## 2 Literature Review  
### Research on Carbon Emission Efficiency  
Currently, research on carbon emission efficiency abroad mainly focuses on the selection of appropriate indexes that reflect carbon emission efficiency and the methods for quantifying it. The selection of appropriate indexes mainly focuses on either a single element or full elements. A single element is the ratio of carbon emissions to a certain element. For example, [PERSON] et al. [11] and [PERSON] [12] defined carbon emission efficiency as the amount of carbon emissions per unit of GDP. [PERSON] et al. believed that it is more realistic to measure the carbon emission efficiency of developing countries in regard to the amount of carbon emissions per unit of energy consumption [13]. [PERSON] also thought that energy consumption intensity is the best indicator to measure a country's carbon emission efficiency [14]. However, the single-element method applied to measure carbon emission efficiency is prone to certain errors and uncertainties, and thus cannot objectively and comprehensively reflect the results of carbon emission efficiency. Consequently, researchers have proposed all-element indexes to quantify carbon emission efficiency. For example, [PERSON] proposed an assessment framework of energy consumption, economic development, and carbon emissions [15]. [PERSON] et al. developed the concept of a comprehensive index system for measuring carbon emission efficiency, and provided environmental efficiency assessment indices for the Organization for Economic Co-operation and Development (OECD) countries [16]. [PERSON] et al. evaluated the carbon emission efficiency of manufacturing industries in OECD countries on the basis of DEA models (with undesired output included) [17]. [PERSON] et al. considered slack variables in their assessment framework to measure the efficiency of all-element carbon emissions, and measured the overall carbon emission efficiency of OECD countries [18]. Currently, the main methods used to measure carbon emission efficiency abroad include DEA and stochastic frontier analysis (SFA). DEA has been widely used; for example, [PERSON] et al. applied DEA to analyze the carbon emission efficiency in South Africa [19]. Moreover, [PERSON] et al. used DEA to assess the carbon emission efficiency of some industries and regions [20]. [PERSON] et al. successfully applied DEA to measure the carbon emission efficiency of various countries [21]. However, traditional DEA has certain limitations, and thus is unable to eliminate some uncertain elements. SFA, by contrast, can make up for the shortcomings of the traditional DEA method, and represents another important method for measuring carbon emission efficiency. For example, [PERSON] et al. measured the carbon emission efficiency of 170 countries based on the SFA method [22] and [PERSON] et al. applied SFA to measure the carbon emission efficiency in Pakistan [23]. Research on carbon emission efficiency in China is consistent with the international trend, with specific focuses on the selection of measurement indices that reflect carbon emission efficiency and the measurement methods that can help quantify it. The selection of appropriate indices is mainly concentrated on the question of whether to choose single-element or all-element methods. The single-element method used to measure carbon emission efficiency is mainly based on carbon productivity and carbon emission intensity. For example, [PERSON] measured the carbon productivity in China at the provincial level [24]. [PERSON] et al. [25], [PERSON] et al. [26], and [PERSON] et al. [27] considered carbon emission intensity as an important indicator reflecting the carbon emission efficiency of the manufacturing industry or in a region. In regard to the all-element method used to measure carbon emission efficiency, [PERSON] et al. evaluated the low-carbon economic development performance of 31 provinces in China on the basis of the all-element Malmquist productivity index [28]. [PERSON] et al.