and the negative adjustment is zero. Therefore, we have $\sum\nolimits_{j = 1}^{n} {\left| {\delta_{aj}^{u} } \right|} { = 2}\sum\nolimits_{j = 1}^{n} {\alpha_{aj}^{u} }$ for models (3) and (A1). Further, it is easy to demonstrate that the objective function of model (A1) is twice that of model (3), as $\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{a = 1}^{l} {w_{a} \left| {\delta_{aj}^{u} } \right|} } { = }\sum\nolimits_{a = 1}^{l} {\sum\nolimits_{j = 1}^{n} {w_{a} \left| {\delta_{aj}^{u} } \right|} } = 2\sum\nolimits_{a = 1}^{l} {\sum\nolimits_{j = 1}^{n} {w_{a} \alpha_{aj}^{u} } } { = 2}\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{a = 1}^{l} {w_{a} \alpha_{aj}^{u} } }$. Under the premise that the constraints remain unchanged (the feasible region remains unchanged), if the objective function has a proportional transformation, the optimal solution does not change.

Letting $\delta_{aj}^{^{\prime}u} = w_{a} \delta_{aj}^{u}$, the constraints of model (A1) change into linear constraints as in model (A2):

$$ \begin{array}{*{20}l} {Min\;\sum\nolimits_{j = 1}^{n} {\sum\nolimits_{a = 1}^{l} {\left| {\delta_{aj}^{^{\prime}u} } \right|} } } \hfill & {} \hfill \\ {s.t.\;\sum\nolimits_{r = 1}^{s} {u_{r} y_{rj} - \sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} - \sum\nolimits_{a = 1}^{l} {(w_{a} f_{aj}^{u} + \delta_{aj}^{^{\prime}u} )} } + \mu_{0} } = 0} \hfill & {\forall j} \hfill \\ {\sum\nolimits_{j = 1}^{n} {\delta_{aj}^{^{\prime}u} = 0} } \hfill & {\forall a} \hfill \\ {\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} { + }\sum\nolimits_{a = 1}^{l} {(w_{a} f_{aj}^{u} + \delta_{aj}^{^{\prime}u} )} } \ge C} \hfill & {\forall j} \hfill \\ {w_{a} f_{aj}^{u} + \delta_{aj}^{^{\prime}u} \ge 0} \hfill & {\forall a,j} \hfill \\ {u_{r} ,v_{i} ,w_{a} \ge 0} \hfill & {\forall r,i,a} \hfill \\ {\delta_{aj}^{^{\prime}u} \;and\;\mu_{0} \;are\;free} \hfill & {} \hfill \\ \end{array} {\kern 1pt} $$

(A2)

In model (A2), C is a prespecified constant that ensures the $\sum\nolimits_{i = 1}^{m} {v_{i} x_{ij} + \sum\nolimits_{a = 1}^{l} {w_{a} (f_{aj}^{u} + \delta_{aj}^{u} )} }$ in model (A1) is a positive number (it is used in a denominator). The following Proposition 1 shows that the value of C will not affect the optimal solution of $\delta_{aj}^{^{\prime}u}$ in model (A2).

Proposition 1.

If $\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is an optimal solution to the model (A2) with one given positive constant C, then, $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is also an optimal solution of the model (A2) with any other positive constant $C^{\prime}$ in the same round optimization.

Proof.

First, we prove that $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is a feasible solution for model (A2) with a given random positive constant $C^{\prime}$. For the convenience of writing, we set ${{\varphi }} = \frac{{C^{\prime}}}{C}$ $\left( {{{\varphi }} > 0} \right)$. Then, according to this optimal solution $\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$, we bring $\left( {{{\varphi }}u_{r}^{*} ,{{ \varphi }}v_{i}^{*} ,{{ \varphi }}w_{a}^{*} ,{{ \varphi }}\delta_{aj}^{{^{\prime}u^{*} }} ,{{ \varphi }}\mu_{0}^{*} } \right)$ into all the constraints when the model (A2) has any positive constant C' as follows.

$$ \begin{aligned} & \sum\limits_{r = 1}^{s} {\varphi u_{r}^{*} y_{rj} - \sum\limits_{i = 1}^{m} {\varphi v_{i}^{*} x_{ij} - \sum\limits_{a = 1}^{l} {(\varphi w_{a}^{*} f_{aj}^{u} + \varphi \delta_{aj}^{{{\prime }u^{*} }} )} } + \varphi \mu_{0}^{*} } \\ & \quad = \varphi \left( {\sum\limits_{r = 1}^{s} {u_{r}^{*} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i}^{*} x_{ij} - \sum\limits_{a = 1}^{l} {(w_{a}^{*} f_{aj}^{u} + \delta_{aj}^{{{\prime }u^{*} }} )} } + \mu_{0}^{*} } } \right) = 0 \\ & \sum\limits_{j = 1}^{n} {\varphi \delta_{aj}^{{{\prime }u^{*} }} } = \varphi \sum\limits_{j = 1}^{n} {\delta_{aj}^{{^{\prime}u^{*} }} } = 0{\kern 1pt} \\ & \sum\limits_{i = 1}^{m} {\varphi v_{i}^{*} x_{ij} { + }\sum\limits_{a = 1}^{l} {\left( {\varphi w_{a}^{*} f_{aj}^{u} + \varphi \delta_{aj}^{{{\prime }u^{*} }} } \right)} } \\ & \quad = \varphi \left( {\sum\limits_{i = 1}^{m} {v_{i}^{*} x_{ij} { + }\sum\limits_{a = 1}^{l} {(w_{a}^{*} f_{aj}^{u} + \delta_{aj}^{{{\prime }u^{*} }} )} } } \right) \ge \varphi C = C^{{\prime }} \\ & \varphi w_{a}^{*} f_{aj}^{u} {\kern 1pt} + \varphi \delta_{aj}^{{{\prime }u^{*} }} = \varphi (w_{a}^{*} f_{aj}^{u} + \delta_{aj}^{{{\prime }u^{*} }} ) \ge 0 \\ {\kern 1pt} & \varphi u_{r}^{*} ,\varphi v_{i}^{*} ,\varphi w_{a}^{*} \ge 0 \\ & \varphi \delta_{aj}^{{{\prime }u^{*} }} ,\;\varphi \mu_{0}^{*} \;are\;free \\ \end{aligned} $$

It is clear that $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ satisfies all the constraints of model (A2) with a given random positive constant $C^{\prime}$, therefore, the feasibility of the solution is proved.

Next, we prove that $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is an optimal solution for model (A2) with $C^{\prime} > 0$. Assuming that the above feasible solution is not the optimal solution, then we can find at least a feasible $\left( {{{\varphi }}u_{r} ,{{ \varphi }}v_{i} ,{{ \varphi }}w_{a} ,{{ \varphi }}\delta_{aj}^{^{\prime}u} ,{{ \varphi }}\mu_{0} } \right)$ for model (A2) with $C^{\prime} > 0$ such that $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{^{\prime}u} } \right|} } < \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } \right|} }$. Accordingly, it is easy to verify that $\left( {u_{r} ,{ }v_{i} ,{ }w_{a} ,{ }\delta_{aj}^{^{\prime}u} ,{ }\mu_{0} } \right)$ is also a feasible solution for model (A2) with C.

For $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{^{\prime}u} } \right|} }$, we have $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{^{\prime}u} } \right|} } { = }\left\{ \begin{gathered} {2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{^{\prime}u} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\delta_{aj}^{^{\prime}u} \ge 0)} \hfill \\ { - 2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{^{\prime}u} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\delta_{aj}^{^{\prime}u} < 0)} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}$. Similarly, for $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } \right|} }$, we have $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } \right|} } { = }\left\{ \begin{gathered} {2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\delta_{aj}^{{^{\prime}u^{*} }} \ge 0)} \hfill \\ { - 2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\delta_{aj}^{{^{\prime}u^{*} }} < 0)} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}$. According to the first constraint of model (A2), we have $\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{^{\prime}u} } { = }\sum\limits_{r = 1}^{s} {\varphi u_{r} y_{rj} - \sum\limits_{i = 1}^{m} {\varphi v_{i} x_{ij} - \sum\limits_{a = 1}^{l} {\varphi w_{a} f_{aj}^{u} } } + \varphi \mu_{0} }$ and $\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } { = }\sum\limits_{r = 1}^{s} {\varphi u_{r}^{*} y_{rj} - \sum\limits_{i = 1}^{m} {\varphi v_{i}^{*} x_{ij} - \sum\limits_{a = 1}^{l} {\varphi w_{a}^{*} f_{aj}^{u} } } + \varphi \mu_{0}^{*} }$. We then take all $DMU_{j}$ with $\delta_{aj}^{^{\prime}u} \ge 0$ and all $DMU_{j}$ with $\delta_{aj}^{{^{\prime}u^{*} }} \ge 0$, and we have $\begin{gathered} \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{^{\prime}u} } \right|} } { = 2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{^{\prime}u} } } { = 2}\varphi \sum\limits_{j = 1}^{n} {{(}\sum\limits_{r = 1}^{s} {u_{r} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} - \sum\limits_{a = 1}^{l} {w_{a} f_{aj}^{u} } } + \mu_{0} } ){\kern 1pt} } \hfill \\ < \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } \right|} } = 2\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } } { = 2}\varphi \sum\limits_{j = 1}^{n} {(\sum\limits_{r = 1}^{s} {u_{r}^{*} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i}^{*} x_{ij} - \sum\limits_{a = 1}^{l} {w_{a}^{*} f_{aj}^{u} } } + \mu_{0}^{*} } )} \hfill \\ \end{gathered}$. However, $\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is an optimal solution of model (A2) with C, so taking all $DMU_{j}$ with $\delta_{aj}^{^{\prime}u} \ge$ and all $DMU_{j}$ with $\delta_{aj}^{{^{\prime}u^{*} }} \ge 0$ we have $\begin{gathered} \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\delta_{aj}^{{^{\prime}u^{*} }} } \right|} } = 2\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\delta_{aj}^{{^{\prime}u^{*} }} } } { = 2}\sum\limits_{j = 1}^{n} {(\sum\limits_{r = 1}^{s} {u_{r}^{*} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i}^{*} x_{ij} - \sum\limits_{a = 1}^{l} {w_{a}^{*} f_{aj}^{u} } } + \mu_{0}^{*} } )} \hfill \\ < \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\delta_{aj}^{^{\prime}u} } \right|} } { = 2}\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\delta_{aj}^{^{\prime}u} } } { = 2}\sum\limits_{j = 1}^{n} {{(}\sum\limits_{r = 1}^{s} {u_{r} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} - \sum\limits_{a = 1}^{l} {w_{a} f_{aj}^{u} } } + \mu_{0} } ){\kern 1pt} } \hfill \\ \end{gathered}$. This contradicts our previous assumption $\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{^{\prime}u} } \right|} } < \sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left| {\varphi \delta_{aj}^{{^{\prime}u^{*} }} } \right|} }$. Therefore, $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is an optimal solution for model (A2) with $C^{\prime} > 0$.

The above Proposition 1 shows that the value of C does not affect the optimal solution of $\delta_{aj}^{^{\prime}u}$ in model (A2). Besides, it shows that under the condition of any given constant C, the $\delta_{aj}^{{u^{*} }}$ obtained by solving model (A2) is also the optimal solution for model (A1). Assume that $\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is an optimal solution of model (A2) based on an arbitrary positive number C. Then, according to Proposition 1, $\frac{{C^{\prime}}}{C}\left( {u_{r}^{*} ,{ }v_{i}^{*} ,{ }w_{a}^{*} ,{ }\delta_{aj}^{{^{\prime}u^{*} }} ,{ }\mu_{0}^{*} } \right)$ is also an optimal solution for model (A2) based on C'. Therefore, we can get $\delta_{aj}^{{u^{*} }} { = }\frac{{\delta_{aj}^{{^{\prime}u^{*} }} }}{{w_{a}^{*} }} = \frac{{\frac{{C^{^{\prime}} }}{C}\delta_{aj}^{{^{\prime}u^{*} }} }}{{\frac{{C^{^{\prime}} }}{C}w_{a}^{*} }}$. That is to say, $\delta_{aj}^{{u^{*} }}$ is the optimal value of model (A1) and is not affected by the value of the positive number C.

To facilitate the solution, let $a_{aj} = \frac{1}{2}\left( {\left| {\delta_{aj}^{^{\prime}u} } \right| + \delta_{aj}^{^{\prime}u} } \right),b_{aj} = \frac{1}{2}\left( {\left| {\delta_{aj}^{^{\prime}u} } \right| - \delta_{aj}^{^{\prime}u} } \right)$. It is obvious that $a_{aj} \ge 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b_{aj} \ge 0$, and we have $\left| {\delta_{aj}^{^{\prime}u} } \right| = a_{aj} + b_{aj} ,\delta_{aj}^{^{\prime}u} = a_{aj} - b_{aj} \left( {\forall a,j} \right)$. Therefore, model (A2) can be transformed into the following model (A3):

$$ \begin{array}{*{20}l} {Min\;\sum\limits_{j = 1}^{n} {\sum\limits_{a = 1}^{l} {\left( {a_{aj} + b_{aj} } \right)} } } \hfill & {} \hfill \\ {s.t.\;\sum\limits_{r = 1}^{s} {u_{r} y_{rj} - \sum\limits_{i = 1}^{m} {v_{i} x_{ij} - \sum\limits_{a = 1}^{l} {(w_{a} f_{aj}^{u} + a_{aj} - b_{aj} )} } } + \mu_{0} = 0} \hfill & {\forall j} \hfill \\ {\sum\limits_{j = 1}^{n} {(a_{tj} - b_{tj} ) = 0} } \hfill & {\forall a} \hfill \\ {\sum\limits_{i = 1}^{m} {v_{i} x_{ij} + \sum\limits_{a = 1}^{l} {(w_{a} f_{aj}^{u} + a_{aj} - b_{aj} )} } \ge C} \hfill & {\forall j} \hfill \\ {w_{a} f_{aj}^{u} + a_{aj} - b_{aj} \ge 0} \hfill & {\forall a,j} \hfill \\ {u_{r} ,v_{i} ,w_{a} ,a_{aj} ,b_{aj} \ge 0} \hfill & {\forall r,i,a,j} \hfill \\ {\mu_{0} \;is\;free} \hfill & {} \hfill \\ \end{array} {\kern 1pt} $$

(A3)

Thus, model (3) is translated into a linear programming problem.