Dynamic energy efficiency of slack‐based measure in high‐income economies

Mankind is constantly pursuing economic growth and social development, and while these bring convenience to people, they involve large amounts of greenhouse gas emissions under the heavy use of energy, which produces quite serious air pollution, which not only affects people's health but also leads to ecological environmental damage. Nowadays, the impact of the environment and the negative impacts of the world's economies while promoting economic growth are the most important issues for economies to achieve balanced development. This study uses the dynamic slack‐based data envelopment analysis (DEA) model to assess the environmental energy efficiency of high‐income economies (including China) and explore the negative impacts on the environment, to obtain a basis for energy‐saving emission reduction methods or configurations by using 48 high‐income economies (including China) from 2010 to 2014. The interperiod (carryover) variable is gross domestic product (GDP). The empirical results show that economies with high energy efficiency have a large consumption of energy and are unable to reduce carbon dioxide emissions. We also find that there exists a difference in the results depending on whether GDP is considered as the carryover variable. European economies are more efficient in energy consumption than others, and Asian economies are the most inefficient. In order to pursue GDP growth, economies need to consider reducing their energy consumption and CO2 emissions and improve their energy usage efficiency, further implementing sustainability.

efficiency are generally classified as parametric and nonparametric. 9 Parametric methods such as stochastic frontier analysis (SFA) estimate a production (cost) function, and deviations in the function form affect the results of such models. In contrast, it is not necessary to estimate the production (cost) function when using nonparametric methods, such as data envelopment analysis (DEA). Relative to SFA, DEA is more popular for investigating energy efficiency because it can easily consider undesirable outputs 10,11 .
The DEA analytical framework uses mathematical programming models to compute the distance between each decisionmaking unit (DMU) and the best practice frontier constructed by the DMUs, based upon which the DMU efficiency scores can be calculated. With its methodological advancements, DEA has also received increasing attention in energy and environmental studies, [12][13][14][15][16][17][18] in which energy efficiency measurement has been identified as an important application. 17 Earlier studies dealing with this topic include Boyd and Pang 19 and Ramanathan. 20 Later, Hu and Wang 21 developed a total-factor energy efficiency index and applied it to evaluate China's regional energy efficiency. Wei et al 22 conducted an empirical analysis of energy efficiency in the iron and steel sector in China by using the Malmquist index. Honma and Hu 23 employed the total-factor energy efficiency model to evaluate regional energy efficiency in Japan. Zhang et al 24 used DEA window analysis to investigate dynamic trends in the total-factor energy efficiency of a sample of developing economies. In addition to evaluating regional energy efficiency, Mukherjee 25,26 employed DEA to assess manufacturing energy use efficiency in the United States and India.
A common feature of the above studies is that they evaluated energy efficiency ignoring undesirable outputs. However, fossil fuel-based energy use will inevitably produce undesirable outputs, such as CO 2 emissions. 27 Zhou and Ang 28 first developed several DEA models for evaluating energy efficiency based on environmental DEA technologies by incorporating undesirable outputs. Since then, more and more studies have focused on analyzing energy efficiency with both desirable and undesirable outputs. The more attention in energy efficiency is about the relationship in energy usage, economic growth, and environmental pollution, such as Wang and Feng. 29 Some scholars, such as Sueyoshi and Goto, 4 Dogan and Tugcu, 30 and Asayesh and Raad, 31 use region or industries to evaluate energy efficiency. For instance, Mandal 32 applied DEA to estimate the energy efficiency of the Indian cement industry, and indicated that biased energy efficiency scores resulted from neglecting undesirable outputs. Shi et al. 6 extended the DEA model by treating undesirable outputs as inputs to evaluate industrial energy efficiency in China. Yeh et al 33 compared the totalfactor energy efficiency between China and Taiwan by using DEA with undesirable outputs based on data translation. Some studies regard the CO 2 , SO 2 , etc. as the undesirable output to estimate its impact on energy efficiency. 5,[34][35][36] Also, some studies apply the directional distance function introduced by Chung et al 37 to explore the impact of undesirable output on energy efficiency, such as Färe et al, 18 Aparicio et al. 38 However, as dynamic DEA model introduced by Färe and Grosskopf, 39 more studies have been widely applied and extended this dynamic DEA model in many fields. [40][41][42][43][44][45][46] Sengupta 47 proposed the dynamic DEA model to analyze risk and output fluctuation on the dynamic production frontier by using the adjustment cost method, including the shadow value of quasi-fixed input and its optimal path in the analysis of linear programming. Färe et al 18 introduced several kinds of intertemporal variables into realistic multi-output production processes across periods. Dynamic DEA has been widely applied and extended in many fields. [40][41][42][43][44][45][46] Tone and Tsutsui 48 applied the slack-based model to measure overall efficiency and period efficiency considering carryovers as the link between two periods. Jafarian-Moghaddam and Ghoseiri 49 developed a fuzzy dynamic multi-objective DEA model to assess the performance of railways. Soleimanidamaneh 50 provided a new technique to gain an algorithm with computational advantages, using dynamic DEA models to estimate returns to scale. Sueyoshi and Sekitani 42 dealt with dynamic DEA to assess the environmental performance of US coal-fired power plants during [1995][1996][1997][1998][1999][2000][2001][2002][2003][2004][2005][2006][2007] and found that it is necessary for the United States to extend the scope of the Clean Air Act (CAA) to control the amount of CO 2 emissions.
Although there are many papers and a lot of research focusing on energy and environmental efficiencies such as CO 2 emissions and hazardous wastes, 16,17,[51][52][53][54][55] one limitation of these papers and research is that they evaluated undesirable output (eg, CO 2 ) efficiency within cross-sectional data and not over time, or only consider the dynamic efficiency analysis and ignore the slacks. Furthermore, according to International Energy Agency, 56 energy demand is increasingly going to be exploited due to the growth in population and income. It is expected that in 2035, the world's population will be 8.7 billion, GDP will increase more than 100%, and CO 2 emissions produced by energy will increase to 34.8 billion tons. 57 Ravallion et al 58 indicate that CO 2 emissions are affected by income distribution, because high-income economies tend to consume more fossil fuels than low-income economies. Therefore, in the case of the distribution of income in different economies, CO 2 emissions derived from energy consumption are different. Furthermore, based on Table 1, from 2010 to 2014, the CO 2 emissions share of highincome economies (including China) was 82.75%, 82.12%, 81.64%, 81.39%, and 80.94% (according to standard income levels from the World Bank in 2016; USD 12 235 per person per year). The reason we include China is that it will be very important in world economic development in the future. Therefore, it is a significant and important issue to investigate energy consumption in high-income economies.
Therefore, this study applies the dynamic slack-based measure DEA model by combining the SBM model and dynamic DEA model to explore energy efficiency of the research object by year and overall energy efficiency by using high-income economies (including China) from 2010 to 2014, and compares the results between static SBM and DSBM models. Moreover, this research follows Seiford and Zhu 59 to promote CO2 as an undesirable output and considers real GDP as a variable of carryover in the dynamic models 48 to give some suggestions for energy policies. The method in the literature for processing efficiency generates overestimated scores when the dynamic essence is ignored. This shows a dynamic analysis whenever data are available. The data can be decomposed into efficiency variance elements, and the variance can be solved. We find that there exists a difference in the results depending on whether GDP is considered as the carryover variable. European economies are more efficient in energy consumption than others, and Asian economies are the most inefficient.
The following study, organized in Section 2, shows the dynamic DEA model. Section 3 presents the results and a discussion of the results. Section 4 offers conclusions and policy implications.

| METHODS
DEA is a means of measuring the relative efficiency of a set of decision-making units (DMUs) that apply multiple inputs to produce multiple outputs for a given time period. There are some methods for measuring efficiency deviations over time, for example, window analysis and the Malmquist index. 60 Window analysis was proposed by Klopp, 61 while the Malmquist index was established by Färe et al 62 Even if these models can take into account the time change effect, they usually ignore carryover activities between 2 consecutive terms and only focus on separate time periods independently, training local optimization in a single period. In the real business world, long-term planning and investment is a subject of great distress for business growth. The dynamic DEA model proposed by Färe and Grosskopf 39 is the first innovative contribution for such a purpose. They introduced the dynamic aspects of production into the conventional DEA model when multiple outputs are involved. They formulated several intertemporal models, which became the origin for many later studies on dynamic DEA. Later, Chen, 63 Nemoto and Goto, 64  The causing model is nonoriented and it can process inputs and outputs individually. This means that the model is proper for nonuniformly distributed inputs and outputs and different weights can be assigned to the inputs and outputs depending on their degree of position. Tone and Tsutsui divided carryovers into 4 types to analyze the foundation of dynamic DEA models: (a) desirable (good), (b) undesirable (bad), (c) discretionary (free), and (d) nondiscretionary (fixed).The DEA model variables can be divided into 3 categories: input, output, and nonoriented. SBM can be used to identify the optimum solution.
Additionally, according to the characteristics of carryovers, we classify real GDP into desirable carryovers resembling profit carried forward to the next term. The main idea of this study is that governments in all economies must consider reducing CO 2 emissions in the process of economic growth. Therefore, GDP performance in this period will have an influence on the efficiency of the next period. We used dynamic SBM models, which can evaluate the overall efficiency of DMUs for whole terms as well as term efficiencies. For these cases, a single period optimization model does not fit for performance evaluation. To cope with a long-term point of view, the dynamic DEA model incorporates carryover activities into the model and supports measurement of period-specific efficiency based on long-term optimization during the whole period. The calculations of the system and period efficiencies under dynamic conditions are demonstrated. One important finding is that the method for calculating system efficiency in the literature produces overestimated scores when the dynamic nature is discounted. This makes it necessary to conduct a dynamic analysis whenever data are available. 66 Hence, another main consideration for the study is to choose a dynamic model for calculation. This study utilized the model established based on the expectations of Tone and Tsutsui, 48 which included T periods and n DMUs, each of which has different inputs, outputs, and carryovers in period t and period t links to the next period, t + 1. The particulars of the model in Figure 1 are as follows: Following is the nonoriented model: (1) Equation (2) shows the connection equation of t and t + 1: Here is the most efficient solution:

| RESULTS
The research sample covers 48 high-income economies during 2010-2014 according to United Nations (UN) data, World Bank data, the BP Statistical Review of World Energy, the Taiwan Bureau of Energy and Ministry of Economic Affairs, and the Taiwan Directorate-General of Budget, Accounting, and Statistics, Executive Yuan, in 2017 (Table 2). We used three input variables, fossil fuel energy consumption, real capital stock, and labor force; one output variable, CO 2 emission; and one carryover variable, real GDP. Table 3 presents descriptive statistics of the input, output, and carryover variables' data results as follows:

Research Results
The overall average efficiency value generated by the static SBM was 0.7241 (Table 4), and the differences among the continents indicate that the European high-income economies have the best energy efficiency, with an average value of 0.7759, the Asian economies have worse energy efficiency, model. Let n DMUs (j = 1 …, n) over T terms (t = 1, …, T). There are m inputs (i = 1, k, m) of the DMUs. F: nondiscretionary (fixed) inputs (i = 1, k, p). S: output (i = 1, k, s). P: nondiscretionary (fixed) outputs (i = 1, k, r). z: link (carryover) has good, bad, free, and fixed categories. w: weight with an average value of 0.6336, the American economies' average efficiency value was 0.7199, and the Oceania economies' value was 0.7139. The overall efficiency value estimated by DSBM is 0.6869 (Table 5), and when the economies are divided by four continents, the average efficiency values of Oceania, Asia, Europe, and America are 0.8106, 0.6007, 0.7191, and 0.7308, respectively. Therefore, the overall efficiency of the evaluation results under the dynamic SBM model is lower than that of the static SBM model, showing that the empirical results of the dynamic model will not produce overall efficiency overestimation, which also conforms to the view of objectivity of Tone and Tsutsui 48 's model evaluation.
From 2010 to 2014, the economies that were efficient (efficiency value of 1) as estimated by the nonorientation DSBM model under the assumption of variable returns to scale (VRS) were China, United Kingdom, Italy, Luxembourg, Malta, the Netherlands, Norway, Uruguay, the United States, and Switzerland. The empirical results of taking GDP as a crossover variable show that the efficiency values of the Netherlands and Italy are different due to the consideration of the carryover variable. Similarly, Trinidad and Tobago, South Korea, the Czech Republic, and Hungary performed worse than the others under the DSBM model. They also show differences in the efficiency value when considering the carryover variable. Therefore, the dynamic measurement of efficiency value is more accurate on the whole.

Income Economies (Including China)
This study employs the dynamic nonorientation SBM model of VRS to analyze various input and output adjustment ranges of high-income economies (including China) from 2010 to 2014 through target values that are regarded as the reference for economies to achieve energy and environmental efficiency corresponding to their benchmark in the frontier. The results are sorted as follows.

| Annual adjustment of energy consumption input of high-income economies (Including China) from 2010 to 2014
As shown in Table 6, from 2010 to 2014, the inputs in energy consumption of high-income economies (including China) should have been reduced by 20.68% on average, and these economies (Switzerland, Italy, United Kingdom, Hong Kong, Japan, Luxembourg, Malta, the Netherlands, Norway, Uruguay, the United States, and China) had no energy

Means by Continents
The United States

| Annual adjustment in inputs of capital storage in high-income economies (Including China) from 2010 to 2014
As shown in

Means by Continents
The United States

Means by Continents
The

| Annual adjustment of labor input in high-income economies (Including China) from 2010 to 2014
As shown in Table 8

| Adjustment range of carbon dioxide output from 2010 to 2014
As shown in Table 9

| Adjustment range and improvement of crossover variable GDP
Besides the original characteristics of the data envelopment analysis method, the dynamic DEA method adds the concept of linking to the period of existence (carryover) in order to fully analyze the impact of cross-year time variables when evaluating overall efficiency. It also provides the adjustment rate during existence, which can be used as a reference for each DMU to achieve relative efficiency. The period of existence selected by this research is GDP, which is the economic basis for economies to be connected to the next year across the period. The original value of GDP, the target value of efficiency boundary, and the adjustment ratio of the carryover period of high-income economies (including China) from 2010 to 2014 are summarized as follows.
The overall average GDP of the 48 high-income economies (including China) from 2010 to 2014 was 73.62, among which the adjustment ranges in 2011 and 2012 were larger than average. Among the economies requiring adjustment, Trinidad and Tobago reached 422.92% in 2014, and France reached 0.01% in 2012. Among them, 25 economies needed to adjust their GDP in 2013, and the average adjusted value was 72.48%.
A total of 10 economies (Switzerland, Germany, United Kingdom, Italy, Luxembourg, the Netherlands, Norway, Uruguay, the United States, and China) did not need to adjust their GDP from 2010 to 2014. The remaining economies would have had to increase their GDP to achieve efficiency improvement, as shown in Table 10.

| Slack variable analysis in highincome economies (Including China) divided by four continents
As shown in Table 11, the reduction in energy consumption in Asia is the largest at −35.53%, while that in Europe is the smallest at −11.97%. The reduction of capital stock in Asia is the largest at −40.06%, while that in Oceania is the smallest at −11.01%. The largest reduction in labor force is −20.67% in Asia, with no adjustment required in the Americas. In terms of CO 2 emission reduction, the maximum is 0.91% in Asia and the minimum is 0.10% in Europe. GDP is forecast to increase by 105.02% in the Americas (an average of 416.32% in Trinidad and Tobago) and by 5.73% in Oceania.

IMPLICATIONS
This study used mainly DSBM and applied DEA-Solver Professional 9.0 software to investigate the energy and environment efficiency of high-income economies (including China) from 2010 to 2014. As sustainable economic development is increasing the demand for energy, in order to promote sustainable development on the Earth, governments will need to use social and environmental resources more effectively to achieve the maximum output with the least input. Based on the comprehensive analysis of the DEA model, the empirical results can be summarized as follows: (a) 12 economies (the United States, China, Uruguay, Luxembourg, Switzerland, Norway, Malta, Kuwait, Iceland, Ireland, Swiss, and United Kingdom) have an efficiency value of 1 in the past five years, implying their energy is used most efficiently in these high-income economies. (b) The results from DSBM show that 10 economies (the United States, China, Uruguay, Norway, Switzerland, United Kingdom, Italy, Luxembourg, Malta, and the Netherlands) achieved an energy efficiency value of 1 in 2010-2014, which means these economies efficiently use their resources. The average efficiency between 2010 and 2014 has an increasing trend. (c) It is found that the European average efficiency value is best, the Asian average efficiency value is the worst, the American average efficiency value is 0.7199, and the Oceania average efficiency value is 0.7139. 4) We also find 9 economies (Switzerland, Luxembourg, Malta, the Netherlands, Norway, Uruguay, Swiss, the United States, and China) do not need to adjust their inputs including energy consumption, capital, and labors. And 8 economies (Switzerland, Italy, Luxembourg, the Netherlands, Norway, Uruguay, the United States, and China) perform most efficient in using the energy consumption, capital, and labors to produce the GDP and undesirable output CO 2 .
Therefore, first, it can be conducted that China, the United Kingdom, Italy, Luxembourg, Malta, the Netherlands, Norway, Uruguay, the United States, and Switzerland, all of which have efficient energy use, can continue to control emission reduction in accordance with the original energy use planning policy. Second, there are 10 economies (the United Arab Emirates, Canada, the Czech Republic, Israel, South Korea, Kuwait, Poland, Qatar, Saudi Arabia, and Taiwan) should decrease waste in their resources, and also need to strengthen the management of carbon dioxide emission reduction and management policies; formulate various industrial initiatives such as carbon dioxide emissions cap regulations; levy pollution taxes on the lock industry, metal manufacturing, mining, and other chemical industries; widen promotion of energy saving to households to encourage investing in energy-saving equipment; emphasize energy efficiency to reduce emissions; and generally reduce sources of pollution emission to effectively implement policies to reduce carbon dioxide pollution. Third, 14 high-income economies (the United Arab Emirates, Bahrain, Canada, Estonia, Finland, South Korea, Kuwait, New Zealand, Oman, Qatar, Saudi Arabia, Sweden, Trinidad and Tobago, and Taiwan) should strengthen efforts to reduce energy consumption, and encourage development of the energy industry to improve energy efficiency, then lock in an effective energy development mode and promote continuing to reduce energy consumption and improve the ratio of energy use, to achieve the purpose of reducing energy consumption, while reducing carbon dioxide emissions at the same time.