DEA environmental assessment in a time horizon: Malmquist index on fuel mix, electricity and CO2 of industrial nations

Energy Economics 40 (2013) 370–382

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DEA environmental assessment in a time horizon: Malmquist index on fuel mix, electricity and CO2 of industrial nations Toshiyuki Sueyoshi a,⁎, Mika Goto b a b

New Mexico Institute of Mining & Technology, Department of Management, 801 Leroy Place, Socorro, NM 87801, USA Central Research Institute of Electric Power Industry, 1-6-1, Otemachi, Chiyoda-ku, Tokyo 100-8126, Japan

a r t i c l e

i n f o

Article history: Received 30 March 2013 Received in revised form 15 July 2013 Accepted 21 July 2013 Available online 3 August 2013 JEL classification: C60 C68 M52 Keywords: Environmental assessment DEA Time series Electricity

a b s t r a c t Climate change and global warming become a major policy issue in the world. Economic activities produce not only desirable outputs (e.g., electricity) but also undesirable outputs (e.g., CO2 emission). The important policy issue is how each nation can balance between economic development and environmental protection to attain a sustainable society. In attaining the sustainable society, environmental assessment is increasingly important because it can serve as an initial step toward the green growth of each nation. For the purpose, this study proposes a new use of DEA (Data Environment Analysis) for environmental assessment in a time horizon. The proposed use of DEA incorporates Malmquist index to examine the degree of a frontier shift among multiple periods. The frontier shift indicates a technology progress and/or managerial innovation during an observed period. The index is conceptually separated into six subcomponents, which are further divided into twelve different subcomponents (six subcomponents × two disposability concepts) under the natural and managerial disposability. In the index measurement, it is necessary for us to consider a frontier crossover among different periods because technology innovation usually has a time lag until it really appears. As an empirical application, this study utilizes the proposed approach to identify the relationship among fuel mix, electricity and CO2 of ten industrial nations. This study finds three important empirical findings. First, there is a time lag in technology innovation on electricity generation and CO2 emission reduction. Consequently, it is necessary to consider the existence of a frontier crossover in assessing the electric power industry. Second, nuclear generation, as found in France, as well as hydro and renewable energy, as found in Netherlands, are important for the development of a sustainable society although the former is associated with a very high level of risk and the latter has a limited generation capacity. Finally, the electric power industry has been making a corporate effort to reduce the amount of CO2 emission by utilizing nuclear and renewable energy. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Climate change and global warming become a major policy issue in the world. The climate change implies an increase in average global temperature regarding air, sea and land on the earth. Natural events and human activities, including all industrial and economic activities, contribute to an increase in average global temperature. The change is primarily caused by an increase in “greenhouse” gases such as carbon dioxide (CO2). The change may occur with the development of global economy in which economic activities in all nations closely link together. In combating the climate change and global warming, all nations need to consider both their economic growth and environmental protection so that they direct themselves toward a sustainable society. To establish the sustainable society with a green growth, this study proposes a new use of DEA (Data Envelopment Analysis) as a methodology ⁎ Corresponding author. E-mail addresses: [email protected] (T. Sueyoshi), [email protected] (M. Goto). 0140-9883/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.eneco.2013.07.013

for environmental assessment which evaluates operational and environmental performance of organizations in a time horizon. It is widely known that the DEA has already established its reputation as one of very popular performance evaluation methodologies in management science & operations research, economics, public policy and other social sciences (e.g., accounting and marketing). In addition to the conventional use of DEA1 for performance evaluation, many researchers applied it for environmental assessment. For example, Zhou et al. (2008) summarized more than 100 articles in environment and energy studies. A contribution of DEA environmental studies was that they found the importance of output separation into 1 The research of Sueyoshi and Sekitani (2009) has discussed strengths and drawbacks of various radial and non-radial DEA models. This study does not discuss the conventional use of DEA, rather focusing upon a description on DEA environmental assessment. The use of DEA is not limited on performance analysis. In the history of DEA, it had many different types of applications. For example, Sueyoshi and Goto (2011d, 2012k) applied DEA for rank analysis by combining it with DEA-DA (Discriminant Analysis). Sueyoshi and Goto (2013b) and Sueyoshi et al. (2009) discussed DEA financial assessment. Sueyoshi and Goto (2013c) provided pitfalls and remedies on the use of DEA.

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desirable and undesirable outputs because production activities usually produced not only desirable but also undesirable outputs. The output separation was a contribution, indeed. See Glover and Sueyoshi (2009) for the history of DEA, returning to the development of L1 regression developed in the 18th century. Both were closely linked together in their developments. See Ijiri and Sueyoshi (2010), as well. The previous DEA environmental assessment in the past decade included Dyckhoff and Allen (2001), Korhonen and Luptacik (2004), Kumar (2006), Liang et al. (2004), Oude Lansink and Bezlepkin (2003), Pasurka (2006), Picazo-Tadeo et al. (2005), Ramanathan (2002), Sueyoshi and Goto (2009a, 2009b, 2010a, 2010b, 2011a, 2011b, 2011c, 2012a, 2012b, 2012c, 2012d, 2012e, 2012f, 2012g, 2012h, 2012i, 2012j, 2013a, 2013d), Sueyoshi et al. (2013), Triantis and Otis (2004), Zaim (2004), Zhou et al. (2008, 2012) and many other articles. Acknowledging the importance of previous DEA contributions, however, this study needs to mention that they have not yet documented a methodology to assess the unified performance of organizations, which have produced both desirable and undesirable outputs, in a time horizon. Of course, this study knows that there were several exceptions in DEA environmental assessment. For example, Zhou et al. (2010) incorporated the Malmquist index in DEA environmental assessment in order to examine an occurrence of a frontier shift among multiple periods. The index was separated into four subcomponents along with the frontier shift. It is indeed true that they have made a contribution in DEA. To specify the position of this study in DEA literatures, we need to describe two differences between the study (Zhou et al., 2010) and this research. One of the two differences is that they have not made any linkage with disposability concepts, such as natural and managerial disposability, that describe corporate strategies for adapting a change in environmental regulation on undesirable outputs. In contrast, this study incorporates such two disposability concepts into the proposed approach. As a result, this study uses twelve subcomponents for Malmquist index, all of which are differently expressed under the natural and managerial disposability and then proposed to measure a frontier shift of efficiency during multiple periods. The approach of Zhou et al. (2010) has considered only four components for Malmquist index. Thus, there is a difference between the two studies in terms of the number of subcomponents. The other difference is that the Malmquist index measurement needs to consider a frontier crossover among different periods. The occurrence of the frontier crossover was not considered in their study. In contrast, this study incorporates the frontier crossover into the proposed index measurement. The purpose of this study is to combine between the Malmquist index measurement and the concept of natural and managerial disposability in order to measure twelve index subcomponents related to a shift of an efficiency frontier in a time horizon. This type of research has been never explored in the previous studies on DEA environmental assessment. The reminder of this study is organized as follows: Section 2 reviews strategic concepts related to disposability. This section extends the disposability concepts (Sueyoshi and Goto, 2012a) by incorporating a time horizon into them. Section 3 discusses how to unify desirable and undesirable outputs under natural disposability. Section 4 shifts our description to managerial disposability. Section 5 applies the proposed approach to investigate the relationship among fuel mix, electricity and CO2 in ten industrial nations. Section 6 concludes this study along with future research extensions. 2. Previous studies and strategic concepts 2.1. Previous studies on DEA in a time horizon Two groups of research were closely related to DEA in a time horizon. One of the two groups was due to an innovative work of Malmquist (1953). His study documented how to measure production indexes

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among different regions. Other production economists, including Bjurek (1996), Bjurek and Hjalmarsson (1995), Färe and Grosskoph (1992) and Grifell-Tatje and Lovel (1995), applied the Malmquist index measurement to DEA in a time horizon. The index measurement proposed in the previous studies had methodological strengths and drawbacks. A contribution of this group was that the index measurement was introduced and separated into four subcomponents. Consequently, it was possible for us to measure both a shift of efficiency frontiers and the influence of each subcomponent on the frontier shift. That was a contribution, indeed. In contrast, the index measurement had a drawback. That is, the previous measurement depended upon an assumption that production activity was characterized by a functional form(s). Furthermore, it was expected that production technology always shifted an efficiency frontier toward better performance in an observed period. However, many applications did not satisfy such underlying assumptions. Thus, it is necessary for us to consider the index measurement in which a frontier shift may not occur because technology development has a time lag until its influence really appears. In the worst case, an efficiency frontier often retreats itself because of inappropriate management or a lack of investment on technology innovation. In addition to the economic approach, management scientists (e.g., Bowlin, 1987) were interested in the development of “DEA window analysis”. The methodology examined how much an efficiency score was changed by shifting a combination of adjacent periods, often referred to as “window”. As an extension of the window analysis, Thore et al. (1994) and Goto and Tsutsui (1998) combined between the window analysis and the Malmquist index measurement. The combined measure was referred to as “Malmquist productivity index”. An important feature of the window analysis was that it could avoid the assumption that an efficiency frontier did not retreat. Even if a frontier crossover occurred between different periods, the window analysis pooled observations for a few consecutive periods into a window in which we can identify a new efficiency frontier. Consequently, a group of efficiency scores was smoothed over time and these efficiency scores were determined by comparing their performance with a newly established efficiency frontier within the window. That was a contribution. A drawback of window analysis was that it lacked an analytical scheme to decompose its subcomponents, as found in the Malmquist index measurement. The conventional combination between the index measurement and the window analysis could be found in Sueyoshi and Goto (2001) and Sueyoshi and Aoki (2001). Position of this study: All the previous studies on DEA in a time horizon considered only desirable outputs and inputs, so not paying attention to an existence of undesirable outputs. Since DEA environmental assessment needs to unify both desirable and undesirable outputs, this study attempts to develop a new analytical framework that combines the Malmquist index measurement and the concept of natural and managerial disposability. Then, the combination is incorporated into a time horizon. That is a new research effort for DEA environmental assessment. 2.2. Strategic concepts: Natural and managerial disposability Before discussing DEA environmental assessment in a time horizon, it is necessary for this study to mention two strategic concepts on desirable and undesirable outputs from environmental protection (Sueyoshi and Goto, 2012a). The first strategic concept, referred to as “natural disposability”, indicates that an organization decreases a directional vector of inputs to decrease a directional vector of undesirable outputs. Given the reduced vector of inputs, the organization increases a directional vector of desirable outputs as much as possible. For example, let us consider a coal-fired power plant where CO2 emission is produced by coal combustion. The coal is used as an input for the operation of a coal-fired power plant. Consequently, if the coal-fired power plant reduces the amount of coal combustion, the reduction immediately

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decreases the amount of CO2 emission at the level that it can satisfy the amount requested by governmental regulation. Given the amount of coal combustion, the coal-fired power plant maximizes the amount of electricity generation. That is a “natural disposability”. In this case, the coal-fired power plant may attain the reduction of CO2 emission with a limited corporate effort. This study considers the natural disposability as negative adaptation to a change on environmental regulation. A cost concern associated with the natural disposability is an operational cost. It is easily expected that a total operation cost decreases but an average operation cost increases by the natural disposability because of less generation. This type of strategy has been long supported by many economists (e.g., Palmer et al., 1995). The second strategic concept, referred to as “managerial disposability”, indicates an opposite case of the natural disposability. In the disposability, a firm increases a directional vector of inputs to decrease a directional vector of undesirable outputs by utilizing technology innovation on undesirable outputs. Given the increased input vector, the firm increases a directional vector of desirable outputs as much as possible. For example, a coal-fired power plant increases the amount of coal combustion so that it can increase the amount of electricity generation. Here, even if the power plant increases the amount of coal combustion, the increase can reduce the amount of CO2 emission by a managerial effort such as using high quality coal with less CO2 emission and/or an engineering effort to utilize new generation technology (e.g., clean coal technology). Management of the power plant considers a change on environmental regulation as a business opportunity. The challenge of the power plant is a corporate effort for environmental protection, so implying “managerial disposability”. This study considers the managerial disposability as positive adaptation to the change on environmental regulation. A cost concern associated with the managerial disposability is an opportunity cost, implying that consumers do not purchase electricity from a dirty-imaged old power plant. Rather, they prefer purchasing electricity from a clean-imaged power plant even if the electricity price is more expensive than that of the dirty-imaged power plant. Thus, the opportunity cost really poses a risk to the management of power plants. It is easily imagined that a total operation cost may increase because of technology investment but an average operational cost decreases by managerial disposability because of more generation. Moreover, an opportunity cost associated with environmental pollution is much larger than an operation cost in modern business so that corporate leaders pay serious attention to various environmental issues. This type of strategy has been long supported by corporate strategists in U.S. business schools (e.g., Porter and van der Linde, 1995). To describe the concept of natural and managerial disposability by an axiomatic expression, let us consider X ∈ Rm + as an input vector, G ∈ Rs+ as a desirable output vector and B ∈ Rh+ as an undesirable output vector. All of them are column vectors whose components are all positive. The concept of natural and managerial disposability is specified by the following vectors of production factors, respectively, under constant RTS (Returns to Scale) and constant DTS (Damage to Scales: corresponding to RTS on undesirable outputs), respectively: n o Xn Xn Xn n P ðX Þ ¼ ðG; BÞ : G≤ G λ ; B≥ B λ ;X≥ X λ ; λ j ≥0 ð j ¼ 1; …; nÞ & j¼1 j j j¼1 j j j¼1 j j n o X X X n n n m G λ ; B≥ B λ ;X≤ X λ ; λ j ≥ 0 ð j ¼ 1; …; nÞ ; P ðX Þ ¼ ðG; BÞ : G≤ j¼1 j j j¼1 j j j¼1 j j

where the subscript (j) indicates the j-th organization (j = 1, … , n). The λ stands for an unknown variable. Hereafter, this study refers to the organization as “DMU (Decision Making Unit)”. The difference between the two concepts on disposability is that production technology under natural disposability, or Pn(X), has the inequality constraints on inputs: X ≥ ∑ nj = 1Xjλj. In contrast, the production technology of the managerial disposability, or Pm(X), has X ≤ ∑ nj = 1Xjλj as the input constraints.

2.3. A frontier shift in a time horizon To extend the concept of natural and managerial disposability into a time horizon, this study prepares four figures to intuitively describe the measurement of Malmquist index. The four figures serve as a visual basis for computing twelve subcomponents for Malmquist index under the two disposability concepts. Fig. 1 depicts a frontier shift from the t−1 th period to the t th period under natural disposability in which DMUs attempt to enhance their operational performance as the first priority and their environmental performance as the second priority. For our visual convenience, Fig. 1 considers a single input (x) and two desirable outputs (g1 and g2). The visual description is easily extendable to more inputs and desirable outputs than the simple case i.e., (a single input and a single desirable output), as found later in the proposed DEA formulations. In Fig. 1, at−1 stands for the observed performance of a DMU at the t−1 th period. Meanwhile, ct indicates the observed performance of the DMU at the t th period for t = 2, … , T. The symbol (o) stands for an origin in the figure. A frontier shift occurs toward the north-east direction under the assumption that technology innovation on desirable outputs occur between the two periods, where the north-east direction is because of the natural disposability. Fig. 1 assumes that the amount of undesirable outputs is the same between the two periods. In Fig. 1, the performance of at−1 is projected to aet−1 on the efficiency frontier in the t-1 th period and aet on the efficiency frontier in the t th period, respectively, where the superscript (e) implies an efficiency frontier. In the two cases, this study assumes a unique projection and a unique reference set as visually discussed in the figure. Under e the same assumption, the performance of ct is projected to ct−1 on the e efficiency frontier for the t−1 th period and ct on the efficiency frontier for the t th period, respectively. This type of projection, as depicted in Fig. 1, occurs under constant RTS to avoid an infeasible solution on a computer code. A geometric mean, which is widely used to measure the average of the two line segments (aet − aet−1 and cet − cet−1), indicates the degree of a frontier shift between the t−1 th and t th periods. The geometric mean is referred to as a “Malmquist index” in the DEA community. The mean is expressed by the following notation:

t

IN t−1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0aet 0cet sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e u 0at 0ct 0at−1 0ct ¼ ¼u u e e ; 0aet−1 0cet−1 t 0at−1 0ct−1 0at−1 0ct

ð1Þ

where INtt−1 stands for the Malmquist index between the t−1 th and t th periods under natural disposability. Shifting our description from natural disposability to managerial disposability, Fig. 2 depicts a frontier shift from the t−1 th period to the t th period in which each DMU attempts to enhance the environmental performance as the first priority and the operational performance as the second priority. For our visual description, Fig. 2 considers a single input (x) and two undesirable outputs (b1 and b2). In Fig. 2, dt−1 stands for the observed performance of a DMU in the t−1 th period. Meanwhile, qt indicates the observed performance of the DMU in the t th period. A frontier shift occurs toward the south-west direction under the assumption that technology innovation (e.g. clean coal technology) on undesirable outputs occur between the two periods. The south-west direction is due to managerial disposability. For our visual convenience, the amount of desirable outputs is the same between the two periods as found in Fig. 2. The performance of dt−1 can be projected to det−1 on the efficiency frontier for the t−1 th period and det on the efficiency frontier for the t th period, respectively, in Fig. 2. Meanwhile, the performance of qt

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g2 x

ate

d t −1

b2 x

d te−1

ate−1

cte

at − 1 cte−1

qte−1

d te

ct

t-1 th period

t-1 th period

qt qte

t th period

t th period 0

0

b1 x

g1 x

Fig. 1. Frontier shift under natural disposability. (a) An efficiency frontier shifts toward an increase in desirable outputs without any frontier crossover between the two periods. The natural disposability implies that the first priority is operational performance and the second priority is environmental performance. The figure assumes that undesirable outputs are same on all DMUs, so dropping their influences on the frontier shift. (b) The figure assumes that DEA does not suffer from an occurrence of multiple projections and reference sets. (c) A unique projection onto the two efficiency frontiers is for our visual convenience. This study fully understands that real projections in DEA are more complicated than the ones depicted in the figure.

can be projected to qet−1 on the efficiency frontier of the t−1 th period and qet on the efficiency frontier of the t th period, respectively. This type of projection occurs under constant DTS to maintain a computational feasibility. The Malmquist index, which measures the geometric mean of the two line segments (det−1 − det and qet−1 − qet ), is expressed by the following notation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0dt−1 0qt−1 e e u 0d 0q 0d 0q t t−1 t−1 t−1 t ¼u IM t−1 ¼ ð2Þ u e e ; 0det 0qet 0qt t 0dt 0dt−1 0qt where IMtt−1 stands for the Malmquist index between the t−1 th and t th periods under managerial disposability. 2.4. A crossover between efficiency frontiers Fig. 3, corresponding to Fig. 1, depicts a possible occurrence of the frontier crossover between the t−1 th and t th periods. The performance of DMUs in Fig. 3 is measured by natural disposability. In Fig. 3, an efficiency frontier retreats between the two periods. As a result, it is necessary to combine the two frontiers to shape a new efficiency frontier, or the dotted line in Fig. 3. To attain the status of efficiency, at−1 and ct need to shift their locations to aet−1 &t and cet−1 &t on the newly shaped efficiency frontier. The geometric mean measures the average of t he two line segments (aet−1 &t − aet−1 and cet−1 &t − cet ), indicating the degree of a frontier shift between the t−1 th and t th periods. The geometric mean in this case is expressed by the following notation:

t INC t−1

373

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u0at−1&t 0ct−1&t e e u 0at−1&t 0ct−1&t u 0at−1 0ct ¼ ¼u ; e 0aet−1 0cet 0cet t 0at−1 0at−1 0ct

Fig. 2. Frontier shift under managerial disposability. (a) An efficiency frontier shifts toward a decrease in undesirable outputs without any frontier crossover between the two periods. The managerial disposability implies that the first priority is environmental performance and the second priority is operational performance. The figure assumes that desirable outputs are same on all DMUs, so dropping their influences on the frontier shift. (b) See the note of Fig. 1.

In a similar manner, Fig. 4, corresponding to Fig. 2, visually describes a frontier crossover between the t−1 and t th periods. In this figure, dt − 1 and qt need to shift their projected points, det−1 &t and qet−1 &t, on the newly shaped efficiency frontier, or the dotted line for the enhancement of their environmental performance. The geometric mean in this case measures the average of the two line segments (det−1 − det−1 &t and qet − qet−1 &t), indicating the degree of a frontier shift between the t−1 and t th periods. The geometric mean is expressed by the following notation:

t

IMC t−1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e 0qet sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0dt−1 e e u 0d 0dt−1 0qt 0qt ¼ ¼ u et−1 ; e 0det−1&t 0qet−1&t u t 0dt−1&t 0qt−1&t 0dt−1 0qt

ð4Þ

where IMCtt−1 stands for the Malmquist index between the t−1 th and the t th periods under managerial disposability, considering a possible occurrence of the frontier crossover between the two periods. It is important to note the following two concerns: One of the two concerns is that the performance of all DMUs at the t th period depends upon only that of the t−1 th period, not the other periods, in

g2 x

ate−1&t ate−1 at − 1

cte

cte−1&t

ct t-1 & t th periods

ð3Þ t – 1 th Period 0

t stands for the Malmquist index between the t−1 th and where INCt−1 t th periods under natural disposability, considering a possible occurrence of the frontier crossover between the two periods.

t th period

g1 x

Fig. 3. Frontier crossover between two periods under natural disposability. (a) See the note of Fig. 1.

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Figs. 3 and 4. Such a relationship is used for our visual convenience. It is possible to increase the size of a window in formulating the proposed DEA approach. In this case, we can deal with an occurrence of the frontier crossover not only between the adjacent (t−1 th and t th) periods but also among all other period combinations. The other concern is that Eqs. (3) and (4) have a computational benefit, compared with Eqs. (1) and (2). That is, we can measure the four components related to Eqs. (3) and (4) as efficiency scores because they incorporate an occurrence of a frontier crossover between two periods. Meanwhile, Eqs. (1) and (2) consist of two efficiency scores and two index measures because they do not incorporate a likelihood of the frontier crossover. Finally, this study uses the term “efficiency” when a performance measure exists between 0% (full inefficiency) and 100% (full efficiency). Meanwhile, the other term “index” is used to express the measure that may be more than unity (100%). When the index is more than unity, it indicates a possible occurrence of the frontier shift due to technology innovation. In contrast, an opposite case (i.e., a frontier retreat) can be found if it is equal to or less than unity.

column vector of desirable outputs (Gjt) but also a column vector of undesirable outputs (Bjt) at the t th period where Xjt = (x1jt,x2jt,…,xmjt)Tr, Gjt = (g1jt,g2jt,…,gsjt)Tr and Bjt =(b1jt, b2jt,…,bhjt)Tr. The superscript “Tr” indicates a vector transpose. It is assumed that Xjt N 0, Gjt N 0 and Bjt N 0 for all j = 1, … , n for computational convenience. All components of each column vector maintain the inequality. To compare the performance of all DMUs (J) among multiple periods, this study adds subscripts (t) to J so that Jt stands for all DMUs in the t th period. Returning to Fig. 1, which depicts a frontier shift under technology progress, the Malmquist index between the two periods can be reorganized as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u u u 0at−1 0ct 0aet 0cet sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u u u 0ae u 0a 0cet−1 0aet 0cet 0ct t t−1 INt−1 ¼ ¼u ¼u u t−1 u e e e e 0at−1 0ct−1 t 0at−1 0ct−1 t 0at−1 0ct 0aet 0cet 0at−1 0ct sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UENt−1 IUIN t→t−1 t ⇒ INt−1 ¼ IUINt−1→t UEN t ð5Þ

3. Formulations for natural disposability t t The six components, used to measure INt−1 and INCt−1 , are specified by the following manner:

(a) UENt−1: The measure stands for unified efficiency under natural disposability in the t−1 th period. (b) UENt: The measure stands for the unified efficiency in the t th period. (c) IUINt−1 → t: The measure stands for inter-temporal unified index from the t−1 th period to the t th period under natural disposability. (d) IUINt → t−1: The measure stands for the index from the t th period to the t−1 th period. (e) IUENt−1 → t−1 &t: The measure stands for inter-temporal unified efficiency from the t−1 th period to the t−1 & t th periods under natural disposability. (f) IUENt → t−1 &t: The measure stands for the efficiency from the t th period to the t−1 & t th periods.

where UENt−1 and UENt are the two efficiency scores that belong to a range between zero (full inefficiency) and one (full efficiency). Meanwhile, IUINt → t−1 and IUINt−1 → t may become larger or smaller than unity, depending upon an occurrence of a frontier crossover between the two periods. These measures can be formulated by the four DEA models, each of which is specified as follows: UENt: The degree of UENt on the k th DMU in the t th period is measured by the following model under natural disposability:

Maximize ξ þ ε s:t:

X

x R i¼1 i

x

di þ

Xs

x

xijt λjt þ di

g R r¼1 r

g

dr þ

Xh f ¼1

b

b

Rf df

i

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

j ∈ Jt

X

g

grjt λjt − dr − ξg rkt

¼ grkt ; ð∀ k ∈ J t & r ¼ 1; …; sÞ;

j ∈ Jt

X

3.1. Without frontier crossover

hX m

b

bfjt λjt þ d f þ ξbfkt

ð6Þ

¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ;

j ∈ Jt x

To measure the Malmquist index under natural disposability in a time horizon (t = 2, … , T), this study considers n DMUs (j = 1, … , n) whose operational and environmental achievements are relatively compared by each other in their performance assessment. The j th DMU uses a column vector of inputs (Xjt) in order to yield not only a

b2 x

d t −1

d te− 1 d te− 1& t

q te qte−1&t 0

qt t-1 th period t th period

b1 x

Fig. 4. Frontier crossover between two periods under managerial disposability. (a) See the note of Fig. 2.

λjt ≥ 0 ð j ¼ 1; …; n and t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;

where dxi (i = 1, … , m), dgr (r = 1, … , s) and dbf (f = 1, … , h) are all slack variables related to inputs as well as desirable and undesirable outputs, respectively. λt = (λ1t,…,λnt)Tr is a column vector of unknown variables, often referred to as “structural” or “intensity” variables, that are used for connecting the input and output vectors by a convex combination. An unknown scalar value (ξ), which is unrestricted (URS), stands for an inefficiency score. The inefficiency score corresponds to a distance between an efficiency frontier and an observed vector of desirable and undesirable outputs, as depicted in Fig. 1. Another scalar value (ε) is also incorporated into Model (6) and it stands for a small number to indicate the relative importance between the inefficiency score and the total amount of slacks. This study sets ε = 0.0001 for our computation. The selection is subjective. It is possible for us to set ε = 0. In the case, it is necessary to depend upon a method for multiplier restriction (e.g., cone ratio or incorporation of SCSCs: Strong Complementary Slackness Conditions) because dual variables may become zero.

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375

That is problematic in DEA because corresponding production factors are not utilized in the efficiency assessment. The data ranges (R) in Model (6) are determined by the upper and lower bounds of production factors.2 These upper and lower bounds are specified by

IUINt → t−1: The degree of IUINt → t−1 regarding the k th DMU that shifts from the t th period to the t−1 th period is determined by the following model:

n  o n  −1   ¼ ðm þ s þ hÞ max xij  j∈J t−1 ∪J t − min xij  j∈J t−1 ∪J t gÞ  n  o n  −1   g −1 Rr ¼ ðm þ s þ hÞ max g rj j∈ J t−1 ∪ J t − min g rj  j∈J t−1 ∪J t gÞ and  n  o n  −1   b −1 R f ¼ ðm þ s þ hÞ max b f j j∈ J t−1 ∪ J t − min bfj j∈ J t−1 ∪J t gÞ :

s:t:

−1

x Ri



Maximize ξ þ ε X

hX m

x R i¼1 i

x

di þ

Xs

g R r¼1 r

g

dr þ

Xh

b

f¼1

b

Rf df

i

xijt−1 λjt−1 þ di

x

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

g

¼ g rkt ; ð∀ k ∈ J t & r ¼ 1; …; sÞ;

b

¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ;

j ∈ J t−1

X

g rjt−1 λjt−1 − dr − ξgrkt

j ∈ J t−1

X

ð9Þ bfjt−1 λjt−1 þ d f þ ξbfkt

j ∈ J t−1 x

λjt−1 ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ;

Here, ∪ stands for a union set. Thus, Jt−1 ∪ Jt stands for all DMUs in the t−1 th and t th periods.

g

The degree of UENkt of the k-th DMU in the t th period is determined by UENkt

h X i Xs Xh m  x x g g b b ; ¼ 1− ξ þ ε R d þ R d þ R d i¼1 i i r¼1 r r f¼1 f f

ð7Þ

where the inefficiency score and all slack variables are determined on the optimality of Model (6). The equation within the parenthesis, obtained from the optimality of Model (6), indicates the level of unified inefficiency under natural disposability. The unified efficiency is obtained by subtracting the level of inefficiency from unity. The superscript (*) indicates optimality. The degree of UENkt is always less than unity, indicating an existence of some level of inefficiency, or equal unity for the status of full efficiency. UENt−1: The degree of UENt−1 regarding the k-th DMU in the t−1 th period is measured by replacing t by t−1 in Model (6). IUINt−1 → t: The degree of IUINt−1 → t regarding the k th DMU from the t−1 th period to the t th period is determined by the following model: Maximize ξ þ ε s:t:

X

hX

m x R i¼1 i

x

di þ

Xs

g R r¼1 r

x

xijt λjt þ di

g

grjt λjt − dr − ξg rkt−1

f ¼1

b

b

Rf df

i

ð8Þ

b

bfjt λjt þ d f þ ξbfkt−1

¼ bfkt − 1 ; ð∀ k ∈ Jt−1 & f ¼ 1; …; hÞ;

j ∈ Jt

x

λjt ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

where an efficiency frontier consists of DMUs in the t−1 th period and a DMU to be examined is selected from a group of DUMs in the t th period. The degree of the index is measured by Model (7) where the inefficiency score and all slack variables are determined on the optimality of Model (9). 3.2. With frontier crossover Returning to Fig. 3, the conceptual framework for Eq. (3) under natural disposability can be expressed by the following measures in the case of a frontier crossover: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e u 0a 0ct sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u t−1 u 0at−1&t 0ct−1&t u u e e u 0a 0c 0a 0c 0cet u 0aet−1 t t−1&t t−1&t t−1 t u u ¼u INC t−1 ¼ ¼ t 0at−1 0ct 0aet−1 0cet 0aet−1 0cet t 0aet−1&t 0cet−1&t 0at−1 0ct sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UEN t−1 UENt t ; ⇒INC t−1 ¼ IUENt−1→t−1&t IUENt→t−1&t

ð10Þ

¼ grkt − 1 ; ð∀ k ∈ J t−1 & r ¼ 1; …; sÞ;

j ∈ Jt

X

Xh

¼ xikt − 1 ; ð∀ k ∈ J t−1 & i ¼ 1; …; mÞ;

j ∈ Jt

X

g

dr þ

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;

where Malmquist index, or INCtt−1, is separated into four subcomponents, all of which are used to measure a frontier shift between the two periods (i.e., UEN t−1 , UENt , IUENt−1 → t−1 &t and IUENt → t−1 &t ). Among the four subcomponents, this study has already discussed the measurement of UENt−1 and UENt in Section 3.1, so focusing upon the measurement of IUENt−1 → t−1 &t and IUENt → t−1 &t, hereafter. IUENt−1 → t−1 &t: The degree of IUENt−1 → t−1 &t regarding the k th DMU between the two periods is determined by the following model: Maximize ξ þ ε

where an efficiency frontier consists of DMUs in the t th period and a DMU to be examined is selected from a group of DUMs in the t−1 th period. The degree of the index is measured by Eq. (7) where the inefficiency score and all slack variables are determined on the optimality of Model (8).

hX

X

s:t:

m x R i¼1 i

x

di þ

Xs

g R r¼1 r

x

xijt λjt−1&t þ di

g

dr þ

Xh

b

f ¼1

b

Rf df

i

¼ xikt−1 ; ð∀ k ∈ J t−1 & i ¼ 1; …; mÞ;

j ∈ J t−1&t

X

j ∈ J t−1&t

X

g

g rjt λjt−1&t − dr − ξgrkt−1 ¼ grkt − 1 ; ð∀ k ∈ J t−1 & r ¼ 1; …; sÞ;

ð11Þ

b

bfjt λjt−1&t þ d f þ ξbfkt−1 ¼ bfkt−1 ; ð∀ k ∈ J t−1 & f ¼ 1; …; hÞ;

j ∈ J t−1&t x

λjt−1&t ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ; 2

It is possible to eliminate the data ranges in the proposed DEA models. In the case, the incorporation of SCSCs makes positive dual variables. See, for example, Sueyoshi and Goto (2012e, 2012g). An occurrence of zero on dual variables is problematic because its corresponding production factor is not used in DEA assessment. Moreover, it implies zero RTS or zero DTS, so indicating an occurrence of congestion. This study is not interesting in the occurrence because we consider it as technology innovation on undesirable outputs.

where an efficiency frontier consists of DMUs in the t−1 & t th periods and a DMU to be examined by Model (11) is selected from a group of DUMs in the t−1 th period. The degree of IUENt−1 → t−1 &t on the k th DMU is determined by Eq. (7) where the inefficiency measure and all slack variables are determined on the optimality of Model (11).

376

T. Sueyoshi, M. Goto / Energy Economics 40 (2013) 370–382

IUENt → t−1 &t: The degree of IUENt → t−1 &t on the k th DMU between the two periods is determined by the following model: Maximize ξ þ ε

hX m

X

s:t:

x x R d i¼1 i i

Xs

g g R d r¼1 r r

þ

þ

Xh f¼1

b b

Rf df

4.1. Without frontier crossover The conceptual framework on Eq. (2), or IMtt−1 under managerial disposability, can be expressed by the following measures as depicted in Fig. 2:

i

xijt λjt þ di

x

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

g

¼ grkt ; ð∀ k ∈ J t & r ¼ 1; …; sÞ;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0dt−1 0qt−1 e e u 0d 0q 0d 0q t t−1 t−1 t−1 t IM t−1 ¼ ¼u u e 0det 0qet 0qet t 0dt 0dt−1 0qt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UEMt−1 IUIM t→t−1 t ⇒ IMt−1 ¼ IUIM t−1→t UEM t

j ∈ J t−1&t

X

g rjt λjt − dr − ξg rkt

j ∈ J t−1&t

X

ð12Þ b

b f jt λjt þ d f þ ξbfkt

¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ;

j ∈ J t−1&t x

λjt ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; TÞ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;

where an efficiency frontier consists of DMUs in the t−1 & t th periods and a DMU to be examined is selected from DUMs in the t th period. The degree of IUENt → t−1 &t regarding the k th DMU in the t th period is determined by Eq. (7) where the inefficiency measure and all slack variables are determined on the optimality of Model (12). At the end of this section, it is necessary to describe two differences between index measures in Section 3.1 and these corresponding efficiency scores in Section 3.2. One of the two differences is that as a result of incorporating a frontier crossover in Models (11) and (12), the two models always produce efficiency scores, not indexes, which belong to a range between 0 (full inefficiency) and 1 (full efficiency). Such a feature cannot be found in Models (8) and (9). The other difference is t that INt−1 may be more than, equal to or less than unity, but INCtt−1 is more than or equal to unity and it cannot be less than unity. If a DMU exhibits unity in INCtt−1, then it indicates that there is a frontier crossover between the t−1 th and t th periods and the achievement of the DMU locates on the three efficiency frontiers for the t−1 th, t th and the combined t−1 & t th periods. In other words, there is no technological progress between the two periods. In contrast, if a DMU exhibits more unity in INCtt−1, then it indicates that the efficiency frontier of the t th period locates differently from that of the t−1 th period, as depicted in Fig. 3. Thus, it is possible for us to identify whether the DMU has an operational progress (e.g., due to technological development) or an operational retreat (e.g.,, due to poor business environment) between the two periods by examining INCtt−1. Finally, it is important to note that the description here is based upon the assumption that a data set to be examined does not contain an outlier because it drastically changes the location of an efficiency frontier. 4. Formulations under managerial disposability To describe the six components of IMtt−1 and IMCtt−1 from DEA environmental assessment, this study specifies them more clearly in the following manner: (a) UEMt−1: The measure stands for unified efficiency under managerial disposability in the t−1 th period. (b) UEMt: The measure stands for the unified efficiency in the t th period. (c) IUIMt−1 → t: The measure stands for inter-temporal unified index from the t−1 th period to the t th period under managerial disposability. (d) IUIMt → t−1: The measure stands for the index from the t-th period to the t−1 th period. (e) IUEMt−1 → t−1 &t: The measure stands for inter-temporal unified efficiency from the t−1 th period to the t−1 & t th periods under managerial disposability. (f) IUEMt → t−1 &t: The measure stands for the efficiency from the t th period to the t−1 & t th periods.

ð13Þ

As formulated in Eq. (13), IMtt−1 is separated into four subcomponents, all of which are measured under constant DTS. UEMt: The degree of UEMt of the k th DMU in the t th period is measured by the following model: Maximize ξ þ ε s:t:

X

hX m

x R i¼1 i

x

di þ

Xs

g R r¼1 r

g

dr þ

Xh

b

f ¼1

b

Rf df

i

x

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

g

¼ grkt ; ð∀ k ∈ J t & r ¼ 1; …; sÞ;

b

¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ;

xijt λjt − di

j ∈ Jt

X

g rjt λjt − dr − ξg rkt

j ∈ Jt

X

ð14Þ bfjt λjt þ d f þ ξbfkt

j ∈ Jt x

λjt ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ:

The degree of UEMkt regarding the k th DMU in the t th period determined by h X i Xs Xh m  x x g g b b ; ð15Þ UEMkt ¼ 1− ξ þ ε R d þ R d þ R d i i r r f f i¼1 r¼1 f¼1 where the inefficiency score and all slack variables are determined on the optimality of Model (14). The equation within the parenthesis, obtained from the optimality of Model (14), indicates the level of unified inefficiency under managerial disposability. The unified efficiency is obtained by subtracting the level of inefficiency from unity. Thus, the unified efficiency always belongs to the range between 0 (full inefficiency) and 1 (full efficiency). UEMt−1: The degree of UEMt−1 regarding the k th DMU at the t−1 th period is measured by replacing t by t−1 in Model (14). IUIMt−1 → t: The degree of IUIMt−1 → t of k th DMU from the t−1 th period to the t th period is determined by the following model: Maximize ξ þ ε s:t:

X

hX

m x R i¼1 i x

j ∈ Jt

X

Xs

g R r¼1 r

g

dr þ

Xh

b

f ¼1

b

Rf df

i

¼ xikt − 1 ; ðk ∈ J t−1 & i ¼ 1; …; mÞ;

xijt λjt − di

j ∈ Jt

X

x

di þ

g

g rjt λjt − dr − ξg rkt − 1 ¼ grkt−1 ; ðk ∈ J t−1 & r ¼ 1; …; sÞ;

ð16Þ b

bfjt λjt þ d f þ ξbfkt−1 ¼ bfkt−1 ; ðk ∈ J t−1 & f ¼ 1; …; hÞ;

j ∈ Jt x

λjt ≥ 0 ð j ¼ 1; …; n & t ¼ 1; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; ::; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;

where an efficiency frontier consists of DMUs in the t th period and a DMU to be examined is selected from DUMs in the t−1 th period. The

T. Sueyoshi, M. Goto / Energy Economics 40 (2013) 370–382

degree of IUIMt−1 → t of the k th DMU is measured by Eq. (15) where the inefficiency score and all slack variables are determined on the optimality of Model (16). IUIMt → t−1: The degree of IUIMt → t−1 of the k th DMU from the t th period to the t−1 th period is determined by the following model:

IUEMt → t − 1 &t: The degree of IUEMt → t − 1 &t of the k th DMU in the t th period is determined by the following model: Maximize ξ þ ε

Maximize ξ þ ε X

s:t:

x R i¼1 i

Xs

x

g R r¼1 r

di þ x

xijt−1 λjt−1 − di

g

dr þ

Xh

b

f ¼1

b

Rf df

j ∈ J t−1 & t

i

X

j ∈ J t−1 & t

X

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

j ∈ J t−1

X

x R i¼1 i

x

di þ

Xs

g R r¼1 r x

xijt−1 & t λjt−1 & t − di

g

dr þ

Xh f¼1

b

b

Rf df

i

¼ xikt ; ð∀ k ∈ J t & i ¼ 1; …; mÞ;

g

grjt−1 & t λjt−1 & t − dr − ξgrkt ¼ grkt ; ð∀ k ∈ J t & r ¼ 1; …; sÞ;

ð20Þ b

j ∈ J t−1 & t

g

grjt−1 λjt−1 − dr − ξgrkt ¼ grkt ; ð∀ k ∈ Jt & r ¼ 1; …; sÞ;

j ∈ J t−1

X

hX m

X

s:t:

hX m

377

bfjt−1 & t λjt−1 & t þ d f þ ξbfkt ¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ; x

λjt−1 & t ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ;

ð17Þ b

g

bfjt−1 λjt−1 þ d f þ ξbfkt ¼ bfkt ; ð∀ k ∈ J t & f ¼ 1; …; hÞ;

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ:

j ∈ J t−1 x

λjt−1 ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; TÞ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ:

where an efficiency frontier consists of DMUs in the t−1 th period and a DMU to be examined is selected from a group of DUMs in the t th period. The degree of IUIMt−1 → t of the k th DMU between the two periods is measured by Eq. (15) where the inefficiency score and all slack variables are determined on the optimality of Model (17). 4.2. With frontier crossover By shifting our description to the case of a frontier crossover, the conceptual framework on Eq. (4), or IMCtt − 1, under managerial disposability, can be expressed by the following unified measures as depicted in Fig. 4: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e e 0qt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0dt−1 e e u 0d 0q 0d 0q t t−1 t t ¼ u et−1 IMC t−1 ¼ e 0det−1&t 0qet−1&t u t 0dt−1&t 0qt−1&t 0dt−1 0qt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UEM UEM t t−1 t : ⇒IMC t−1 ¼ IUEMt−1→t−1&t IUEM t→t−1&t

ð18Þ

where an efficiency frontier consists of DMUs in the t−1 & t th periods and a DMU to be examined is selected from a group of DUMs in the t th period. The degree of IUEMt → t−1 &t is measured by Eq. (15) where the inefficiency score and all slack variables are determined on the optimality of Model (20). As a result of incorporating an occurrence of a frontier shift in Models (19) and (20), the two models can always produce an efficiency score on a range between 0 (full inefficiency) and 1 (full efficiency). Such a feature cannot be found in Models (16) and (17). The proposed Malmquist index measurement under natural and managerial disposability has two important concerns to be discussed here. One of the two concerns exists on the assumption that all DEA models in Sections 3 and 4 can uniquely produce a DEA solution. However, the assumption is not always true because they may suffer from an occurrence of multiple solutions (e.g., multiple projections and reference sets). If such a problem occurs, it is necessary to incorporate SCSCs to the proposed DEA models for a time horizon. See Sueyoshi and Goto (2012f, 2012g) on how to incorporate SCSC into DEA formulations under two disposability concepts although they do not incorporate the time horizon. The other concern is that this study considers only two periods (t−1 and t) in discussing the occurrence of a frontier crossover between two periods. It is possible for us to extend the size of a window (or a length of periods) to more than three periods (Jp ∪ ⋯ ∪ Jt−1 ∪ Jt), where the subscript (p) indicates a starting period of the window. In the case of multiple (more than two) periods, the data ranges (R) are determined by the upper and lower bounds of production factors as follows: n  o n  −1   max xij  j∈J p ∪⋯∪J t−1 ∪ J t − min xij  j∈J p ∪⋯∪J t−1 ∪ J t gÞ n  o n  −1   g −1 Rr ¼ ðm þ s þ hÞ max grj j∈J p ∪⋯∪J t−1 ∪J t − min grj  j∈J p ∪⋯∪J t−1 ∪ J t gÞ and  n  o n  −1   b −1 R f ¼ ðm þ s þ hÞ max bfj j∈J p ∪⋯∪J t−1 ∪J t − min bfj j∈J p ∪⋯∪J t−1 ∪J t gÞ : x

−1

Ri ¼ ðm þ s þ hÞ

UEMt−1 and UEMt: The measurement of UEMt−1 and UEMt can be measured by Model (14). IUEMt−1 → t−1 &t: The degree of IUEMt−1 → t−1 &t of k th DMU in the t−1 th period is determined by the following model:

Maximize ξ þ ε

hX m

X

s:t:

j ∈ J t−1 & t

X

j ∈ J t−1 & t

X

j ∈ J t−1 & t

x R i¼1 i

x

di þ

Xs

g R r¼1 r

g

dr þ

x

Xh

b

f ¼1

b

Rf df

i

¼ xikt − 1 ; ðk ∈ J t−1 & i ¼ 1; …; mÞ;

xijt−1 & t λjt−1 & t − di

g

g rjt − 1 & t λjt − 1 & t − dr − ξg rkt−1 ¼ grkt − 1 ; ðk ∈ J t−1 & r ¼ 1; …; sÞ;

ð19Þ

b

bfjt − 1 & t λjt − 1 & t þ d f þ ξbfkt−1 ¼ bfkt − 1 ; ðk ∈ J t−1 & f ¼ 1; …; hÞ; x

λjt−1 & t ≥ 0 ð j ¼ 1; …; n & t ¼ 2; …; T Þ; ξ : URS; di ≥ 0 ði ¼ 1; …; mÞ; g

b

dr ≥ 0 ðr ¼ 1; …; sÞ; and d f ≥ 0 ð f ¼ 1; …; hÞ;





Finally, it is important to note that many production economists (e.g., Zhou et al., 2012) have used a directional distance function to express the analytical structure for inefficiency measurement. This study does not follow such a conventional expression because the analytical description has a difficulty in mathematically describing a frontier crossover between difference periods. Furthermore, the use of conventional distance functions has implicitly assumed a unique solution on DEA. In contrast, this study clearly specifies DEA models as linear programming, so that they may suffer from an occurrence of multiple solutions in their primal and dual models. See Sueyoshi and Goto (2012f, 2012g) for an occurrence of multiple solutions and a description regarding how to handle such a difficulty. 5. An application on energy mix in industrial nations

where an efficiency frontier consists of DMUs in the t−1 & t th periods and a DMU to be examined is selected from DUMs in the t−1 th period. The degree of IUEMt−1 → t−1 &t of the k-th DMU is measured by Eq. (15) where the inefficiency score and all slack variables are determined on the optimality of Model (19).

This study has sampled ten industrial nations from OECD (Organization Co-operation and Development). Table 1 summarizes the industrial nations along with the average of three inputs (i.e., the net electrical capacity of main producer plants with respect to combustible, nuclear,

378

T. Sueyoshi, M. Goto / Energy Economics 40 (2013) 370–382

Table 1 Average of production factors (1999–2009). Inputs and outputs

Input 1

Input 2

Input 3

Desirable output

Undesirable output

Combustible

Nuclear

Hydro + renewables

Electricity

CO2

1000 GWh

Million metric tons CO2

11.99 63.25 21.28 47.26 16.12 1.37 0.47 7.47 11.77 99.40

67.19 26.07 26.76 45.89 4.50 11.31 1.01 27.79 4.79 111.32

589.40 534.15 566.52 1031.58 348.80 220.30 95.12 258.11 370.70 3959.94

116.37 27.76 255.97 369.66 147.66 106.98 22.82 90.88 162.66 2168.78

Unit

1000 MWe

Canada France Germany Japan Korea Mexico Netherlands Spain United Kingdom United States

32.66 20.71 69.05 139.25 38.97 32.12 17.28 34.76 57.50 667.72

(a) Renewables include wind, solar, geothermal, tide, wave and ocean energies. (b) The unit of the three inputs is 1000 MWe.

hydro and renewable energies), a single desirable output (i.e., the total net production of electricity) and a single undesirable output (i.e., the amount of CO2 emission from fuel combustion of main electricity plants). In preparing Table 1, this study considers two empirical difficulties in the data selection. One of the two difficulties is an occurrence of “zero” in a data set. It is not appropriate for us to apply radial models to a data set that contains zero because they do not have the property of “translation invariance”. The property indicates that it is possible to add a small number to a data set so that all data are positive, but DEA efficiency scores are not influenced by the data shift. Only RAM (Range-Adjusted Measure) has the desirable property among all DEA models, currently available to us. See Sueyoshi and Sekitani (2009) for the mathematical proof on RAM's translation invariance. The other difficulty is an occurrence of “an outlier” in a data set. An important feature of DEA is that it relatively evaluates the performance of a DMU by comparing it with those of the others. In the relative comparison, if there is a DMU, containing an outlier in their production factors, then the outlier dominates the computation process of DEA so that the outlier exhibits unity, but the other DMUs are very low in their efficiency measures. That is mathematically acceptable, but managerially unacceptable. See Sueyoshi and Goto (2013c). The proposed approach for environmental assessment belongs to DEA radial measurement. Therefore, it is difficult for us to handle zero in a data set. As a result, clearly acknowledging that OECD has other member nations (e.g., Australia, Italy, Norway and Turkey), this study excludes these nations because they do not have any nuclear power plant during the observed period so that these member nations contain zero in their data sets. This study also excludes other member nations (e.g., Sweden) because their production factors are very small, so becoming an outlier in this DEA assessment. The two concerns may indicate a drawback of DEA as an empirical methodology. The data sources used in this study are OECD statistics such as (a) IEA Electricity Information Statistics, OECD-Net electrical capacity, Total main activity producer plants, containing inputs, (b) IEA Electricity Information Statistics, OECD-Electricity/heat supply and consumption, Electricity, Total net production, containing a desirable output, and

(c) IEA CO2 Emissions from Fuel Combustion Statistics, Detailed CO2 estimates, Main activity electricity plants, Total, containing an undesirable output (IEA CO2 emission). The two units, MWe and GWh, stand for Megawatt electric and Gigawatt hour, respectively. Table 2 documents unified efficiency scores (UEN) under natural disposability from 1999 to 2009. The ranges for DEA computation are determined by information on data for two annual periods in 1999–2009. As summarized in Table 2, the ten industrial nations are separated into two groups. One of two groups contains efficient nations, including Canada, France, Korea, Mexico, and Netherlands. The other group consists of Germany, Japan, Spain, United Kingdom and United States that exhibit some level of inefficiency under natural disposability (operational performance: first priority and environmental performance: second priority). Table 3 summarizes the Malmquist indexes (INtt−1) of the ten nations which are measured under natural disposability along with the assumption that a frontier crossover has not occurred in the observed periods (2000–2009). In a similar manner, Table 4 summarizes the indexes (INCtt−1) under the assumption that a frontier crossover has occurred between adjacent annual periods. The index measures provide two interesting findings. One of the two findings is that the electric power industry is a huge process industry whose investment in new technology requires a large amount of capital. The speed of technology innovation on generation (as a desirable output) is relatively slow so that the Malmquist index measures fluctuate around unity until a generation system can completely utilize the new technology. Consequently, Table 3 (under the assumption of no frontier crossover) cannot catch the influence of technology innovation. Meanwhile, Table 4 (under an occurrence of the crossover between two adjacent periods) can identify such a frontier shift with the indexes more than or equal to unity, although the speed and degree of technology innovation are still very modest. Policy implication: Three efficient nations (Canada, France and Netherlands) have not exhibited a technology progress on generation, but two efficient nations (Korea and Mexico) have showed the

Table 2 Unified efficiency under natural disposability (1999–2009).

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 0.8786 0.8633 0.8027 0.8571 0.8893 0.8873 0.8845 0.8375 0.8260 0.7616

0.8794 0.8488 0.8130 0.8059 0.7917 0.8013 0.8030 0.8297 0.8183 0.7935 0.7932

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.7707 0.7472 0.7892 0.7230 0.7638 0.7404 0.6977 0.7303 0.6957 0.7582 0.7680

1.0000 0.9312 0.8986 0.9490 0.9474 0.9286 1.0000 1.0000 0.8837 0.8552 0.8455

1.0000 1.0000 0.8174 0.7952 0.7350 0.7234 0.7425 0.7727 0.7305 0.7209 0.6759

T. Sueyoshi, M. Goto / Energy Economics 40 (2013) 370–382

379

Table 3 Malmquist index under natural disposability (2000–2009): No crossover.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

1.0183 0.9708 1.0191 0.9563 0.9585 1.0159 0.9826 1.0154 1.0114 0.9378

1.0191 1.0576 0.9801 0.9922 1.0144 1.0193 1.0495 0.9808 0.9804 0.9018

1.0706 1.0109 1.0037 1.0074 0.9954 1.0150 0.9715 1.0374 0.9774 0.9369

1.0374 1.0198 1.0109 0.9952 1.0169 1.0061 0.9789 1.0306 0.9795 0.9558

1.1030 0.9764 1.0244 1.0161 1.0108 1.0234 0.9164 1.0093 0.9658 0.9844

1.0403 0.9986 0.9665 0.9356 1.0536 1.0378 0.9851 1.0111 1.0261 0.9835

0.9915 1.0109 0.9221 0.9115 0.9812 0.9433 0.9320 1.0285 0.9451 1.0122

1.0544 0.9881 1.0150 0.9648 1.0301 1.0416 1.0127 1.0111 1.0341 1.0445

1.0706 1.0256 0.9992 1.0213 1.0015 0.9388 0.9947 0.9096 0.9376 0.9788

1.0587 0.9479 0.9968 1.0175 1.0277 0.9828 0.9529 1.0822 0.9717 0.9872

Table 4 Malmquist index under natural disposability (2000–2009): Crossover.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

1.0000 1.0095 1.0022 1.0143 1.0000 1.0012 1.0030 1.0016 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0038 1.0000 1.0444

1.0398 1.0163 1.0014 1.0162 1.0110 1.0149 1.0154 1.0183 1.0106 1.0402

1.0184 1.0171 1.0052 1.0090 1.0153 1.0116 1.0116 1.0191 1.0100 1.0250

1.0242 1.0000 1.0000 1.0025 1.0000 1.0069 1.0066 1.0000 1.0025 1.0000

1.0113 1.0000 1.0112 1.0309 1.0265 1.0149 1.0020 1.0000 1.0110 1.0070

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0263 1.0191 1.0071 1.0193 1.0150 1.0214 1.0111 1.0068 1.0180 1.0229

1.0321 1.0121 1.0025 1.0133 1.0160 1.0528 1.0021 1.0133 1.0228 1.0170

1.0203 1.0208 1.0029 1.0169 1.0220 1.0257 1.0246 1.0410 1.0148 1.0177

technology progress. Meanwhile, all inefficient nations (Germany, Japan, Spain, United Kingdom and the United States) have exhibited the technology progress where they have a technological space to improve their efficiency scores. Table 5 documents unified efficiency scores (UEM) under managerial disposability, where environmental performance is the first priority and operational performance is the second priority, from 1999 to 2009. As summarized in Table 5, France and Netherlands were efficient in 1999–2009. The other eight nations have exhibited some level of inefficiency in the annual periods. The empirical finding indicates the following implication. Policy implication: France has been utilizing nuclear energy to generate electricity and to reduce CO2 emission. Netherlands uses hydro and renewable generations. The performance of France indicates that the nuclear energy is important in developing a sustainable society which needs to balance between economic progress and environmental protection. It is indeed true that we need to pay attention to renewable energy sources, but their generation outputs are significantly small, compared with that of nuclear generation. A shortcut to establish the sustainable society may depend upon a use of nuclear energy until we can access advanced technology on renewable generation. This concern on electricity generation does not imply that this study denies the importance of renewable energy

sources. As found in Netherlands, it is possible for us to fully utilize a combination between hydro-generation and renewable energy (e.g., wind) for electricity generation and reduction of CO2 emission. It is easily envisioned that the use of hydro and renewable energy will be able to contribute to the development of a sustainable society. However, it is important for us to consider geographic features and historical processes regarding their generation capacities and fuel mixes, both are specific to each nation. t Table 6 summarizes the Malmquist indexes (IMt−1 ) of the ten nations which are measured under managerial disposability along with the assumption that a crossover of efficiency frontiers has not occurred in the observed periods (2000–2009). In a similar manner, Table 7 summarizes t the indexes (IMCt−1 ) under the assumption that a frontier crossover has occurred between two adjacent (t−1 th and t th) periods. The index measures in Tables 6 and 7 provide two interesting findings. One of the two findings is that the electric power industry has a time lag to establish technology innovation on CO2 reduction because the electricity generation belongs to a capital-intensive industry so that the technology innovation needs a large amount of capital investment to reduce the amount of CO2 emission which is required by the government of each nation. Such a unique feature influences the index measures in such a manner that they fluctuate around unity, as in Table 6, and the technology innovation, or a

Table 5 Unified efficiency under managerial disposability (1999–2009).

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

0.8093 0.6015 0.4505 0.5423 0.5809 0.5876 0.6969 0.5576 0.5553 0.6342 0.8198

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.3926 0.3572 0.2586 0.2887 0.3322 0.3203 0.3751 0.3332 0.3362 0.3699 0.4232

0.5628 0.5237 0.4344 0.4466 0.4691 0.4442 0.4934 0.4430 0.4170 0.4485 0.5336

0.4862 0.3548 0.2746 0.3101 0.3327 0.2970 0.3537 0.3020 0.3371 0.3601 0.3417

0.3554 0.3096 0.2514 0.3003 0.3684 0.3848 0.4116 0.4016 0.4347 0.4682 0.4768

1.0000 1.0000 0.7523 0.8271 0.8385 0.8000 1.0000 1.0000 1.0000 1.0000 1.0000

0.4220 0.3436 0.2809 0.2887 0.4412 0.4322 0.5529 0.5453 0.5785 0.7114 0.9219

0.5130 0.4643 0.3636 0.3943 0.4172 0.4030 0.4653 0.4050 0.4461 0.4487 0.5479

0.3251 0.2642 0.2935 0.3628 0.4162 0.3883 0.4397 0.4020 0.4245 0.4200 0.4914

380

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Table 6 Malmquist index under managerial disposability (2000–2009): No crossover.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

1.2223 1.3170 0.8953 0.9169 1.0588 0.8773 1.2951 0.9979 0.9372 0.9059

1.1516 1.2622 0.9208 0.9363 1.0565 0.8509 1.2093 0.9834 0.9251 0.9054

1.0881 1.2583 0.9101 0.9100 1.0776 0.8390 1.1493 0.9391 0.9964 0.9354

1.0736 1.2377 0.9101 0.9100 1.0776 0.8557 1.1350 0.9342 1.0141 0.9366

1.0924 1.2746 0.9101 0.9100 1.0776 0.8316 1.1564 0.9417 0.9940 0.9319

1.0927 1.2589 0.9101 0.9100 1.0776 0.8584 1.1309 0.9304 1.0274 0.9375

1.0203 1.1874 0.9101 0.9100 1.0776 0.8884 1.1052 0.9203 1.0632 0.9435

1.1655 1.3317 0.9101 0.9100 1.0776 0.8203 1.1655 0.9435 0.9783 0.9376

1.0667 1.2311 0.9101 0.9100 1.0776 0.8624 1.1271 0.9292 1.0328 0.9392

1.1079 1.2565 0.9101 0.9100 1.0776 0.8715 1.1203 0.9268 1.0400 0.9400

frontier shift, becomes very modest, as found in Table 7. The other finding is summarized as the following policy implication: Policy implication: The electric power industry has been paying attention to the reduction of CO2 emission by utilizing nuclear power and renewable energies. The empirical evidence is consistent with the previous study (Sueyoshi and Goto, 2012d). Their study indicates that the Japanese electric power industry has operated more efficiently than the other manufacturing industries in terms of CO2 reduction. This study confirms that their finding may be extendable to the electric power industry of the other industrial nations.

6. Conclusion and future extensions This study proposed a new use of the Malmquist index to measure a frontier shift among different periods. A contribution of this study was that it made an analytical linkage between the index measurement and the concept of natural and managerial disposability. The Malmquist index measure, discussed in this study, was separated into twelve different subcomponents (6 subcomponents × 2 disposability concepts) in this study. All subcomponents provided us with information on how much the frontier shift occurred among different periods. The proposed DEA approach identified the three empirical findings, all of which were useful in the development of a green growth to establish a sustainable society. First of all, the electric power industry had a time lag until technology innovation really influenced the amount of generation and the reduction of CO2 emission. That was because the industry needed a large capital investment to introduce new technology for generation and environmental protection. Therefore, it was necessary to consider the existence of a frontier crossover under the assumption that the frontier is formulated by a wider range of combinations of advanced generation technology in a time horizon. Second, nuclear generation, as found in France, and hydro & renewable energy, as found in Netherlands, were important for the development of a sustainable society although the former was associated with a very high level of risk and the latter had a limited generation capability. Finally, the industry had been making a corporate effort to reduce the amount of CO2 emission

by utilizing nuclear power and renewable energy although the degree and speed of technology innovation were relatively modest. This research has three tasks to be overcome in the future. First, this study has documented only an application related to the electric power industry from 1999 to 2009. The time horizon in the data set should be expanded to obtain more long-term policy implications on the industry. That is a drawback of this study. Second, this study needs to develop a new computational framework for DEA by incorporating the concept of cost and future energy uncertainty. This study does not consider the research issue on a better fuel mix from a perspective of cost burden for the development of a sustainable society. The stochastic DEA will be a promising approach for the research task. Finally, it is possible to measure the type of RTS and the type of DTS in a time horizon. The proposed DEA approach assumes constant RTS and constant DTS to obtain computational feasibility. The determination of inter-temporal RTS and DTS in a time horizon will be further explored as a future extension of this study. Policy implications to industrial nations (including Japan): The most interesting finding was that France was efficient because of nuclear power generation and Netherlands was efficient because of hydro and renewable generation when the performance of industrial nations was measured under managerial disposability. It is widely known that the nuclear power generation is always associated with very high risk, as found in the disaster at Fukushima Daiichi in March, 2011. It is a major policy issue for all nations to pay serious attention to various types of security issues in the operation of nuclear power plants. Meanwhile, the utilization of hydro power is limited in its growth because many industrial nations have already developed dams which can generate a large amount of electricity. The space for developing large hydro power plants is now limited in many industrial nations. As a result, many nations recently pay attention to renewable energy. To promote the generation by renewable energy technologies, Japan introduced FIT (Feed-in Tariff) program in July 2012, like other nations in Europe, in which electric power companies needed to purchase all electricity produced by renewable energy sources (e.g., wind power and photovoltaic generations) by a fixed price. It is easily expected

Table 7 Malmquist index under managerial disposability (2000–2009): crossover.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Canada

France

Germany

Japan

Korea

Mexico

Netherlands

Spain

United Kingdom

United States

1.1056 1.1476 1.0568 1.0443 1.0290 1.0676 1.1380 1.0010 1.0330 1.0506

1.0416 1.0812 1.0258 1.0189 1.0178 1.0379 1.0627 1.0010 1.0253 1.0344

1.0429 1.1201 1.0482 1.0483 1.0381 1.0930 1.0725 1.0316 1.0397 1.0322

1.0363 1.1133 1.0482 1.0483 1.0381 1.0820 1.0653 1.0341 1.0364 1.0329

1.0395 1.1283 1.0482 1.0483 1.0381 1.0973 1.0753 1.0307 1.0383 1.0361

1.0442 1.1231 1.0482 1.0483 1.0381 1.0801 1.0642 1.0368 1.0366 1.0328

1.0099 1.0878 1.0482 1.0483 1.0381 1.0596 1.0504 1.0418 1.0313 1.0287

1.0769 1.1524 1.0482 1.0483 1.0381 1.1022 1.0796 1.0293 1.0453 1.0328

1.0329 1.1094 1.0482 1.0483 1.0381 1.0768 1.0620 1.0374 1.0353 1.0315

1.0494 1.1329 1.0482 1.0483 1.0381 1.0713 1.0586 1.0386 1.0346 1.0312

T. Sueyoshi, M. Goto / Energy Economics 40 (2013) 370–382

that the FIT will promote the generation by renewable energy. However, the energy density of renewable energy is low and the cost of renewable energy is expensive. Furthermore, the energy is influenced by external factors (e.g., a weather change). The low energy density of renewable technologies implies that they need to overcome a technological difficulty to efficiently produce a large amount of electricity as a main generation source. All renewable energy technologies have not yet overcome such a difficulty to become a large generation source. As a result of technological immaturity on renewable energy, this study understands that most of industrial nations, in particular Japan whose energy self-sufficiency is only 4%, depend upon nuclear power generation. However, it is important to mention that the generation is always associated with a technical issue regarding how to handle nuclear waste disposal. No country, promoting nuclear power generation, has a clear answer on the nuclear waste disposal. The issue will be a major policy problem to all nations in future, which are currently operating or planning nuclear power plants. Finally, it is necessary to discuss the importance of balancing between economic prosperity and environment protection in order to direct each nation toward a sustainable society which can attain its green growth by paying serious attention to environmental protection. To attain the sustainable society, this research looks forward to seeing future technology development on renewable energy. It is also expected that the technology development produces an economic benefit for renewable energy in such a manner that it can compete with the other conventional generation technologies without relying on any financial support such as FIT. Until the development of such a sustainable society, it is necessary for us to utilize the nuclear power generation as an important component of a fuel mix under a rigorous safety standard in a short time horizon, not a long term horizon where it is hoped that the renewable energy will be more efficient than the nuclear energy. In conclusion, it is hoped that this study makes a contribution for DEA environmental assessment. We look forward to seeing future research extensions as discussed in this article. Acknowledgments This work is supported by JSPS Grant-in-Aid for Scientific Research (C) 24530287. The authors thank two reviewers whose constructive comments have improved the quality of this study. References Bjurek, H., 1996. The Malmquist total factor productivity index. Scand. J. Econ. 98, 303–3013. Bjurek, H., Hjalmarsson, L., 1995. Productivity in multiple output public services: a quadratic frontier function and the Malmquist index approach. J. Public Econ. 56, 447–460. Bowlin, W.F., 1987. Evaluating the efficiency of US air force real-property maintenance activities. J. Oper. Res. Soc. 38, 127–135. Dyckhoff, H., Allen, K., 2001. Measuring ecological efficiency with data envelopment analysis (DEA). Eur. J. Oper. Res. 132, 312–325. Färe, R., Grosskoph, S., 1992. Malmquist productivity indexes and fisher ideal indexes. Econ. J. 102, 158–160. Glover, F., Sueyoshi, T., 2009. Contributions of Professor William W. Cooper in operations research and management science. Eur. J. Oper. Res. 197, 1–16. Goto, M., Tsutsui, M., 1998. Comparison of productive and cost efficiencies among Japanese and US electric utilities. Omega 26, 177–194. Grifell-Tatje, E., Lovel, C.A.K., 1995. A note on the Malmquist productivity index. Econ. Lett. 47, 167–175. Ijiri, Y., Sueyoshi, T., 2010. Accounting essays by Professor William W. Cooper: revisiting in commemoration of his 95th birthday. Abacus 46, 464–505. Korhonen, P.J., Luptacik, M., 2004. Eco-efficiency analysis of power plants: an extension of data envelopment analysis. Eur. J. Oper. Res. 154, 437–446. Kumar, S., 2006. Environmentally sensitive productivity growth: a global analysis using Malmquist–Luenberger index. Ecol. Econ. 56, 280–293. Liang, L., Wu, D., Hua, Z., 2004. MES-DEA modeling for analyzing anti-industrial pollution efficiency and its application in Anhui province of China. Int J Global Energy 22, 88–98. Malmquist, S., 1953. Index number and indifferences surfaces. Trab. Estat. 4, 209–2042. Oude Lansink, A., Bezlepkin, I., 2003. The effect of heating technologies on CO2 and energy efficiency of Dutch greenhouse firms. J. Environ. Manage. 68, 73–82.

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