### 1. Introduction

There is a global consensus that climate change is being driven by an increase in atmospheric greenhouse gases, most notably CO

_{2}emissions. Along with the global climate and environment situation becoming more serious, as main element of greenhouse gas, CO_{2}emissions are concerned.To address this issue, Chinese government proposed targets that CO

_{2}emissions per unit gross domestic product (GDP) should be cut by 40–45% in 2020 compared to 2005. As a restriction index, this target has been included in the future medium-and-long term plans for national economic and social development. In order to achieve the goal of carbon emission reduction, the choice of carbon dioxide emissions allocation method and initial quotas confirmation must be solved as important topic of research.Data envelopment analysis is effective for a non-parametric approach. It has been widely used in solving the problem of resource allocation. Moreover, DEA-relative models have been proposed in the presence of undesirable outputs especially CO

_{2}emissions. However, dated from the increasing CO_{2}emissions, these models implied a reduction in performance. Thus, these existing models are not used to CO_{2}emissions reallocation on account of the assumption of constant total sum. In this respect, ZSG-DEA model shows much more priority to deal with sum-constant reallocation problem, especially the interactive combination with Environment Production Technology. Consequently, ZSG environmental production technology allocation model is proposed to equitably and economically distribute CO_{2}reduction responsibility among China’s provinces. Particularly, the Chinese central government released a new regulation guiding the trade of carbon emission quotas in December 2014. This paper can help facilitate the implementation of this regulation by allocating the appropriate regional CO_{2}emissions.From an academic viewpoint, the allocation of CO

_{2}emissions has been widespread studied. Holmberg et al. [1] to allocate CO_{2}emissions and fuel costs by using the energy, exergy and market. A system and quantitative method is proposed to implement the “common but differentiated responsibilities” rule of distribution of carbon dioxide emissions by Wei et al [2]. Pan et al. [3] emphasized “Equitable Access to Sustainable Development” for per capita cumulative CO_{2}emission rights allocation schemes. Morini et al. [4] raised a method for the optimal demand allocation among combined heat and power (CHP) and renewable energy systems to minimize the primary energy consumption. Hasan et al. [5] presented a benefit-based allocation method by using a Shapley value approach. Wang et al. allocated CO_{2}emission quotas to major countries using different regimes for a sample period of 2011–2100 [6]. Levihn compared different allocation methods and discusses both advantages and disadvantages of each method [7]. There are a number of papers to use GDP to analyze irrigation future of agricultural water management in Africa [8–10].Different from these previous studies, this paper attempts to employ DEA to allocate CO

_{2}emissions. As a non-parametric approach, DEA has been widely used in the resource allocation problem studied by Fang and Zhang [11]. especially the allocation problem with a fixed total amount of input or output. It is considered that Cook and Kress proposed the first model [12], under the DEA framework, that deals with the fixed input allocation problem. Cook and Kress’s approach was based on output-oriented version of the CCR-DEA model, in which the objective of DEA model is to minimize the weighted combination of input variables with the constraint. Cook and Zhu extended this method to cases [13] that the input-oriented CCR-DEA model was utilized. Lin R. presented several DEA models to solve the same fixed input resources allocation problem [14]. Aparicio and Monge et al. also conducted research on such problems of the emission permits allocation [15]. In their study, a centralized point of view was adopted in DEA method to correspond to the three objectives: maximizing aggregated desirable production, minimizing the consumption of input resources, and minimizing undesirable total emissions.By introducing the zero sum game concepts in to the DEA method, Gomes and Lins developed a ZSG-DEA model which was used to reallocate CO

_{2}emissions allowance among the Annex I parties and Non-Annex I countries of Kyoto Protocol. [16] Also by using ZSG-DEA model, Serrao proposed a model to efficiently reallocate agricultural greenhouse gas emissions among 15 EU countries [17]. Since the DEA based method has been successfully and effectively applied in the resource allocation problem, in this paper we choose a DEA based approach for the CO_{2}emissions allowance allocation over the provinces in China.One of the key issues related to the CO

_{2}emissions allowance allocation under the DEA framework is how to deal with the CO_{2}should be be minimized. There are several approaches to modelling such types of undesirable outputs in the DEA context, for instance, dealing the undesirable outputs through a weak disposability reference technology by assuming the undesirable outputs and desirable outputs are generated in the same production process proposed by Färe et al and Arita Duasa et al [18–19]. Feng G, Färe et al and Lozano and Gutierrez applying the directional distance function to simultaneously increase the desirable outputs and decrease the undesirable outputs [20–22]. Cheng and Liu [23] translating the undesirable outputs into desirable outputs mathematically under the classification in variance and treating the undesirable outputs as inputs by Zhou et al and Zhang et al [24–25]. Furthermore, Sueyoshi et al and Goto proposed a DEA model using the range adjusted measure which combined the undesirable and desirable outputs in a unified treatment [26–28]. Since the regional CO_{2}emissions allowance is a sub-divided quota of the total emission control target of China, which can essentially be considered as a distribution of the resource to each region, the approach proposed in this paper therefore realistically treats the undesirable outputs of the CO_{2}emissions allowance as inputs.### 2. Materials and Methods

### 2.1. CCR-DEA Model and Environmental ZSG-DEA

Assuming that there are

*G*decision making units (DMUs) which convert*s*inputs into*t*outputs. Let*x**denote the i-th input and*_{ig}*y**denote the j-th output for DMUg. The classic output-oriented CCR model which was proposed by Charnes et al. [29] for calculating the technical efficiency of DMUg can be expressed in Eq. (1).*_{jg}##### (1)

$$\begin{array}{c}\text{Max\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{h}_{g}\\ s.t.\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{x}_{ig}\le {x}_{ig},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}i=1,\hspace{0.17em}\cdots ,s\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{y}_{jg}\ge {h}_{g}\hspace{0.17em}{y}_{jg,},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}j=1,\hspace{0.17em}\cdots ,t\\ {\lambda}_{g}\ge 0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}g=1,\hspace{0.17em}\cdots ,G\end{array}$$Once Eq. (1) is solved, the ZSG-DEA model can be constructed by using the efficiency scores derived from Eq. (1). The ZSG-DEA model was first proposed by Lins et al. [30] in order to estimate the winning efficiency of different countries in Olympics. The idea is that the total amount of an input (output) is fixed so that a decrease in the input (output) for one DMU can lead to an increase in the input(output) for another DMUg. As discussed by Gomes and Lins [16], the output-oriented ZSG-DEA model can finally be formulated in Eq. (2).

##### (2)

$$\begin{array}{c}\text{Max\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{h}_{rg}\\ {h}_{rg}\hspace{0.17em}{y}_{ig}\le \sum _{p}{\lambda}_{p}\hspace{0.17em}{y}_{ip}\left|1-\frac{{y}_{ip}({h}_{p}-1)}{{\displaystyle \sum _{p\ne k}}{y}_{ip}}\right|\\ \sum _{p}{\lambda}_{p}\hspace{0.17em}{x}_{jp}\le {x}_{jg}\\ {\lambda}_{p}\ge 0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for\hspace{0.17em}}\forall \hspace{0.17em}p\end{array}$$Where hp represents expansion factor for DMUp that can be calculated from Eq. (1), and

*h**represents expansion factor of DMU*_{rg}_{g}evaluated by ZSG-DEA. The ZSG-DEA model has been widely used for the cases with single input or output variables in resource allocation.In this paper, we constructed a ZSG-DEA model for CO

_{2}emission allocation based on the assumption that CO_{2}emissions are weakly disposable. Similar to many previous studies such as Zhou et al. [31], we choose capital stock (*K*), population (*P*) and energy consumption (*E*) as three inputs, GDP (*Y*) as a single desirable output and CO_{2}emissions (*C*) as a single undesirable output. Then the macro-production process can be modeled by the following Environmental Production Technology exhibiting constant returns to scale:##### (3)

$$\begin{array}{c}T=\{(E,\hspace{0.17em}K,\hspace{0.17em}P,\hspace{0.17em}Y,\hspace{0.17em}C):\sum _{k=1}^{k}{\lambda}_{k}\hspace{0.17em}{E}_{k}\le E\\ \sum _{k=1}^{k}{\lambda}_{k}\hspace{0.17em}{K}_{k}\le K\\ \sum _{k=1}^{k}{\lambda}_{k}\hspace{0.17em}{P}_{k}\le P\\ \sum _{k=1}^{k}{\lambda}_{k}\hspace{0.17em}{Y}_{K}\le Y\\ \sum _{k=1}^{k}{\lambda}_{k}\hspace{0.17em}{C}_{k}\le C\\ {\lambda}_{k}\ge 0,\hspace{0.17em}k=1,\hspace{0.17em}2,\hspace{0.17em}\cdots ,\hspace{0.17em}k\}\end{array}$$The environmental production technology T, also referred to as environmental DEA technology, can satisfy both weak disposability and null-jointness assumptions. The weak disposability assumption indicates that it is possible to reduce CO

_{2}emissions and GDP proportionally, while the null-jointness assumption implies that removing all the CO_{2}emissions must be at the cost of zero GDP. Based upon Eq. (3), we can easily formulate the following undesirable output orientation DEA model for calculating the CO_{2}emission expansion coefficient. This study focuses on the CO_{2}emission allocation and treats CO_{2}emission as the undesirable output, the emission expansion coefficient is the technical efficiency in our model, which means a larger expansion coefficient reflects higher technical efficiency.##### (4)

$$\begin{array}{c}\text{Max\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{h}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{E}_{g}\le {E}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{K}_{g}\le {K}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{POP}_{g}\le {POP}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}C{O}_{2g}\le {h}_{p}\hspace{0.17em}C{O}_{2p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{GDP}_{g}\le {GDP}_{p}\\ {\lambda}_{g}\ge 0,\hspace{0.17em}g=1,\cdots ,\hspace{0.17em}G\end{array}$$With the consideration of research findings in Gomes et al. [16] and the basis of environmental DEA technology, this paper formulate the ZSG-DEA model in Eq. (5) the total amount of CO

_{2}emissions for all the DMUs is fixed. Provided that the CO_{2}emissions for DMUg are decreased, the portion reduced by the DMU will be allocated to other DMUs, proportional to their actual amounts of CO_{2}emissions.##### (5)

$$\begin{array}{c}\text{Max\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{h}_{rp}\\ {h}_{rp}\hspace{0.17em}C{O}_{2p}=\sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}C{O}_{2g}\left|1-\frac{C{O}_{2p}\hspace{0.17em}({h}_{rp}-1)}{{\displaystyle \sum _{g\ne p}}C{O}_{2g}}\right|\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{E}_{g}\le {E}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{K}_{g}\le {K}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{POP}_{g}\le {POP}_{p}\\ \sum _{g=1}^{G}{\lambda}_{g}\hspace{0.17em}{GDP}_{g}\le {GDP}_{p}\\ {\lambda}_{g}\ge 0,\hspace{0.17em}g=1,\hspace{0.17em}\cdots ,\hspace{0.17em}G\end{array}$$Lins et al. [30] and Gomes and Lins [16] showed that there exists a relationship between

*h**and the ZSG-DEA*_{p}*h**. Following Gomes and Lins, we derive the following Eq. (6) for solving*_{rp}*h**from*_{rp}*h**:*_{p}##### (6)

$${h}_{rp}=\left[1+\frac{{\displaystyle \sum _{j\in W}}C{O}_{2jp}({\theta}_{nm}{h}_{rp}-1)}{{\displaystyle \sum _{j\notin W}}C{O}_{2jp}}\right]$$where W represents the set of cooperative DMUs with hp higher than unity and

*θ**=*_{nm}*h**/*_{n}*h**. The DMUs belonging to W need to decrease their CO*_{m}_{2}emissions by $\sum _{j\in W}}{C}_{j}({\theta}_{nm}{h}_{rp}-1)$, which will be allocated to the DMUs that do not belong to W.### 2.2. Grey Model

The grey system was proposed by Deng [32]. The grey system theory has been successfully applied in many fields such as management, economy, engineering, finance, etc. There are three types of systems-white, black, and grey. A system is called a white system when its information is totally clear. When a system’s information is totally unknown, it is called a black system. If a system’s information is partially known, then it is called a grey system. The grey forecasting model adopts the essential part of the grey system theory. The Grey Model GM (1,1) grey forecasting model can be used in circumstances with relatively little data, and it can use a first-order differential Eq. to characterize an unknown system. So the GM (1,1) grey forecasting model is suitable for forecasting the competitive environment where decision makers can reference only limited historical data. The GM (1,1) procedure can be separated into five steps:

Step 1: Collect original data and build a data sequence. The observed original data are defined as

*X*^{(0)}(*i*), where*i*is*i*th sample. The raw sequence of*n*samples is defined as:##### (7)

$$\begin{array}{c}{X}^{(0)}=({x}^{(0)}(1),\hspace{0.17em}{x}^{(0)}(2),\hspace{0.17em}\cdots ,\hspace{0.17em}x(0)(n)),\hspace{0.17em}{x}^{(0)}(k)>0,\\ k=1,2,\hspace{0.17em}\cdots ,n.\end{array}$$Step 2: Transform the original data sequence into a new sequence. A new sequence

*X*^{(1)}generated by the accumulated generating operation (AGO), where##### (8)

$${X}^{(1)}=({x}^{(1)}(1),\hspace{0.17em}{x}^{(1)}(2),\hspace{0.17em}\cdots ,\hspace{0.17em}{x}^{(1)}(n)).$$The

*x*^{(1)}(*k*) is derived as follows:##### (9)

$${x}^{(1)}(k)=\sum _{i=1}^{k}{x}^{(0)}(i),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}k=1,2,\hspace{0.17em}\cdots ,n.$$Step 3: Build a first-order differential Eq. of the GM (1,1) model. By transforming the original data sequence into a first-order differential Eq., the time series can be approximated by an exponential function. The grey differential model is obtained as

where a is a developing coefficient and b represents the grey input. According to Eqs. (8) and (10), the parameters a and b can be estimated by the least-squares method. Parameter

*â*is represented asWhere

##### (12)

$$B=\left[\begin{array}{cc}-{z}^{(1)}(2)& 1\\ -{z}^{(1)}(3)& 1\\ \xb7& \xb7\\ \xb7& \xb7\\ -{z}^{(1)}(n)& 1\end{array}\right],\mathrm{\hspace{0.17em}\u200a\u200a}{Z}^{(1)}=({z}^{(1)}(2),\hspace{0.17em}{z}^{(1)}(3),\hspace{0.17em}\cdots ,\hspace{0.17em}{z}^{(1)}(n)).$$The background value for formula (12) is represented as

##### (13)

$${z}^{(1)}(k)=\frac{1}{2}({x}^{(1)}(k)+{x}^{(1)}(k-1)),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}k=2,3,\hspace{0.17em}\cdots ,n$$And

##### (14)

$$Y=\left[\begin{array}{c}{x}^{(0)}(2)\\ {x}^{(0)}(3)\\ \xb7\\ \xb7\\ {x}^{(0)}(n)\end{array}\right]$$
Eq. (10), we get the grey forecasting predictor as follows:

##### (15)

$$\begin{array}{c}{\stackrel{\u2322}{x}}^{(1)}(k+1)=\left({x}^{(1)}(0)-\frac{b}{a}\right){e}^{-ak}+\frac{b}{a}\\ \text{where\hspace{0.28em}}k=1,2,\hspace{0.17em}\cdots ,n.\end{array}$$Step 5: Check the process: after a new sample is obtained, Steps 1 to 4 are followed to predict the behavior of the process until unusual conditions occur. If a new predicted point is plotted beyond the upper or lower limits, it means that the process may be out-of-control and that an investigation will be started; otherwise, the process continues to be monitored.

### 3. Results and Discussion

### 3.1. Variables and Data Description

We apply the ZSG-DEA model by using Eqs. (4)–(6) to investigate how to efficiently allocate China’s CO

_{2}emissions to different provinces by 2020. All data covering the period of 2004–2012 were obtained from the Chinese Statistical Yearbook. Data on capital stock were obtained by using the perpetual inventory method described in Shan [33], while data on other variables came from various issues of China’s Energy Statistics Yearbooks and China Statistical Yearbooks.The data on CO

_{2}emissions at province level are not available. With reference to the 2006 IPCC National Greenhouse Gas Inventories, energy-related CO_{2}emissions can be calculated as Eq. (16),where

*I*denotes total CO_{2}emissions,*K**is carbon emission coefficient of the kind of primary energy,*_{i}*E**refers to the*_{i}*i**kind of primary energy consumption, and 44/12 is ratio of molecular weights of CO*_{th}_{2}and*C*. Primary energy carbon emission coefficients are recommended by the*Energy Research Institute of Chinese National Development and Reform Commission*. Coefficients for coal, fossil oil, natural gas, and nonfossil energy are 0.7476, 0.5825, 0.4435 and 0 respectively (ton C/ton standard coal).The prediction of China’s economic growth, capital stock, population, energy consumption and CO

_{2}emissions by 2020 are given as follows.First of all, according to the research of the

*Development Research Center of the State Council of China*(Wang, 2005) [34] and the EIA (2009), annual GDP growth rate of China in 2011–2020 is 5.3% and 6.4% (in the low economic growth scenarios) and 7.4% and 8% (in the high economic growth scenarios) respectively. Here, annual GDP growth rate during 2010–2020 is assumed as 8%.Secondly, according to the research of the United Nations Department of Economic and Social Affairs, China’s population by 2020 will be 1.43 billion (UNDESA, 2009) [35]. We use this population projection.

Thirdly, according to the research of physical capital stock grow at average rate of 14% and 10. 96% of the depreciation in china during 2013–2020 proposed by Lin [36], China’s capital stock by 2020 is calculated based on the perpetual inventory method and to adjust in 1952.

Furthermore, energy consumption forecasting model is a small sample data trend modeling problem. Grey forecasting model has a strong ability to deal with the information of small sample. We can directly use GM (1,1) model to predict for energy consumption in China during 2004–2012 and analysis error of historical actual data and predicted value.

As can be seen in Table 1, the error is below 5%. The forecasted results show that the method is feasible and the forecasted results well match the actual measured data. Thus, energy consumption of each province is calculated using the gray forecast in 2020.

Finally, we take 2004–2012 data of all provinces except Tibet as samples, using GM (1,1) model to forecast carbon dioxide emissions of 30 provinces in China in 2004–2012. By comparing the predicted results with real numerical values from Table 2, we can reach that the prediction of carbon dioxide emissions which uses the GM (1,1) model is credibility. Thus, the carbon dioxide emissions of 30 provinces by 2020 can be obtained.

Table 3 lists the descriptive statistical results of the five variables in China by 2020. Unfortunately, due to the lack of data, Tibet, Hong Kong, Macau and Taiwan were not included in this study, while other 30 provinces, autonomous regions and provincial municipalities (such as Beijing, Shanghai, Tianjin and Chongqing) were included.

### 3.2. Empirical Results

We first calculate the CO

_{2}emission*h**scores by using Eq. (4), which is listed in Table 4. It is clear that in 2020 several less developed provinces, e.g. Hebei, Shanxi, Inner Mongolia, Liaoning and Ningxia had*_{p}*h**score of unity, indicating that the amounts of CO*_{p}_{2}emissions in these provinces were too large to be further expanded. On the contrary, their CO_{2}emissions need to be reduced in order to keep consistency with their current input and output levels. Table 4 also lists that some provinces or provincial leveled municipalities (such as Beijing and Hainan) had larger*h**scores, which could be explained by their higher CO*_{p}_{2}emission efficiency. Most of the province’s expansion coefficient is 1, which means that the level of carbon emissions in most of the provinces in China is relatively high. We need to pay attention to it in the future development process and improve at the harmonious development of the energy, economic and social development.We further apply the ZSG-DEA model to derive the ZSG-DEA

*h**scores by using Eq. (6) and the optimal amounts of CO*_{rp}_{2}emissions for each province. Table 5 lists the results for the year 2020, which suggests that provinces with*h**higher than 1 can be allowed increasing their CO*_{rp}_{2}emissions so as to reach ZSG-DEA frontier. For example, in 2020 Beijing could be allowed increasing CO_{2}emissions by 24.13 million tons. On the other hand, those provinces with*h**lower than unity should reduce their CO*_{rp}_{2}emissions in order to reach ZSG-DEA frontier. For instance, Hebei province had to reduce 35.85 million tons of CO_{2}emissions. Based on the previous analysis we find that when total amount of CO_{2}emission in China remains no changes, all the provinces become more efficient after appropriate reallocation.We also present our CO

_{2}emission allocation results in different regions in Fig. 1, showing that the comparison between the results of the ZSG-CO_{2}emission and Initial CO_{2}emission in 2020. There are 13 provinces in the allocation of ZSG exceeded the actual amount of emissions, while other 17 provinces of ZSG allocation amount is less than the actual emissions. In the 13 provinces, Guangdong, Sichuan and Chongqing have been promoted and made large utilization of hydropower, nuclear power and other clean energy. While Heilongjiang, Qinghai and Xinjiang are major agricultural provinces, the other 17 provinces are mostly based on industry. This phenomenon shows a high instability and uncertainty in the future.From the perspective of economics analysis, some provinces are facing higher pressures on CO

_{2}emission reduction. If reducing CO_{2}emissions only from the perspective of reducing energy consumption blindly, economic development and people’s living fields are inevitably deteriorated. It is not rational to limit their energy consumption or even cut energy supply to the general public in order to achieve the national emission reduction targets. Therefore, efforts should focus on how to optimize local energy structure and promote energy saving in the whole societies, especially in the local industries. Renewable or cleaner energy sources, such as natural gas, wind power, solar power, and geothermal power, should be fully supported by considering the local energy endowments. Also, innovative efforts, such as eco design, energy audits, cleaner production, and industrial symbiosis, should be initiated especially in the local industries. In addition, economic instruments, such as carbon trading, carbon tax, carbon cap, should be applied so that provinces with more reduction potentials have adequate drivers to facilitate their carbon reduction efforts, while funds can be collected for supporting related research & development activities.### 4. Conclusions

Along with China’s status of being the greatest energy consumer and CO

_{2}emitter in the world, there is a huge necessity for China to achieve emission-cutting target through regional allocation of emission allowance equally. This paper developed a non-radial DEA model to measure CO_{2}emissions to measure technical efficiency (the CO_{2}emission expansion coefficient). The definition of technical efficiency was introduced in the methodology section. We calculate the technical efficiency through Eq. (4) and construct efficient allocation mechanism through Eqs. (5) and (6). After the ZSG allocation, every DMU’s CO_{2}emission is on the ZSG frontier, indicating the overall “Pareto Optimality”. We treat CO_{2}emission as one undesirable output to modify previous method in which CO_{2}emissions are treated as the input variable.After that, we employ empirical analysis to quantify the efficient allocation of CO

_{2}emissions between different provinces using China’s provincial data for a sample in 2020. Our results show that the actual CO_{2}emissions in some provinces (especially those energy-abundant provinces) were higher than their maximal CO_{2}emission allowances calculated from the ZSG-DEA model, indicating that these provinces are facing great pressures on CO_{2}emission reduction. Our results have a certain strategic significance for policy making. The level of ZSG-CO_{2}emissions may be used as an indicator for monitoring the harmony between CO_{2}emissions and other factors such as capital investment and economic development.The inconsistency between ZSG-CO

_{2}emissions and the actual CO_{2}emissions in different regions shows that regions react quite differently to this “dilemma” problem (to achieve both economic development and energy conservation and emission reduction). In order to achieve the proposed national emission reduction targets, different regions should collaborate through innovative efforts, such as the use of effective economic instruments, capacity building and technology transfer.The conclusion drawn by this study is important for the government to adopt relative strategies and enrich the low-carbon-economy system in China. However, the research is still preliminary and worthy of further study, such as method improvement, in-depth analysis of variable relationship.