1 Introduction

Since mainland China’s economic reform and opening policy started in 1978, China’s economy has seen remarkable development. According to the National Bureau of Statistics of China (NBSC), the average annual growth rate of China’s gross domestic product (GDP) has increased 15.73 % from 1979 to 2013. This increasing trend has made China the second largest economy in the world following the United States (Bi et al. 2012). With the rapid development of its economy, China’s energy consumption is also growing rapidly. For example, Wang (2010) indicated that China has already overtaken the United States to become the world’s largest energy consumer. The rapidly increasing energy consumption inevitably has caused serious environmental problems (Wang et al. 2007, 2013a; Song et al. 2014; Wu et al. 2015). Therefore, the energy shortage and environmental problems have already become significant difficulties for the economic growth and societal development of China (Zhou et al. 2008a; Wu et al. 2013, 2014a, b; Saharidis 2015).

In order to balance rational utilization of energy, environmental pollution, and sustainable development, many energy and environmental regulations have been strengthened by the Chinese government. For example, for the 11th 5-year plan effective from 2006 to 2010, China’s central government set a target of reducing the energy consumption of per unit of GDP by 20 % and the main pollutant total emissions by 10 %. Evaluating energy and environmental performance has attracted increasing interest in recent years since it is considered a crucial method to save energy, protect the environment, and mitigate global climate change (Wang et al. 2013a). Since different regions of China may have different energy consumption structures and different environmental protection policies, the regional energy and environmental performances within China may have big differences across different regions (Wang et al. 2013b). Hence, in order to save energy and protect the environment, it is necessary to evaluate China’s regional energy and environmental efficiency.

Traditionally, there are two main approaches for measuring efficiency. One is the parametric stochastic frontier analysis (SFA) approach; the other is the nonparametric data envelopment analysis (DEA) approach (Coelli et al. 2005). Wu et al. (2014a, b) indicated that the SFA approach is suitable only for one-output scenarios and the results largely depend on the predicted forms of the production functions. Therefore, incorrect results may be obtained due to using an incorrect form of production function. Developed by Charnes et al. (1978), DEA is a programming-based technique for measuring the relative efficiency of a group of homogenous decision making units (DMUs) (Cooper et al. 2007; Song et al. 2013; Cook and Seiford 2009; Sharma and Yu 2013; Yang et al. 2015a, b). As a nonparametric technique, DEA does not need any prior functional form of the production frontier and can effectively measure the efficiency of a system with multiple inputs and multiple outputs (Smirlis et al. 2012; Panta et al. 2013; Huang et al. 2015; Wu et al. 2009). In the past years, DEA has been extensively applied in the performance evaluation and benchmarking of hospitals (e.g. Karagiannis and Velentzas 2012; Dimas et al. 2012), supply chains of enterprises (e.g. Chen et al. 2006), and other entities (e.g. Halkos et al. 2015; Shabani et al. 2012; Ibanez and McCalley 2011; Tsolas and Charles 2015). Since any economic activity is a joint production process using energy resources (coal, oil, etc.) and other non-energy resources (capital, labor, etc.) to produce desirable outputs (GDP) and undesirable outputs (CO2, wastes, etc.), DEA is used to construct the framework of total-factor efficiency evaluation in this paper.

Recently, DEA has been widely applied to studying energy and environmental efficiency. As for energy efficiency, Ramanathan (2000) compared the energy efficiencies of alternative transport modes by using DEA. Hu and Wang (2006) applied DEA models to analyze the total factor energy efficiency index of 29 administrative regions in China for the period 1995–2002. Hu and Lee (2008) employed DEA methods to measure the energy utility efficiency of China. Yeh et al. (2010) measured the technical efficiency of energy utilization in the Chinese mainland and Taiwan by using the traditional BCC model (Banker et al. 1984). Wu (2012) presented several DEA models to evaluate industrial energy efficiency in 28 of China’s regions.

As for environmental efficiency, Färe et al. (1989) made the first attempt to deal with undesirable outputs while using nonlinear programming. Seiford and Zhu (2002) applied a radial DEA model, considering undesirable outputs, to improve the environmental efficiency by decreasing undesirable outputs and increasing desirable outputs. Using DEA techniques, Zhou et al. (2008b) divided outputs into desirable and undesirable to calculate the carbon emission efficiency of eight world regions. Song and Wang (2014) calculated China’s regional environmental efficiency for 1992, 1999, 2007, and 2012, and then decomposed the provincial environmental efficiency from the perspective of government regulation and technological progress. Yang et al. (2015a, b) measured the environmental efficiency of China based on an environmental super-efficiency DEA model using data of 30 provinces in China during the period 2000–2010.

In addition, there is also some literature that directly addresses combined energy and environmental efficiency. For example, Bian and Yang (2010) employed several DEA models for estimating resource efficiencies and environment efficiencies simultaneously. Shi et al. (2010) measured the energy and environmental overall technical efficiency, pure technical efficiency, and scale efficiency of 28 administrative regions in China based on three extended DEA models. Wang et al. (2012) established several DEA efficiency models based on environmental production technology to estimate the environmental efficiency, economic efficiency, and economic-environmental efficiency of 28 provinces in China. Wang et al. (2013a) applied the range-adjusted measure based on a nonparametric approach to evaluate the regional energy and environmental efficiency of China over the period 2006–2010. Wang et al. (2013b) utilized a DEA window analysis technique to measure the energy and environmental efficiency in cross-sectional and time-varying data of 29 administrative regions of China during the period 2000–2008.

In this paper, following Bian and Yang (2010) and Wang et al. (2013b), we propose an improved DEA model to measure the total-factor energy and environmental efficiency of 30 provincial-level regions in mainland China during the 11th 5-year plan period (2006–2010). In particular, our proposed total-factor energy and environmental efficiency measure considers a joint production framework of non-energy inputs (labor and capital) and energy input (total energy consumption), as well as a desirable output (GDP) and an undesirable output (waste gas). In addition, in order to reflect the dynamic efficiency and productivity change considering the undesirable output and energy input, a DEA-based Malmquist index method is utilized in our paper. The Malmquist index is further classified into the change in technical efficiency (TEC) and the change in technical progress (TPC) in order to reflect the trends of those two aspects.

The structure of this paper is as follows. Section 2 presents the methodology of our study. Then the energy and environmental efficiency and Malmquist index of different regions and areas in China during the period 2006–2010 are analyzed in Sect. 3. Finally conclusions are shown in Sect. 4.

2 Methodology

In this section, we firstly present a non-radial DEA model to evaluate the total-factor energy and environmental efficiency. Then we explore the total-factor energy and environmental efficiency by applying DEA-based Malmquist indices to analyze the efficiency and productivity changes over time.

2.1 DEA model for evaluating the energy and environmental performance

Suppose that there are n DMUs denoted by DMU j (j = 1,…,n), and each of them represents an administrative region of China. Each DMU j (j = 1,…,n) consumes m non-energy inputs denoted by \( {\mathbf{X}} = (X_{1j} ,X_{2j} , \ldots ,X_{mj} )^{T} \) and d energy inputs denoted by \( {\mathbf{E}} = (E_{1j} ,E_{2j} , \ldots ,E_{dj} )^{T} \) to produce s desirable outputs denoted by \( {\mathbf{Y}} = (Y_{1j} ,Y_{2j} , \ldots ,Y_{sj} )^{T} \) along with p undesirable outputs denoted by \( {\mathbf{F}} = (F_{1j} ,F_{2j} , \ldots ,F_{pj} )^{T} \).

In the DEA literature, there have been a variety of approaches to dealing with undesirable outputs, which can be mainly divided into two categories (Song et al. 2012; Chen and Delmas 2012). The first one is based on the weak disposability assumption of undesirable outputs (Färe et al. 1989, 2005; Zhou et al. 2012) and the second category on the strong disposability assumption on undesirable output. Two ways for treating undesirable output under this latter assumption have been proposed. One way treats undesirable outputs as inputs (Tyteca 1997; Shi et al. 2010; Macpherson et al. 2013); the other way is a transformation method which contains a non-linear or linear monotonic decreasing transformation (Scheel 2001; Seiford and Zhu 2002). In this paper, the weak disposability assumption of undesirable outputs is chosen for building our new DEA models since it is prevalent in DEA literature. The corresponding DEA production technology T exhibiting constant returns to scale (CRS) can be characterized as follows:

$$ \begin{array}{*{20}l} {T = \{ {(}{\mathbf{X,E,Y,F}})| } \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij} \le {\mathbf{X}}_{{\mathbf{i}}} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj} \le {\mathbf{E}}_{{\mathbf{k}}} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj} \ge {\mathbf{Y}}_{r} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj} = {\mathbf{F}}_{g} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {\left. {j = 1, \ldots ,n} \right\}} \hfill \\ \end{array} $$
(1)

In this study, the undesirable outputs are generated by fuel combustion during the industrial production process. The corresponding undesirable outputs should be reduced if energy consumption is reduced. Therefore, following Shi et al. (2010) and Wang et al. (2013b), we first provide the following radial DEA-based model for evaluating the DMU0’s total-factor energy and environmental performance as

$$ \begin{array}{*{20}l} {Min} \hfill & \theta \hfill & {} \hfill \\ {{\kern 1pt} s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij} \le X_{i0} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj} \le \theta E_{k0} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj} \ge Y_{r0} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj} = \theta F_{g0} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} $$
(2)

Note that model (2) proportionally decreases the amounts of energy inputs and undesirable outputs as much as possible for a given level of non-energy inputs and desirable outputs. Obviously, the index θ for energy and environmental efficiency is between 0 and 1. The larger the index, the better the corresponding region performs both in saving energy and protecting the environment.

Once the optimal solutions \( (\theta^{ * } ,\lambda_{j}^{ * } ) \) are obtained by solving model (2), we can determine each DMU’s efficient targets for all inputs/outputs as \( \left( {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} X_{ij} } ,\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} E_{kj} } ,\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} Y_{rj} } ,\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} F_{tj} } } \right) \).

The above total-factor energy and environmental efficiency measure is a kind of radial DEA efficiency index. Zhou et al. (2007) and Zhou and Ang (2008) indicated that this radial measure may have weak discriminating power in energy and environmental efficiency comparisons. Therefore, following Bian and Yang (2010) and Wang et al. (2013b), we extend the radial energy and environmental efficiency measure to the following non-radial measure.

$$ \begin{array}{*{20}l} {Min} \hfill & {\frac{1}{2}\left( {\frac{ 1}{d}\sum\limits_{k = 1}^{d} {\theta_{k} } + \frac{ 1}{p}\sum\limits_{g = 1}^{p} {\theta_{g} } } \right)} \hfill & {} \hfill \\ {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij} \le X_{i0} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj} \le \theta_{k} E_{k0} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj} \ge Y_{r0} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj} = \theta_{g} F_{g0} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} $$
(3)

Model (3) can evaluate the energy and environmental efficiency by using different non-proportional adjustments for any energy inputs and undesirable outputs. In other words, this measure can account for energy input mix effects and undesirable output mix effects when measuring DMU energy and environmental efficiency. In addition, we should point out that the energy and environmental efficiency is measured by using different non-proportional adjustments and the unified efficiency is calculated through a decision maker specified weight assigned to each of these two efficiency scores. Similar to Bian and Yang (2010) and Wang et al. (2013b), the weights are both set to 1/2 in this paper, but the decision makers also could assign different weights to these scores to represent different preferences between energy utilization performance and environmental protection performance.

2.2 DEA-based Malmquist index for evaluating the dynamic performance

In this study, we evaluate the energy and environmental efficiency of different regions in China not only for a single year but for the 11th 5-year plan from 2006 to 2010, which may be considered a dynamic evaluation and could provide us with more useful information about efficiency changes. Developed by Malmquist (1953), the Malmquist index has been widely applied in DEA literature for evaluating dynamic efficiency (e.g. Färe et al. 1994; Mahadevan 2002; Chen and Ali 2004; Camanho and Dyson 2006). Therefore, we extend the Malmquist index analysis for use in the dynamic evaluation of energy and environmental performance.

Let \( ({\mathbf{X}}^{t} {\mathbf{,E}}^{t} {\mathbf{,Y}}^{t} {\mathbf{,F}}^{t} ) \) denote the production process of period t, t = 1,…,T. Then the Malmquist index can be defined using the results of the following four steps.

  1. 1.

    Compare \( ({\mathbf{X}}_{0}^{t} {\mathbf{,E}}_{0}^{t} {\mathbf{,Y}}_{0}^{t} {\mathbf{,F}}_{0}^{t} ) \) to the empirical production frontier (EPF) at time t, i.e., calculate \( {\kern 1pt} {\kern 1pt} {\kern 1pt} \theta^{t}_{0} ({\mathbf{X}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{E}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{Y}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{F}}^{{\mathbf{t}}}_{{\mathbf{0}}} ) \) via the following linear program:

$$ \begin{aligned} {\kern 1pt} \theta^{t}_{0} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right) = Min\frac{1}{2}\left( {\frac{ 1}{d}\sum\limits_{k = 1}^{d} {\theta_{k} } + \frac{ 1}{p}\sum\limits_{g = 1}^{p} {\theta_{g} } } \right) \hfill \\ \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X^{t}_{ij} \le X^{t}_{i0} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E^{t}_{kj} \le \theta_{k} E^{t}_{k0} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y^{t}_{rj} \ge Y^{t}_{r0} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F^{t}_{gj} = \theta_{g} F^{t}_{g0} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} \hfill \\ \end{aligned} $$
(4)
  1. 2.

    Compare \( ({\mathbf{X}}_{0}^{{t{ + }1}} {\mathbf{,E}}_{0}^{{t{ + }1}} {\mathbf{,Y}}_{0}^{{t{ + }1}} {\mathbf{,F}}_{0}^{{t{ + }1}} ) \) to the empirical production frontier (EPF) at time t + 1, i.e., calculate \( {\kern 1pt} {\kern 1pt} {\kern 1pt} \theta^{{t{ + }1}}_{0} ({\mathbf{X}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{E}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{Y}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{F}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ){\kern 1pt} \) via the following linear program:

$$ \begin{aligned} {\kern 1pt} \theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right) = Min\frac{1}{2}\left( {\frac{ 1}{d}\sum\limits_{k = 1}^{d} {\theta_{k} } + \frac{ 1}{p}\sum\limits_{g = 1}^{p} {\theta_{g} } } \right) \hfill \\ \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij}^{{t{ + }1}} \le X_{i0}^{{t{ + }1}} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj}^{{t{ + }1}} \le \theta_{k} E_{k0}^{{t{ + }1}} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj}^{{t{ + }1}} \ge Y_{r0}^{{t{ + }1}} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj}^{{t{ + }1}} = \theta_{g} F_{g0}^{{t{ + }1}} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} \hfill \\ \end{aligned} $$
(5)
  1. 3.

    Compare \( ({\mathbf{X}}_{0}^{t} {\mathbf{,E}}_{0}^{t} {\mathbf{,Y}}_{0}^{t} {\mathbf{,F}}_{0}^{t} ) \) to the empirical production frontier (EPF) at time t + 1, i.e., calculate \( {\kern 1pt} {\kern 1pt} {\kern 1pt} \theta^{{t{ + }1}}_{0} ({\mathbf{X}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{E}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{Y}}^{{\mathbf{t}}}_{{\mathbf{0}}} ,{\mathbf{F}}^{{\mathbf{t}}}_{{\mathbf{0}}} ){\kern 1pt} \) via the following linear program:

$$ \begin{aligned} {\kern 1pt} \theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right) = Min\frac{1}{2}\left( {\frac{ 1}{d}\sum\limits_{k = 1}^{d} {\theta_{k} } + \frac{ 1}{p}\sum\limits_{g = 1}^{p} {\theta_{g} } } \right) \hfill \\ \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij}^{{t{ + }1}} \le X_{i0}^{t} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj}^{{t{ + }1}} \le \theta_{k} E_{k0}^{t} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj}^{{t{ + }1}} \ge Y_{r0}^{t} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj}^{{t{ + }1}} = \theta_{g} F_{g0}^{t} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,{\kern 1pt} } \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} \hfill \\ \end{aligned} $$
(6)
  1. 4.

    Compare \( ({\mathbf{X}}_{0}^{{t{ + }1}} {\mathbf{,E}}_{0}^{{t{ + }1}} {\mathbf{,Y}}_{0}^{{t{ + }1}} {\mathbf{,F}}_{0}^{{t{ + }1}} ) \) to the empirical production frontier (EPF) at time t + 1, i.e., calculate \( {\kern 1pt} {\kern 1pt} {\kern 1pt} \theta^{t}_{0} ({\mathbf{X}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{E}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{Y}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ,{\mathbf{F}}^{{{\mathbf{t}}{ + }1}}_{{\mathbf{0}}} ) \) via the following linear program:

$$ \begin{aligned} \theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right) = Min\frac{1}{2}\left( {\frac{ 1}{d}\sum\limits_{k = 1}^{d} {\theta_{k} } + \frac{ 1}{p}\sum\limits_{g = 1}^{p} {\theta_{g} } } \right) \hfill \\ \begin{array}{*{20}l} {s.t.} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} X_{ij}^{t} \le X_{i0}^{t + 1} ,} } \hfill & {i = 1, \ldots ,m} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} E_{kj}^{t} \le \theta_{k} E_{k0}^{t + 1} ,} } \hfill & {k = 1, \ldots ,d} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} Y_{rj}^{t} \ge Y_{r0}^{t + 1} ,} } \hfill & {r = 1, \ldots ,s} \hfill \\ {} \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} F_{gj}^{t} = \theta_{g} F_{g0}^{t + 1} ,} } \hfill & {g = 1, \ldots ,p} \hfill \\ {} \hfill & {\lambda_{j} \ge 0,} \hfill & {j = 1, \ldots ,n} \hfill \\ \end{array} \hfill \\ \end{aligned} $$
(7)

Then the Malmquist index is defined as:

$$ {\kern 1pt} M_{0} = \left[ {\frac{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}}\frac{{\theta_{0}^{t + 1} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{{\kern 1pt} \theta_{0}^{t + 1} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}}} \right]^{1/2} $$
(8)

The Malmquist index M 0 measures the efficiency and productivity changes between periods t and t + 1. Productivity declines if M 0 < 1, remains unchanged if M 0 = 1, and improves if M 0 > 1.

According to Färe et al. (1994), the Malmquist index M 0 can be divided into two components:

$$ {\kern 1pt} M_{0} = \frac{{\theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}}\left[ {\frac{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{\theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}\frac{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}}{{\theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}} \right]^{1/2} $$

where the first component on the right hand side measures the change in technical efficiency (TEC) between periods t and t + 1, denoted as follows.

$$ TEC_{0} = \frac{{{\kern 1pt} \theta_{0}^{{t{ + }1}} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}} $$
(9)

Technical efficiency declines if TEC 0 < 1, remains unchanged if TEC 0 = 1, and improves if TEC 0 > 1. The second component, which is the geometric mean, measures the change in technical progress (TPC) between periods t and t + 1, denoted as follows.

$$ TPC_{0} = \left[ {\frac{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}{{{\kern 1pt} \theta_{0}^{t + 1} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{{\mathbf{t}}{ + }1}} } \right)}}\frac{{\theta_{0}^{t} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right){\kern 1pt} }}{{\theta_{0}^{t + 1} \left( {{\mathbf{X}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{E}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{Y}}_{{\mathbf{0}}}^{{\mathbf{t}}} ,{\mathbf{F}}_{{\mathbf{0}}}^{{\mathbf{t}}} } \right)}}} \right]^{1/2} $$
(10)

Technical progress declines if TPC 0 < 1, remains unchanged if TPC 0 = 1, and improves if TPC 0 > 1.

3 Empirical studies

3.1 Data description

This section describes how to use our developed approach to evaluate the energy and environmental efficiencies of 30 provincial-level regions in mainland China during the 11th 5-year plan period (2006–2010). As Golany and Roll (1989) indicated, the number of evaluated DMUs should be more than five times the total selected number of inputs and outputs, otherwise the validity and credibility of the results will be seriously compromised. Hence, following many studies (Wang et al. 2013a, b; Li et al. 2013), our study selects five factors as inputs and outputs. We employ labor and investment of fixed assets (namely capital) as two non-energy inputs, energy consumption as the only energy input, GDP as one desirable output, and industrial waste gas emissions as one undesirable output. The input–output measures that are used in this study are summarized in Table 1.

Table 1 Variables of inputs and outputs
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We use the data of the 11th 5-year plan period (2006–2010). Unfortunately, the data for the Tibet Autonomous Region are incomplete and so it is not considered in the current study. The data are derived from the “China Statistical Yearbook” and “China Energy Statistical Yearbook”. A statistical description of data is shown in Table 2.

Table 2 Descriptive statistics of the raw data
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3.2 Results and discussion of regional energy and environmental efficiency

We apply our methodology provided in Sect. 2 to calculate the energy and environmental efficiencies of 30 provincial-level regions in mainland China during the 11th 5-year plan period (2006–2010). By firstly employing model (3), the evaluated results are listed in Table 3. In Table 3, columns 3–7 show the energy and environmental efficiencies of the 30 regions from 2006 to 2010 and column 8 lists the average energy and environmental efficiencies for these 5 years.

Table 3 The energy and environmental efficiency of 30 regions of China
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It should be noted that the higher the value of energy and environmental efficiency is, the more efficient the region is. From Table 3, the following conclusions can be drawn. Firstly, Beijing and Hainan performed well from 2006 to 2010. Their energy and environmental efficiencies were all equal to 1, i.e. they were all efficient in these 5 years. Secondly, Ningxia performed the worst since the energy and environmental efficiencies for the 5 years were 0.157, 0.156, 0.159, 0.160 and 0.131, and its average efficiency over these 5 years was 0.152. Thirdly, more than half of the 30 regions did not perform well during the 11th 5-year plan. For example, there are 20 regions which had average energy and environmental efficiency below 0.5, including Hebei with 0.258, Shanxi with 0.19, Inner Mongolia with 0.246, Liaoning with 0.32, and Jilin with 0.463. Fourthly, the energy and environmental efficiency trend was not promising since the energy and environmental efficiency of most regions did not show any obvious increasing trend during the 5 years. In fact, they actually had a decreasing trend in some regions. Take Xinjiang for example. Its energy and environmental efficiency diminished from 0.355 to 0.330 (2006–2007), from 0.330 to 0.3 (2007–2008), from 0.3 to 0.265 (2008–2009), and from 0.265 to 0.26 (2009–2010). Finally, we know that the developed regions generally performed better than those less developed regions. For example, the average energy and environmental efficiencies of developed regions like Beijing, Shanghai, Jiangsu, and Guangdong were 1, 0.685, 0.706, and 0.846 receptively. This result shows us that the energy and environmental efficiency of provincial-level regions in China are not cause for optimism and so more actions need to be taken, by both national and regional governments, to practically handle the problems of energy shortage and environmental pollution.

To analyze the energy and environmental efficiency on a relatively larger scale, we group the 30 provincial-level regions into three categories: eastern area, central area, and western area. These areas and their constituent regions are listed in Table 4.

Table 4 Areas of China and constituent provincial-level regions
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From Table 4, we know there are 11, 10, and 9 regions in the east, center, and west of China respectively. In the past 30 years, the eastern area has experienced the fastest economic growth in China. Along with a dense population, this area is home to the most light industry and foreign trade firms. Due to its convenient transportation and developed infrastructure, this area also attracts the most foreign investment and the best technology. The central area has a total population of 361 million, which accounts for 28 % of the national population. This area shows a lower level of development than the eastern area. The western region accounts for 71 % of the total area of China. Economic development in this area is lagging behind that of the eastern area and central area. Next we illustrate the average energy and environmental efficiencies of the three areas and 30 regions of China in Fig. 1.

Fig. 1
figure 1

Regional energy and environmental efficiency of China

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From Fig. 1, the following conclusions can be drawn. Firstly, in the eastern area, 3 out of 11 regions (Beijing, Guangdong, and Hainan) are highly energy and environmentally efficient with average efficiency scores above 0.8, and 7 out of 11 regions (Tianjin, Liaoning, Shandong, Shanghai, Jiangsu, Zhejiang, and Fujian) have average efficiency scores between 0.3 and 0.8, with only one region (Hebei) having average efficiency score below 0.3. Secondly, in the central area, Jiangxi has the highest average efficiency and its score is no more than 0.8, which means no region in this area has high energy and environmental efficiency. All the average efficiency scores of the other regions in this area are below 0.6, and Shanxi has the lowest average efficiency score of 0.190, followed by Inner Mongolia with 0.246. Thirdly, in the western area, all of the 9 regions have low efficiencies below 0.5.

We also calculate the average efficiencies of the three larger areas and the whole country for each year from 2006 to 2010; the results are shown in Table 5.

Table 5 Energy and environmental efficiency of the three big areas
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In order to clearly reflect the difference of the three areas, we illustrate the energy and environmental efficiencies of Table 5 in the following Fig. 2. From Fig. 2, we have the following conclusions. Firstly, from an area perspective considering each year during 2006–2010, the eastern area achieved the highest average energy and environmental efficiency score, followed by the central area and then the western area, but the efficiency scores of the central and western areas are both below the average efficiency at the whole country level. Secondly, all of these three areas had similar increasing and decreasing trends during the years 2007–2010. Finally, the energy and environmental efficiency of the whole country had a decreasing trend from 2006 to 2009. After that, the efficiency began to present an increasing trend from 2009 to 2010.

Fig. 2
figure 2

Average energy and environmental efficiency of China and its three areas

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3.3 Results and discussion of Malmquist index

In order to dynamically analyze the productivity changes for each region during our study period and give a more detailed and clear demonstration, we show 30 provincial-level regions’ Malmquist indices for 2006–2007, 2007–2008, 2008–2009, and 2009–2010. Applying models (4)–(8) to the data in Table 2, the Malmquist indices M can be calculated. Table 6 reports the Malmquist index of 30 regions of China over the 5-year period.

Table 6 Malmquist indices of 30 regions of China
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From Table 6, we can draw the following conclusions. Firstly we know that the values of the Malmquist index of the 30 regions are mostly greater than 1 for the reported period, which indicates that most regions’ productivity improved during each year of the period 2006–2010. Consider, for example, Beijing. Its productivity improved 6.3 % for 2006–2007, 29.3 % for 2007–2008, 14.4 % for 2008–2009, and 2.6 % for 2009–2010. A few regions have a Malmquist index lower than 1, as seen by the Malmquist index for Hebei being 0.959 for 2008–2009 and for Shanghai being 0.92 for 2006–2007. These rare Malmquist indices lower than 1 indicate the decline of productivity in these few regions from 2006 to 2010. Secondly, the maximum value of Malmquist index for the 30 regions during 2006–2010 is 1.433 for Sichuan for 2007–2008, which had also the minimum value of Malmquist index (0.841) for 2006–2007. Thirdly, the average values of Malmquist index for the 30 regions are all greater than 1 during 2006–2010, which indicates the improvement in productivity of each region during the 11th 5-year plan of China. Fourthly, the average Malmquist indices of the eastern area, central area, and western area are 1.084, 1.078, and 1.075 respectively, which means the productivity in those areas during 2006–2010 improved 8.4, 7.8, and 7.5 % respectively. This result indicates that the eastern area had the highest improvement of productivity during the 11th 5-year plan, followed by central area and then the western area. Lastly, we find that the values of Malmquist index of all the 30 regions does not show any obvious increasing or decreasing trend during the four periods. Moreover, we know that the productivity of most regions had an increasing trend during 2006–2008 and 2008–2010 since the values of Malmquist index of most regions for 2007–2008 (2009–2010) are bigger than the values for 2006–2007 (2008–2009). However, the productivity of most regions had a decreasing trend during 2007–2009 since the values of Malmquist index of most regions for 2008–2009 are smaller than the values for 2007–2008. Take Hainan for example. All of its values of Malmquist index are bigger than 1, that is, its productivity improved each year, however its improving trend is not increasing since its values of Malmquist index are 1.05 (2006–2007), 1.166 (2007–2008), 1.135 (2008–2009), and 1.23 (2009–2010). To be more precise, Hainan’s productivity had an increasing trend during 2006–2008 (from 1.05 to 1.166) and 2008–2010 (from 1.135 to 1.23), but a decreasing trend during 2007–2009 (from 1.166 to 1.135).

Applying models (9) and (10) from Sect. 2, we divide the Malmquist index into the change in technical efficiency (TEC) and the change in technical progress (TPC). The corresponding results are shown in Table 7.

Table 7 TEC and TPC of Malmquist index
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From Table 7, we have the following conclusions. Firstly, during each year of the period 2006–2010, the values of TECs for most regions are lower than 1, while the values of TPCs for most regions are higher than 1. For example, there are 22 regions which have the values of TEC lower than 1 and just 8 regions with the values of TEC higher than 1 for 2006–2007. In contrast, there are only two regions (Shanghai and Jiangsu) which have values of TPC lower than 1 for 2006–2007. These results indicate a decline of technical efficiency and an improvement of technical progress. Secondly, from the average values of TEC and TPC during the period 2006–2010, there are 23 regions which have a declining trend of technical efficiency (that is the value of TEC is lower than 1) while all the 30 regions have an increasing trend of technical progress (that is the value of TPC is higher than 1). Thirdly, combining Tables 6 and 7, we know which values of TEC and TPC led to the higher or lower Malmquist index. Take Anhui for example. Its value of Malmquist index for 2006–2007 is 0.979 which indicates that the productivity of Anhui declined 2.1 %. This result was caused by the changes of technical efficiency, where the value of TEC is lower than 1 in corresponding time period, whereas the technical progress improved since the value of TPC is greater than 1 for 2006–2007. Finally, considering the various trends of TECs, we know that the technical efficiency did not have an increasing or decreasing trend for most regions during the period 2006–2010, that is, it does not present any regular changes. Although the technical progress also did not have an increasing or decreasing trend for most regions during the whole period 2006–2010, the regions did have identical trends during parts of the period. For example, the technical progress of all regions has an increasing trend for 2006–2008 (2008–2010) since the values of TPCs of all regions for 2007–2008 (2009–2010) are all higher than their values for 2006–2007 (2008–2009). In addition, the technical progress of all regions has a decreasing trend for 2007–2009 since the values of TPCs of all regions for 2008–2009 are all lower than their values for 2007–2008.

4 Conclusions

During the last three decades, the economy of China has been increasing rapidly. The fast economic progress has led to the growth of GDP accompanied by serious problems such as energy shortage and environmental pollution. Naturally, it is desirable to evaluate the energy and environmental performance of provincial-level regions of China.

Based on the above-mentioned context, in this paper we apply DEA models to evaluate the total-factor efficiency of energy and environment of 30 provincial-level regions and three areas in mainland China during China’s 11th 5-year plan period (2006–2010). In particular, this study measures a joint production framework of non-energy inputs (labor and capital) and an energy input (total energy consumption), as well as a desirable output (GDP) and an undesirable output (waste gas). In addition, this study applies a DEA-based Malmquist index to evaluate the dynamic productivity change considering the undesirable output and energy input.

The major empirical results of this study show that: (1) Two regions, namely Beijing and Hainan, had the highest total-factor energy and environmental efficiency during the period 2006–2010. In addition, most regions had poor energy and environmental efficiency. Thus, these two regions could be seen as the benchmarks for inefficient regions to bring about energy and environmental efficiency improvement. (2) From an area perspective, China’s eastern area had the highest energy and environmental efficiency, followed by the central area, with the efficiency of the western area being the worst. The efficiency differences of the three areas may arise from the imbalance of economic development. (3) From the Malmquist index, we know that productivity in most regions improved during each year of the period 2006–2010. The eastern area had the highest improvement of productivity during the 11th 5-year plan, followed by the central area and then the western area. (4) Dividing the Malmquist index into the change in technical efficiency (TEC) and the change in technical progress (TPC), we know that most regions had a declining trend in technical efficiency while most regions had an increasing trend in technical progress.