Energy Intensity, Energy Efﬁciency and Economic Growth among OECD Nations from 2000 to 2019

: This study examines the energy intensity (EI), energy efﬁciency (EE), and economic growth, measured by the type of returns to scale (RTS), of 37 nations in the Organization for Economic Co-operation and Development (OECD) from 2000 to 2019. We apply a non-parametric approach to estimate the three measures from their consumption of four primary energy sources, such as coal, gas, oil, and zero emission (e.g., renewable and nuclear power) as inputs and gross domestic product (GDP) as an output. In this study, we have the two types of efﬁciency measures over time: window-based and cross-sectional-based measures. Three ﬁndings are identiﬁed from our empirical study. First, the operationally efﬁcient group, including France, Iceland, Japan, Switzerland, UK, and USA, presented a stable status of full efﬁciency in the window-based efﬁciency measure. Iceland and Switzerland were also in the higher efﬁciency group based on the cross-sectional measure. Their efﬁciencies were high and stable over the observed periods. Second, zero-carbon-emission (e.g., renewable and nuclear) energies outperformed other energy sources (coal, gas, and oil) in terms of a potentiality of EI/EE improvement. In other words, OECD nations can improve on their EI/EE measures by reducing fuel consumption of coal, gas, and oil while maintaining their high GDP levels. Finally, four industrial nations (France, Japan, UK, and USA) had a status of unity in their EI/EE measures for zero-carbon-emission energies with decreasing RTS. These nations would increase zero-carbon emission for energy consumption to increase GDP while keeping optimal EI/EE because such changes in consumption would not largely affect EI/EE due to their constant RTS status. Iceland showed increasing RTS. The nation may improve the EI level by increasing zero-carbon-emission energy consumption and economic size. The four nations can increase zero-emission energy consumption to achieve further economic growth without observing a large deterioration of EI/EE because it is very close to constant RTS. The examination of RTS provides policy directions for the improvement of EI and EE. Switzerland showed decreasing RTS and may deteriorate the EI/EE by increasing energy consumption and the size of each economy. The remaining countries, whose degree of EI/EE measures was less than unity, showed increasing or decreasing RTS. The examination of RTS provides important implications for energy policy to enhance the degree of EI/ EE.


Introduction
Reducing the energy intensity (EI) of the economies is vital to achieving a carbonneutral society with net-zero emissions (NZE) of carbon dioxide (CO 2 ).A recent International Energy Agency report [1], titled "Net zero by 2050: A roadmap for the global energy sector", details the global pathway to net-zero emission (NZE) by 2050 and requires all governments to significantly strengthen and successfully implement their energy and climate policies.The report says, "minimizing energy demand growth through improvements in energy efficiency makes a critical contribution in the NZE, and indicate key global milestones for energy efficiency for the NZE".The report sets the annual EI improvements (megajoule (MJ) per USD GDP) at −4.2% over 2020-2030 and −2.7% over 2030-2050 to achieve NZE [1].
The importance of EI in combating global warming and climate change is evident from the fact that approximately 73% of global greenhouse gas (GHG) emissions originate from our energy use in the electricity, heat, and transportation sectors (Our World in Data: https://ourworldindata.org/emissions-by-sector). The modern society accepts that economic growth is closely linked to energy use, and for most advanced economies, a decoupling between energy use with GHG emissions and economic growth is increasingly important (International Energy Agency: https://www.iea.org/news/decouplingof-global-emissions-and-economic-growth-confirmed),not only through technological advancement to improve energy efficiency but also using policy instruments to promote energy saving.Furthermore, to attain the immediate goal of CO 2 emission reduction, it is critical to shift the energy sources from fossil fuel to zero-carbon-emission ones, such as renewable and nuclear power.
To partly respond to the global challenge, this study first considers that EI is part of energy efficiency (EE).Later, we prepare a figure, which visually describes the relationship between EI and EE.Then, we will analyze the relationship between energy use and economic enhancement in 37 OECD nations.In this research effort, more specifically, we first measure the degree of EI by the quantity of energy consumption and the amount of output from economic activities, that is, the gross domestic product (GDP).As a measure of energy use, EI is calculated from the amount of energy consumption required to produce a unit of GDP.A high EI level implies high energy consumption to produce a unit of GDP, thereby denoting an energy-inefficient economy.Such measure is also influenced by a nation's industrial structure, economic size, and industrial development.This fact implies that we need to evaluate energy efficiency by comparing similar peer economies.Furthermore, it is important to investigate the EI value from a perspective of fuel mix and a ratio of actual and optimal values of EI.This aspect leads to the necessary investigation of EE measurement between EI and GDP.
It is important to mention that EI is a special case of EE, whereby both measures use GDP as a single output.The EI compares the magnitude of multiple inputs (i.e., energy consumption), given the same amount of GDP.Meanwhile, the amount of GDP is not fixed in the EE measurement.Thus, the EI measure is a special case of the EE measure.
In addition to measuring the degree of EI and EE, this research examines the level of returns to scale (RTS) as a scale measure to understand the relationship among EI, EE, and the size change of GDP in each country.The RTS measurement provides us with information on "how the unit change in EI influences the size of GDP".Therefore, this study also needs to examine the relationship between the size of GDP and the degree of EI and EE, thus expanding the research horizon, in contrast to previous studies on EI.
Here, note that a general implication of GDP is discussed by GDP = consumption (C) + government expenditures (G) + investments (I) + exports-imports.The main contributor to GDP is consumption (C), which is usually more than half of the countries.Some countries have a larger contribution from exports than consumption, e.g., The Netherlands, Slovakia, Slovenia, Lithuania.
To attain the research purpose, this study methodologically proposes a new type of non-parametric measure, or data envelopment analysis (DEA), to assess the holistic relationship among EI, EE, and GDP.In this study, we use a data set on energy consumption and GDP related to the countries of OECD.Specifically, the GDP in this DEA study serves as an output measure, while the inputs include the level of consumption of coal, gas, oil, and zero-carbon-emission energy sources, which comprise renewable and nuclear power.
Energies 2023, 16,1927 3 of 29 A unique feature of the proposed method (i.e., DEA) is that it is a non-parametric approach, which can avoid the specification of a function form (e.g., linear regression) among energy consumption and GDP.We conduct the holistic analysis in a time horizon to examine the type and quantity of EI reduction and economic enhancement by applying DEA-based EE.See Sueyoshi et al. [2], Glover and Sueyoshi [3], and Sueyoshi and Goto [4] for the methodological benefit and history of DEA.The proposed approach is necessary for this study because no previous research work has discussed the analytical linkage between EI and GDP in DEA-based EE and RTS.
The remainder of this paper is organized as follows: Section 2 reviews previous contributions on EI.Section 3 describes our methodological concepts of efficiency improvement, which originate from a reduction in energy consumption and economic enhancement.Section 4 discusses how DEA handles the proposed assessment in a time horizon.Section 5 describes RTS measures from the EI perspective.Section 6 summarizes and concludes the research while providing future extensions.

EI and EE in OECD Nations
Among the previous research efforts in OECD countries, we select the following six studies on EI and EE.(a) First, Azhgaliyeva, Liu, and Liddle [5] studied the determinants of EI using empirical methods and cross-country data from 44 countries over the period 1990-2016.They found that both GDP per capita and economy-wide energy prices were negatively associated with EI.They provided evidence that several policy instruments were effective in reducing EI.(b) Second, Chakraborty and Mazzanti [6] attempted to determine a long-term relationship between green energy innovation and EI by considering 21 OECD countries over 1975-2014 and using various estimators to address key econometric issues.They found long-and short-term relationships between EI and green energy innovation.(c) Third, Filippini and Hunt [7] tried to isolate core energy efficiency for a panel of 29 OECD countries by combining the approaches used in energy demand modeling and frontier analysis.Their specifications on energy demand controlled for income, price, climate, country-specific effects, area, industrial structure, and an underlying energy demand trend in order to obtain a measure of efficiency.(d) Fourth, Voigt, De Cian, Schymura, and Verdolini [8] analyzed EI trends and drivers in 40 major economies using a logarithmic mean by applying the Divisia index to a unique data set of input-output table time series, accompanied by environmental satellite data.They decomposed efficiency changes to either changes in technology or changes in the structure of the economy, studied trends in global EI between 1995 and 2007, and highlighted sectoral and regional differences.(e) Fifth, Wurloda and Noailly [9] analyzed the impact of green innovation on EI in a set of 14 industrial sectors in 17 OECD countries over the 1975-2005 period.They created a stock of green patents for each industrial sector and estimated a translog cost function to measure the impact of green innovation on EI.They found that green innovation has contributed to the decline in EI in many sectors.(f) Sixth, Fidanoski, Simeonovski, and Cvetkoska [10] used DEA and calculated the efficiency scores on minimizing energy use and losses as well as environmental emissions for a sample of 30 OECD member states during the period from 2001 to 2018.They showed that taking care of the environment does not affect efficiency in general, while the reliance on energy produced from renewable sources does slightly reduce it.Table 1 summarizes recent studies on the EI and economic growth.Pan, Chen, Ying, Zhang [15] Analyzed the concept of environmental Kuznets curve, and analyzed the relationship between energy efficiency and economic development.Showed that labor has a significant negative impact on energy efficiency and the increase in labor input will reduce energy efficiency.

Convergence of EI and EE
Three studies have focused on the convergence of EI.(a) First, Mulder and de Groot [25] evaluated EI developments across 18 OECD countries and 50 sectors over the period 1970-2005.They found that across countries, EI levels tend to decrease mostly in the manufacturing sectors, while the service sector showed more diverse trends across sub-sectors.They also conducted convergence analysis and revealed that only after 1995, cross-country variation in aggregate EI levels clearly tended to decrease, driven by a strong and robust trend break in manufacturing and enhanced convergence in services sectors.(b) Second, Liddle [26] examined world convergence of EI using two large data sets: a 111-country sample spanning 1971-2006 and a 134-country sample spanning 1990-2006.The results from both data sets revealed continued convergence.Their investigation of geographical differences revealed that the OECD and Eurasian countries have shown considerable, continued convergence, while the Sub-Saharan African countries have converged among themselves but at a slower rate than the OECD and Eurasian countries.(c) Finally, Meng, Payne, and Lee [27] examined the convergence of per capita energy use among 25 OECD countries over 1960-2010, using unit root tests in which two endogenously determined structural breaks were employed.The results indicated significant support for per capita energy use convergence among OECD countries.

DEA Applied Energy
Several studies applied a network DEA and a dynamic DEA to analyze OECD countries' energy efficiency.These DEA models assumed a network structure or a dynamic resource allocation system in production processes.These included works by Ouyang and Yang [28] and Guo, Lu, Lee, and Chiu [29] investigating OECD countries.In addition, Keskin [30] used a slack-based network DEA approach to measure the social prosperity efficiency of OPEC member countries.Moreover, there are studies that have analyzed OECD countries from the perspectives of EI and EE and sustainable economic and financial development (Ziolo, Jednak, Savić, and Kragulj [31]) and environmental sustainability of road transport (Mo and Wang [32]).Table 2 summarizes recent research efforts, which apply DEA for energy studies.

Position of This Study
The above literature review reveals that previous studies have not explored the relationship between EE (including EI) and GDP from a perspective of relative performance measurement based on DEA efficiency and fuel mix issues.We know that several research efforts have examined EI, EE, and the economy/economic growth in relation to health care, finance, renewable energy, and innovation.Other research works have conducted research on EI and temporal convergence among countries/regions.DEA is often applied to EI and EE studies.There are some other related research topics, e.g., EE measurement and green finance and innovation.They usually utilize econometric models and simple index measures, paying attention to input-specific EI improvement, based on actual values of data.
In contrast, this study is interested in conducting a holistic assessment of how energy is consumed to produce higher GDP by applying DEA and considering an efficient fuel mix from the perspective of EI.That is, we measure EI and the economies of scale from the perspective of efficient energy allocation.There are no previous studies that have explored the research interest.We compute the linkage for 37 industrial countries belonging to the OECD from 2002 to 2019.DEA is a holistic measure of efficiency, which uses multiple components without assuming a functional form between inputs and outputs [33].
This study has two unique features to be noted.One of the two features is that we measure a potential improvement in EI, EE, and RTS through comparison between actual and optimal values (on efficiency) by fully utilizing DEA computational capability.The proposed DEA approach considers a possibility that all nations may become either efficient or inefficient.The method measures the level of inefficiency and then identifies an efficiency frontier.Based upon the frontier, we examine the level of EI and EE and the type of RTS on an efficiency frontier on which all countries exist without inefficiency.The other feature is that we compute the three measures without assuming that all countries are efficient.The efficiency assumption is widely found in economics on which conventional productivity measures in economics have depended.In contrast, this study does not make such an assumption, so that we need to measure the level of efficiency/inefficiency status.

Underlying Concepts
To increase readability, this section starts by specifying the nomenclatures used in Sections 3 and 4, which are summarized as follows: d: slack variables of production factors; G: a column vector of outputs (g); i: a subscript related to the type of inputs (x); j: a subscript of the j th DMU (i.e., country); k: a subscript indicating a specific DMU; m: the number of inputs; n: the number of DMUs; r: a subscript related to the type of outputs; R: data range adjustments on G and X; s: the number of output types; t: a subscript indicating a specific period; T: the number of periods; U: a column vector of dual variables related to outputs; V: a column vector of dual variables related to inputs; X: a column vector of inputs; λ: an unknown column vector of intensity (or structural) variables; ε s : a very small prescribed number, which is set at 0.0001 in this study; σ: an intercept of the supporting hyperplane; and ξ: an inefficiency score.

Energy Consumption Reduction and Economic Efficiency Enhancement
Figure 1 visually describes the relationship between EI reduction and GDP enhancement.In this study, energy is separated into four types of consumption: coal, gas, oil, and zero-emission (renewable and nuclear) power.Usually, a decrease in energy consumption is expected to lead to a decrease in GDP if there is no change in technology development and fuel mix.Meanwhile, we expect that this energy consumption may lead to GDP enhancement through technology development and energy mix strategy.We need to reduce the amount of energy consumption based on coal, gas, and oil to reduce CO2 emission.Renewable and nuclear power generations can reduce the amount of CO2 emission.However, they suffer from another type of difficulty.For example, renewable energy has high setup and maintenance costs and low energy density.It is also influenced by weather conditions, so its intermittency avoids it to become a base load.Nuclear energy suffers from the problem of "nuclear waste" after its generation.Hence, in this study, we believe that a reduction in energy consumption is better.Of course, we understand the importance of such renewable energy sources in reducing the emission of greenhouse gas.(c) "Electricity" is a secondary energy, so that we do not include it as energy consumption.
The figure considers that renewable and nuclear power generations have important roles in achieving a carbon-neutral society because both of them are zero-emission energy sources.Therefore, they are different from the others.Moreover, there is a noticeable movement to re-evaluate nuclear power generation.For example, the French President Emmanuel Macron announced in November 2021 that nuclear power plant construction would resume.Another example is the European Commission proposal in February 2022 for a taxonomy for climate change mitigation and climate change adaptation.Both have included the utilization of nuclear power generation as a sustainable investment.Such a change in policy direction is evident; however, in reality, it is not easy to reduce total energy consumption because human and economic activities employ energy uses.It is  We need to reduce the amount of energy consumption based on coal, gas, and oil to reduce CO 2 emission.Renewable and nuclear power generations can reduce the amount of CO 2 emission.However, they suffer from another type of difficulty.For example, renewable energy has high setup and maintenance costs and low energy density.It is also influenced by weather conditions, so its intermittency avoids it to become a base load.Nuclear energy suffers from the problem of "nuclear waste" after the generation.Hence, in this study, we believe that a reduction in the energy consumption is better.Of course, we understand the importance of such renewable energy sources in reducing the emission of greenhouse gas.(c) "Electricity" is a secondary energy, so that this study does not include it as energy consumption.

Size of Economy
The figure considers that renewable and nuclear power generations have important roles in achieving a carbon-neutral society because both of them are zero-emission energy sources.Therefore, they are different from the others.Moreover, there is a noticeable movement to re-evaluate nuclear power generation.For example, the French President Emmanuel Macron announced in November 2021 that nuclear power plant construction would resume.Another example is the European Commission proposal in February 2022 for a taxonomy for climate change mitigation and climate change adaptation.Both have included the utilization of nuclear power generation as a sustainable investment.Such a change in policy direction is evident; however, in reality, it is not easy to reduce total energy consumption because human and economic activities employ energy uses.It is also not easy to replace fossil fuel with zero-emission energy because of higher cost, intermittency, and technological and social risk.Indeed, there is no perfect way to reduce EI and increase GDP simultaneously.Therefore, considering such feature of the zero-emission energy source (renewable and nuclear power), this study incorporates it as an input for efficiency measurement, as depicted in Figure 1.
Methodological importance.Figure 1 presents a methodological development, where we need to discuss the linkage between energy consumption reduction and GDP enhancement in a framework of DEA operational efficiency, or energy efficiency, in this context.It has long been considered that DEA can provide such a holistic measure.However, note that the straightforward use of standard DEA does not have such methodological capability.For example, it can measure the level of efficiency (θ * ) of the specific k th DMU.The superscript symbol (*) denotes the optimal solution of the variable obtained from the DEA model.The level of efficiency is referred to as "OE (θ * : operational efficiency)", along with its related slacks (d x * i and d g * r ) in this study.Conventionally, it has been conventionally referred to as "technical efficiency".See Sueyoshi and Goto [4] for a historical rationale on the use of OE.Later, OE becomes EE in this study.The DEA community uses OE, but it indicates EE in energy studies.Since EE is the case of a single output in the OE measurement, we fully utilize DEA to measure the level of OE.
The level of the input-based OE (θ * ) is measured for optimality in the following manner [4]: This type of efficiency measure originates from reducing the level of energy consumption, such as oil and gas, which serve as inputs in this study.The reduction is an input-based measure to improve the level of OE measures.Meanwhile, the output-oriented OE is measured by an inefficiency measure (τ * ) as follows [4]: This type of efficiency measure originates from economic (GDP) enhancement.The GDP serves as a single output in this study.The increase leads to OE enhancement.Conventional difficulty.Both Equations ( 1) and ( 2) are derived from the standard DEA models.One difficulty of the two standard models is that "they have different efficiency measures [i.e., OE(input) * = OE(output) * ]".As a result, we encounter a difficulty in evaluating the performance of OECD nations.Different input/output models produce different OE measures.Therefore, we wonder what is our final answer on OE?In contrast to the difficulty of previous DEA measures, this study unifies the input-oriented (energy consumption) and output-oriented (GDP) measures to identify a single measure as a new alternative to the standard OE measures.This is the first methodological contribution.
Visual Description.Figure 2 consists of an input (x: energy consumption) on a horizontal axis and an output (g: GDP) on a vertical axis.The figure depicts the input-oriented projection, implying energy consumption reduction, and the output-oriented projection, implying GDP enhancement, both of which increase the DEA-based OE measures, or energy efficiency.In the figure, a frontier consists of efficient decision-making units, or DMUs {A-B-C-D-F}, while {F} is "inefficient" because the DMU, an OECD nation in this study, is located within the efficiency frontier.Here, let us explain the three possible projections from inefficiency to efficiency.First, the input-oriented model uses a projection, which shifts {F} to H on the frontier for efficiency enhancement.This type of projection is measured in Equation (1) for "energy consumption reduction", projection {F} to H.The projection implies that given a fixed amount of GDP, it indicates an amount of energy consumption reduction to attain the status of OE.Second, the output-oriented model projects from {F} to G, which focuses on "economic (GDP) enhancement".This projection implies that given an amount of energy consumption, it enhances the amount of GDP to attain the status of full OE.This type of projection is measured by Equation (2).In the third case (in the proposed approach), if we can project {F} to {C}, then the projection attains both energy consumption reduction and GDP enhancement.As depicted in Figure 2, all the projections may change the status of inefficiency to efficiency on the frontier.The approach proposed in this study uses the third type of projection, different from Equation (1) (energy consumption reduction) and Equation (2) (economic (GDP) enhancement).

A Scale Effect on Energy Intensity and Energy Efficiency
Figure 3 visually describes a rationale on why the RTS measurement is necessary to understand both EI and EE (e.g., OE in this study).They are clearly influenced by the size of each nation.For example, the two measures of the USA are structurally different from a small OECD nation.The GDP (as a scale of the economy) of the USA is the largest in the world.It is easily imagined that there is correlation between EI and EE, even if both are different measures.The EI is measured by a single energy source consumption divided by GDP, while the EE is measured by GDP divided by all energy source consumptions.They are clearly influenced by the size of each nation and its energy and industrial structures.Therefore, as a scale measure, we need to measure the RTS of all OECD nations in terms of energy consumption.No previous study has documented the research concern.

A Scale Effect on Energy Intensity and Energy Efficiency
Figure 3 visually describes a rationale on why the RTS measurement is necessary to understand both EI and EE (e.g., OE in this study).They are clearly influenced by the size of each nation.For example, the two measures of the USA are structurally different from a small OECD nation.The GDP (as a scale of the economy) of the USA is the largest in the world.It is easily imagined that there is correlation between EI and EE, even if both are different measures.The EI is measured by a single energy source consumption divided by GDP, while the EE is measured by GDP divided by all energy source consumptions.They are clearly influenced by the size of each nation and its energy and industrial structures.Therefore, as a scale measure, we need to measure the RTS of all OECD nations in terms of energy consumption.No previous study has documented the research concern.
a small OECD nation.The GDP (as a scale of the economy) of the USA is the largest in the world.It is easily imagined that there is correlation between EI and EE, even if both are different measures.The EI is measured by a single energy source consumption divided by GDP, while the EE is measured by GDP divided by all energy source consumptions.They are clearly influenced by the size of each nation and its energy and industrial structures.Therefore, as a scale measure, we need to measure the RTS of all OECD nations in terms of energy consumption.No previous study has documented the research concern.

Strengths and Shortcomings of DEA
The strengths and shortcomings of DEA are summarized as follows.First, DEA is able to incorporate multiple inputs and multiple outputs, where we do not assume any functional form that specifies the relationship between them.It is not impossible but difficult for us to handle multiple outputs by using standard economics.Second, DEA can measure the degree of OE by a percentile expression (i.e., 0 is full inefficiency and 1 is full efficiency), so that we can assess the performance of public and private entities.A drawback of DEA is that we cannot use it for forecasting, as found in statistics and econometrics.Third, DEA estimates a production function by piece-wise linear approximation, which connects the observed data on an efficiency frontier (e.g., {A}-{B}-{C}-{D}-{E} depicted in Figure 2).Finally, DEA does not assume any error distribution (e.g., a normal distribution).It is true that such DEA capability is useful and important for performance assessment.However, DEA is not capable of any statistical inferences and tests.We must use DEA along with statistical analysis.This is another shortcoming of DEA.

Formulations for Measurement in a Time Horizon
All formulations discussed in this research are relatively new, particularly when applied to energy research.One exception is a study by Sueyoshi et al. [2].The study prepared by Sueyoshi and Goto [4] has documented almost all DEA research efforts applied to energy which were listed in the Science Citation Index (SCI) and the Social Science Citation Index (SSCI), from 1978 to 2017.See also Sueyoshi et al. [39] for a literature review on 693 research efforts on DEA and DEA-based environmental assessment.To the best of our knowledge, all formulations in this research are different from them, fitting the scope of this research purpose.Here, we note that the proposed approach has three methodological extensions.First, the previous assessment separates production factors into input-oriented and output-oriented measures.Both produce different estimates of efficiency in DEA measurement.The proposed approach unifies the two measures and produces a unique efficiency measure.Second, it incorporates a time horizon through which we can capture the changes in time among multiple annual periods.Finally, we measure the type of RTS to examine how the size of energy consumption and GDP influence the EE of each nation.

Cross-Sectional Operational Efficiency (CSOE)
To extend the proposed OE (= EE) measurement into a time horizon, we use two different measures.First, we pool all of their observations on DMUs in a cross-sectional structure.This type of OE measure is a straightforward extension, which includes a time horizon.Therefore, we refer to it as "CSOE".The measure is used as the basis for a relative comparison; for example, it investigates the statistical difference among different periods and groups by using a rank-sum test.This is the methodological benefit of CSOE.
To describe the CSOE measurement, let us consider each entity as a DMU to be evaluated.Regarding each DMU, the energy with m components (X: energy consumption) is transformed by the production technology into an economic performance measure with a single output (g: GDP).Any functional form is not assumed in this research to specify the relationship between X and g.DEA connects them by multipliers, or weights, among them.Thus, the approach estimates weights (so-called multipliers), not parameters; it is therefore "non-parametric" and avoids any specification error of a functional form.
Projection.This study utilizes a new type of DEA, which determines the level of OE of the specific k th DMU at the specific t th period (t = 1, . . ., T).An important feature of the proposed measurement is that it evaluates the performance of each DMU given observed X kt and g kt .The observation (X kt and g kt ) serves as the directional vector on which each DMU changes the status from inefficiency to efficiency, as shown in the projection from {F} to {C} in Figure 2. The standard DEA models did not have such directional vector, as discussed previously.
Before discussing the formulations used for this research, we note two concerns.One of the two concerns is that we use the subscript (j) to specify all DMUs (j = 1, . . ., n), while the subscript (k) indicates a specific DMU to be evaluated.The other concern is that this study also uses the subscript (z) to indicate all periods (z = 1, . . ., T), while the subscript (t) indicates a specific period to be evaluated.
As the first formulation, this study measures the degree of cross-sectional operational efficiency (CSOE v kt ) of the specific k th DMU at the specific t th period using the following formulation, where the superscript (v) indicates variable RTS.See Sueyoshi et al. [2].
An efficient frontier of DMUs for all periods (z = 1, . . ., T) is formulated by the left-hand side of Model (3).A crossover may be contained in the efficiency frontier between or among multiple periods, and thereby, the frontier of this model consists of the best performers in all T periods.On the other hand, the right-hand side indicates the performance of the specific k th DMU in a specific period (t) to be examined.The observation (x ikt and g kt ) serves as a directional vector in which Model (3) maximizes the level of inefficiency for calculation.As conventional DEA models do not have the directional vector, their efficiency measures are input-oriented or output-oriented, as specified in Equations ( 1) and (2).Meanwhile, Model (3) incorporates the directional vector to produce an efficiency measure, which combines both Equations ( 1) and (2).
The distance measures (+d x it + ξ kt x ikt and −d g t − ξ kt g kt ) indicate differences between the observed performance of each DMU and the efficiency frontier in multiple (m + 1) dimensional factors (e.g., inputs and outputs).The unified inefficiency (ξ kt ) indicates the degree of a directional vector, which should be arranged toward the efficiency frontier.The slacks (d x it and d g t ) express the remaining parts of the difference, which we cannot specify in measuring the level of inefficiency.The small number (ε s ), expressing the relationship Energies 2023, 16, 1927 14 of 29 between inefficiency and total slacks, indicates a number that we need to prescribe for the operation of Model (3).
Strengths.Model (3) has an important feature, in that an efficiency score does not depend upon the energy (input) or economic (output) orientation, as found in Models ( 1) and ( 2).This feature is different from standard ratio models, which depend upon the orientation.As discussed previously, input-oriented efficiency is different from outputoriented efficiency.Model (3) does not hold such difficulty because all factors (inputs and outputs) are unified in a single inefficiency measure (ξ kt ), and there is no difference between them.Note that the standard models originated from the ratio model; therefore, they suffered from the methodological difficulty.
The degree of CSOE v kt of the k th DMU in the t th period is measured by Here, superscript (v) implies "variable" RTS in Model (3).The optimality (*) of Model ( 3) determines the inefficiency measure (ξ) and all slack variables (d x it and d g t ), and the degree is obtained by subtracting the level from unity, as specified in Equation ( 4).
Strength.The important feature of Model ( 3) is that it puts an inefficiency measure (ξ) before inputs and outputs.The optimization attempts to minimize the level of inefficiency.Such two features cannot be found in the previous DEA studies, except in Ref [2].As a result, the projection is uniquely determined.Such are the strengths of Ref [3].
Drawbacks.Both Equations ( 3) and ( 4) incorporate an assumption that DMUs at the t th period (past) can access the future technology in the z periods.Such assumptions are a methodological shortcoming.The other difficulty of the CSOE v kt , measured by Model (3), is that all DMUs in the future are assumed to access an efficiency frontier (therefore, technology) in the past.An opposite case is also true.This is problematic.For example, the structure of Model (3) incorporates an assumption that DMUs in 2020 may utilize technology in the past, e.g., 2000, for their operations.Such assumption is often not practical to implement in most industries because they must consider business competition in the global market.This indicates a drawback of Model (3).In reality, DMUs mostly utilize technology that is as recent as possible to meet the global competition.
At the end of the CSOE v kt description, it is necessary for us to specify the following data ranges on X and g used in Model (3).Here, R x i is a data range on the i th input (m components), and R g is a data range on the output (a single component: GDP).
The data ranges are applied to all DMUs (j = 1, . . ., n) in all periods (t = 1, . . ., T) in the proposed DEA models, so that the DEA results are able to avoid a possible occurrence of zero in dual variables (i.e., multipliers).Such a possible occurrence implies that corresponding production factors (X and g) are not fully utilized in DEA assessment, which is problematic in DEA applications.To avoid the difficulty, this study incorporates the data ranges ( 5) and (6) to fully utilize the available information on X and g.

Window-Based Operational Efficiency (WOE)
To overcome this drawback of Model (3), this study utilizes the concept of "window analysis", which combines recent periods into a time horizon to be examined.We use the following window-based approach to measure the level of window-based operational efficiency (WOE v kt ) on the k th DMU at the specific t th period. Maximize Here, W t (the t th window) indicates a group of recent periods (e.g., {t − 2, t − 1, t}∈ W t ) before the specific t th period.The left side term (i.e., ∑ of Model ( 7) indicates an efficiency frontier, which consists of DMUs in W t , comprising variable RTS by incorporating ∑ z∈W t ∑ n j=1 λ jz = 1 as a side constraint.Note that Model ( 7) is a general form of Equation ( 3).If W t covers all periods, then Model (7) becomes Model (3).The difference between the two models is the periods to be covered.Model (7) covers partial periods, while Model (3) covers all period combinations.This type of application incorporated in Model ( 7) corresponds to the "moving average" in forecasting (statistics).
We measure the degree of WOE v kt of the k th DMU at the t th period by where the inefficiency measure (ξ) and all slack variables (d x it and d g t ) within the parenthesis are determined on the optimality (*) of Model (7).Equation ( 8) determines the degree of efficiency by subtracting the level from unity.A historical review can be found in Refs [3,4,39].
To extend Model (7) to the scale-related (i.e., RTS) measurement of the k th DMU, we need to consider the dual formulation of Model (7), which becomes The first group contains constraints related to W t , which includes t − 2, t − 1, and t th periods, in Model (9).
Using Model (9), we measure the degree of WOE v kt of the k th DMU at the t th period by Here, the optimality (*) of Model (9) determines the dual variables (v i and u).Both Equations ( 8) and (10) produce the same degree of efficiency on optimality.

DEA-Based EI
The concept of EI (as part of EE) is originally considered, as one input/GDP with both observed values.This research measures the two values based upon those in an efficient frontier.Such treatment eliminates the existence of inefficiency in practicality.
To measure EI, we first discuss how to measure the level of DEA-based optimal EI.This study uses Model (7) for our measurement.For each year, +d x * it + ξ * kt x ikt and −d g * t − ξ * kt g kt indicate the differences (inefficiency) from the observed performance (x ikt and g kt ) of the k th DMU at the t th period.On an efficiency frontier, the observations are adjusted by both of which are adjusted on the optimality of Model (7).We determine the EI of the optimal values of the i th input (energy consumption) of the k th DMU at the t th period by Meanwhile, the EI of observed data (EIO) is specified by We use the ratio between the two EI scores (REI) measured by Equations ( 11) and ( 12) as follows: The ratio indicates a potential to improve the status of EI, more generally EE.The degree, different from unity, indicates that a gap exists between optimal and actual measures on EI.The former (optical EI) is measured on an efficiency frontier, so that it outperforms the observed one.Thus, the degree of REI is less than or equal to unity.The status of unity indicates that the specific k th DMU is empirically the best performer in EI.Similarly, if it is less than unity, the DMU needs to improve the status of EI/EE.

Return to Scale (RTS)
There are two ways to enhance the level of EI/EE.Each nation has to decrease the amount of energy consumption (x) under the given level of GDP (g), or each nation has to increase the amount of GDP (g), given x.The previous studies in Section 2 mainly discussed the first case but did not directly address the second one.Moreover, the input vector has multiple energy consumptions in this study, while the previous studies only pay attention to a single energy consumption in the EI computation.While acknowledging the contributions of these studies, we describe a new research direction on EI, i.e., a change of x components for an increase in GDP (g) using the economic concept of RTS.
To begin with, we describe the concept of "scale elasticity".The degree of scale elasticity (e g ) is conceptually specified as e g = marginal product/average product = (dg/dx)/(g/x).
As specified in Equation ( 14), scale elasticity has the opposite structure of EI; that is, the degree of EI is measured by x/g, while the e g is measured by g/x and dg/dx in a simple case.
Using elasticity, we classify the concept of RTS of each DMU by the following rule: (a) e g > 1 ⇔ Increasing RTS, (b) e g = 1 ⇔ Constant RTS, and (c) e g < 1 ⇔ Decreasing RTS.
To discuss a relationship for RTS in a time horizon, we start from a simple case with time, where an input (x t ) is used to produce an output (g t ) in the t th period.We incorporate the subscript (t) to express each period and assume that a supporting hyperplane is expressed by v t x t − u t g t + σ t = 0 or g t = (v t /u t )x t + σ t /u t .A mathematical requirement to identify the supporting hyperplane is that u t should be positive in the sign.The variable (σ t ), indicating a constant term of the supporting hyperplane, is unrestricted (URS) in the sign.
Note that if u t is zero, it is difficult to determine the location of a supporting hyperplane because v t /u t and σ t /u t become "infinite".The ratio ( dg t /dx t = v t /u t ) indicates a "marginal rate of transformation".The variable (u t ), related to the t th output, characterizes a supporting hyperplane in a data domain of x t and g t .
Returning to Model (9), we discuss the type of RTS measured by Model ( 9) because it is a general form of Model (3).In the optimal solution V * kt , u * kt , σ * kt of Model ( 9), each V * kt is considered as a virtual input price vector of X kt .Similarly, u * kt is a virtual output price for g kt .Each component of these vectors indicates a slope of a supporting hyperplane(s) on a production possibility set.Both should be positive.
As the intercept of a supporting hyperplane(s) is directly related to RTS, we determine RTS by examining a sign of the dual variable (σ * kt ).The optimal solution of Model (9) identifies the following types of RTS on the k th DMU at the t th window: (17)

Data
The data set used in this study consists of an output and four inputs from 37 OECD countries during the annual periods over 2000-2019.The output data indicate real GDP in constant 2010 million US dollars.The data set was obtained from World Bank Open Data (https://data.worldbank.org/).All input data were obtained from Our World in Data, Energy section (https://ourworldindata.org/energy).Table 3 summarizes the descriptive statistics of the data.
Since this study is interested in the DEA-based EE, EI, and RTS of the OECD countries, the GDP here represents the total value added in a country as an output, and the four inputs are primary energy consumptions of coal, gas, oil, and zero-carbon (carbon-neutral) energy sources.All were measured in terawatt hours (TWh), as provided in the data source.The fourth input of energy consumption is derived from carbon-neutral energy sources from nuclear power and renewable energy, such as biofuel, solar power, wind power, hydro power, and other sources.

OE, EI, and RTS Measures
As an illustrative example, Table 4 summarizes the CSOE measures of the 37 OECD nations in 2000 and 2019 (both are illustrative examples) and the total average from 2000 to 2019.We computed the efficiency measures using Model (3), and the degree of these efficiencies was determined by Equation (4). Figure 4 visually describes the average of these efficiency measures of 37 nations from 2000 to 2019.Their efficiency (CSOE) scores, based upon a cross-sectional structure, were measured by Model (3), where we treated all data sets from 2000 to 2019 as a single data set.The left-hand side (under avg.) of Table 4, along with Figure 4, shows that Denmark, Iceland, Israel, Luxembourg, Sweden, and Switzerland have exhibited high (more than 0.9) annual average of CSOE.The result implies that they showed higher CSOE performance than other OECD nations.Next, changing the OE measurement from CSOE (cross-sectional) to WOE (window), Table 5 summarizes the energy efficiency measures from the window analysis of the 37 OECD nations from 2002 to 2019.This type of efficiency measure (WOE) assumes an annual shift, which is more clearly identified than CSOE. Figure 5 visually describes the average of annual efficiency changes of 37 nations from 2002 to 2019.The efficiency scores were measured by Model (7).Table 5 clearly indicates a group of nations whose average efficiencies are unity, comprising France, Iceland, Japan, Switzerland, the UK, and the USA.These countries present a stable status of full efficiency in moving across the window periods.Both Iceland and Switzerland are included in the higher efficiency score group in CSOE and are also classified as full efficiency countries in WOE, as their efficiencies are high and stable over the periods.Next, changing the OE measurement from CSOE (cross-sectional) to WOE (window), Table 5 summarizes the energy efficiency measures from the window analysis of the 37 OECD nations from 2002 to 2019.This type of efficiency measure (WOE) assumes an annual shift, which is more clearly identified than CSOE. Figure 5 visually describes the average of annual efficiency changes of 37 nations from 2002 to 2019.The efficiency scores were measured by Model (7).Table 5 clearly indicates a group of nations whose average efficiencies are unity, comprising France, Iceland, Japan, Switzerland, the UK, and the USA.These countries present a stable status of full efficiency in moving across the window periods.Both Iceland and Switzerland are included in the higher efficiency score group in CSOE and are also classified as full efficiency countries in WOE, as their efficiencies are high and stable over the periods.Table 6 summarizes the REI measures of each energy consumption source in 37 OECD nations in 2019, for example, because the year is the latest annual period.It is interesting to note that, as listed at the bottom of Table 6, zero-carbon energy has outperformed the other energy sources (coal, gas, and oil) in terms of REI in 2019.In other words, the observed EI of zero-carbon energy is close to the optimal values, which are determined Table 6 summarizes the REI measures of each energy consumption source in 37 OECD nations in 2019, for example, because the year is the latest annual period.It is interesting to note that, as listed at the bottom of Table 6, zero-carbon energy has outperformed the other energy sources (coal, gas, and oil) in terms of REI in 2019.In other words, the observed EI of zero-carbon energy is close to the optimal values, which are determined by the WOE models.Table 7 shows France, Iceland, Japan, Switzerland, the UK, and the USA attaining unity in the degree of REI for zero-carbon energy.That is, these nations have DEA-based optimal EI values.The ratio (REI) indicates a potential to improve the status of EI of each energy source.The degree, different from unity, indicates that a gap exists between the optimal and actual measures of EI.The former is measured on an efficiency frontier, so that it outperforms the observed one.The degree of REI is less than or equal to unity.The status of unity indicates that the specific k th DMU is the best performer on EI.On the other hand, if it is less than unity, the DMU needs to improve the status of EI.The REI in Table 6 implies that the zero-emission energy (0.65) has more influence than the other three energy sources in terms of their EI measures.See the bottom and right-hand side of Table 6.

Conclusions and Future Extensions
This study discussed how to measure the DEA-based EE, EI, and RTS of 37 OECD nations during 2000-2019.We applied a new type of DEA approach with cross-sectional and window analyses to a data set consisting of four energy consumption sources (coal, gas, oil, and zero-carbon-emission energies) to produce the GDP.The new DEA models incorporated a directional vector to identify the degree of EE of these industrial nations.After examining their efficiency scores, we discussed the measurement of EI and REI of the OECD nations.Finally, we re-examined their EE and EI statuses based on their RTS statuses.The combined use of REI and RTS provides the policy direction on future energy consumption for GDP enhancement.
After completing the methodology development, we obtained the following three implications from the application to the 37 OECD nations.First, the efficient group of six countries-France, Iceland, Japan, Switzerland, the UK, and the USA-presented a stable status of full efficiency in WOE.Iceland and Switzerland were also in the higher efficiency group based on CSOE.Their operational efficiencies were high and stable over the observed periods.Second, zero-carbon-emission (renewable and nuclear) energies outperformed the other energy sources (coal, gas, and oil) in terms of REI, a potentiality of EI improvement.In other words, OECD nations can improve further on total EI by reducing fuel consumption of coal, gas, and oil while maintaining GDP levels.Finally, four industrial nations-France, Japan, the UK, and the USA-had a status of unity in their REI measures for zero-carbon-emission energy with decreasing RTS.These nations would increase zero-carbon-emission energy consumption to increase the GDP while keeping optimal EI because such changes in consumption would not largely affect the REI due to their constant RTS.Iceland showed increasing RTS and may improve its EI level by increasing zero-carbon-emission energy consumption and economic size.Switzerland showed decreasing RTS and may deteriorate EI by increasing energy consumption and the size of the economy.The remaining countries, whose REI measures were less than unity, showed increasing or decreasing RTS.The examination of RTS type provides interesting implications for potential improvement of EI and energy policy direction to enhance the degree of REI and economic growth.
This study proposed a holistic approach for measuring EI, EE, and RTS by applying DEA from the perspective of efficient energy allocation or an efficient fuel mix.Further, we demonstrated that the proposed approach is practical for assessing EI measures considering RTS and economic growth.No previous studies explored the research issue in the context of RTS.Among the previous studies, Refs [10,13,24,30,31,[34][35][36][37][38] in Tables 1 and 2 employed DEA to investigate EI or energy consumption issues, as in this study, but they did not incorporate RTS measures in the examinations.Therefore, quantitatively comparing the results of EI obtained in this study with those of the previous studies may be limited in providing policy implications because of the utilization of different methods and data.On the other hand, this study indicated from the above three implications that economic growth and efficient EI with optimal fuel mix for sustainability can be simultaneously attained by OECD nations.For example, Refs [5,8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24]31,34,35,38] have handled the research issue of EI and the economy/economic growth, whereas 19,22,31,35] directly/indirectly have showed consistency between EI improvement and economic growth, as we suggest in this study.Interestingly, although Refs [18,34] have revealed a negative relationship between EI and economic growth at present, the former recommends revised policy for EI.The economic growth may be used as a remedial measure for environmental recovery by enhancing investment in the renewable energy sector, energy efficiency, and structural transformation of the industrial sector, while the latter finds three important ways for economic growth pressure to restrain the improvement of energy efficiency, which provide further policy implications on how to improve energy efficiency and promote high-quality economic development under the economic growth target management.These findings inspire us with regard to future research direction in examining energy consumption and economic growth.

27 Figure 1 .
Figure 1.Research structure of energy intensity and GDP.Note: (a) The authors prepared this figure for this research.(b)We need to reduce the amount of energy consumption based on coal, gas, and oil to reduce CO2 emission.Renewable and nuclear power generations can reduce the amount of CO2 emission.However, they suffer from another type of difficulty.For example, renewable energy has high setup and maintenance costs and low energy density.It is also influenced by weather conditions, so its intermittency avoids it to become a base load.Nuclear energy suffers from the problem of "nuclear waste" after its generation.Hence, in this study, we believe that a reduction in energy consumption is better.Of course, we understand the importance of such renewable energy sources in reducing the emission of greenhouse gas.(c) "Electricity" is a secondary energy, so that we do not include it as energy consumption.

Figure 1 .
Figure 1.Research structure of energy intensity and GDP.Note: (a) The authors have prepared the figure for this research.(b)We need to reduce the amount of energy consumption based on coal, gas, and oil to reduce CO 2 emission.Renewable and nuclear power generations can reduce the amount of CO 2 emission.However, they suffer from another type of difficulty.For example, renewable energy has high setup and maintenance costs and low energy density.It is also influenced by weather conditions, so its intermittency avoids it to become a base load.Nuclear energy suffers from the problem of "nuclear waste" after the generation.Hence, in this study, we believe that a reduction in the energy consumption is better.Of course, we understand the importance of such renewable energy sources in reducing the emission of greenhouse gas.(c) "Electricity" is a secondary energy, so that this study does not include it as energy consumption.

Energies 2023 , 27 Figure 2 .
Figure 2. Energy consumption reduction and economic (GDP) enhancement.Note: (a) The authors prepared this figure for this research.(b)Observed DMUs are listed with { }, while projected observations are not listed with { }.For example, {A} is an observed DMU, and H is a projected performance of {F}.(c) The degree of EI = energy consumption/GDP.The output-oriented DEA projects {F} to G. The input-oriented DEA projects {F} to H.The EI reduction needs both the reduction in energy consumption and GDP enhancement.Therefore, we look for the projection from {F} to {C}.In this case, the degree of EI becomes H′-{F} divided by G′-{F}.(d) The figure indicates the case of a single input and a single output.Meanwhile, we use multiple energy sources as inputs and GDP as an output in this study.In a case containing multiple inputs and outputs, this research attempts to explore the issue by fully utilizing the analytical capability of DEA.

Figure 2 .
Figure 2. Energy consumption reduction and economic (GDP) enhancement.Note: (a) The authors have prepared the figure for this research.(b) Observed DMUs are listed with { }, while projected observations are not listed without { }.For example, {A} is an observed DMU, and H is a projected performance of {F}.(c) The degree of EI = energy consumption divided by GDP.The output-oriented DEA projects {F} to G. The input-oriented DEA projects {F} to H.The EI reduction needs both the reduction in energy consumption and GDP enhancement.Therefore, we look for the projection from {F} to {C}.In this case, the degree of EI becomes H -{F} divided by G -{F}.(d) The figure indicates the case of a single input and a single output.Meanwhile, we use multiple energy sources as inputs and GDP as an output in this study.In a case containing multiple inputs and outputs, this research attempts to explore the issue by fully utilizing the analytical capability of DEA.

Figure 3 .
Figure 3. Relationship among EI, EE, and RTS.Note: (a) The authors have prepared the figure for this research.The "energy intensity (EI)" is measured as a single energy consumption divided by GDP.Conversely, the "energy efficiency (EE)", corresponding to operational efficiency (OE) in DEA, is measured by GDP divided by the consumption of all energy sources.The "returns of scale (RTS)" are measured by a change of GDP divided by a unit change of all energy source consumptions.(b) GDP is an output in this study.The type of RTS is measured for the single output (GDP).The DEA/RTS may handle a general case of EE.