Measuring the Efficiency of Energy and Carbon Emissions: A Review of Definitions, Models, and Input-Output Variables

Abstract

The importance and urgency of improving energy and carbon emissions efficiency in mitigating climate change and achieving carbon neutrality have become an increasingly relentless focus in recent years. Assessing the performance of energy saving and carbon emissions reduction is a significant necessity to achieve sustainable economic development. Therefore, from the perspective of production economics, this paper presents a review of the definition, models, and input-output variables for measuring total-factor energy efficiency and total-factor carbon emissions efficiency. Relevant literature in this field, published between 2006 and 2021, has been systematically analyzed using CiteSpace software, which includes a quantitative and visual review of a large body of published literature. This review found that the current definitions of total-factor energy efficiency and total-factor carbon emissions efficiency are confusing and misleading. Furthermore, future research on energy saving and carbon emissions reduction should incorporate subject areas such as economics, energy, and ecology.

1. Introduction

In December 2015, the 21st United Nations Climate Change Conference adopted the Paris Agreement, the first ever-global agreement to reduce carbon emissions. With the aim of holding the global average temperature rise to well below 2 °C above pre-industrial levels and endeavoring to limit warming to 1.5 °C [1], this agreement focused all nations on a common cause for the first time in an ambitious effort to combat climate change, primarily caused by massive carbon emissions, and to adapt to its impacts. Several studies have explored ways to reduce carbon emissions, including reducing anthropogenic carbon dioxide (CO₂) emissions through decarburization of energy structure [2,3,4], improvement of energy efficiency [5,6], and promotion of low-carbon-consumption behaviors [7,8]. Other studies have explored ways of achieving net zero or even negative carbon emissions through carbon-dioxide-removal technologies [9]. These include potential solutions, such as forest vegetation carbon sink [10], carbon capture, utilization and storage technologies [11,12], and bioenergy with carbon capture and storage technologies [13,14]. Based on the research and analysis of these carbon-reduction pathways, many countries have announced net zero emission commitments and have adopted relevant actions to achieve this.

After six years of efforts, however, the latest report by the UN Intergovernmental Panel on Climate Change [15] highlights that massive greenhouse gas emissions are exacerbating climate deterioration. Global warming will exceed 1.5 °C and 2 °C in the 21st century unless there are significant reductions in CO₂ and other greenhouse gas emissions in the coming decades. Although global commitments and actions are increasing, they are still far from what is needed to limit the global temperature rise to 1.5 °C and avert the worst effects of climate change [16]. According to the report, it is necessary to restrain cumulative CO₂ emissions to at least net zero and combine this with substantial, rapid, and sustained reductions in other greenhouse gas emissions.

Among all the pathways, improving energy efficiency is the first step to reduce the environmental impacts of production processes and achieve sustainable development [17], and this is the core foundation to meet the needs of the world’s growing and increasingly affluent population and achieve net zero emissions by 2050 [16]. Although the use of clean energy is gradually increasing, according to BP [18], fossil fuels still accounted for 83% of global primary energy consumption in 2020. As the main fuel in a sustainable global energy system [19], energy efficiency plays an essential role in achieving global climate and sustainable development goals and has been widely discussed and studied by governments and researchers. This paper reviews efficiency from the perspective of production economics to provide a reference for sustainable economic development.

In recent years, studies and literature on energy efficiency have grown considerably. However, when we conducted keyword clustering on total-factor energy efficiency and total-factor carbon emissions efficiency, we found that the carbon emissions efficiency’s cluster#2 is “energy efficiency” and that there is not enough differentiation between these two terms. Carbon emissions efficiency is the most direct carbon reduction approach but is often confused with energy efficiency and has not received enough research attention in its own right. In response to a call for scholars to analyze energy efficiency in an historical context [20], this paper investigates the origins of efficiency and compares different variables of energy efficiency and carbon emissions efficiency from the perspective of production economics. By sorting out the latest research results from basic information, cited papers, and keywords, this paper further distinguishes and clarifies energy efficiency and carbon emissions efficiency.

Existing papers on total-factor energy efficiency and total-factor carbon emissions efficiency mainly focus on the extension and application of methodologies [21,22,23], but few studies provide an overview, comparison, and evaluation of the models. Therefore, this paper fills this research gap through an in-depth review and comparison of basic models. Some social scientists criticize the main technical and economic approaches to measure energy efficiency as being too narrowly focused and call for observation studies of more complex social backgrounds and practices [24]. In the measurement of efficiency, the reflection of society and production is mainly reflected in the selection of input and output variables, which is often taken for granted and seldom paid attention to [25]. Different selections of variables will lead to different efficiency results, and these affect reality in different ways. In order to ensure the evaluated efficiency is more in line with reality, it is necessary to improve the existing input-output variables.

Given the identified research gap mentioned above, the main purpose of this paper is to systematically review the definition of energy efficiency and carbon emissions efficiency from an economic perspective, compare the basic models, and review the selection of input-output variables. Based on such a comprehensive review, the most appropriate model and input-output variables are proposed for conducting further research on measuring efficiency of energy and carbon emissions to achieve climate change mitigation and carbon neutrality.

This paper is structured in five parts. The Section 2 presents overview of the key literature on total-factor energy efficiency and total-factor carbon emissions efficiency. Then, Section 3 explains the concept and measurement of energy efficiency and carbon emissions efficiency. Section 4 then compares the basic models, while Section 5 reviews the selection of input-output variables, and this is followed by the concluding section.

2. Total-Factor Efficiency Publications

CiteSpace software was used to systematically comb the relevant literature on total-factor energy efficiency and total-factor carbon emissions efficiency so as to explore the basic situation, hot topics, and future trends for research in these areas.

2.1. Research Method and Data

Commonly used visual analysis tools for extracting the information in the publications include HistCite, RefViz, SATI, and CiteSpace [26]. By comparing the characteristics of each software, it can be found that CiteSpace is easy to operate and can perform multiple analyses, such as co-citation graphs and timeline graphs. Therefore, we chose version 5.8.R2 of the CiteSpace software as the main tool for a comprehensive analysis of the selected papers.

On the Web of Science Core Collection database, a topic search was conducted for “total-factor energy efficiency (TFEE)” and “total-factor carbon emissions efficiency (TFCE)”. This included 467 and 315 publications, respectively, from 2006 to 2021.

Through CiteSpace, we first sorted out the basic information from the TFEE- and TFCE-related searches, including authors, countries, institutions, etc. Secondly, through co-citation analysis, we identified the most important citations in this field. Furthermore, we use the co-occurrence and cluster analysis of “keywords” to determine the frontier research and hot topics in this field at different stages of development. Based on these analyses, the development trend of TFEE and TFCE research was explicated, and the problems and breakthroughs that may require attention in the future are also discussed.

2.2. Basic Situation Analysis

2.2.1. Published Papers

A total of 457 publications, from 2006 to the present, with the keyword “total-factor energy efficiency (TFEE)” were analyzed. And published papers in this study have been collected from the WOS database, which we believe covers most existing literatures about the topic. In 2006, Hu and Wang [27] first proposed the concept of TFEE. In the following 8 years, the number of published literatures about TFEE remained stable, at around 10 papers on average per year. Although the number of publications remained small during the earlier years, they laid a theoretical foundation for subsequent studies. Since 2013, the number of published papers has increased rapidly, indicating that scholars are increasingly interested in TFEE analysis.

The “total-factor carbon emissions efficiency (TFCE)” was used as the keyword search to analyze 315 publications from 2006 to 2021. Figure 1b shows a similar trend to that of TFEE. The number of publications on TFCE increased slowly from 2006 to 2014 and then grew rapidly from 2014, indicating scholars’ attention to TFCE increased in line with the increasingly prominent environmental problems.

2.2.2. Cooperation Networks

(1) TFEE: The cooperation network for countries and institutions related to TFEE literature was analyzed to determine which ones have a strong influence and cooperation capacity. A total of 457 published papers came from 55 countries and 264 research institutions. It can be found from Figure 2 that China, the United States, and the United Kingdom are the most prolific contributors. The most published research institutions are Xiamen University, National Chiao Tung University, Tsinghua University, Georgia Institute of Technology, and University of Sussex.

An analysis of authors found that more than 400 scholars have studied TFEE, and 6 authors have published more than 3 papers on the topic. Among these 6 authors, 22 papers have been published by Boqiang Lin, 9 by Jinli Hu, and 4 by Honma. In addition, the authors’ research team also published two papers on TFEE [28,29].

In the co-occurrence analysis of subjects, the topics most covered by TFEE are Energy (Count = 264, Centrality = 0.56), Environmental Sciences and Ecology (Count = 279, Centrality = 0.45) and Economics (Count = 133, Centrality = 0.27), indicating that TFEE is a topic involving economy, environment, and energy, and the research of TFEE is most meaningful only when these three aspects are considered simultaneously.

(2) TFCE: The cooperation network among countries and institutions related to TFCE literature was also analyzed to demonstrate countries and institutions with strong influence and cooperation capacity. A total of 315 publications came from 50 countries and 249 research institutions. China, the United States, the United Kingdom, and Australia are the most prolific contributors. The highest contributing research institutions are the Chinese Academy of Sciences, Sichuan University, and Inha University.

More than 400 scholars have carried out research on TFCE, with 5 authors publishing more than 3 papers on the topic; 15 papers were published by Boqiang Lin, 5 by Ping Zhou, and 4 by B.W. Ang.

In addition, the topics most covered by TFCE are Environmental Sciences and Ecology (Count = 260, Centrality = 0.59), Energy (Count = 104, Centrality = 0.5), and Economics (Count = 99, Centrality = 0.32), respectively.

2.3. Knowledge Base Analysis

2.3.1. Cited Papers

Figure 3a shows the distribution of authors with the highest centrality in TFEE-cited literatures. Centrality is an indicator to measure the importance of nodes in the network, which is used in CiteSpace to discover and measure the importance of literature. It can be found from Figure 3 that Fare (2010) [30], Zhang (2013) [31], Hu and Wang (2006) [27], and Lin (2017) [32] contributed the most important literature. Figure 3b shows the distribution of authors with the highest centrality in TFCE-cited literatures. It can be seen that Zhang (2013) [33], Shi (2010) [34], and Ang (2011) [35] contributed the most important literatures.

2.3.2. Keyword Clustering

Keywords can be clustered by using the log-likelihood ratio (LLR) algorithm of CiteSpace software. The quality of clustering is generally determined by two values; one is Modularity (Q); when Q > 0.3, the clustering structure is significant. The other one is Mean Sihouette (S); when S > 0.5, the cluster is reasonable, and when S > 0.7, the clustering results are convincing. By clustering the data obtained from WOS database, Q = 0.4101 > 0.3, and S = 0.7751 > 0.7, indicating that the clustering result is reliable.

(1)

TFEE

A timeline map was obtained by keyword clustering of TFEE-related literature (as shown in Figure 4). The timeline map can help us analyze the key nodes of clustering in each period. As can be seen from Figure 4, the keywords mined by the WOS database are clustered into 6 categories. Cluster#0 is data envelopment analysis, cluster#1 is environmental regulation, cluster#2 is cogeneration, cluster#3 is energy productivity, cluster#4 is industrial energy efficiency, and cluster#5 is LMDI (Logarithmic Mean Divisia Index).

Keywords with a frequency less than 3 were removed. As can be seen from Figure 4, the first landmark paper of each cluster appeared between 2006 and 2008. In 2013, the research on region and SBM (slacks based measure) model was the hot spot in the “data envelopment analysis” cluster.

The frequency of keywords in cluster#0, #1, and #4 is always high, indicating that research in these three areas has remained the focus of TFEE research since 2006. The keywords at the beginning of this area of research included performance, productivity, etc., but gradually changed to environmental efficiency, green productivity, etc.

(2)

TFCE

A timeline map was obtained by keyword clustering of TFCE-related literature (as shown in Figure 5). The keywords mined by the WOS database are clustered into 6 categories. Cluster#0 is carbon emission intensity, cluster#1 is technology gap, cluster#2 is energy efficiency, cluster#3 is carbon sequestration, cluster#4 is greenhouse gas emission, and cluster#5 is environmental performance.

As can be seen from Figure 5, from 2007 to 2012 keywords with the highest frequency were CO₂ emissions and environmental, etc. Energy efficiency and economic growth began to appear in 2013 and economy in 2017. It shows that the focus of TFCE research has experienced the process of environment—ecology and energy—ecology, energy, and economy.

In the “carbon sequestration” cluster, high-frequency keywords began to appear in 2013 and gradually increased, indicating that carbon neutrality is getting more and more attention, which is also one of the hot spots of current research.

In addition, the keywords of TFCE are clustered, and cluster#2 is energy efficiency, indicating that existing literature does not distinguish between TFCE and TFEE but incorrectly mixes them up. Therefore, in this paper, both concepts are treated separately to show the differences between them.

3. Defining Energy Efficiency and Carbon Emissions Efficiency

3.1. The Origin of Energy Efficiency and Carbon Emissions Efficiency

From an economic perspective, efficiency is defined as the full and most efficient use of limited and scarce resources to satisfy people’s wants and needs given the technology [36], that is, produce more with less. If there is no way to make someone better off and nobody worse off, then the situation is Pareto efficient [37]. Extending this concept to production economics, a 100% Pareto–Koopmans efficiency is achieved if and only if no inputs or outputs of any decision-making unit (DMU) can be improved without worsening the other inputs or outputs. However, in most management and social science applications, the theoretically possible Pareto efficiency is unknown. Therefore, it is replaced by relative efficiency [38], which is fully efficient if and only if other DMUs cannot improve inputs or outputs without worsening some of its other inputs or outputs on the basis of empirically available evidence. In this case, measuring production efficiency is actually evaluating whether there is waste of input by comparing the minimum input with the actual input while the output is unchanged or evaluating whether there is an output shortage by comparing the actual output with the maximum output with the input unchanged.

3.2. Single-Factor Indicator of Energy Efficiency and Carbon Emissions Efficiency

3.2.1. Energy Intensity

Energy efficiency was first recorded in 1888 [39] and is known as the thermal economy of a machine [40]. Patterson [41] was the first to elaborate on energy efficiency in economic terms. He defined energy efficiency as the ratio of GDP to energy consumption, commonly referred to as energy productivity. Energy intensity, as the reciprocal of energy productivity, is widely used to measure energy efficiency [42], and is used by many international organizations and countries as a strictly binding target for carbon reduction [43,44].

3.2.2. Carbon Intensity

Actual production and energy consumption will inevitably produce environmental pollution as a by-product of economic growth, of which CO₂ emissions have the greatest impact on climate deterioration. Like energy intensity, carbon intensity, measured as CO₂ emissions per unit of output, is also widely used as a performance indicator of carbon emissions [45,46]. Goldemberg suggested using energy intensity to analyze national trends in energy consumption and carbon intensity to analyze climate change [47,48]. With the increasing severity of climate change, in addition to energy intensity, the International Energy Agency [49] has also incorporated CO₂ intensity into its statistics to assess carbon emissions and sustainability targets.

3.2.3. Inadequacy of Single-Factor Indicator

Since only a single-input factor is considered in addition to one output, energy intensity and carbon intensity are often referred to as single-factor indicators [50,51]. Although these indicators are easy to understand and convenient to measure [34,52], they do have some shortcomings. On the one hand, the measurement of intensity does not conform to the definition of efficiency in economics. Intensity has no comparison between the optimal and actual energy consumption or carbon emissions. Blindly reducing the intensity may cause economic development to deviate from the optimal state. On the other hand, intensity only focuses on the goal of realizing energy saving, carbon emissions reduction, and output growth but pays little attention to other factors, such as capital and labor and structural changes in production [53,54]. It also fails to reflect the change in technical efficiency [55]. Energy intensity may decrease due to labor substitution or organizational structure improvements rather than technological advancement in energy efficiency [56], and it reflects energy consumption rather than energy efficiency [57]. Carbon intensity separates the intrinsic connections between CO₂ emissions, energy consumption, and economic growth [58,59], making it a measure of decarburization and an assessment of energy policies at a national level rather than carbon efficiency [60]. Improving energy efficiency is the most effective way to reduce the resource reallocation cost of energy saving and emission reduction.

3.3. Total-Factor Indicator of Energy Efficiency and Carbon Emissions Efficiency

Through the previous analysis, it is not difficult to find that the definition of single-factor is inconsistent with economic theory, which is divorced from reality and not in line with the production process. In view of the deficiencies of single-factor indicators, total-factor indicators are more convincing in measuring energy saving and carbon emissions reductions.

In measuring the efficiency of DMUs with multiple inputs and multiple outputs, Farrell extends the Pareto–Koopmans property to inputs and outputs, avoiding the necessity of resource prices or other weight assumptions and formal relations between inputs and outputs. This makes it possible to determine relative efficiency by taking the performance of other DMUs to evaluate the overall behavior of each DMU. The obtained result is known as Farrell efficiency [61], which can estimate the amount of input that can be saved or the amount of output that can be increased without worsening any input and output. With the assumption of equal access to inputs by all DMUs, Farrell efficiency is restricted to expressing technical efficiency. However, Farrell’s empirical work was limited to cases of a single output and failed to take into account other non-zero slack, which is an important source for mix inefficiencies in both inputs and outputs.

3.3.1. Total-Factor Energy Efficiency

On the basis of Farrell efficiency, Charnes, Cooper, and Rhodes [62] further formalized the relative efficiency, made up for the shortcomings of Farrell’s work, and developed a dual pair of linear programming model data envelopment analysis (DEA) to identify the best-practice frontier and estimate relative efficiency. The production function is generally regarded as the basis for economics. By constructing the frontier with the possibility of multiple production functions, DEA closely relates to the theory of production economics. Compared with other methods for measuring relative efficiencies, DEA does not need to set a specific function form and strict assumptions and has less data constraints [63,64]. Therefore, since the initial study of Charnes, Cooper, and Rhodes, DEA has become widely accepted and rapidly applied in a number of fields, including the nonprofit, public, and private sectors. It is often used to estimate production efficiency and to examine the overall performance of operations and management.

With more and more extensive application, in addition to deeper development of methodologies, DEA is also used to measure various efficiency values. As the catastrophic problems caused by man-made greenhouse gas emissions become more and more serious, reducing global energy consumption and CO₂ emissions has attracted great interest and become a primary focus of international attention. Ramanathan [65] was the first to introduce DEA in carbon reduction and to measure the amount of carbon emissions that each country can reduce, but he did not measure specific carbon efficiency. Considering that the prominent driving factor of CO₂ emissions lies in huge energy consumption, Hu and Wang [27] further proposed the concept of total-factor energy efficiency (TFEE) by employing a DEA model. This has been widely applied since it was proposed and has vigorously supporting research on energy efficiency and carbon emissions reduction.

Taking into account the role of capital, labor, and energy consumption in the production process and their contribution to GDP, TFEE is defined and measured as:

$$0\le TFEE=\frac{Target Energy Input}{Actual Energy Input}\le 1$$

The target energy input is the minimum energy input under the best practice. Comparing the minimum energy input with the actual energy input, $TFEE$ conforms to the definition of efficiency in economics. A higher $TFEE$ implies less redundancy of energy and more efficiency in energy used. The difference between the actual input and the target input is the total adjustment, which is initially referred to as radial adjustment, conforming to Farrell efficiency, and later incorporates non-zero slacks to satisfy Pareto efficiency [66]. The total adjustments of energy input can be used to measure the inefficient proportion of actual energy consumption and measure the energy-saving target ratio (ESTR) [67]. The relation between $TFEE$ and $ESTR$ is as follows:

$$0\le ESTR=1-TFEE\le 1$$

The higher $ESTR$ is, the lower $TFEE$ is, and the greater the amount of energy consumption that can be saved under the same economic growth level.

3.3.2. Total-Factor Carbon Emissions Efficiency

The scientific definition and measurement of total-factor energy efficiency is also applicable to total-factor carbon emissions efficiency. As a byproduct of economic growth generated by energy consumption, CO₂ emission is generally incorporated as an undesirable output into the input-output variable [68,69]. With reference to $TFEE$, $TFCE$ is calculated as follows:

$$0\le TFCE=\frac{Target C{O}_2 Emissions}{Actual C{O}_2 Emissions}\le 1$$

The technical efficiency measured by Charnes, Cooper, and Rhodes (1978) is the minimum efficiency value of each input and output, reflecting the overall performance of each DMU relative to inputs and outputs, so it is usually used to measure environmental or ecological efficiency [70,71]. Compared with overall efficiency, $TFEE$ and $TFCE$ only indicate the performance of energy saving and CO₂ emissions, so they are more convincing in the measurement of specific sustainable development goals. In addition, on the basis of TFEE and TFCE, some new indicators for measuring energy consumption and carbon emissions have been developed, such as energy performance index, defined as the ratio of actual energy efficiency to target energy efficiency, and carbon performance index, measured as the ratio of target carbon intensity to actual carbon intensity [64,72].

4. Efficiency Measurement Model Based on DEA

The concept and idea of data envelopment analysis (DEA) was first proposed by Farrell [61] and is a commonly used benchmark tool to measure the production efficiency of decision-making units (DMUs) and to reflect their performance. DEA is a method that is easy to operate. It does not need to make any restrictive assumptions about relevant functions and can deal with multiple input and output variables of different units.

4.1. Radial Model

4.1.1. CCR

In 1978, Charnes, Cooper, and Rhode [62] first proposed an input-oriented data envelopment analysis, which is based on Constant Return to Scale (CRS) in the intersection of mathematics, operational research, mathematical economics, and management science. Subsequently, this method has attracted widespread attention and application. An input-oriented CRS-DEA model can be expounded as follows:

In the case of the production, technology was defined as $T=\left\{\left(x,q\right):q\le Q\lambda , x\ge X\lambda \right\}$. Suppose that there are I DMUs, and each DMU has N inputs and M outputs. Measuring the efficiency of the ith DMU is to solve the following mathematical programming problem:

$$\begin{gathered} mi{n}_{\theta ,\lambda } \theta \\ s.t.-q_i+Q\lambda \ge 0 \\ \theta x_i-X\lambda \ge 0 \\ \lambda \ge 0 \end{gathered}$$

(1)

where θ is a scalar, and λ is a I × 1 vector of constants. The column vectors $x_i$ and $q_i$ represent the ith DMU’s input and output, respectively. The N × I input matrix X and the M × I output matrix Q represent the data for all I DMUs. The value of θ is the efficiency score for the ith DMU. The DMU with a score = 1 is the efficient DMU, which means the DMU is on the frontier.

According to Figure 6, SS’ is a piece-wise linear isoquant determined by all the DMUs in the sample, where the radial contraction of the input vector is ($X\lambda , Q\lambda )$, the radial adjustment is AA’, and the constraints in Equation (1) ensure that the projected point cannot lie outside the feasible set. There are four firms, A, B, C, and D, where firms using input combinations C and D are the two efficient firms that define the frontier, and firms A and B are inefficient firms. Based on the measure of technical efficiency, the efficiency of A and B can be represented as OA′/OA and OB′/OB, respectively. However, it is not certain if A′ is an efficient point because when the use of input $x_2$ is reduced (that is, CA’), it still produces the same output. For firm B, it is effective when B moves to B′, as its input combinations change, i.e., efficiency equals one.

Under the framework of total-factor production, Hu and Wang [27] first added capital, labor, and energy as input factors and applied the CRS-DEA method to measure the ratio of optimal energy input to actual energy input. Zhao et al. [73] used the CCR model to analyze the changes in TFEE in Chinese industrial sectors and provinces. The results show that the TFEE for the industrial sector in the eastern provinces is generally higher than that of other provinces. The conclusion is consistent with most studies. Kim et al. [74] used the CCR model to assess the efficiency of three new and renewable energy technologies, and their results show that wind power is the most efficient renewable energy source in South Korea. Li and Lin [75] used a super-efficiency CRS-DEA model to measure energy-adjusted total-factor productivity and energy and carbon dioxide emissions-adjusted total-factor productivity (denoted as TFEE and TFCE, respectively). The results showed that China’s economic growth does not improve both TFEE and TFCE. Chang [76] employed a new computation model to solve the problem that TFEE is not workable under a meta frontier framework, to conduct an empirical analysis on the 28 member countries in the EU.

In addition, several studies combine the CCR model with other models or approaches. Ewertowska [77] combined the Life Cycle Assessment and CCR-DEA model to analyze the environmental performance and eco-efficiency of the power of major European economies. Aydın [78] combined CCR with Tobit to find the influencing factors of energy efficiency. Yang and Wei [79] used a CRS-DEA model to analyze the TFEE of key regions along the Belt and Road from 2005 to 2015 and used the Malmquist index to decompose. Similar studies have been carried out by other scholars, such as Cui et al. [80] and Liu et al. [75].

4.1.2. BCC

Considering that in the case of imperfect competition, government regulation, financial constraints, etc., the CRS assumptions will no longer pertain, Banker, Charnes, and Cooper [81] proposed to improve the CRS-DEA method by adding convexity constraint $I{1}^{\prime }\lambda =1$ into the CRS-DEA model to explain Variable Return to Scale (VRS).

In order to explain VRS, the CRS linear programming problem can be easily modified by adding the convexity constraint to Equation (2):

$$\begin{gathered} mi{n}_{\theta ,\lambda } \theta \\ s.t. -q_i+Q\lambda \ge 0 \\ \theta x_i-X\lambda \ge 0 \\ I1’\mathsf{\lambda }=1 \\ \lambda \ge 0 \end{gathered}$$

(2)

where $I1$ is the I × 1 vector with element 1. This approach forms a convex hull of intersecting planes that more closely encloses data points than the conical hull of CRS. Therefore, the measured technical efficiency value is greater than or equal to that obtained by the CRS model.

Similar to Figure 6, in Figure 7, an output-orientated DEA with two outputs could also be represented by a piece-wise linear production possibility curve. When point P is projected to the point P′, it is on the frontier but not on the efficient frontier because the production of $q_i$ could be increased by the amount AP’ without using any more inputs.

Zhang et al. [82] used a VRS-DEA model based on DEA window analysis to investigate TFEE in 23 developing countries from 1980 to 2005. China’s effective energy policies play a crucial role in improving energy efficiency. Fang et al. [83] utilized a VRS-DEA model to assess the pure technical efficiency, total-factor energy efficiency, and environment-adjusted total-factor energy efficiency of service sectors in Taiwan, China over the period 2001–2008. The results demonstrate that Taiwan’s service industry still has a 5–18% potential improvement on input resource savings. Honma and Hu [84] used a VRS-DEA model to measure the TFEE of industry in 14 developed countries from 1995 to 2005 in order to compare the gap between Japan’s industrial energy efficiency with other developed countries. In this study, Germany, the United Kingdom, and the United States frequently appear as benchmarks for inefficient Japanese industries.

A model-orientation approach indicates whether the objective is to minimize inputs or maximize outputs. Some studies defined the scale efficiency as overall technical efficiency/pure technical efficiency, in case there is a statistically significant difference in efficiency between CCR and BCC. [77,85].

4.2. Non-Radial Model

4.2.1. Traditional Slacks-Based Measure (SBM) Model

In the traditional CRS/VRS-DEA model, the inefficient decision-making unit achieves the effective state by reducing all inputs or expanding all outputs in the same proportion. Therefore, the traditional DEA model is also referred to as a radial model. When the DMU is inefficient, there is a certain distance between the current state and the optimal target. In the radial model, this distance is equal to the sum of the radial improvement and the slack improvement, but the latter is not reflected in the efficiency measurement of the radial model.

In this context, Tone (2001) [86] proposed a non-radial and non-angular Slacks-Based Measure model, which directly introduced the slack variable into the objective function. The SBM model can find the target energy input more effectively, avoid the influence and error caused by radial and angle differences, better reflect the essence of efficiency measurement, and improve the reliability and accuracy of efficiency measurement.

The SBM model integrates the input and output of each DMU and provides an effective solution to the relaxation problem. It is assumed that there are n DMUs, with m inputs and s outputs. In this case, the production possibility set was defined as $\mathrm{P}=\left\{\left(x,y\right)\right|x\ge X\lambda , y\le Y\lambda , \lambda \ge 0\}$, where $\lambda =\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right)$ is an n × 1 nonnegative vector, $X=\left[x_1,x_1,\dots ,x_n\right]$ is an m × n matrix of input vectors, and $Y=\left[y_1,y_2,\dots ,y_n\right]$ is an s × n matrix of output vectors. Measuring the efficiency of the ith DMU is to solve the following mathematical programming problem:

$$\begin{gathered} \min \rho =\frac{1-\frac{1}{m}{{\displaystyle \sum }}_{i=1}^m{s}_i^{-}/x_{i0}}{1+\frac{1}{s}{{\displaystyle \sum }}_{r=1}^s{s}_r^{+}/y_{r0}} \\ s.t.x_0=X\lambda +s^{-} \\ {y}_0=Y\lambda -s^{+} \\ \lambda \ge 0, s^{+}\ge 0,s^{-}\ge 0 \end{gathered}$$

(3)

where ρ represents the technical efficiency, $x_0$ and $y_0$ are the input vector and output vector of each DMU, $s_i^{-}\in R_m$ means the input excess, and $s_r^{+}\in R_s$ means the output shortfall.

4.2.2. Slacks-Based Measure (SBM) Model with Undesirable Output

Tone (2004) [87] constructed the SBM model by taking undesirable outputs into account. Therefore, compared with other DEA models, the SBM model with undesirable output can better reflect the essence of efficiency evaluation.

Suppose that there are n DMUs, and each $DM{U}_j\left(j=1,2,\dots ,n\right)$ has m inputs $x_j={\left(x_{1j},x_{2j},\dots ,x_{mj}\right)}^{’}$, $s_1$ desirable outputs $y_j^g={\left(y_{1j}^g,y_{2j}^g,\dots ,y_{s1j}^g\right)}^{’},$ and $s_2$ undesirable outputs $y_j^b={\left(y_{1j}^b,y_{2j}^b,\dots ,y_{s2j}^b\right)}^{’}$. The production possibility set was defined as $\mathrm{P}=\left\{\left(x,y^g,y^b\right)\right|x\ge X\lambda ,y^g\le Y^g\lambda ,y^b\ge Y^b\lambda ,\lambda \ge 0\}$. The specific model is expressed as follows:

$$\begin{gathered} \min \rho =\frac{1-\frac{1}{m}{{\displaystyle \sum }}_{i=1}^m\frac{s_i^{-}}{x_{i0}}}{1+\frac{1}{s_1+s_2}({{\displaystyle \sum }}_{r=1}^{s_1}\frac{s_r^g}{y_{r0}^g}+{{\displaystyle \sum }}_{r=1}^{s_2}{s}_r^b/y_{r0}^b)} \\ s.t.x_0=X\lambda +s^{-} \\ {y}_{r0}^g=Y^g\lambda -s^g \\ {y}_{r0}^b=Y^b\lambda -s^b \\ \lambda \ge 0,s^{-}\ge 0,s^g\ge 0,s^b\ge 0 \end{gathered}$$

(4)

where $s^{-}\in R^m$, $s^b\in R^{s_2},$ and $s^g\in R^{s_1}$ are all slacks. $0<\rho \le 1$, when $\rho =1$ means the DMU is on the production frontier, which is completely efficient.

4.2.3. Application of the SBM Model

The SBM model has been widely used since it was put forward. We used “slack-based measure” as the keyword to retrieve more than 4000 literature sources in the JSTOR database, which were mainly divided into theoretical model innovation and application literature.

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The development of the SBM model:

Tone [88] proposed the super-efficiency SBM model in 2002. The advantage of this model lies in the combination of the super-efficiency DEA model and the SBM model. When multiple DMUs are effective, the relative efficiency of each DMU can be further distinguished. For the measurement of relative efficiency of multi-input and multi-output DMU, Tone [89] constructed a network-SBM model to measure departmental efficiency and the overall efficiency of decision-making units in 2009.

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The application of the SBM model:

Choi et al. [90] used a SBM model to analyze the reduction potential, efficiency, and abatement cost of CO₂ emissions in China, and the results showed that China’s per capita CO₂ emissions reduction can reach 56.1 million tons, and the national per capita CO₂ emissions reduction can reach 16.83 million tons. The CO₂ emissions efficiency varies from 0.146 to 1, with an average of 0.645. Camioto et al. [91] analyzed total-factor energy efficiency in the BRICS group and the G7 group by using a SBM model. Yang and Li [79] put undesirable output on the output side of the SBM model rather than on the input side and then analyzed the total-factor efficiency of water resource and TFEE in China. Gao et al. [92] extended the input-output model to the field of environmental economics and used the SBM model to estimate the carbon emissions efficiency in China, measuring the embodied carbon emissions efficiency and direct carbon emissions efficiency, respectively.

4.3. Comparison of Radial and Non-Radial Models

As mentioned above, the traditional DEA model is a radial model. However, in real life, not all inputs and outputs change in the same proportion. One shortcoming of the traditional radial DEA model is that it does not take the input/output slack into account when measuring the efficiency of the DMU. In many cases, we find the existence of non-radial slack; that is to say, in addition to radial proportional improvement, there may also be non-radial slack improvement in the decision making units.

In Section 4.1 and Section 4.2, we have clearly uncovered the advantages of the SBM model. Figure 8 shows the difference of slack between the radial model and the non-radial model. In Figure 8a, point A is an inefficient decision-making unit that can be adjusted to the frontier by reducing the radial AA’. However, it is evident that point A’ still does not achieve optimal efficiency. A′ can continue to reduce the input to point B while holding the amount of output constant; that is, BA′ is slack. However, in Figure 8b, SBM is a non-radial and non-angular model, and point A can move directly to B and achieve the optimal efficiency level. This can avoid the influence and error caused by radial and angle differences and improve the reliability and accuracy of efficiency measurement.

Compared with the traditional DEA model, the SBM model is more discriminative in estimating efficiency and can tap the potential to the maximum extent. Therefore, the SBM model is the most suitable model for estimating total-factor energy efficiency and total-factor carbon emissions efficiency.

5. Input-Output Variables for Efficiency Measurement

The seemingly straightforward measurements have their underlying theoretical and empirical assumptions [93]. Efficiency can be defined and measured in various forms, and the results obtained can affect socio-economic development in different ways, such as placing an unfair burden of carbon reduction on consumers and isolating humans from the natural world. In efficiency measurement, social scientists call for more observation of complex social contexts and practices and the dissecting of efficiency from conceptual foundations, practical applications, and sociology [20,94,95]. This is mainly reflected in the selection of input and output variables of efficiency measurement. Existing papers primarily focus on the methodological development of the DEA method but pay little attention to whether the selected variables can properly reflect the process under study to the greatest extent. In this case, it is necessary to improve the existing input-output variable to provide reference for future research on TFEE and TFCE. It is vital that the accurate selection of variables is related to the measurement results of the model.

5.1. Production Possibility Set

According to production economics [96,97], DMU must ensure the technological feasibility in the production process of transforming inputs into outputs. The state of technology determines and restricts the possibility of inputs to produce outputs. The most general way to express this constraint is to think of the DMU as having a production possibility set, $Y\subset R^n$, where each vector $y=\left(y_1,\dots ,y_n\right)\in Y$ is a technologically feasible production plan, observing the convention of $y_j<0$ if resource j is consumed as input, and $y_j>0$ if it is produced as an output. In this way, the set Y can fully describe the technological possibilities facing the DMU.

In DEA, production technology set Y is set to describe a multi-input and multi-output production technology [98]. Assuming that x and q denote a non-negative $N\times 1$ input vector and a non-negative $M\times 1$ output vector, respectively, the set S is then defined as:

$$S=\left\{\left(x,q\right):x can produce q\right\}$$

Consisting of all input-output vectors $\left(x,q\right)$, the set S can be equivalently defined using the output set $P\left(x\right)$ of all output vectors q that can be produced by the input vector x or the input set $L\left(q\right)$ of all input vectors x that can produce a given output vector q [99]. In addition, $P\left(x\right)$ is also the basis for describing the production possibility curve of two-dimensional output vectors and is sometimes referred to as the production possibility set related to various input vectors x.

5.2. Input-Output Variables Selection

As an improvement of single-factor energy efficiency, total-factor energy efficiency overcomes the one-sidedness of single-factor energy efficiency to a certain extent and takes into account the coordination and substitution of factors. However, in the existing literature, the selected input-output variables are repeated or omitted, and this does not conform to the theory of production economics and the actual production situation.

In the current research on TFEE, the most commonly selected variables are capital, labor, and energy as input variables and value added as output variable [50,53,100]. Some scholars also chose the value of output as an output variable [101]. On this basis, some scholars consider the impact of the environment and add undesirable outputs into the output variable [102]. Some scholars have also considered other intermediate inputs [84]. However, these studies have a certain degree of repetition or omission in the selection of variables.

The input-output variables of total-factor energy efficiency (TFEE) are also applicable to the measurement of total-factor carbon emissions efficiency (TFCE). Pang (2015) chose employment, capital stock, and energy use as inputs; GDP as desirable output; and CO₂ emissions as undesirable output, analyzing the impact of clean energy utilization on TFEE and TFCE, respectively. Based on the measurement of TFEE, Li and Lin (2017) also added CO₂ emissions as an undesirable output and measured the TFCE.

Based on a comprehensive review of previous studies, this paper indicates an improved measurement method for total-factor efficiency and proposes a more accurate and complete measurement input-output variables of energy efficiency, which provides a scientific method for measuring energy efficiency and carbon emissions efficiency. In order to ensure the accuracy and reliability of results for energy efficiency measurement, it is necessary to select the most appropriate input-output variables. Input-output variables must be consistent with and reflect the production process as close to reality as possible.

5.2.1. Output Variables

(1) Desirable output: Most of the existing studies use GDP (value added) as the desirable output, but this does not include the value of energy inputs and ignores the importance of intermediate inputs. The value of output as the desirable output is proposed based on System of National Accounts and is consistent with economic theory.

The System of National Accounts was developed by the United Nations. After adjusting many times, more than 100 countries and regions now use the system for accounting. Based on this, it holds that material products created and labor activities providing services are value-creating activities. Compared with GDP, we believe that the value of output, which measures total production activities, is the most appropriate proxy variable for desirable output.

In addition, some scholars also use the value of output as the desirable output to replace GDP in their studies but do not take other intermediate materials and services into consideration, which is also inaccurate.

(2) Undesirable output: In order to consider the impact of environmental pollution, it is proposed that the variable of greenhouse gas emissions should be used as undesirable output considering the environmental impact.

5.2.2. Input Variables

It is proposed that capital, labor, energy consumption, and other intermediate materials and services except energy should be selected as input variables.

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Capital stock: First, the capital stock in the base year is obtained by dividing the investment of fixed assets by the depreciation rate. Then, according to Goldsmith (1951), the perpetual inventory method is selected to measure the subsequent capital stock.

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Labor: The numbers employed in the production.

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Energy consumption: Energy consumption refers to the final energy consumption.

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Other intermediate materials and services except energy, which is equal to intermediate consumption minus energy value.

5.3. Curse of Dimensionality

Although DEA is widely used in measuring efficiency, it is sensitive to the input-output variables system. The more input and output variables selected, the greater the chance that some inefficient DMUs will be classified as the efficient [103] and the less discerning the DEA analysis is [104]. This state is referred to as the “curse of dimensionality” [105,106]. In order to ensure the discriminatory power between efficient and inefficient DMUs, the existing literature indicates some empirical rules about the number of DMUs and the number of input and output variables as listed in

Table 1. Empirical rules of the minimum number of DMUs.

Author	Minimum Number of DMUs
Golany and Roll [107]; Homburg [108]	$n\ge 2\left(m+s\right)$
Nunamaker [109]; Banker et al. [81]; Friedman and Sinuany-Stern [110]; Bowlin [111]; Raab and Lichty [112]	$n\ge 3\left(m+s\right)$
Boussofiane et al. [113]	$n\ge m\times s$
Dyson et al. [114]	$n\ge 2\left(m\times s\right)$
Cooper et al. [115]	$n\ge \max \left\{m\times s; 3\left(m+s\right)\right\}$
Kohl and Brunner [116]	$n\ge 20+\frac{m+s-1}{2}\left(-10+10\left(m+s-3\right)\right)$

, where n is the number of DMUs, m is the number of input variables, and s is the number of output variables.

All of these rules of thumb require a relatively large number of DMUs, which are difficult to meet for many studies. To overcome this drawback and ensure the discriminatory power between efficient and inefficient DMUs, some scholars choose to expand production technology by incorporating weight restrictions in multiplier DEA models [117,118]. Others choose to collapse input or output variables into single scores by using an entropy weight method [119,120] or a pure DEA model [121,122] and then measuring the efficiency score to rank DMUs. In practical applications, appropriate methods should be adopted to measure and analyze the efficiency based on the characteristics of the research object. At present, there is no unified conclusion on the relationship between the number of variables and the number of DMUs, so scholars can choose appropriate input-output variables according to their own research needs. In general, $n\ge 3\left(m+s\right)$ is commonly used.

6. Conclusions

The International Energy Agency (IEA, 2013) believes that energy efficiency should be taken as the first fuel rather than a hidden fuel and regards it as “a key tool for boosting economic and social development” (IEA, 2014). Sociologists often criticize the current measures of energy efficiency as too narrowly focused. Most of the existing studies have focused on measuring efficiency of energy and carbon emissions efficiency from the perspective of different methods but ignored the correct choice of input-output variables. Furthermore, the total-factor energy efficiency and total-factor carbon emissions efficiency have not been clearly distinguished in the existing literature.

This paper presents a comprehensive review of total-factor energy efficiency and total-factor carbon emissions efficiency from the perspective of production economics and attempts to distinguish them in terms of their definitions, measurement methods, models, and input-output variables, respectively, in order to provide reference for future studies on sustainable development and carbon emissions reduction. This has been achieved by a quantitative and visual review of academic progress in the field by employing CiteSpace software as a tool for a systematic analysis of a large number of publications from 2006 to 2021. The main conclusions are as follows:

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From the perspective of research subjects, China and the United States have studied total-factor energy efficiency and total-factor carbon emissions efficiency the most. However, compared with research at the country and industry level, there is still scarcity of literature on enterprise energy efficiency. Furthermore, we propose that the most meaningful research should consider economy, energy, and ecology at the same time.

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From the perspective of definitions, compared with the single-factor index of binding targets for national and international organizations, the definitions of total-factor energy efficiency and total-factor carbon emissions efficiency are more consistent with the essence of economics.

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From the perspective of the model, compared with the traditional radial DEA model, the non-radial Slacks-Based Measure model has the advantage of being more differentiated. This can minimize the input, increase the output, and extract the maximum potential, and therefore, it is more suitable as a standard model to measure efficiency. Moreover, the Slacks-Based Measure model with undesired outputs can deal with the undesired outputs, such as carbon dioxide, more simply and with higher efficiency.

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From the perspective of the input-output variables, the problem with most existing literature is that the selection of input-output variables is either repetitive or represents a flawed omission, which does not conform to the theory of production economics or the actual production situation. This paper proposes to select accurate and complete input-output variables for energy efficiency measurement, such as the value of output as the desired output variable and the impact of other intermediate materials and services. The new set of input-output variables is identified as the most suitable one, but it is also consistent with the production process.

Some research gaps in the existing literature have been identified by this comprehensive review. Firstly, existing studies on sustainable development and carbon emissions reduction are only analyzed from the single perspective of energy efficiency or carbon emissions efficiency, without systematic and comprehensive consideration of their joint contribution to carbon emissions reduction. The relationship and role of energy efficiency and carbon emissions efficiency is not clearly defined. In the literature analyzing carbon emissions reduction and sustainable development, the role of improving energy efficiency is often considered, but the most direct method of carbon emissions reduction is ignored. Secondly, the total-factor energy efficiency and total-factor carbon emissions efficiency are mostly studied in developed countries (e.g., the United States, the European Union) and some developing countries with relatively fast-growing economies (e.g., China), while research from other countries is relatively scarce. Carbon emissions reduction and sustainable development are common tasks for all countries in the world, so it is necessary to study energy consumption and carbon emissions for all countries. In addition, there is a lack of research on efficiency at enterprise level. Future research can be refined to enterprise level to obtain more accurate performance of energy consumption and carbon emissions reduction. Finally, value of output is proposed as the desirable output in this paper, and other intermediate inputs and services except energy are included in the input variables. It is worth noting that double counting is incorrect only when it is used for allocation (such as measuring GDP). When it is used for production measurement (such as measuring value of output), double counting is no longer a problem.

This paper has systematically reviewed total-factor energy efficiency and total-factor carbon emissions efficiency from the perspective of production economics, which provides useful and inspiring reference for further research on carbon emissions reduction and sustainable development. Future research should fully consider social variables, such as social systems and industrial structures, and better reflect the complexity of true efficiency by taking account of the interactions between social, economic, and environmental measures.