# Assessing carbon dioxide emission reduction potential using a cost approach

Kassoum Ayouba, Jean-Philippe Boussemart, Raluca Parvulescu

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Kassoum Ayouba, Jean-Philippe Boussemart, Raluca Parvulescu. Assessing carbon dioxide emission reduction potential using a cost approach. European Journal of Operational Research, 2025, 10.1016/j.ejor.2025.10.018. hal-05321991

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Innovative Applications of O.R.

# Assessing carbon dioxide emission reduction potential using a cost approach

Kassoum Ayouba<sup>a, *,</sup> Jean-Philippe Boussemart<sup>b</sup> Raluca Parvulescu<sup>c, D</sup>

## ARTICLE INFO

### JEL codes:

Q52

Q40

D24

C61

### Keywords:

Data envelopment analysis

Carbon dioxide emissions

Emission-generating technologies

Pollution cost

Energy use

Activity model

## ABSTRACT

Departing from traditional approaches that treat carbon dioxide (CO2) emissions as a bad output, therefore relying on the weak disposability assumption, this paper considers CO2 emissions as a cost to minimise, thus preserving the materials balance condition. In particular, we extend the pollution cost approach (Coelli et al., 2007) by evaluating peer countries on their carbon intensity per total energy consumption, in addition to their energy use. The proposed methodology is then applied to estimate the extent to which a selection of 33 OECD and BRICS countries can reduce their CO2 emissions, given their gross domestic products and populations over the 2001–2019 period. Our results indicate that the period mean reduction potential for CO2 emissions of 60 % (i.e. an efficiency level of 40 %) can be decomposed into a 33 % reduction in energy intensity and a 40 % decrease in the carbon intensity of energy (i.e. efficiencies of 67 % and 60 %, respectively).

## 1. Introduction

The world's most crucial and pressing challenges involve global warming and climate change, which are primarily caused by the direct release of carbon dioxide (CO2) and other greenhouse gases (GHG) into the atmosphere. The need for urgent actions for a liveable world was highlighted during the 28th Conference of the Parties of the United Nations (UN) Framework Convention on Climate Change (COP 28) held in Dubai (United Arab Emirates, November 30–December 12, 2023) by UN Secretary-General António Guterres. He also urged world leaders to plan a total phase-out of fossil fuels, pointing out the fact that 2023 was expected to be the warmest year on record and that the window for maintaining the 1.5 °C temperature threshold was rapidly closing.

A crucial point for effective climate policies consists of identifying the underlying main drivers behind emissions. Such information can provide policymakers with guidelines on available pathways for reducing emissions and mitigating the environmental impact of human activities. To this end, an interesting starting point is the Kaya identity (1989, 1990), which expresses emissions (CO2) as a function of population (POP), gross domestic product (GDP), and energy (E) as follows:

$$CO2 = \left(\frac{CO2}{E}\right) \left(\frac{E}{GDP}\right) \left(\frac{GDP}{POP}\right) POP \quad (1)$$

Emissions are driven by four main factors: the carbon intensity of energy or the CO2 emission coefficient (CO2/E or e), the energy intensity of the economy (E/GDP), GDP per capita (GDP/POP) and population (POP). Given the size of their populations, one of the challenges faced by many developed countries is how to scale down emissions at the maximum possible whilst preserving (or limiting the decline of) economic activity. Following Kaya, a possible solution already followed by many developed countries—albeit at various speeds—is the decarbonisation of their energy systems (i.e. to act on CO2/E). This entails exploring opportunities to increase the share of renewables and nuclear in the energy mix whilst reducing the share of fossil fuels, such as coal, petroleum and gas. Another path is the reduction of the energy intensity of the economy (E/GDP) by promoting, for example, the energy-efficient renovation of dwellings and the use of energy-efficient machines. The Kaya equation can thus be reduced as follows:

$$\frac{CO2}{GDP} = e^{-\left(\frac{E}{GDP}\right)} \quad (2)$$

<footnote><sup>a</sup> CESAER, INRAE, Institut Agro, F-21000, 26, bd Docteur Petitjean 21000 Dijon, France</footnote>

<footnote><sup>b</sup> Univ. Lille, CNRS, IESEG School of Management, UMR 9221 – LEM – Lille Economie Management, F-59000, 3, rue de la Digue 59000 Lille, France</footnote>

<footnote><sup>c</sup> IESEG School of Management, Univ. Lille, CNRS, UMR 9221 – LEM– Lille Economie Management, F-59000, 3, rue de la Digue 59000 Lille, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: kassoum.ayouba@inrae.fr (K. Ayouba), jean-philippe.boussemart@univ-lille.fr (J.-P. Boussemart), r.parvulescu@ieseg.fr (R. Parvulescu).</footnote>
In the present paper, we explore the extent to which a selection of 33 OECD and BRICS countries could reduce their CO2 emissions given their GDP and population levels over the 2001–2019 period by aligning their energy intensity and energy mix with the best practices observed. Considering Kaya’s equation, this entails identifying possible decreases in both *E* and *e*. Here, we use a nonparametric production frontier framework based on activity models to estimate such potential decreases. When it comes to estimating the possible reduction of resources without decreasing production, nonparametric frameworks present the advantage of doing away with the definition of a functional form for production technology and its associated pollution cost frontier. These methods derive an efficient cost frontier based on the use of linear programming techniques to construct a surface that envelops observations. The distance of a given observation to this frontier provides an estimation of feasible reductions. To estimate the level of feasible CO2 emissions reduction while considering the current GDP and/or population, most studies in the nonparametric production frontier literature treat emissions either as undesirable outputs and use the frameworks of Färe et al. (1989) and Färe and Grosskopf (2004) or as by-products using the framework of Murty et al. (2012) and Murty and Russell (2018).

In the first approach, production technology modelling assumes the weak disposability of undesirable outputs and the null-jointness of the desirable and undesirable outputs. Intuitively, this means that the latter cannot be reduced without affecting the production of the former. Using this framework, Lin et al. (2013) measured environmental efficiency in 63 countries over the period 1981–2005 and tested whether adopting the Kyoto Protocol had a positive effect on environmental efficiency. They computed environmental efficiency using a directional output distance function, which allowed the simultaneous expansion of GDP (desirable output) and the contraction of CO2 (undesirable output). They found that high-income countries achieved the highest progress in their average environmental efficiency, whereas lower-middle- and low-income countries recorded negative growth in their average environmental efficiencies. Similarly, Zhou et al. (2006) proposed slacks-based efficiency measures to study the CO2 emissions of 30 OECD countries from 1998 to 2002. Using a selection of developed countries from 1990 to 2004, Lozano and Gutierrez (2008) estimated the minimum level of GHG emissions compatible with given levels of population, GDP, and so on. Following this approach, Boussemart et al. (2017) investigated 119 countries on all continents and estimated their shadow carbon prices using a robust nonparametric framework. Their empirical results revealed an annual increase of about 2.24 % in the global shadow carbon price, although there were significant regional disparities over the period analysed. They also found a substantial sigma convergence process in carbon prices across countries over the period 1990–2007. However, a divergence movement prevailed following the global financial crisis. Finally, they analysed the relationship between shadow carbon prices and the implementation of the Kyoto Protocol.

In the second approach (i.e. the by-production approach), two sub-technologies are considered: one generating the desired outputs and another generating the by-products. Operational and environmental efficiencies can be estimated with regard to the specific corresponding subtechnology frontier. This approach also adequately enables suitable trade-offs amongst outputs (desired and undesired) and inputs. Within this framework, Ray et al. (2018) used data on an unbalanced panel of 64 countries from 1986 to 2011 to compute for each country's opportunity costs of a targeted CO2 emission (by-product) reduction in terms of the amount of GDP reduction (desired output). Extending the by-production approach, Boussemart et al. (2020) analysed performance across the economic, environmental, and social pillars of 20 OECD countries. By linking together three subtechnologies, they modelled a global technology that combined the objectives of these three pillars. The practicality of their approach was illustrated by the analysis of policy trade-offs amongst different economic, environmental, and social objectives.

A criticism addressed by Coelli et al. (2007) to the undesirable

outputs framework modelled with the weak disposability assumption (Färe et al., 1989), is that it does not respect the materials balance principle (MBP). The MBP is a physical law stating that matter contained in inputs cannot disappear, but will either be contained in the desired outputs or discharged to the environment as waste or residuals (i.e. undesired outputs; see Ayres & Kneese, 1969). While the by-production approach respects the MBP, this method depends heavily on researchers' arbitrary choices as to the weights of the different objectives linked to each subtechnology in the global objective function (maximising the desirable outputs and minimizing the undesirable outputs). A third possible way in the literature is to approach the problem of pollution-generating technologies (i.e. the joint production of undesired and desired outputs) as a cost minimisation problem, following Coelli et al. (2007).

Specifically, this framework does not incorporate undesirables as inputs or outputs in production technology modelling. Instead, they use materials flow coefficients to identify the input mix that results in the minimal materials inflow required to produce a given bundle of desirable outputs. Their materials inflow minimisation problem mimics the well-known cost minimisation problem in economics. Lauwers (2009) provided further arguments to justify the adaptation of traditional frontier-based models to comply with the MBP. Hoang and Coelli (2011) extended the initial efficiency-based frame to estimate the intertemporal total factor productivity (TFP) performance of farms. The nutrient total factor productivity index (NTFP) also respects the MBP. More recently, Rødseth (2016) complemented this eco-efficiency estimation via the cost-polluting approach by considering whether additional inputs could be used to control pollutant emissions. By adding the minimisation of these pollution control inputs to the initial model of Coelli et al. (2007), Rødseth (2016) showed that the new efficiency measure can help improve the environmental performance rankings of the evaluated decision-making units (DMUs).

Our contributions are threefold. First, we extend Coelli et al.'s (2007) work by proposing a model that endogenises the materials flow coefficients, which are the CO2 emission coefficients in our application. This gives countries two levers to minimise emissions: reducing energy consumption and lowering the CO2 emission coefficient. Notably, Eder (2022) proposed another generalisation of the Coelli et al. (2007) model. In particular, their framework allowed for the estimation of environmental efficiency scores when the materials flow coefficients were nondiscretionary and heterogeneous across decision-making units. Contrary to our approach, the model proposed by Eder (2022) does not consider materials flow coefficients as a lever to decrease pollution; instead, they are considered nondiscretionary (uncontrollable). Furthermore, there is no direct comparison (for the potential decrease) of materials flow coefficients across all decision-making units (DMUs), and the heterogeneity of materials flow coefficients is considered by restricting the set of possible reference units of a given DMU (weight restrictions). At the same time, our generalisation relies on fewer sets of assumptions.

Second, we show how this extended model could be estimated by adapting the methodology proposed by Boussemart et al. (2022) in which DMUs minimise their costs whilst considering both input quantity and input price optimisations. In our extended Coelli et al. (2007) model, endogenous emission coefficients play the role of unit prices in the pollution cost calculation.

Third, our analysis uses a balanced sample of 33 OECD and BRICS countries spanning the period 2001–2019 to examine their energy efficiency and decarbonisation potential through the transition to renewable and low-carbon energy sources. This study is especially pertinent in the context of the current energy crisis triggered by the Ukraine War, which has underscored the need for countries reliant on imported fossil fuels to enhance their energy security. In fact, this target can be achieved by improving energy intensity and diversifying energy mixes. The insights offered by our analysis suggest valuable options for these 33 countries to decarbonise their energy systems and reduce their economic
energy intensity.

The remainder of the paper is organised into sections. In the next section, we describe our cost approach for modelling pollution with endogenous emission coefficients and present a methodology for its estimation. Section 3 presents the data used, the adaptations made to our general model to stick to the available data and the obtained results. Finally, Section 4 presents our concluding remarks.

## 2. Methodology: A cost approach for modelling pollution

Our work aims to enhance the comprehension of the different levers of action to reduce CO2 emissions and is based on a previous study (Coelli et al., 2007). The proposed methodology relies on activity models, such as data envelopment analysis (DEA), and treats pollution as a cost that must be minimised. In line with standard approaches, the Coelli et al. (2007) model assumes that emission coefficients are given. The first subsection deals with this general approach. The proposed extension allows for endogenous emission coefficients. This hypothesis makes sense for specific models, such as the one studied here, where energy is considered at an aggregate level, implying a possibly different mix of energy sources amongst DMUs and potentially different emission coefficients. Consequently, we propose to evaluate, aside from the efficiency of energy use, the emission coefficient efficiency. The adaptation of the existing model to encompass this assumption is done in the second subsection.

### 2.1. Traditional production technology with exogenous emission coefficients

Consider the case of a DMU producing an output quantity vector $\mathbf{Y} \in \mathbb{R}_+^M$, with $M = \{1, ..., m, ..., M\}$ out of a vector of input quantities $\mathbf{X} \in \mathbb{R}_+^I$, with $I = \{1, ..., i, ..., I\}$. The corresponding technology can then be defined by:

$$T = \{ (\mathbf{X}, \mathbf{Y}) \in \mathbb{R}_+^I \times \mathbb{R}_+^M : \mathbf{X} \text{ can produce } \mathbf{Y} \} \quad (3)$$

Assumptions regarding technology are standard and refer to no free lunch, boundedness, closure, free disposability and convexity (see Banker et al., 1984). If we observe $N$ DMUs with corresponding production plan vectors $(\mathbf{X}_n, \mathbf{Y}_n)$, $\forall n = 1, ..., N$, with $\mathbf{X}_n = (X_{1,n}, ..., X_{I,n}, ..., X_{I,n})$ and likewise, $\mathbf{Y}_n = (Y_{1,n}, ..., Y_{m,n}, ..., Y_{M,n})$, then the technology $T_1^{CRS}$ could be estimated by DEA under constant returns to scale (CRS) as follows:

$$T_1^{CRS} = \left\{ (\mathbf{X}, \mathbf{Y}) : X_i \ge \sum_{n=1}^{N} \mu_n X_{i,n}, \forall i \in I; Y_m \le \sum_{n=1}^{N} \mu_n Y_{m,n}, \forall m \in M; \mu_n \ge 0, \forall n \in N \right\} \quad (4)$$

In what follows, we associate with the input quantity vector $(\mathbf{X})$ a vector of emission coefficients $\mathbf{e} \in \mathbb{R}_+^I$, which leads to a total level of pollution (or emissions) defined as $P = \mathbf{X}\mathbf{e}^I$.<fnref n="1" /> Coelli et al. (2007) showed that pollution is minimised when aggregate emissions related to input use are also minimised. Thus, decreasing the pollution level for a given DMU $a$ parallels the classical cost minimisation problem (under CRS), i.e. $P = C^{CRS}(\mathbf{Y}, \mathbf{e})$, with $C^{CRS}$ defined in Eq. (5). As shown by Coelli et al. (2007), the main advantage of this method, compared with the undesirable output approach (e.g. Färe & Grosskopf, 2004; Färe et al., 1989),

is that in a cost minimisation problem, the MBP is always respected. For the technology expressed in Eq. (4), we refer to Coelli et al. (2007) and define the pollution (cost) function as follows:

$$C^{CRS}(\mathbf{Y}, \mathbf{e}) = \min_{\mathbf{X} \ge 0} \{ \mathbf{X}\mathbf{e}^I : (\mathbf{X}, \mathbf{Y}) \in T_1^{CRS} \}, \forall (\mathbf{Y}, \mathbf{e}) \in \mathbb{R}_+^M \times \mathbb{R}_+^I \quad (5)$$

In the pollution minimisation problem shown in Eq. (5), emission coefficients are given for DMUs and do not represent a lever towards the estimation of the minimum pollution level. Whilst they are not necessarily identical across DMUs, they can be considered exogenous, thus reducing the problem to an input quantity allocation. Indeed, under this approach, DMUs seek only the optimal input quantities, given the emission coefficients. Their strategy to reduce emissions is based uniquely on an 'input quantity effect', whilst completely ignoring the possibility of acting on their emission coefficients and on their energy mixes. Related to traditional cost minimisation and allocative efficiency models, the limitation of analyses based on exogenous input prices has been emphasised several times in the literature (Ayouba et al., 2019; Boussemart et al., 2022; Camanho & Dyson, 2008; Portela & Thalanousis, 2014).

In this work, we propose to extend the existing pollution cost model (i.e. Coelli et al., 2007) based on exogenous emission coefficients by considering that DMUs, when choosing different input quantity mixes, end up having different emission coefficients for those mixes. In other words, the emission coefficients will be endogenised in our approach. In the Appendix, we present a comparison between our results and the ones that would have been obtained by adapting and applying the methodology in Coelli et al. (2007) to our data set.

### 2.2. Proposition of a pollution cost approach with endogenous emission coefficient

The standard framework of analysis overlooks variations in emission coefficients. As a result, it is unable to assess DMUs' efficiency in this regard. Emission coefficients for each homogenous type of energy are assumed to be exogeneous for each DMU and not subject to cross-comparison amongst DMUs.<fnref n="2" /> However, for some combined forms of energy, e.g. fossil fuels, there can be different categories (coal, petroleum, natural gas), each of them associated with a different emission coefficient. Therefore, when dealing with energy use and CO2 emissions, the set $I$, defining the input dimension is to be understood as an aggregation of different forms of energy sources. As a result, the associated emission coefficient is a weighted average of emission coefficients from these different energy sources. This leads to a form of heterogeneity amongst the DMUs, both from the perspective of the input quantities used and their respective mixes, as revealed by the potentially different weighted emissions factor. Consequently, DMUs dispose of two levers to reduce their pollution costs: change the quantity of polluting inputs and/or choose a different mix of polluting inputs, resulting in different emission coefficients. From a practical perspective, this approach has the advantage of giving DMUs more room for action compared with the traditional approach.

In the recent literature, Boussemart et al. (2022) proposed a standard cost minimisation method in which DMUs minimise their costs while considering both input quantities and input prices as optimisation

<footnote n="1">¹ Similarly, we can assume $e_Y \in \mathbb{R}_+^M$ to be a vector of technical coefficients associated with the output. From this perspective, the methodology developed here, based on the cost minimization problem, can be extended to include both input and output emission coefficients in a profit maximization approach.</footnote>

<footnote n="2">² When analysing a detailed nomenclature grid of a polluting input like energy, which includes various types of fossil fuels (e.g., coal, petroleum, natural gas, oil shales, bitumens, tar sands and heavy oils), it is reasonable to assume that the emission factor for each fuel type is identical based on physicochemical laws. Thus, comparing DMUs on this dimension would not be pertinent.</footnote>
levers.<fnref n="3" /> In adapting such a methodology to the problem studied in the present work, we must first define a technology that encompasses the emission coefficients and is defined under variable return to scale, $T_2^{VRS}$. This new technology is obtained by imposing two additional constraints in Eq. (4): the first one is related to the emission coefficients, and the second deals with the sum of activity variables. The resulting technology is defined as follows:

$$T_2^{VRS} = \left\{ \begin{array}{l} (\mathbf{X}, \mathbf{e}, \mathbf{Y}) : \mathbf{X}_i \ge \sum_{n=1}^{N} \mu_n \mathbf{X}_{i,n}, \quad \forall i \in I; \quad e_i \ge \sum_{n=1}^{N} \mu_n e_{i,n}, \quad \forall i \in I; \\ \mathbf{Y}_m \le \sum_{n=1}^{N} \mu_n \mathbf{Y}_{m,n}, \quad \forall m \in M; \quad \sum_{n=1}^{N} \mu_n = 1; \quad \mu_n \ge 0, \quad \forall n \in N \end{array} \right\} \quad (6)$$

This methodology can easily be adapted to study pollution cost, as shown below.

$$C^{VRS}(\mathbf{Y}) = \min_{\mathbf{x},e \ge 0} \{\mathbf{X}\mathbf{e}^T : (\mathbf{X}, \mathbf{e}, \mathbf{Y}) \in T_2^{VRS}\}, \quad \forall \mathbf{Y} \in \mathbb{R}_+^M \quad (7)$$

Coelli et al. (2007) assume that the cost minimisation problem is realised under constant returns to scale (see Eq. (5)), thus allowing for a rescale of input and output quantities for a given set of emission coefficients. However, the model proposed by Boussemart et al. (2022) was developed under variable returns to scale. Indeed, this more general assumption prevents the rescale of all corresponding variables, including prices, which would have been irrelevant in this specific dimension (prices). In our current approach, a more accurate framework should allow for a rescale in the quantity dimensions (input and output), but not in the emission coefficient one. This issue is solved by showing that a total cost minimisation problem in CRS is equivalent to an average cost minimisation problem in VRS. Using previous results in the literature (Ray, 2004, 2015; Ray et al., 2021) and adapting them in our study, we define the VRS average cost minimization problem as $AC^{VRS}(\mathbf{y}, \mathbf{e})$. To obtain it, one can simply refer to Eq. (5) where the equivalent CRS cost minimization was stated and adopt the following transformation:

$$C^{CRS}(\mathbf{Y}, \mathbf{e}) = AC^{VRS}(\mathbf{y}, \mathbf{e}) \quad (8)$$

where, for a chosen fixed output $\mathbf{Y}_{m'} \in \mathbb{R}_+^1$ ($m' \in M$), $\mathbf{y} = \frac{1}{\mathbf{y}_{m'}} \mathbf{Y}$, $\mathbf{x} = \frac{1}{\mathbf{y}_{m'}} \mathbf{X}$.

These rescaled vectors are generally referred to as the “input and output intensities” relative to the chosen fixed output. Thus, the technology defined in Eq. (4) has a VRS equivalent (i.e. $T_1^{VRS}$) obtained by replacing the input and output quantity variables with their relative measures obtained by the transformation in Eq. (8).

$$T_1^{VRS} = \left\{ \begin{array}{l} (\mathbf{x}, \mathbf{y}) : \mathbf{x}_i \ge \sum_{n=1}^{N} \mu_n \mathbf{x}_{i,n}, \quad \forall i \in I; \quad \mathbf{y}_m \le \sum_{n=1}^{N} \mu_n \mathbf{y}_{m,n}, \quad \forall m \in M; \\ \qquad \sum_{n=1}^{N} \mu_n = 1; \quad \mu_n \ge 0, \quad \forall n \in N \end{array} \right\} \quad (9)$$

With these notations, the associated VRS cost function is defined as follows:

$$AC^{VRS}(\mathbf{y}, \mathbf{e}) = \min_{\mathbf{x} \ge 0} \{\mathbf{x}\mathbf{e}^T : (\mathbf{x}, \mathbf{y}) \in T_1^{VRS}\}, \quad \forall (\mathbf{y}, \mathbf{e}) \in \mathbb{R}_+^M \times \mathbb{R}_+^I \quad (10)$$

The problem in Eq. (10) corresponds to the minimisation of the average cost for producing one unit of the fixed output. By referring to these clarifications and returning to Eq. (7), we are now able to provide the VRS version of the average cost minimisation problem that deals

with endogenous emission coefficients. For this, we define $T_1^{VRS}$ as the transformed technology that includes the emission coefficient.

$$T_{1, \text{full}}^{\text{VRS}} = \left\{ \begin{array}{l} (\mathbf{x}, \mathbf{e}, \mathbf{y}) : \mathbf{x}_i \ge \sum_{n=1}^{N} \mu_n \mathbf{x}_{i,n}, \quad \forall i \in I; \quad e_i \ge \sum_{n=1}^{N} \mu_n e_{i,n}, \quad \forall i \in I \\ \mathbf{y}_m \le \sum_{n=1}^{N} \mu_n \mathbf{y}_{m,n}, \quad \forall m \in M; \quad \sum_{n=1}^{N} \mu_n = 1; \quad \mu_n \ge 0, \quad \forall n \in N \end{array} \right\} \quad (11)$$

The optimisation of the average cost value with respect to both input intensity ($\mathbf{x}$) and emission coefficient ($\mathbf{e}$) vectors is given by:

$$AC^{VRS}(\mathbf{y}) = \min_{\mathbf{x},e \ge 0} \{\mathbf{x}\mathbf{e}^T : (\mathbf{x}, \mathbf{e}, \mathbf{y}) \in T_{1,\text{full}}^{VRS}\}, \quad \forall \mathbf{y} \in \mathbb{R}_+^M \quad (12)$$

Eq. (12) naturally leads to solving a nonlinear additive problem because the cost minimisation solution results from the product between the optimal input quantity and the optimal emission coefficient. Specifically, the average pollution cost is minimised by activating two levels: the first one is the pollution generating input quantity, and the second one is the emission coefficient associated with those inputs. The solutions from this nonlinear additive model could be obtained using the algorithm proposed by Ray (2004, pages 124–125). This algorithm is based on Taylor’s first order series approximation that transforms the nonlinear objective function into a linear one.<fnref n="4" />

Note that the piecewise linear technologies defined so far assume that marginal products are always decreasing (see Banker & Maindiratta, 1986). In our estimation strategy, detailed in the next section, we have opted for a more flexible approach (based on piecewise log-linear technology), which better fit the data<fnref n="5" /> as it allows for both locally increasing and/or decreasing marginal products, depending on the specific characteristics of each country. The piecewise log-linear technology is derived from a set of fundamental properties that are essentially comparable to those of piecewise linear technologies. As emphasized by Banker and Maindiratta (1986), the key difference lies in the convexity assumption, which is geometric rather than linear. A recent publication by Ferreira et al. (2021) traces back the literature on this subject and also explicitly gives a list of the advantages of this model over the standard linear one (Section 3.2 in their paper).

The associated technology is derived from the technology presented in Eq. (11) and is defined as follows:

$$T_{1, \text{geom.full}}^{\text{VRS}} = \left\{ \begin{array}{l} (\mathbf{x}, \mathbf{e}, \mathbf{y}) : \mathbf{x}_i \ge \sum_{n=1}^{N} \mu_{i,n}^{\mu_n}, \quad \forall i \in I; \quad e_i \ge \sum_{n=1}^{N} \mu_{i,n}^{\mu_n}, \quad \forall i \in I; \\ \mathbf{y}_m \le \sum_{n=1}^{N} \mu_{m,n}^{\mu_n}, \quad \forall m \in M; \quad \sum_{n=1}^{N} \mu_n = 1; \quad \mu_n \ge 0, \quad \forall n \in N \end{array} \right\} \quad (13)$$

Following Banker and Maindiratta (1986), the log-linearization of the geometric convex technology in Eq. (13) is presented as follows:

$$T_{1, \text{loglin,full}}^{\text{VRS}} = \left\{ \begin{array}{l} (\ln(\mathbf{x}), \ln(\mathbf{e}), \ln(\mathbf{y})) : \\ \qquad \ln(\mathbf{x}_i) \ge \sum_{n=1}^{N} \mu_n \ln(\mathbf{x}_{i,n}), \quad \forall i \in I; \quad \ln(\mathbf{e}_i) \ge \sum_{n=1}^{N} \mu_n \ln(\mathbf{e}_{i,n}), \quad \forall i \in I \\ \qquad \ln(\mathbf{y}_m) \le \sum_{n=1}^{N} \mu_n \ln(\mathbf{y}_{m,n}), \quad \forall m \in M; \quad \sum_{n=1}^{N} \mu_n = 1; \quad \mu_n \ge 0, \quad \forall n \in N \end{array} \right\} \quad (14)$$

We provide in the following section a detailed presentation of our estimation strategy based on a piecewise log-linear estimation of cost-

<footnote n="3">³ In their approach and contrary to the standard approach, DMUs are evaluated not only with regard to their input quantities, while considering prices as given (an acceptable hypothesis for perfectly competitive input markets), but also with regard to their observed prices (related to the assumption that input markets are not perfectly competitive).</footnote>

<footnote n="4">⁴ We thank the referee for suggesting this algorithm that allows the nonlinear problem to be estimated more easily.</footnote>

<footnote n="5">⁵ See Banker and Maindiratta (1986), who demonstrated that a piecewise log-linear technology better fits the data than a traditional piecewise linear technology.</footnote>
efficient frontier for CO2 emissions across OECD and BRICS countries.

### 3. Estimation strategy, data and results

In this empirical section, we estimate the CO2 emission reduction potential for a panel of 33 OECD and BRICS countries. This section is divided into three parts: an outline of our estimation strategy, a presentation of the data employed for the estimations and a discussion of our main findings.

#### 3.1. Estimation strategy

As we position our work within the reduced Kaya framework (see Eq. (2)), our model is based on one output, GDP, and one input, Energy (E), where Energy = fossils + nuclear + renewables.<fnref n="6" /> Consequently, the output and input vectors are defined as scalars: $Y = GDP$ and $X = E$. The rescaled output and input variables with regard to the output (GDP) are equal to $y = (\frac{GDP}{GDP}) = (1)$ and $x = (\frac{E}{GDP})$. The rescaled output corresponds naturally to the unit, whereas the rescaled input gives the quantity of energy used to produce one unit of GDP, which is called ‘energy intensity’.<fnref n="7" /> The pollution cost is given by $C = Ee$, with $e$ is the emission coefficient associated to E.

By applying the log-linear minimization problem in Eq. (12) to the case studied here, we get:

$$ \ln(AC^{\text{VRS}}(1)) = \min_{\ln(E/GDP), \ln(e) \ge 0} \left\{ \ln\left(\frac{E}{GDP}\right) + \ln(e) \right. \\ \left. \qquad : \left(\ln\left(\frac{E}{GDP}\right), \ln(e), \ln(1)\right) \in T_{1, \log_{\text{fin}}(1)}^{\text{VRS}} \right\} \quad (15) $$

The cost problem in Eq. (15) can be estimated with the log-linear program (see LP1), which can be solved using traditional linear solver algorithms. As explained above, the pollution cost is minimised by activating two levers: the first one is for the energy intensity (E/GDP), and the second one is the energy-related emission coefficient (e). Consequently, for an evaluated DMU $a$, $\sigma$ represent the change in the energy intensity that is necessary to achieve the lowest pollution cost. Likewise, $\omega$ is the change in the emission coefficient that is necessary to achieve the same objective. The observed pollution cost is then $(E_a/GDP_a) \times e_a$, whereas the minimum achievable pollution cost by this DMU is $[\omega \times (E_a/GDP_a)] \times [e_a \times \sigma]$, where $\sigma \times \omega$ is the overall cost pollution decrease (with $\sigma \times \omega \le 1$). With a log-linear technology, our objective function for pollution cost minimization is $\ln([\omega \times (E_a/GDP_a)] \times [e_a \times \sigma]) = \ln([\omega \times (E_a/GDP_a)]) + \ln([e_a \times \sigma]) = \ln(\omega) + \ln(E_a/GDP_a) + \ln(\sigma) + \ln(e_a)$. Given that for a DMU $a$, $E_a/GDP_a$ and $e_a$ are the observed values, the cost minimization problem then reduces to minimizing $\ln(\omega) + \ln(\sigma)$ as in LP1 below. In the same way, the specific constraints for $E_a/GDP_a$, and respectively for $e_a$, are expressed, under the log-linear technology in additive form, as in (c1) and (c2) in LP1. Finally, note that the constraint on the output (GDP) is dropped from LP1 as it provides no new information for the estimation.

$$ \text{Indeed, } \sum_{i=1}^{N} \tau_n \ln \left( \frac{GDP_a}{GDP_r} \right) \ge \ln \left( \frac{GDP_a}{GDP_r} \right) \Leftrightarrow 0 \ge 0. $$

$$ \left\{ \begin{array}{l} \min_{\sigma, w, \tau} (\ln(\sigma) + \ln(w)) \\ \text{s. t.} \\ \sum_{i=1}^{N} \tau_n \ln\left(\frac{E_n}{GDP_n}\right) \le \ln(\sigma) + \ln\left(\frac{E_a}{GDP_a}\right) \quad (c1) \\ \\ \sum_{i=1}^{N} \tau_n \ln(e_n) \le \ln(w) + \ln(e_a) \quad (c2) \\ \\ \sum_{i=1}^{N} \tau_n = 1 \quad (c3) \\ \\ \tau_n \ge 0, \forall n \in N \quad (c4) \end{array} \right. \qquad (\text{LP1}) $$

LP1 provides the optimal CO2 emissions by unit of GDP, which can be found through the (log) linear combinations of existing countries, thus making it a unique solution (a global optimum). Theoretically speaking, several potential combinations of the pollution cost components, $(\frac{E}{GDP})$ and $(e)$, can satisfy the objective function. However, by using the same vector for the intensity variables in both constraints (constraints c1 and c2 in LP1), the referent countries for the optimal level of energy intensity are identical to those found for the emission coefficient. Therefore, from a computational perspective, the number of available combinations for the optimal values of the two scores, $\sigma$ and $\omega$, is probably very limited. In particular, it seems highly unlikely that there are any multiple combinations of optimal solutions for each component. Thus, to verify this, we ran two alternative versions of LP1. In the first version, we added a constraint whereby the energy intensity score should be lower than or equal to the emission coefficient score ($\sigma \le \omega$). In the other version, we introduced the converse alternative: $\sigma \ge \omega$. We then checked whether the respective optimal solutions of these two versions were equal. The results revealed that they were unequal and that the lowest value was obviously equal to the solution found by the original programme, LP1. This finding suggests that the optimal solutions found for the two scores are unique.

We enriched LP1 in two directions to obtain the final log-linear program (LP2) used in our analysis to estimate the minimum possible pollution emission per country. First, we consider that the environmental policies of countries focus on the impacts in terms of pollution caused by quantities and mixes of energy, whilst economic activity and demography are likely to be a matter of economic and social policies. In this context, we assume that GDP and population levels are exogenous variables that are not directly linked by energy choices. In other words, the minimum emission cost per unit of GDP is obtained for a given level of GDP per capita. As a result, the observed GDP/POP is treated as a control variable, and a specific constraint with equality must be added, thereby ensuring that the optimal solution respects the evaluated country's standard of living (constraint c3 in LP2).

Second, we argued that the optimal solution should maintain roughly the same economic structure as the evaluated country, which means that the shares for the main economic sectors in the GDP should be respected. For this, we explicitly consider in the new linear programme (LP2) the corresponding shares of the primary, secondary and tertiary sectors in the GDP of each country. We introduce the following notations: let $k$ be the economic sector concerned, with $k = 1$ for the agricultural sector, $k = 2$ for the secondary sector and, finally, $k = 3$ for the tertiary sector. For any country $a$, let $s_a^k$ be the observed share of the sector $k$ in the GDP of that country. The sum of these shares, for the optimal solution must obviously be equal to 100 % (constraint c4 in LP2).

Furthermore, let $h_a^k$ and $h_a^{-k}$ be the potential increase and decrease of the share of each sector within the GDP of the evaluated country $a$. Note that $h_a^k$ and $h_a^{-k}$ are endogenous to the optimisation problem, and they each represent the structural changes needed so that a country can

<footnote n="6">⁶ The resulting weighted average emission coefficient will likely vary amongst DMUs (i.e. countries) as they are confronted with an aggregate mix of polluting inputs.</footnote>

<footnote n="7">⁷ In line with the KLEM standard framework, this analytical frame can be extended to include, besides energy, other input variables, such as capital and labor. However, such an extension poses several challenges in terms of data reliability, notably the capital stock variable for the BRICS and Eastern European countries. Nevertheless, even if such data were reliable, one should keep in mind that the emission coefficients associated with these extra inputs can be assumed as being null (i.e. they do not contribute to CO2 emissions). Consequently, these extra input variables are excluded from the objective function and are not considered decision variables.</footnote>
reach the frontier in terms of efficient CO2 emissions per unit of GDP (constraint c5 in LP2).

From a short-to mid-run point of view, it is reasonable to assume that evaluated countries have limited flexibility to adopt structural changes; thus, they should be limited to a given threshold. Obviously, this will also have an impact on the CO2 emission reduction that can be obtained. For this, we introduce the Hamming distance, denoted as HD, which refers to a scalar that is arbitrarily fixed by the analyst to control for the degree of flexibility in the reallocation of resources.<fnref n="8" /> In the short-run scenario presented, we assume HD = 10%, which means that the evaluated country has a latitude to adjust its economic structure by ±5% (constraint c6). Thus, the final linear programme is presented below.

$$
\left\{
\begin{array}{l}
\min_{\sigma,w,\tau} (\ln(\sigma) + \ln(w)) \\
\text{s. t.} \\
\sum_{i=1}^{N} \tau_n \ln\left(\frac{E_n}{\text{GDP}_n}\right) \le \ln(\sigma) + \ln\left(\frac{E_a}{\text{GDP}_a}\right) \quad (c1) \\
\sum_{i=1}^{N} \tau_n \ln(e_n) \le \ln(w) + \ln(e_a) \quad (c2) \\
\sum_{i=1}^{N} \tau_n \ln\left(\frac{\text{GDP}_n}{\text{POP}_n}\right) = \ln\left(\frac{\text{GDP}_a}{\text{POP}_a}\right) \quad (c3) \\
\sum_{k=1}^{3} \sum_{i=1}^{N} \tau_n s_n^k = 100 \quad (c4) \\
\sum_{i=1}^{N} \tau_n s_n^k = s_a^k + h_a^{+k} - h_a^{-k}, \forall k = \{1, 2, 3\} \quad (c5) \\
\sum_{k=1}^{3} h_a^{+k} - h_a^{-k} \le HD \quad (c6) \\
\sum_{i=1}^{N} \tau_n = 1 \quad (c7) \\
h_a^{+k}, h_a^{-k} \ge 0, \forall k = \{1, 2, 3\} \quad (c8) \\
\tau_n \ge 0, \forall n \in N \quad (c9)
\end{array}
\right.
\qquad (LP2)
$$

## 3.2. Data and variables

For our analysis of CO2 emissions, we analysed data from the selected 33 countries<fnref n="9" /> (28 OECD countries + 5 BRICS countries) over the period 2001–2019. In line with the estimation strategy outlined earlier, we sourced relevant variables for estimating LP2. Our data came from the World Bank and the US Energy Information Administration, as detailed in Table 1. GDP is measured in constant 2011 international US dollars, adjusted for purchasing power parity (PPP), while CO2 emissions are reported in millions of tonnes. Total energy consumption, recorded in quadrillion Btu, was obtained as the aggregation of fossil fuels (coal, gas and oil), nuclear and renewable energy sources.

Over the last two decades and relative to the world total, these 33 countries have accounted for 76 % of total GDP, 60 % of the global population, 79.5 % of energy consumption and >80 % of CO2 emissions. An analysis of the trends in the selected variables indicates that GDP per capita grew at an annual rate of 2.40 % due to a faster increase in GDP than in population. At the same time, energy intensity (E/GDP) decreased at a rate of -1.17 % annually due to a differential in growth in favour of GDP compared with overall energy consumption. Finally, the CO2 emission coefficient (e = CO2/E) improved slightly over the period, with a trend decrease of -0.10 %.

Fig. 1a displays the indices (and growth rates) of key variables in level, as shown in the first four rows of Table 2. Complementing this, Fig. 1b presents the respective evolutions of GDP relative to population, energy intensity, and the CO2 emission coefficient corresponding to the last three rows in Table 2.<fnref n="10" /> An analysis of both figures leads to the observation that a significant rise in GDP per capita and a moderately increasing global population are the primary drivers of CO2 emissions. However, the observed reductions in energy intensity and emission coefficients, achieved partly by shifting towards carbon-free energy sources, have not been sufficient to decrease overall CO2 pollution effectively.

Algebraically, the relationships between the related growth rates of the variables in the Kaya identity (trends for 2001–2019 in %) can be expressed as follows:

$$
\begin{align*}
\text{CO2} &= e\left(\frac{E}{\text{GDP}}\right)\left(\frac{\text{GDP}}{\text{POP}}\right)\text{POP} \\
&\Rightarrow \\
\frac{d\text{CO2}}{\text{CO2}} &= \frac{de}{e} + \frac{d\left(\frac{E}{\text{GDP}}\right)}{\left(\frac{E}{\text{GDP}}\right)} + \frac{d\left(\frac{\text{GDP}}{\text{POP}}\right)}{\left(\frac{\text{GDP}}{\text{POP}}\right)} + \frac{d\text{POP}}{\text{POP}} \\
1.95\% &= -0.10\% - 1.17\% + 2.40\% + 0.82\%
\end{align*}
$$

## 3.3. Results and discussion

To gauge the collective ability of the 33 selected countries to reduce their CO2 emissions, we applied LP2 to the aggregate production plan.<fnref n="11" />

<footnote n="8">⁸ The concept of the Hamming distance (HD), originally from fuzzy set theory (Kaufmann, 1975), was initially introduced within a DEA context by Dervaux et al. (1997) and then applied later by Boussemart et al. (2016).</footnote>

<footnote n="9">⁹ The selected countries include Australia, Austria, Belgium, Brazil, Canada, China, Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, India, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Republic of Korea, Russia, Slovakia, South Africa, Spain, Sweden, Switzerland, Turkey, the UK and the USA.</footnote>

<footnote n="10">¹⁰ Note that the Kaya identity in Equation (1) was obtained based on these three ratios.</footnote>

<footnote n="11">¹¹ The aggregate production plan is developed in two stages. First, we calculated the total GDP, energy consumption, and population for the entire sample. Then, these aggregates were used to derive the components of the production technology required for the estimation in LP2, namely energy intensity, emissions coefficient, and GDP per capita. Similarly, the share of each sector was determined following a comparable approach.</footnote>
<table><caption>**Table 1**
Definition, unit of measurement and source of the variables.</caption><thead><tr><td>Variables</td><td>Definition</td><td>Unit</td><td>Source</td></tr></thead><tbody><tr><td>GDP</td><td>GDP, PPP constant 2011 international $</td><td>106 US $ 2011</td><td>https://data.worldbank.org/indicator</td></tr><tr><td>POP</td><td>Total population</td><td>10<sup>6</sup></td><td>https://data.worldbank.org/indicator</td></tr><tr><td>E</td><td>Energy consumption</td><td>10<sup>15</sup> Btu</td><td>https://www.eia.gov/international/data/world</td></tr><tr><td>CO2</td><td>CO2 emissions</td><td>10<sup>6</sup> Tons</td><td>https://www.eia.gov/international/data/world</td></tr><tr><td>E/GDP</td><td>Energy intensity</td><td>10<sup>3</sup> Btu/$</td><td></td></tr><tr><td>e</td><td>Emission coefficient</td><td>mg/Btu</td><td></td></tr></tbody></table>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Illustrations of the Evolution of the Indices (and Their Trends) for the Main Variables in the Kaya Identity.</figcaption></figure>
<table><caption>**Table 2**
Descriptive statistics of the variables.</caption><thead><tr><th></th><th>Average 2001–2019</th><th>Total 33 countries</th></tr></thead><tbody><tr><td rowspan="3">GDP</td><td>Level</td><td>74 593 497</td></tr><tr><td>% of Total world</td><td>75.99 %</td></tr><tr><td>Trend 2001–2019 (%)</td><td>3.22 %</td></tr><tr><td rowspan="3">POP</td><td>Level</td><td>4 191</td></tr><tr><td>% of Total world</td><td>60.19 %</td></tr><tr><td>Trend 2001–2019 (%)</td><td>0.82 %</td></tr><tr><td rowspan="3">E</td><td>Level</td><td>410</td></tr><tr><td>% of Total world</td><td>79.51 %</td></tr><tr><td>Trend 2001–2019 (%)</td><td>2.05 %</td></tr><tr><td rowspan="3">CO2</td><td>Level</td><td>25 245</td></tr><tr><td>% of Total world</td><td>80.18 %</td></tr><tr><td>Trend 2001–2019 (%)</td><td>1.95 %</td></tr><tr><td rowspan="3">GDP/POP</td><td>Level</td><td>17 799</td></tr><tr><td>Trend 2001–2019 (%)</td><td>2.40 %</td></tr><tr><td>Level</td><td>5.50</td></tr><tr><td rowspan="2">E/GDP</td><td>Trend 2001–2019 (%)</td><td>-1.17 %</td></tr><tr><td>Level</td><td>61.55</td></tr><tr><td rowspan="2">e</td><td>Trend 2001–2019 (%)</td><td>-0.10 %</td></tr><tr><td>Level</td><td>61.55</td></tr></tbody></table>

As shown in Fig. 2, the annual CO2 score fluctuates between 35 % and 44 %, indicating a potential reduction capacity range of 56 %–65 %. Notably, the first major contributor to pollution cost efficiency is energy intensity (E/GDP), with an efficiency score ranging from 63 % to 70 %. In comparison, the efficiency score related to the emission coefficient, which reflects the energy mix, is comparatively lower, ranging from 55 % to 65 %. Globally, our results indicate that over the entire period of analysis, the mean CO2 emission reduction potential is 60 % (i.e. an efficiency level of 40 %), which can be decomposed into a 33 % reduction in energy intensity and a 40 % decrease in the carbon intensity of energy (i.e. efficiencies of 67 % and 60 %, respectively).

The analysis of efficiency scores derived from the LP2 estimation for each individual country in the dataset revealed a notable potential for CO2 emission reduction across various countries should they adopt the best energy management practices. From Fig. 3, we observe that in terms of the efficiency for CO2 emissions per unit of GDP, Switzerland, Brazil and India are the most efficient countries, followed by Norway, Sweden, Turkey, France, Portugal, Hungary and Mexico. At the other end of the distribution, the least efficient countries are Australia, Canada, South Africa, Russia, the Netherlands, the USA and South Korea. Using an undesirable output framework, Iram et al. (2020) produce this type of ranking based on countries CO2 emissions in 2017. They however consider only OECD countries, which make the comparison of the results not simple. Nevertheless, when we focus on the 24 common countries between the two studies, we note that their eight most efficient countries are the same as ours (Norway, Sweden, France, Hungary, Mexico, Switzerland, Turkey and Portugal), in different order, however. The same observation applies to the four least performers (Australia, USA, Canada, and the Netherlands). Note that, as in our analysis, Australia is the least efficient country in Iram et al. (2020). Wu et al. (2018) also use an undesirable output framework to rank a sample of Asia-Pacific Economic Cooperation (APEC) countries based on their CO2 emissions efficiency over the period 2006–2010. The countries that appear both in our analysis and in Wu et al. (2018) are as follows: Australia, Canada, China, Japan, Mexico, New Zealand, USA, Republic of Korea, and Russia. Considering these countries, while from our estimation the most efficient country is Mexico, followed by New Zealand and Japan respectively, in Wu et al. (2018), the most efficient is New Zealand, followed by Japan and Mexico, respectively. Thus, the three best performers are the same. We also note that, in Wu et al. (2018), Australia and Canada outperform Republic of Korea and China. The exact opposite is observed in our analysis.

Aside from their overall CO2 efficiency scores, countries can be categorised according to a typology that crosses the two efficiency scores, namely, energy intensity and emission coefficient related to the mix of different types of energies. In Fig. 4, we plotted the trends for

each score, enabling a comparison of the dynamics between these two dimensions. Thus, the X-axis shows the trend over the period studied for the score associated with the E/GDP variable (the score σ), whereas the Y-axis represents the trend over the same period for the score associated with the emission coefficient (the score ω). Therefore, countries can be classified along the four frames to which they belong. At the origin of this plot, we find countries for which the two scores observed show completely flat evolutions (Switzerland, Brazil and India). At the same time, countries that have managed to increase their efficiency in both dimensions are plotted in the upper right-hand frame. With regard to the 45° diagonal (the dashed lines), for some countries situated beneath this diagonal, the increase in the efficiency of the energy-to-GDP ratio was stronger than the improvement observed by the emissions rate efficiency (Czech Republic, the UK, Hungary, Mexico, Russia, New Zealand, Poland and Slovakia). The reversed situation was observed for countries in the same frame but above the 45° diagonal (Denmark, Sweden and Turkey). France and Spain observed an increase in their CO2 emission coefficient, whereas their energy intensity efficiency scores followed a rather flat evolution.

In the left-hand, upper frame, one can find the countries for which the improvement in the emission coefficient efficiency was associated with a decrease in the efficiency of the energy-to-GDP ratio. For countries above the 45° diagonal (Belgium, Finland, Greece, Iceland, Italy, Portugal and the USA), the increase in the emission coefficient efficiency was stronger than the deterioration of the efficiency of energy consumption per unit of GDP. The reverse was observed for countries in the same frame but situated below the diagonal (Australia and Norway). For Austria, we noticed that the trends for the two efficiency scores were equal, indicating that they deteriorated at the same pace.

In the bottom left frame, the countries experiencing a negative trend in both efficiencies throughout the study period are presented. In China's case, the decline in energy mix efficiency was more pronounced than the decrease in energy use efficiency per unit of GDP. For the Netherlands, the two respective time trends were identical, demonstrating that the deterioration in energy intensity efficiency was comparable to the decline in CO2 emission coefficient efficiency. Meanwhile, in Canada, the energy intensity efficiency score showed a slightly greater deterioration than the CO2 emission coefficient score.

Finally, in the last frame, except for Japan, the remaining countries showed efforts to improve their energy-to-GDP efficiency, accompanied by a deterioration in their energy mix efficiencies (bottom right-hand side). In particular, this trend was observed in Germany, South Korea, Poland, Slovakia and South Africa. In New Zealand, the two respective trends did not show significant evolutions, indicating that this country showed relatively flat evolutions in both dimensions.

We conducted the analysis by presenting the geometric mean for each score for the entire period of study. As shown in Fig. 5, for each country, the geometric means of these scores are illustrated in reference to the whole sample aggregate production plan mean score. First, we notice that aside from Switzerland, India and Brazil, 11 other countries showed performances above the aggregate production plan, as shown in both scores. These countries demonstrated a combination of policy initiatives, technological advancements and shifts in energy sources, all of which contributed to their good performance in energy efficiency in terms of energy intensity and the choice of energy mixes. In particular, for Switzerland, France and Brazil, this efficient positioning is explained by their good profiles both in terms of energy intensity and choice of energy mixes in favour of renewable and nuclear energy (specifically France). For India, although its emission coefficient is amongst the worst, its overall efficiency is explained not only by a relatively good position in terms of energy intensity, but also by the specificity of having a very low level of GDP per capita. On the one hand, it seems that Austria, Spain, Great Britain, Hungary, Italy and Japan owe their good performance, to a large extent, to their increased efforts in implementing energy efficiency policies that have improved their energy intensity over time. Notably, Mexico appears to be slightly super-efficient from this
<figure><img src="image_5.png" /><figcaption>Fig. 2. Efficiency scores (in %) for the aggregated production plan.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Ranking of countries according to their CO2-to-GDP emission scores (Period Geometric Mean).</figcaption></figure>

perspective. On the other hand, Finland and Norway have made sig-
nificant investments in renewable energy sources. Their energy mixes
are increasingly dominated by hydroelectric, wind and solar power, all
of which contribute to lower CO2 emissions. Notably, in our estimation,
Norway appears to be highly efficient in terms of its CO2 emissions
coefficient.

In the upper left-hand panel, we displayed countries that generally
surpass the aggregate production plan in terms of emission coefficient
efficiency but exhibit weaker performance regarding their energy in-
tensity efficiency. Notably, Iceland and Sweden are highly efficient on
the first criterion (energy mix efficiency). However, despite the devel-
opment of renewable sources, such as hydropower and geothermal en-
ergy, these two countries still face challenges in reducing their overall

energy consumption per unit of output. The energy mix performance of
some countries, such as Russia and the USA, is relatively comparable to
that of the aggregate production plan. Conversely, both countries exhibit
inefficiencies in energy intensity due to several contextual factors. In
Russia, for example, its inefficiency largely stems from its extensive
reliance on energy-intensive industries, such as oil and gas extraction,
and a legacy infrastructure that often lacks modern, energy-efficient
technologies. The harsh climatic conditions further exacerbate energy
consumption for heating purposes. Similarly, in the USA, its inefficiency
in terms of energy intensity can be attributed to its large-scale industrial
sector, widespread use of automobiles and generally high per capita
energy consumption. In comparison, the relatively good performance of
the energy mix efficiencies of Belgium, Canada, New Zealand and
<figure><img src="image_7.png" /><figcaption>Fig. 4. Typology of the countries with respect to the trend for each efficiency score.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Typology of countries by their efficiency profiles (Period Geometric Mean).</figcaption></figure>
Slovakia is mainly due to their efforts in integrating renewable energy sources into their energy portfolios. Belgium and New Zealand, for instance, have made significant strides in incorporating wind and hydroelectric power. Slovakia, with its emphasis on nuclear energy, has managed to reduce its reliance on fossil fuels. Canada, despite its vast fossil fuel reserves, has been diversifying its energy mix with considerable investments in hydropower and other renewable sources. However, the energy intensities in these countries remain inefficient. Specifically, for Canada, this inefficiency is driven by its vast geographical size, leading to high energy consumption in transportation and logistics. In addition, Canada's substantial industrial sector, especially in resource extraction industries such as mining and oil sands, is inherently energy intensive.

Meanwhile, China's performance in terms of energy mix can be explained by its gradual shift towards incorporating renewable energy sources alongside its traditional reliance on coal. This transition includes significant investments in solar and wind energy, which progressively reduce the country's dependence on coal, although it remains the dominant energy source. Regarding energy intensity, China's inefficiency is primarily due to its rapid industrialisation and urbanisation, which have led to substantial increases in energy demand. Yet, despite recent efforts to improve energy efficiency, the country continues to have high energy consumption relative to its GDP due to the sheer scale of its industrial and manufacturing activities, coupled with its growing urban energy needs.

In the left-hand, bottom pane, Australia, South Korea, the Netherlands and South Africa exhibit poorer results, compared with the aggregate production plan, in both energy intensity efficiency and energy mix efficiency. This finding is mainly due to their heavy reliance on fossil fuels and energy-intensive industrial sectors. For example, Australia's inefficiency can be attributed to its dependency on coal for both domestic use and export and its significant potential for renewable energy, which remains underutilised. Similarly, South Korea's industrialised economy, dominated by high-energy-consuming sectors such as electronics and automotive, relies substantially on imported fossil fuels, thus impeding its efficiency. The Netherlands, with its strong industrial base and historic reliance on natural gas, faces challenges in transitioning to a more efficient energy mix. Meanwhile, South Africa's predominantly coal-based energy sector suffers from infrastructural and economic constraints, which are further exacerbated by energy-intensive mining and manufacturing sectors.

Finally, in the right-hand bottom pane, we find the Czech Republic, Germany, Denmark, Greece, Poland and Turkey, whose energy intensity scores outperform the one obtained for the aggregate production plan, whilst their performance in terms of the efficiency of their energy mix shows deterioration compared with the aggregate production plan. All of these countries have made sustained efforts to adopt energy-efficient technologies for different sectors of their economies. However, these efforts have been hampered by an energy mix that is still dominated by coal and fossil fuels.

To reach the efficient frontier in terms of CO2 emissions, countries should not only decrease their overall energy consumption but also change their energy mix. The observed energy mix (left-hand side in Figs. 6–8) is computed as the share of each type of energy (fossils, renewable and nuclear) in the total quantity of energy for each country cumulated over the entire period of study. The optimal energy mix (right-hand side in Figs. 6–8) is defined on the same basis and computed from the optimal shares of each component (in %) of the optimal energy quantity.<fnref n="12" />

Overall, we find that fossil energies would have to decrease their shares from 85 % to 63 %, while renewable energies would have to increase from 9 % to 27 %. Another unsurprising finding is the increase in the share of nuclear power from 6 % to 10 %, which also contributes to the reduction of CO2 emissions (see Fig. 6).

A considerable decrease in the use of fossil fuels in favour of renewable energies can be found in most countries, but even more so in the most polluting ones, with pronounced nuances in the use of nuclear energy. For example, for China, the first contributor to the total CO2 emissions of the sample (33 % of the total CO2 emissions in our sample), the weight of fossil energies should be reduced (from 91 % to 77 %), with nuclear power maintaining a small share of around 1 %–2 %, whilst renewable energies should increase from 8 % to 21 % (see Fig. 7). For the USA, the second contributor to total CO2 emissions (22 % of our sample), there should also be a drastic decrease in the share of fossil fuels (from 83 % to 56 %) in favour of an increase in nuclear power (from 8 % to 21 %) and renewables (from 8 % to 23 %) (see Fig. 8).

## 4. Conclusion

To limit the deterioration of the environment, it has been increasingly acknowledged that past and current energy use patterns should be adjusted. Due to this requirement, developed countries are confronted with the challenge of scaling down CO2 emissions from their energy use to the maximum possible while preserving (or limiting the decline of) economic activity at the same time. In this article, we provide an answer to this crucial question. Specifically, we use a nonparametric pollution cost frontier framework to explore the extent to which a selection of 33 OECD and BRICS countries could reduce their CO2 emissions given their GDP and population levels over the period 2001–2019. We extend the model proposed by Coelli et al. (2007), consistent with the materials balance condition, in which the problem of pollution-generating technologies (i.e. the joint production of undesired and desired outputs) is approached as a cost minimisation problem. In the traditional frame, Coelli et al. (2007) assumed that emission coefficients are given; in comparison, our contribution consists of endogenising them. Indeed, when dealing with heterogeneous energy mixes, the resulting weighted coefficient emission factors can differ across countries. Therefore, in our approach, the overall performance of each selected country related to their CO2 emissions was assessed not only with respect to their energy use per unit of GDP, but also with regard to the carbon intensity of the energy mix they used. We also provide an operational procedure by which to estimate our extended model.

Our results showed that the percentages of CO2 emission reduction potential varied between 56 % and 65 %, depending on the year and on condition that the countries adopted the best energy management practices. This outcome results from a reduction in energy intensity (between 30 % and 37 %) and a reduction in the carbon intensity of energy (between 35 % and 45 %). Interestingly, at the sample aggregate scale, the reduction in the carbon intensity of energy was obtained by decreasing the share of fossil energies (from 85 % to 63 %) and increasing the share of renewable energies (9 % to 27 %) and nuclear energy (6 % to 10 %). As it is increasingly acknowledged that past and current energy use patterns should be adjusted to limit the deterioration of the environment, our results provide interesting insights into the currently available options that can be adopted by these countries.

The improvements in CO2 emissions presented in the current paper have been achieved by considering a short-term perspective, where countries' economic structures remain relatively unchanged, using the shares of the primary, secondary, and tertiary sectors in the GDP as controls. With more disaggregated data on the sectoral composition of the analyzed countries, future research could further control sectoral differences between each evaluated country and its benchmark. For example, one may set a bound on the combined share of some polluting sectors (e.g., the manufacturing, mining, and transportation sectors) in the GDP in the estimation process. Future research could also explore a

<footnote n="12">¹² The optimal energy shares are obtained by applying the optimal activity variables ($r_t$) to the observed shares of each energy component for the referents composing the benchmark of the evaluated country. In this way, LP2 allows for a reallocation of energy types but ensures that the resulting total energy is equal with the total optimal quantity which is a decision variable in the program.</footnote>
<figure><img src="image_9.png" /><figcaption>Fig. 6. Observed and optimal energy mixes (Mean 2001–2019).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Observed and optimal energy mixes - China (Period Mean).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Observed and optimal energy mixes - United States (Period Mean).</figcaption></figure>

long-term perspective, where countries might achieve greater emissions reductions by appropriately modifying their economic structures. In this context, such an extension could be linked to previous studies that address structural effects in the index decomposition literature (see, e.g., Ang & Zhang, 2000; Ang, 2015).

CRediT authorship contribution statement

Kassoum Ayouba: Writing – review & editing, Writing – original draft, Methodology, Formal analysis, Conceptualization. Jean-Philippe Boussemart: Writing – review & editing, Writing – original draft, Methodology, Formal analysis, Conceptualization. Raluca Parvulescu: Writing – review & editing, Writing – original draft, Methodology, Formal analysis, Conceptualization.
Appendix

We compare our results with those obtained by applying the approach developed in Coelli et al. (2007). First, we present LP3, which corresponds to the adaptation of Model (17) in Coelli et al. (2007) to our method based on the (log) linearization of geometric combinations.

$$
\left\{
\begin{array}{l}
\displaystyle \min_{E/GDP, \tau} \left( \ln\left(\frac{E}{GDP}\right) + \ln(e_a) \right) = \left( \ln\left(\frac{E}{GDP}\right) \right) \\
\text{s. t.} \\
\displaystyle \sum_{i=1}^{N} \tau_n \ln\left(\frac{E_n}{GDP_n}\right) \le \ln\left(\frac{E_a}{GDP_a}\right) \\
\displaystyle \sum_{i=1}^{N} \tau_n \ln\left(\frac{GDP_n}{POP_n}\right) = \ln\left(\frac{GDP_a}{POP_a}\right) \\
\displaystyle \sum_{k=1}^{3} \sum_{i=1}^{N} \tau_n s_n^k = 100 \\
\displaystyle \sum_{i=1}^{N} \tau_n s_n^k = s_a^k + \hat{h}_a^{+k} - \hat{h}_a^{-k}, \forall k = \{1, 2, 3\} \\
\displaystyle \sum_{k=1}^{3} \hat{h}_a^{+k} - \hat{h}_a^{-k} \le HD \\
\displaystyle \sum_{i=1}^{N} \tau_n = 1 \\
\displaystyle \tau_n \ge 0, \forall n \in N \\
\displaystyle \hat{h}_a^{+k}, \hat{h}_a^{-k} \ge 0, \forall k = \{1, 2, 3\}
\end{array}
\right.
\qquad (LP3)
$$

In the above model, the goal is to find the optimal level of energy intensity ($\frac{E}{GDP}$) that ensures the lowest level of emissions for the selected countries given the level of emission coefficient ($e_a$). In this model, the focus is on the energy intensity, whilst the emission coefficient is treated as a constant and does not intervene in the optimisation process. A modified version of LP3 can be found in LP3bis, where the potential decrease of energy intensity to reach the benchmark is measured by the parameter $\delta$.
$$
\left\{
\begin{array}{l}
\min_{\delta, \tau} (\ln \delta) \\[1em]
\text{s. t.} \\[1em]
\sum_{i=1}^{N} \tau_n \ln \left( \frac{E_n}{\text{GDP}_n} \right) \le \ln \left( \frac{E_a}{\text{GDP}_a} \right) + \ln(\delta) \\[1em]
\sum_{i=1}^{N} \tau_n \ln \left( \frac{\text{GDP}_n}{\text{POP}_n} \right) = \ln \left( \frac{\text{GDP}_a}{\text{POP}_a} \right) \\[1em]
\sum_{k=1}^{3} \sum_{i=1}^{N} \tau_n s_n^k = 100 \\[1em]
\sum_{i=1}^{N} \tau_n s_n^k = s_a^k + \hat{h}_a^{+k} - \hat{h}_a^{-k}, \forall k = \{1, 2, 3\} \\[1em]
\sum_{k=1}^{3} \hat{h}_a^{+k} - \hat{h}_a^{-k} \le HD \\[1em]
\sum_{i=1}^{N} \tau_n = 1 \\[1em]
\tau_n \ge 0, \forall n \in N \\[1em]
\hat{h}_a^{+k}, \hat{h}_a^{-k} \ge 0, \forall k = \{1, 2, 3\}
\end{array}
\right.
\qquad (\text{LP3bis})
$$

Fig. A1 displays a comparison of our results based on the application of the two models for the aggregated production plan. As can be seen, the potential for CO2 emissions per unit of GDP is much higher when using our model than when using the original model proposed by Coelli et al. (2007).
<figure><img src="image_12.png" /><figcaption>Fig. A1. Comparison between the results obtained with our model and the geometric adaptation of model (17) in Coelli et al. (2007).</figcaption></figure>

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