What position does Spain occupy in the EU-28 (2008-2017) from the standpoint of natural and managerial efficiency?

doi:10.21203/rs.3.rs-1443447/v1

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What position does Spain occupy in the EU-28 (2008-2017) from the standpoint of natural and managerial efficiency?

https://doi.org/10.21203/rs.3.rs-1443447/v1

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The European Union (EU) has become the group of countries most interested in the fight against climate change and is therefore committed to reducing greenhouse gas emissions. This research analyses the Natural and Managerial efficiency of the EU-28 countries for the period 2008–2017. The Data Envelopment Analysis methodology is used and Natural and Managerial Malmquist indices are obtained by assuming that a shift in the efficiency frontier exists and by considering that frontier crossings are possible across the different periods considered. The input variables used are those of Gross Fixed Capital Formation (GFCF), non-emitting energy consumption, and employment. Gross domestic product (GDP) and greenhouse gas (GHG) emissions are taken as the desirable and undesirable output variables, respectively.

The results of the analysis indicate that both Natural and Managerial efficiency have worsened in Spain during the period considered, whereby Spain has dropped in the relevant ranking. With regard to the Malmquist Index (MI), during this entire period, the position of the evolution of Spain in relation to all EU countries was better within the sphere of Managerial efficiency, than in that of Natural efficiency.

JEL Classification: Q532

Efficiency

non-emitting energy

GHG emissions

renewable energy

DEA

Malmquist index

Currently, a large number of countries have prioritised the objective of sustainable development in order to maximise economic growth whilst minimising greenhouse gas emissions. In the specific case of European Union (EU) countries, this objective has been reinforced both from internal and external points of view by following the Paris Climate Agreement and continuing to support the implementation of emission reduction policies, thereby reducing the impact of climate change, not only within the borders of the EU, but also in developing countries (United Nations, 2015).

Air pollution has grown significantly since the early industrial era to the present day, driven largely by economic and population growth (Stern, 2013; Cook et al., 2013; Huaman and Jun 2014; Revesz et al. 2014; Rafaj et al. 2015; Wiedmann et al. 2015). This pollution from greenhouse gas (GHG) emission will produce greater warming and significant alterations in certain elements of the climate model, thereby enlarging the probability of considerable widespread harm to people and ecosystems.

The EU has long been working to reduce this pollution. In 2007, it adopted a climate and energy framework focused on the reduction of GHG emissions, the progressive increase in the use of renewable energy, and a commitment to increased energy efficiency. Specifically, Directive 2009/28 established mandatory national objectives: a reduction of GHG emissions by 20%, renewable energy consumption at 20% of the total energy consumed in the EU, and renewable energy consumption at a minimum of 10% of the energy consumed for transportation by the year 2020.

The 2030 climate policy aims to achieve reductions in greenhouse gas emissions of approximately 45% by 2030, exceeding the commitment under the Paris Agreement to decrease emissions by at least 40% by 2030, compared to 1990 (European Commission, 2018a). The goal of the EU’s long-term strategy is to achieve net zero greenhouse gas emissions by 2050, across a socially just transition in a cost-effective way (European Commission, 2018b).

In 2018, from among the EU-28 Member States, 12 of them had already achieved a rate equal to or above their national 2020 mandatory objectives: Bulgaria (20.5%), Czech Republic (15.2%), Denmark (36.1%), Estonia (30%), Greece (18%), Croatia (28%), Cyprus (13.9%), Italy (17.8%), Latvia (40.3%), Lithuania (24.4%), Finland (41.2%), and Sweden (54.6%) (Eurostat, 2019).

The objective of this paper is to carry out a comparative analysis of Spain’s position within the EU-28, in terms of environmental efficiency. To this end, the natural efficiency approach and the managerial efficiency approach are employed, as is the evolution of progress measured by the Malmquist Index, whereby the production level is maintained, and time windows of different amplitudes are used in order to obtain more information on this dynamic analysis.

In this situation, environmental evaluation is an analytical method that is widely used in the scientific research field, and is connected with the study and prevention of the negative effects of climate change in general, and of pollution, in particular. This article applies an environmental assessment methodology, based on data envelope analysis (DEA), regarding EU countries during the 2008–2017 period.

The DEA, as shown in the review by Zhou et al. (2008), has become one of the most successful methods in the fields of environmental research and analysis on energy efficiency (Mardani et al. 2017). In the field of environmental assessment, Färe et al. (1989) introduced the division of output into two categories, desirable and undesirable, which has been widely used ever since. Liu et al. (2010), Zhou et al. (2010), Sahoo et al. (2011), Wang et al. (2014), Kounetas (2015), Zhang et al. (2015), Zografidou et al. (2016), and Wang et al. (2019). Accordingly, this current work follows the Sueyoshi and Goto model (Sueyoshi and Goto 2013; Sueyoshi and Goto 2015; Sueyoshi, Goto and Wang 2017), whereby EU countries have been used as reference units, in line with Sanz-Díaz et al. (2017).

Through this approach, the intention is to shed light on the importance of the policy followed by EU countries to achieve the global objective of sustainable growth, which focuses on strategies based on the promotion of clean energy (Commission, 2010).

In our model and in line with Woo et al. (2015), employment and gross fixed capital formation (GFCF) are used as non-energy inputs, and the novelty of this paper involves the use of non-emitting energy consumption as an input. The gross domestic product (GDP) is used as a desirable output, and CO₂ emissions are used as an undesirable output. Non-emitting energy consumption includes renewable and nuclear energy; this is one of the bets of the European Commission in 2022, based on the JCR report (Abousahl et al. 2021), which has raised some controversy in the different European governments.

On the basis of these variables, the Natural and Managerial efficiency have been calculated. The first concept indicates that DMUs consider that the input vector must be reduced in order to reduce undesirable outputs, whilst increasing the desirable output vector, if possible (2012a). The second concept describes that the DMU increases the input vector in order to reduce the undesirable output vector, for which it needs to employ innovative technology to produce such an effect, whilst increasing the desirable output vector, if possible.

The Malmquist Index (MI) is also incorporated to examine Natural and Managerial efficiency, from a dynamic point of view, while contemplating the possibility of a frontier shift between two periods caused by technological progress (Rodríguez, Regueiro and Doldán 2020).

This paper is organised as follows: Section 2 describes the data employed and provides an explanation of the methodology used for the DEA, based on the measurement of Natural and Managerial efficiency and the MI, with the corresponding shift in the efficiency frontier, while using several time windows. In Section 3, the results are analysed by comparing the position of Spain with that of the remaining EU countries. Lastly, the conclusions are shown in Section 4

2.1. Methodology

The DEA is a non-parametric mathematical programming technique that calculates the efficiency frontier and indicates which Decision-Making Units (DMUs) are on the frontier, and which are not. In this analysis, a radial model with windows of two or more periods is used.

The DEA technique identifies a frontier that envelopes the data and that is used to evaluate the performance of all of the entities under analysis. The term Decision-Making Unit (DMU) represents any entity that is to be evaluated as part of a homogeneous collection, that is, all DMUs utilises and produce similar inputs and outputs. The result of this process is a set of performance scores that ranges between zero and unity, with the unity value representing the maximum efficiency. Moreover, the DEA identifies the reasons for inefficiency in each input and output for all DMUs. The set of efficient DMUs serves as a benchmark to improve future performance on inefficient DMUs.

This section begins by presenting two concepts associated with the environmental protection assessment that derive from the application of the DEA methodology to the environment, and which have been proposed by Sueyoshi and Goto (2012a, 2012b, 2012c, 2012d, 2012e, 2014, 2015) to measure Natural and Managerial efficiency of various DMUs.

The first concept refers to "Natural disposability", which indicates that DMUs consider that the input vector must be reduced in order to reduce undesirable outputs, whilst increasing the desirable output vector, if possible (2012a, 2012b, 2012c). The second concept, ‘Managerial disposability’, describes the opposite situation, in which the DMU increases the input vector in order to reduce the undesirable output vector, for which it needs to employ innovative technology to produce such an effect, while increasing the desirable output vector, if possible.

In the DEA literature, the concept of “managerial disposability” or “managerial efficiency” exposes the capacity of a DMU to optimize several outputs on a simultaneous basis (eg, increasing or at least maintaining desirable outputs while minimizing undesirable outputs), while inputs are at least maintained or increased (Exposito and Velasco, 2018). The multiple optimization problem applied in DEA (with desirable and undesirableoutputs) is usually achieved through the introduction of “innovation” into the way that inputs are used to obtain outputs, thereby allowing undesirable outputs to be reduced while still augmenting (or at least maintaining) desirable outputs (Sueyoshi and Goto 2011).

To describe the concepts of Natural and Managerial disposability using an axiomatic expression, X∈ ${R}_{m}^{+}$must be considered as the input vector, G ∈ ${R}_{s}^{+}$as the desirable output vector, and B ∈ ${R}_{h}^{+}$as the undesirable output vector. All three are column vectors, whose components are all positive.

The concepts of Natural and Managerial disposability are specified by the following production factor vectors, under constant returns to scale (RTS) and constant damages to scale (DTS), respectively, (where DTS is the parallel economic concept to RTS for the case of undesirable outputs):

$${P}^{N}\left(X\right)=\left\{\left(G,B\right) ;G\le \sum _{j=1}^{n}{G}_{j} {\lambda }_{j}; B\ge \sum _{j=1}^{n}{B}_{j} {\lambda }_{j}; X\ge \sum _{j=1}^{n}{X}_{j} {\lambda }_{j}; {\lambda }_{j}\ge 0, j=1,\dots n\right\}$$

$${P}^{M}\left(X\right)=\left\{\left(G,B\right) ;G\le \sum _{j=1}^{n}{G}_{j} {\lambda }_{j}; B\ge \sum _{j=1}^{n}{B}_{j} {\lambda }_{j}; X\le \sum _{j=1}^{n}{X}_{j} {\lambda }_{j}; {\lambda }_{j}\ge 0, j=1,\dots n\right\}$$

Note that the subscript (j) stands for the j-th DMU and ${\lambda }_{j}$ represents the j-th structural or intensity variable, for j = 1,…,n. The superscript (N) specifies the natural disposability concept, and the superscript (M) denotes the managerial disposability concept. There is a difference between the two concepts of disposability: the production technology under natural disposability, ${P}^{N}\left(X\right)$, has the inequality on inputs, $X\ge \sum _{j=1}^{n }{X}_{j }{\lambda }_{j}$; while the managerial disposability, ${P}^{M}\left(X\right)$, has the inequality $X\le \sum _{j=1}^{n }{X}_{j }{\lambda }_{j}$.

It is important to note that in the radial approach it is possible to incorporate unified inefficiency scores in the computational framework.

In production economics, there are several formulations of indices (such as the Laspeyres and Paasche price indices) as Törmquist and Fisher indices that analyse which is more appropriate when calculating indices of input and output quantities. Total Factor Productivity (TFP) (Coelli 2005) and the Malmquist Caves, Christensen and Diewert 1982) index link efficiency and productivity using a distance function based on productivity measures. The Malmquist Index examines an occurrence of a frontier shift across multiple periods.The Malmquist-Luenberger (M-L) index, introduced by Chung, Färe,, andGrosskopf (Chung, Färe and Grosskopf 1997), is particularly useful when measuring productivity and environmental performance associated with undesirable outputs. The distance function of the M-L index is characterised by the directional distance function (DDF) (Chambers, Chung and Färe 1996).

In this paper, the desirable and undesirable outputs are incorporated in order to address the environment assessment. It is therefore important to unify desirable and undesirable outputs in measurement indices, such as ecotechnology innovation for pollution reduction.

The concept of Natural and Managerial disposability is analysed on a time horizon through the MI values (Sueyoshi, Goto, Wang 2017).

2.1.1. Malmquist Index: Natural Disposability

The Malmquist indexes (MI) allow obtain interperiod changes in relative efficiency to be evaluated. (Sueyoshi et al. 2017). These indexes are estimated for alternative time periods or temporal ‘windows’, which help us to evaluate changes and trends in the medium and long term. This methodology allows for the dynamic assessment of the capacity of each country to achieve specified objectives, compared to the remaining countries

Positive change in the index (MI) under natural disposability signifies potential economic growth.

Firstly, no occurrence of frontier crossover is considered across different periods; that is to say, an efficiency frontier shifts without a frontier crossing occurring between the two periods. The MI between two periods (z-th: base and $t$–th: specific) can be specified using the following expression [24]:

$${IN}_{z}^{t}=\sqrt{\frac{{\text{U}\text{E}\text{N}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}} \frac{{\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{N}}_{t}^{R}}}$$

where ${\text{U}\text{E}\text{N}}_{z}^{R}$ is the Unified Efficiency in the $z$-th base period, and ${\text{U}\text{E}\text{N}}_{t}^{R}$ is the Unified Efficiency in the 𝑡-th base period. ${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}$ is the Inter-temporal Unified Index from the t-th period to the $z$-th period and ${\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}$ is the Inter-temporal Unified Index from the z-th period to the $t$-th period. ${\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}$ and ${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}$ may become more or less than unity, since they depend on a frontier structure (including a crossover) between the two periods. They are therefore intertemporal unified indicators. They are measured under Natural (N) disposability under constant RTS. Here, no occurrence of the frontier crossover between the two (z-th and t-th) periods is assumed.

Secondly, the possible occurrence of frontier crossover is considered and the MI may be specified, using the following expression (Sueyoshi, Goto, Wang 2017):

$${INC}_{z}^{t-1\&t}=\sqrt{\frac{{\text{U}\text{E}\text{N}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to t-1\&t}^{R}} \frac{{\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}}}$$

where ${\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to t-1\&t}^{R}$ is the Inter-temporal Unified Index from the $z$-th period to the $t-1\&t$-th period, and ${\text{U}\text{E}\text{N}}_{\text{t}\to t-1\&t}^{R}$ is the Unified Efficiency from the $t$-th period to the $t-1\&t$-th period.

Frontiers encountered in consecutive periods may intersect for various reasons. These intersections may be due either to low production in subsequent periods or to the implementation of new technologies that elevate environmental efficiency due to time lags in obtaining results.

In order to avoid infeasible solutions on index measurement, it is necessary to assume constant RTS. All efficiency and index measures are obtained by radial measurement under Natural disposability.

Each of the values applied in (1) and (2) are reached by linear programming problems, which are explained below.

No occurrence of frontier crossover

These measures are formulated by the following radial models:

The degree of unified efficiency ${\text{U}\text{E}\text{N}}_{t}^{R}$ of the k-th DMU in the t period $(t=z+1,\dots ,T)$ is measured by the following model under Natural disposability:

(P1)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

where ${d}_{i}^{x-}, 1\le i\le m$, ${d}_{r}^{g}, 1\le r\le s$ and ${d}_{f}^{b}, 1\le f\le h$ are the corresponding slack variables related to inputs, desirable outputs, and undesirable outputs, and these constitute decision variables in (P1). The column vector of unknown variables ${\lambda =(\lambda }_{1},\dots ,{\lambda }_{n})$, called the structural or intensity variables, are considered as decision variables in (P1). The scalar value, $\xi ,$ unrestricted, represents a unified inefficiency measure. The degree of inefficiency is measured as the distance between an efficiency frontier and an observed vector of desirable and undesirable outputs. The scalar value ${ϵ}_{s}$ is an Archimedean number and indicates the relative importance between the inefficiency measure and the sum of all slack variables, and it is taken as a sufficiently small number. The superscript $\vartheta$ has the value 0 for the Natural disposability, and ${J}_{t}$ stands for all DMUs in the $t$-th period.

Adjustments to the data range, R, in model (P1) are determined by the upper and lower bounds of the production factors as follows:

$${R}_{i}^{x}={(m+s+h)}^{-1}{\left(max\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{r}^{g}={(m+s+h)}^{-1}{\left(max\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{f}^{b}={(m+s+h)}^{-1}{\left(max\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

The degree of ${\text{U}\text{E}\text{N}}_{t}^{R}$ of the k-th DMU in the t period is measured by:

$${\text{U}\text{E}\text{N}}_{t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

The level of unified inefficiency under natural disposability is calculated in Eq. (3) between brackets in ${UEN}_{t}^{R}$ and the unified efficiency is attained by subtracting the level of inefficiency from unity. This can either be less than unity, which shows inefficiency, or be unity, thereby indicating full efficiency.

The degree of ${\text{U}\text{E}\text{N}}_{z}^{R}$, with respect to the k-th DMU in the period $z$, is measured by replacing $t$ with $z$ in the Model (P1), where the superscript ($*$) indicates that it is the optimum of (P1).

The degree of ${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}$ with respect to the $k$-th DMU in the $t$-th period, projecting from the $t$-th period to the $z$ period, is determined by the following model:

(P2)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{jϵ{J}_{z}}{x}_{ijz} {\lambda }_{jz}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{z}}{g}_{rjz} {\lambda }_{jz}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{z}}{b}_{fjz} {\lambda }_{jz}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jz}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of the ${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}$ index is measured by:

$${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

The degree of ${\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}$, with respect to the $k$-th DMU in the $z$-th period, projected from the $z$-th period to the $t$-th period, is determined by the following model:

(P3)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{(-1)}^{\vartheta }{d}_{i}^{x-} ={x}_{ikz }; \forall kϵ{J}_{z} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt}-{d}_{r}^{g} - \xi {g}_{rkz} ={g}_{rkz}; \forall kϵ{J}_{z} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkz}; \forall kϵ{J}_{z} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of the ${\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}$ index is measured by:

$${ \text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

Possible occurrence of frontier crossover

The possible occurrence of frontier crossover can now be considered, and for the MI to obtain it, it is necessary to solve the following two models (P4) and (P5):

The degree of ${\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}$ on the k-th DMU in the t-th period is measured by the following model:

(P4)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{jϵ{J}_{t-1\&t}}{x}_{ijt-1\&t} {\lambda }_{jt-1\&t}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t-1\&t}}{g}_{rjt-1\&t} {\lambda }_{jt-1\&t}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t-1\&t}}{b}_{fjt-1\&t} {\lambda }_{jt-1\&t}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt-1\&t}\ge 0; j=1,\dots ,n;specific t; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of ${\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}$, with respect to the $k$-th DMU in the $t$-th period, is determined as follows:

$${\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

The degree of ${\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}$ on the k-th DMU in the t-th period is measured by the following model:

(P5)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{jϵ{J}_{t-1\&t}}{x}_{ijt-1\&t} {\lambda }_{jt-1\&t}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikz }; \forall kϵ{J}_{z} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t-1\&t}}{g}_{rjt-1\&t} {\lambda }_{jt-1\&t}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkz}; \forall kϵ{J}_{z} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t-1\&t}}{b}_{fjt-1\&t} {\lambda }_{jt-1\&t}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkz}; \forall kϵ{J}_{z} ; f=1,\dots ,h$$

$${\lambda }_{jt-1\&t}\ge 0; j=1,\dots ,n;specific t; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of ${\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}$, with respect to the $k$-th DMU in the $z$-th period, is determined as follows:

$${\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

2.1.2. Malmquist Index: Managerial disposability

If there is a shift in the frontier, then potential progress occurs due to technological development and improved management in the periods used. The alteration of the index (MI) under managerial disposability also implies the potential prevention of industrial pollution.

The MI with a frontier shift between two periods may be represented as follows:

$${IM}_{z}^{t}=\sqrt{\frac{{\text{U}\text{E}\text{M}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{M}}_{\text{z}\to \text{t}}^{R}} \frac{{\text{I}\text{U}\text{I}\text{M}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{M}}_{t}^{R}}}$$

$${IMC}_{z}^{t-1\&t}=\sqrt{\frac{{\text{U}\text{E}\text{M}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{M}}_{\text{z}\to t-1\&t}^{R}} \frac{{\text{I}\text{U}\text{I}\text{M}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{M}}_{t\to t-1\&t}^{R}}}$$

It should be noted that, in order to obtain this index (Managerial), it is only necessary to replace the superscript $\vartheta$, giving the value 1 instead of zero (Natural), and to replace all ${d}_{i}^{x-}$ with ${d}_{i}^{x+}$ in the (P1), (P2), (P3), (P4), and (P5).

These models can easily be generalised for windows of three or more periods (Sueyoshi, Goto, Wang 2017). This paper presents data collected with windows of several periods.

For instance, the following models indicate the levels of potential performance, due to the frontier shift. All data is also pooled into a single balanced panel data, for the purpose of measuring the level of unified inefficiency under Natural disposability, and each k DMU in the period $t$.

(P6)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$

$$s.t. \sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{{(-1)}^{\vartheta }d}_{i}^{x+} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt} -{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt} +{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x+}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

where${R}_{i}^{x}={(m+s+h)}^{-1}{\left(max\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$

$${R}_{r}^{g}={(m+s+h)}^{-1}{\left(max\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{f}^{b}={(m+s+h)}^{-1}{\left(max\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

where the superscript $\vartheta$ has the value 1 for the Managerial disposability and ${J}_{t}$ stands for all DMUs in the $t$-th period.

The degree of ${\text{U}\text{E}\text{N}\text{T}}_{t}^{R}$ of the k-th DMU in the t period is measured by the following:

$${\text{U}\text{E}\text{N}\text{T}}_{t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

And similarly, the level of unified inefficiency under Managerial disposability and each k DMU in the $t$-th period.

(P7)$Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]$