A New Metric for CO₂ Emissions Based on the Interaction Between the Efficiency Ratio Entropy/Marginal Product and Energy Use

Abstract

In an era of growing climate concerns and complex environmental policy challenges, novel approaches for accurate carbon emissions measurement are urgently needed. This article introduces an innovative approach for predicting carbon dioxide emissions by analyzing the interaction between energy consumption and production efficiency, measured through an entropy-to-marginal product ratio. Unlike conventional metrics such as Eurostat measurements or the Kaya identity, our framework establishes explicit connections to fundamental physical laws governing energy transformation while offering flexible elasticity parameters that capture non-linear relationships between efficiency improvements and emission reductions. The research combines theoretical modeling with empirical validation across ten European countries, demonstrating how the entropy-based methodology accounts for both production complexity and energy efficiency where traditional linear models fall short. Analysis reveals that energy-efficient countries demonstrate lower entropy maximization under stable conditions, indicating a direct relationship between operational efficiency and environmental impact. Although the model demonstrates strong predictive capabilities with an exceptional accuracy/information cost ratio, limitations exist in achieving accuracy in some country cases. This study concludes by evaluating these strengths and constraints, acknowledging the need for extended time series analysis and sector-specific applications, and providing clear directions for future research that bridge this promising theoretical contribution with practical environmental policy applications.

1. Introduction

The dual challenges of sustaining economic growth and mitigating climate change have become increasingly intertwined in the 21st century. As global economies strive for higher levels of production and prosperity, the concomitant increase in energy consumption has led to a significant rise in carbon dioxide (CO₂) emissions, the primary driver of anthropogenic climate change [1]. This dilemma necessitates a re-evaluation of how we measure and assess the environmental impact of economic activities. Traditional economy-based metrics like Gross Domestic Product (GDP) have long been used as proxies for economic well-being and development. However, these measures fail to account for the environmental externalities associated with production processes, particularly the emission of greenhouse gases like CO₂. As the world grapples with the urgent need to decarbonize economies while maintaining growth trajectories, more nuanced and comprehensive metrics are required. Assessing carbon dioxide emissions in economic production processes requires a comprehensive set of metric methodologies to capture the quantity and impacts of emissions. Various methodological approaches have been developed and refined over time to accurately quantify these emissions from different sources. Let us list the best known without the order of enumeration reflecting relevance.

The first approach is the bottom–up approach. These can be subdivided into two groups. The direct measurement is using sensors or monitoring devices to measure CO₂ concentrations directly at emission sources like power plants or industrial facilities. The second sub-group is related to emission factors, estimating emissions based on activity data (e.g., fuel consumption) and emission factors (e.g., kg CO₂/GJ of fuel). The second is the top–down approach [2]. Likely, these can be split into two sub-categories: the atmospheric inversion using atmospheric CO₂ concentration measurements and transport models to infer surface fluxes. The second sub-group category is satellite remote sensing using satellites like OCO-2 or GOSAT to measure global CO₂ concentrations. The Life Cycle Assessment (LCA) [3] approach evaluates CO₂ emissions throughout a product’s life cycle, from raw material extraction to disposal. The next approach (e.g., [4]) well known by economists is input–output analysis. It uses economic input–output tables to trace CO₂ emissions through supply chains. The carbon footprint approach (e.g., [5]) consists of assessing the total CO₂ emissions associated with an individual, organization, event, or product. The uncertainty analysis approach [1] aims at quantifying and reducing uncertainties in CO₂ emission estimates. Key aspects of uncertainty analysis in CO₂ quantification include the identification of sources of uncertainty and next their quantification [2]. Then, this approach may be helpful to increase the level of precision of already existing estimates. Worthy of enumeration is the approach of the Integrated Assessment Models (IAMs) [6]. These combine economic, energy, and climate systems to assess CO₂ emissions and impacts. The last but not least approach on this list is machine learning and AI, a set of emerging techniques [7] using AI to improve emission estimates or predict future emissions. ML and AI offer transformative potential in quantifying and managing CO₂ emissions across sectors. They can handle complexity, integrate diverse data sources, and provide insights for targeted interventions. However, their responsible development and deployment are crucial to ensure they truly help to reach the intended precision for carbon dioxide quantifying and thereby benefit global climate action.

These methodologies are continually evolving with advancements in technology, data availability, and understanding of carbon cycle processes. The literature in this field is vast and interdisciplinary, spanning atmospheric sciences, economics, engineering, policy studies, and more. To give readers the empirical scope of these approaches,

Table 1. Empirical illustration of different CO₂ metric approaches.

Metric Approach	Sample Empirical Output	Source
Life Cycle Assessment (LCA)	Electric Vehicle Production: 8.8 tonnes CO2e per vehicle (Tesla Model 3 cradle-to-gate)	Tesla Impact Report 2022
Carbon Footprint Analysis	Coca-Cola Company: 3.7 million tonnes CO2e (2022, Scope 1 and 2)	Coca-Cola ESG Report 2022
Input–Output Analysis	US Manufacturing Sector: 1.4 tonnes CO2e per USD 1 M output (2020)	US EPA GHGRP Data 2020
Process Analysis	Cement Production: 842 kg CO2e per tonne of cement (Direct emissions)	Global Cement and Concrete Association (GCCA) Sustainability Report 2023
Hybrid Methods	Global Smartphone Production: 55 kg CO2e per device	Apple Environmental Progress Report 2023 (iPhone 14 lifecycle assessment)

Source: Our own, based on firm reports.

provides examples of sample empirical illustration of selected CO₂ metric approaches.

While current metrics for assessing carbon dioxide emissions in economic production processes provide valuable insights, they also have several limitations. Main limitations:

As far as carbon footprint and Life Cycle Assessment (LCA) are concerned, Wiedmann and Minx [5] point out that carbon footprint calculations often lack consistency in methodology and system boundaries. They argue that there is no clear consensus on whether to include only direct CO₂ emissions or all greenhouse gases expressed as CO₂ equivalents. Majeau-Bettez et al. [8] highlight that LCA studies can suffer from incomplete system boundaries, leading to underestimation of emissions. They also note that data quality and availability can significantly impact results. Concerning input–output analysis limitations, Lenzen [9] discusses how input–output analysis, while useful for assessing economy-wide emissions, often relies on aggregated data that may not accurately represent specific industries or processes. Su et al. [10] point out that input–output tables are typically published with a significant time lag, which can lead to outdated assessments in rapidly changing economies. Next, regarding production-based vs. consumption-based accounting, Peters [11] argues that production-based accounting, which is commonly used in international agreements, fails to account for carbon leakage through international trade. This can lead to misleading assessments of a country’s true carbon footprint. Davis and Caldeira [4] demonstrate how consumption-based accounting can provide a more accurate picture of a country’s carbon responsibility but note that it is more complex to implement and requires more data. Concerning the emissions intensity-based metrics, Ang [12] discusses how emissions intensity metrics (e.g., CO₂ per unit of GDP) can be misleading due to changes in economic structure rather than actual efficiency improvements. The next limitations are related to uncertainty in emission factors. Rypdal and Winiwarter [13] highlight the significant uncertainties in emission factors used to calculate CO₂ emissions, particularly in non-energy sectors like agriculture and land-use change.

Acquaye et al. [14] discuss an important point about the difficulties in accurately measuring embodied carbon in products and services, particularly in complex supply chains that span multiple countries. About temporal and spatial resolution, Gurney et al. [15] argue that many metrics lack sufficient temporal and spatial resolution to accurately capture emissions patterns, particularly in urban areas where emissions can vary significantly over short distances and time periods. Next, Guan et al. [16] highlight the particular challenges of assessing CO₂ emissions in rapidly developing economies, where data quality and availability can be especially problematic.

Matthews et al. [17] point out that the lack of standardization in carbon accounting methods makes it difficult to compare results across different studies and sectors. Concerning the limitations of Shannon entropy-based methods, despite the advantage of representing a holistic approach, this form of entropy is limited to Gaussian systems and struggles with non-linear systems common in economic processes (Chen, et al. [18]).

To conclude this paragraph, while numerous metrics and methods exist for assessing CO₂ emissions in economic production processes, each has its limitations. The main challenges revolve around data quality and availability, system boundary definition, handling of complex and non-linear systems, and the need for more comprehensive approaches that can capture the full lifecycle and global nature of emissions. Future research should focus on developing more integrated and standardized approaches that can overcome these limitations.

Generalized Entropy Metric Approach

Using entropy as a metric to assess carbon emissions from production processes offers a holistic approach that takes into account the overall disorder or randomness of the system. In fact, this is so because a dynamic optimization of entropy under time- and space-constrained environmental conditions will lead to a long-term equilibrium solution. Entropy captures the complexity of the production system, including the multitude of processes, inputs, and outputs involved. High entropy implies greater complexity and potentially higher carbon emissions due to diverse activities and resource utilization. Entropy reduction signifies improved resource efficiency and streamlined production processes, which can lead to lower carbon emissions. By minimizing waste, inefficiencies, and redundant activities, companies can reduce their environmental footprint. Next, entropy can reflect the energy efficiency of production processes. Higher entropy may indicate greater energy dissipation and inefficiency, resulting in higher carbon emissions per unit of output. Entropy, a measure of disorder or dispersal of energy in a system, offers a unique lens through which to view the interplay between energy consumption, economic output, and carbon emissions [19,20]. The entropic metric we propose goes beyond simplistic emission factors or aggregated carbon intensities. It encapsulates the inherent irreversibility and inefficiencies in energy conversion processes that underpin economic activities. By optimizing this entropic function under constraints linking energy consumption to GDP, we aim to provide insights into the most efficient pathways for economic production that minimize CO₂ emissions. Our approach builds upon a rich interdisciplinary literature spanning thermodynamics, ecological economics, and sustainability science. It draws inspiration from the early seminal works on the application of entropy in economics (e.g., [21,22]) and more recent studies on energy–economy interactions (e.g., [23]). We posit that this entropic assessment of CO₂ emissions can offer valuable alternative insights for policymakers, businesses, and researchers striving to navigate the complex trade-offs between economic imperatives and environmental sustainability. For the ergodic systems, well described by the Gaussian class of laws, the solution generated by the Tsallis entropy converges to the Gibbs–Shannon solution (e.g., [24]). The underlying hypothesis is that the economic production system may display complexity, suggesting heavy queue events plausibly with long dynamic correlation.

Following the above, this paper proposes a generalized non-linear version of the Shannon entropic approach in its Tsallis entropy form. This is theoretically best suited to assess CO₂ emissions [25] in the context of complex economic production processes. In the following sections, we will elucidate the theoretical foundations of the Tsallis entropic metric, outline the methodology for its application to real-world economic data, and present case studies demonstrating its utility in guiding policy decisions. That being said, we will avoid going into the theoretical details of this approach in order to avoid drawing readers towards very technical considerations that are not essential to understanding the proposed discussions. Nevertheless, ad hoc references will be proposed. The entropy-based metric introduced in this article contributes an important physical perspective previously underrepresented in emissions analysis. By connecting carbon outputs directly to thermodynamic principles, it bridges a gap between physical science and policy applications, potentially enabling more effective interventions based on a deeper understanding of the fundamental processes driving emissions. Ultimately, as this article will make clear, the proposed measure adds, among other things, an important physical dimension that can guide expectations about what efficiency improvements can actually achieve. Based on what has just been presented in this introduction, let us summarize key elements of the paper. In order to apply a new metric based on the Tsallis entropy metrics for CO₂ emissions, the proposed hypothesis is to consider energy no longer as a raw material feeding the physical capital factor but itself a leading economic production factor (alongside the human factor). This strong hypothesis will make it possible to experiment with the proposed metric by linking the production process with CO₂ emissions. This justifies the rationale for Section 2.1, which should clarify, based on existing literature, the validity of this hypothesis. The remainder of this paper is organized as follows: Section 2 presents the model, beginning with an examination of energy’s key role through a comprehensive literature review linking energy factors with economic production. This section emphasizes energy’s fundamental position as a primary factor in economic production. The subsequent subsection introduces a model for assessing entropy generation through GDP production using energy as a factor, incorporating a power law-based maximum entropy framework. Section 3 presents the model outputs and analysis, concluding with a detailed discussion of the power law-based entropy model’s strengths and limitations in light of the obtained results. A subsection is proposed in this section to compare the entropy-based metric with two others applied on a large scale: that of Eurostat and the Kaya identity. The final Section 4 draws conclusions and identifies potential areas for further research, considering the hypotheses and model proposed in this paper.

2. The Model

2.1. The Uniqueness of Energy in Economic Production Processes

This sub-section emphasizes energy’s leading role as a production input in economic processes. Energy’s fundamental role in economic production represents a significant departure from traditional economics, which typically treats energy as merely a raw material rather than a primary production factor. Under this perspective, economic production (GDP) is fundamentally a derivative of energy use, with energy activating capital stock while generating entropy measurable through CO₂ emissions. Capital stock—encompassing energy conversion devices, information processors, and facilities—functions as an energy multiplier, while labor serves as its manager. This economic system functions as a dissipative structure, continuously incorporating energy, matter, and management as inputs while producing economic output and CO₂ emissions [26,27].

The historical understanding of wealth creation and production factors has evolved significantly since Aristotle first described the concept of energy [27]. The Physiocratic school, led by François Quesnay in 1759 [14], viewed wealth as primarily agricultural in origin. Later, classical and neoclassical economists, including Adam Smith, Karl Marx, and John Maynard Keynes [28,29], focused on traditional production factors like land, labor, and capital, largely overlooking energy’s crucial role. This perspective persisted through twentieth-century economic models, from Solow’s work [30,31] to endogenous growth models [32,33,34,35], which continued to exclude energy as a fundamental factor of production. As Nobel laureate P. Krugman noted, despite these new growth theories, the reasons for varying development success across countries remain somewhat mysterious. More recent models to explain the underdevelopment of some countries have begun to argue for disparities in social capital. Social capital results from specific forms of interpersonal relations, which have their historical and cultural conditions and are usually characteristic of some territory in which they operate. It also includes informal social institutions. Within this framework, the connections between community cohesion, well-being, and local development factors are also emphasized [36,37]. It should be noted that the attempt to incorporate more and more growth factors into modeling, which also takes into account spatial, historical, and environmental aspects, leads to model estimation problems owing, in the best case, to collinearity. This problem has led to the dilemma of describing economic growth through statistical recording, the use of taxonomic methods, or mathematical models of economic growth. During the last decades, economists have begun to realize the central role of energy in the process of economic production and growth. This is probably related to the gradual reduction in fossil fuel energy reserves combined with political limitations of their exploitation for reasons of environmental security plus other socio-political crises like that of the 1970s. These have highlighted the scarcity character of this good and emerged insights of some economists about its central role as a factor of production. The relationship between energy consumption (EC), economic growth, and environmental impact has been extensively studied across various geographical contexts and timeframes. In Belgium, research spanning 1960–2012 revealed a long-term relationship between EC and GDP, with a 17% convergence rate [38]. This study employed ARDL and Toda–Yamamoto approaches, confirming system stability and demonstrating GDP’s positive impact on EC in both the short and long terms. Some researchers have conceptualized the economy as an energy conversion system, where energy flows directly toward producing goods and services. This perspective has enhanced our understanding of labor and capital roles while explaining both the historical stagnation of living standards before 1800 and the subsequent dramatic economic growth [39]. Regional studies have yielded diverse findings. In Shandong Province (1980–2008), researchers identified a long-term relationship between energy consumption and economic growth, with two-way causality [40]. A broader study of 45 countries (1991–2014) confirmed triangular relationships among energy consumption, income, and trade openness [41]. European Union research (2008–2019) covering 34 countries revealed a one-way relationship from economic growth to energy consumption [42]. Asian countries presented varied results: Pakistan showed unidirectional causality from coal to GDP and from GDP to electricity consumption; Nepal demonstrated causality from petroleum to GDP; India showed no causal direction; and Bangladesh and Sri Lanka exhibited causality from GDP to energy consumption [43]. A comprehensive literature review revealed varying support for different hypotheses: growth (43.8%), conservation (27.2%), feedback (18.5%), and neutrality (10.5%) [44]. Regarding environmental impacts, Grossman and Krueger found evidence for an Environmental Kuznets Curve (EKC) for various pollutants, though CO₂ results were less definitive [45]. Dinda’s research showed many cases of monotonically increasing relationships between CO₂ and income [46]. Stern argued that apparent EKC relationships often disappear when controlling factors like trade and structural change [47]. Country-specific studies revealed strong correlations between energy consumption and CO₂ emissions in France [48], while Indonesia demonstrated bidirectional causality between economic growth and CO₂ emissions [49]. The diverse and sometimes contradictory findings in these studies suggest potential methodological limitations, particularly the use of linear models to explain inherently non-linear economic processes. This highlights the need for more sophisticated analytical approaches that better capture the complex relationships between energy consumption, economic growth, and environmental impact. This is exactly the aim of this paper, which uses the entropic model based on the power law, the details of which are presented below.

2.2. The Power Law Based Entropy Model

Let us introduce the Tsallis entropy concept applied in this paper. This generalized version of the traditional Gibbs–Shannon entropy accounts for non-ergodic systems. The Tsallis entropy derives from the power law. Following the above section, the mainstream literature seems still to neglect the relationship between power law (PL) (e.g., [50,51,52]) and economic phenomena processes—probably because the Gaussian family of laws is globally sufficient for time (or space) aggregated data and easy to use and interpret. Nevertheless, recent literature (e.g., [24,53,54,55]) shows that the amplitude and frequency of economic and financial fluctuations do not deviate substantially from many natural or manmade dissipative structures once explained on the same scale of time or space by PL. While concluding his recent paper, the author Gabaix [56] wrote the following: The future of power laws as a subject of research looks very healthy; when datasets contain enough variation in some “size”-like factor, such as income or number of employees, power laws seem to appear almost invariably. In addition, power laws can guide the researcher to the essence of a phenomenon. Fortunately, this quote fits into the second hypothesis posed in the Introduction. The main point to underscore is the fact that when the aggregated data converge to the Gaussian attractor, as typically happens, the outputs of the Gibbs–Shannon entropy [57] (this belongs to the exponential family of laws) coincide with Tsallis entropy derived from Pl (for details see e.g., [58]). At this particular point, the q-Tsallis parameter is equal to unity, while its values may vary between zero and three. This q-Tsallis parameter value is a function of the phenomenon’s statistical structure and complexity. Referring to [56], a PL is the form taken by a remarkable number of regularities in economics and finance, among other things. It is a relation of the following type:

$$y\cong k{x}^{-\alpha }$$

where $y$ and $x$ are variables of interest, α is the PL exponent, and $k$ is typically a scaling constant. Referring to the existing literature, a consensus emerged from different authors underscoring proportional random growth as the key mechanism that explains the formation of power law structures. Nevertheless, the economic study applying PL originated with Yule [59], and it was developed in economics by Champernowne [60] and Simon [61] and studied in depth by Kesten [62]. Concerning the inheritance mechanism for PL, see Jessen and Mikosch [63] for a survey. In this interesting study, the authors have formally shown the relationship between PL and other statistical laws. In particular, among the laws with the same variance, normal law will display the highest entropy. After this short presentation of the power law related Tsallis entropy, the next section proposes through an econometric model formulation a technological process describing economic production for which observed data may display a PL-related entropy. Concerning the link between economic production and power law, see e.g., [60,64,65].

Table 2. Entropy-based model outputs for 10 EU countries. Response variable: GDP per capita. Regressor: consumed energy per capita.

Country	Estimates		Model R2 _Equivalent	Optimal Entropy Values
	Beta	Constant
Denmark	5.443	1.335	0.876	0.435
Belgium	3.836	1	0.918	0.45
Germany	4.867	1.089	0.937	0.488
Netherlands	4.279	1.603	0.334	0.496
Czechia	2.963	0.872	0.955	0.52
Ireland	3.731	2.417	0.565	0.525
France	5.037	0.71	0.951	0.528
Estonia	2.204	0.832	0.943	0.552
Poland	3.065	0.742	0.982	0.552
Bulgaria	2.01	0.575	0.982	0.555

Source: Our own calculations.

has been computed to analyze the direct relationships between GDP and energy input within the entropy production perspective. This is to suggest that besides parameter estimation allowing us to understand quantitative relationships between GDP production and energy, we want to add to this knowledge an additional piece of information about entropy being produced through energy input to produce GDP, having in mind that economic production systems that are more organized will consume less energy per produced unit and relatively less entropy to achieve the equilibrium. Put in similar terms, systems with no constraints will produce the same level of (optimal) entropy, while systems that are more constrained will display lower entropy. The applied Tsallis maximum entropy principle [58] is explained under the next formulation (Equation (1)):

$$\mathrm{Max} \left[H_q\left(p;r\right)\right]=\left.\left\{\left[1-{\displaystyle {\sum }_k{\displaystyle {\sum }_m\alpha \cdot \left.{\left.\left(p_{km}\right.\right)}^q\right]}}\right.\right.+\left[1-{\displaystyle {\sum }_n{\displaystyle {\sum }_j\left(1-\alpha \right)\cdot \left.{\left.\left(r_{nj}\right.\right)}^q\right]}}\right.\right\}\cdot {\left(q-1\right)}^{-1}$$

(1)

subject to

$$Y=X\cdot \beta +e=X\cdot {\displaystyle \sum _{m=1}^M{v}_m}\left({\displaystyle \frac{{p_m}^q}{{\displaystyle \sum _{m=1}^M{p_m}^q}}}\right)+{\displaystyle \sum _{j=1}^J{w}_j}\left({\displaystyle \frac{{r_j}^q}{{\displaystyle \sum _{j=1}^J{r_j}^q}}}\right)$$

(2)

$${\displaystyle {\sum }_{k=1}^K{\displaystyle {\sum }_{m>2\dots M}{p}_{km}}=1}$$

(3)

$${\displaystyle {\sum }_{n=1}^N{\displaystyle {\sum }_{j>2\dots J}^J{r}_{nj}}=1}$$

(4)

where the real $q$, stands for the Tsallis parameter, whose values will vary over the 0–3 interval according to the nature of system complexity or ergodicity.

Above, $H_q\left(p,r\right)$ weighted by the $\alpha$ dual criterion function is non-linear and measures the entropy in the model. The estimates of the parameters (from p) and residual (from r) are sensitive to the length and position of support intervals of the reparametrized β parameters, then behaving as Bayesian prior [57]. Equation (2) explains the relationship between GDP production and energy consumption, while Equations (3) and (4) stand for probability normalization. The expression ${{\rm P}}_m={\displaystyle \frac{{p_m}^q}{{\displaystyle \sum _{m=1}^M{p_m}^q}}}$ in Equation (2) is one of the forms of the Tsallis entropy constraining and is referred to as escort probabilities [58]. Moreover, we have ${{\rm P}}_m\equiv p_m$ for q = 1 (then P is normalized to unity); that is, in the case of Gaussian distribution or ergodic systems. In such a special case, entropy is a function of the data and is therefore extensive. On the contrary, for a parameter q greater than unity, the long-range correlation/interaction between subparts of a system will lead to the non-extensivity of entropy fundamentally well described by the power law related to Tsallis entropy.

In the context of the concrete problem solved in this paper, Equation (2) stands for the linear constraint with respect to entropy maximization given the production of (e.g., GDP) being a function of energy consumed. The sign value of generated entropy will be positive. In fact, the GDP represents the total economic production of a country. This production requires energy consumption for industrial activities, transport, etc. From a thermodynamic point of view, any conversion of energy into productive work is inevitably accompanied by production of positive entropy due to the irreversibility of real processes. The higher the quantity of energy consumed, the greater the productive work potentially carried out, but also the higher the associated entropy production will be. Therefore, to maximize economic production under the constraint of the quantity of energy consumed, we will seek to use this energy in the most efficient way possible, but this will always imply a certain production of positive entropy. In summary, due to the irreversibility inherent in the energy conversion processes necessary for economic production, the optimal value of entropy under the energy/GDP constraint will be a positive value representing the minimum but non-zero entropy production for the level of wealth targeted.

3. Model Outputs and Discussion

3.1. Entropy-Based Model Outputs

The next analysis is set under the angle of entropy production as a result of GDP production through energy consumption, then including energy loss between its generation and the final consumption. Data on GDP per capita and energy consumption per capita have been extracted from Eurostat [66]. All the data cover the period 2001–2019 for ten countries (see Table 2) under analysis. The economic structure of each of these countries is different, albeit to a certain extent, between some countries economies may work at a similar scale. The computations of all models were carried out with the GAMS (General Algebraic Modeling System) code.

Table 2 presents outputs mainly aiming at comparing the optimal entropy level resulting from interactions between GDP and energy as a factor generating it. The first column lists countries under analysis, the second the model estimates, and the third the equivalent of the traditional coefficient of determination comparing explained variance to total variance. The countries on the list have been randomly selected within the set of European countries where the statistical methodology of data collection is homogenized. The last column displays optimal entropy values. To interpret the entropy generated by the GDP per capita as a function of energy consumption per capita (Table 2), we need first to consider the principles of thermodynamics, particularly the second law, and their application in an economic context. This interpretation will provide insights into efficiency, sustainability, and potential for economic growth. Under a thermodynamic perspective, in particular its second law, entropy in an isolated system can never decrease. In real processes, it always increases due to irreversibility property. Next, high-quality (low-entropy) energy like electricity can do more work than low-quality (high-entropy) energy like waste heat. Finally, any conversion of energy (e.g., fuel to work) increases total entropy. If we connect with economic interpretation, we can affirm the next stylized facts. GDP per capita is a measure of economic output or “useful work” per person, while entropy is a measure of the dispersion or “waste” of energy in producing GDP. This process is characterized by a type of entropy generation curve. At very low energy inputs, little economic output is possible. Entropy is low, but so is GDP. Next, as energy input increases, GDP rises. Entropy also rises as energy conversions are not perfectly efficient. Finally, there is an energy level where GDP per unit energy (and per unit entropy) is maximized. This point constitutes the peak efficiency. This represents the most efficient use of energy for economic production. Beyond this peak, additional energy inputs yield smaller GDP increases. Entropy rises faster relative to GDP, indicating more “wasted” energy. The main insight from this point is that energy quality matters: using high-quality energy (e.g., shifting from coal to renewables) can increase GDP without proportionally increasing entropy. Likewise, improving technology (e.g., better engines, insulation) can shift the curve, allowing more GDP for the same energy and entropy. The last insight suggests that, as far as structural changes are concerned, one can say that moving from energy-intensive industries (e.g., heavy manufacturing) to services can lower entropy per GDP. Nevertheless, non-energy factors may also influence GDP. This is the case when factors like human capital, institutions, and natural resources, not just energy, will play a significant role. Let us now comment on the output of our model in line with the regularities. The values of optimal entropy for the 10 countries range from 0.435 in the case of Denmark to 0.555 units in the case of Bulgaria. For the model output comparability purposes, the models implemented were assigned the similar constraining moments and normalization factors (Equations (2) and (3)) to insure relatively identical starting points for this non-linear model system convergency. In particular, since empirical reparameterization requires the use of standard deviation (e.g., [67]) of the variable of interest, that statistic has been computed and applied, respectively, in the case of each of these 10 countries. The only exception concerns Ireland, for which the optimal solution retrieval required around two times the empirical standard deviation of GDP per capita. We note that this country has been characterized by the largest standard deviation of that variable resulting from the highest GDP growth rate contrasts during the analyzed period. The estimates Beta explain the impact of a one kilogram of oil equivalent (KGOE) increase per capita on the change in GDP per capita. If we follow Table 2 above, the highest marginal product is noticed in the case of Denmark, where one KGOE leads on average to an increase in GDP per capita of EUR 5.44 over the period 2001–2019. At the same time, the entropy generated by this process (amounting to 0.435) is the lowest among the remaining nine countries under analysis. If we consider the contrasting case of Bulgaria, the productivity of the same unit of consumed energy leads to a EUR 2.01 increase in GDP per capita. Globally, using a Pearson linear correlation, we found the correlation between the energy marginal product (Beta) and the generated entropy to amount to approximately minus 0.71, which suggests that the higher the marginal product per head, the lower the level of the generated entropy, indicating a higher energy efficiency over the analyzed period. As we will see below, this relationship between the marginal product and the corresponding optimal entropy is not linear and will depend on other factors like the level of management, used technology, etc. Before closing this paragraph, we notice that the Tsallis q parameters generated for each of the ten regression cases have not been presented in the table to avoid overloading the information already in place. Their values vary between 1.15 (for the case of Denmark) and 1.29 (for the case of Poland). All these values of the Tsallis q parameter are significantly greater than unity, which may suggest the non-ergodicity of the relationship between economic production and the related energy consumption. Nevertheless, some caution should be taken when we consider the very small scale of the sample. In this place it would be appropriate to draw attention to the remarkable work stressing non-linearity-induced spurious multifractality versus regular power law distribution [68].

3.2. Entropy and CO₂ Emissions

In the previous section, we have defined the essence of the connection between pro-duction, energy consumption, entropy, and CO₂ emissions to some extent. Let us here extend the discussion and explore a few recent works linking entropy and CO₂ emissions. More recently, some authors explored close relationships between entropy and CO₂ emissions. Moser et al. [69] proposed an entropy-based approach to assess the sustainability of production systems, considering both resource depletion and waste generation, including CO₂ emissions. Warr et al. [70] revisited the connection between entropy, resource depletion, and economic growth, highlighting the role of CO₂ emissions in increasing the entropy of the environment. Feng, C. and co-authors [3] investigated the relationship between entropy and CO₂ emissions in China’s industrial sectors, using a panel data analysis and entropy measures based on energy consumption. Qin, Q., and co-authors [71] examined the relationship between entropy, CO₂ emissions, and economic growth in China, using a panel quantile regression approach to capture potential non-linearities and heterogeneity across regions. Korhonen and co-authors [72] focused on the chemical manufacturing industry, quantifying the relationship between energy consumption, CO₂ emissions, and entropy metrics to identify opportunities for improving energy efficiency and reducing emissions. Casazza and co-authors [73] explored the use of entropy as a measure of circularity in economic systems, highlighting the connection between resource depletion, waste generation, and CO₂ emissions in a circular economy context. These recent works demonstrate ongoing research interest in understanding the thermodynamic foundations of economic production processes, particularly the connection between entropy and environmental impacts, such as CO₂ emissions. The studies employ various methodologies, including panel data analysis, quantile regressions, and industry-specific case studies, to investigate this relationship across different scales and contexts. All these works use the Gaussian Gibbs–Shannon entropy, which, as presented above, is limited to Gaussian systems. Once again, the present work applies entropy based on the power law distribution more suited to dynamic systems, such as those usually characterizing economic production processes. Given the introductory information above, let us now define below an entropic metric to predict CO₂ emissions of a set of countries in a short or medium time horizon.

3.3. Presentation of the Entropy Metric of CO₂ Emissions

I. Theoretical Foundations

After various technical considerations throughout this paper, we now propose the entropic metric briefly presented in the introduction of this paper. We will first present its theoretical foundations followed by an empirical illustration.

A. Model Structure and Parameter Interpretation

The relationship between energy efficiency and CO₂ emissions can be expressed through a mathematical model that incorporates both scale effects and efficiency parameters. It takes the following form:

$${\displaystyle \frac{CO2}{capitai}}={\left({\displaystyle \frac{entropyi}{MPi}}\right)}^{\mu \times \beta }\times {\left({\displaystyle \frac{Energy}{capitai}}\right)}^{\beta }\times \exp \left(ϵi\right)$$

(5)

The formula incorporates several key parameters and variables that are worthy of definition.

Entropy is a measure of system disorder or inefficiency in energy utilization. MP symbolizes the marginal product of energy, representing the additional output generated by one unit of energy input, as seen above. This term symbolizes the productivity of energy consumed in a given economy at a micro or macro level. The elasticity parameter µ modulates the impact of efficiency changes to be explained hereafter. The elasticity parameter β (β ∈ R+) governs both scale effects and efficiency transmission. The variable Energy/capita defines per capita energy consumption in the system. The symbol ϵi represents the random error component with an undefined law but under the power law attractor. The symbol i designates the precise rank of units under analysis in the case of a cross-section model. We could add a time symbol when dealing with a panel model.

As can be easily seen, the model (Equation (5)) can be cast into the production function family, where CO₂ emissions is the response variable and the two terms under exponents are the inputs. As such, the value system exhibits returns to scale. We will have increasing returns to scale (β > 1) when emissions grow faster than energy consumption and constant returns to scale (β = 1) in the case of a linear relationship between emissions and energy use. The system will deal with decreasing returns to scale (β < 1) with emissions growing more slowly than energy consumption. We draw attention to the next fundamental sub-system relationship, the one between energy consumption per capita and its marginal product. As energy consumption increases, MP typically decreases (diminishing returns). This relationship can be expressed as follows:

$$MP=k\times {\left({\displaystyle \frac{Energy}{capita}}\right)}^{-\gamma }$$

(6)

where:

$k$ is a technology parameter.

γ represents the rate of diminishing returns (0 < γ < 1).

Here’s the continuation of the corrected text:

As far as dynamic aspects are concerned, the β scale component’s behavior may change over time as technology and efficiency evolve, suggesting that β might not be constant in long-term analyses. Let us now more closely consider that case.

Efficiency Component Dynamics:

The efficiency component, represented by (entropyi/MPi)^{(α × β)}, provides crucial insights into how system performance affects emissions. The relationship between entropy and MP creates an efficiency measure factor, while the combined elasticity parameter (µ × β) determines the magnitude and direction of efficiency impacts on emissions. Then, the combined effect (entropy/MP) is explained as follows:

$${\displaystyle \frac{Entropy}{MP}}={\displaystyle \frac{entropy}{\left(\frac{1}{k}\times {\left(\frac{Energy}{capita}\right)}^{-\gamma }\right)}}$$

(7)

This ratio may vary following two regimes. Efficient systems will tend to lower entropy while increasing MP. Non-efficient systems will increase disorder and thus entropy while negatively impacting MP. In the long run, despite sustained efficiency, energy productivity will tend to drop down as a result of diminishing returns to scale. At the same time, entropy of the whole system should keep increasing. The magnitude of the exponent (µ × β) in terms of impact on emissions will depend on the trade-off between entropy and technical progress expressed through the term MP. The variations of MP are additionally affected by returns to scale. The exponent (µ × β) determines the overall impact of this efficiency measure. The parameter µ will generally display a negative sign as the numerator (entropy) generally will change in the opposite direction with the denominator (marginal product).

The complete relationships can be rewritten by substituting the MP relationship:

$${\displaystyle \frac{CO2}{capita}}={\left[entropy\times {\displaystyle \frac{{\left(\frac{Energy}{capita}\right)}^{\gamma }}{k}}\right]}^{\alpha \times \beta }\times {\left({\displaystyle \frac{Energy}{capita}}\right)}^{\beta }\times \exp \left(ϵi\right)$$

(8)

The above econometric relation is equivalent to Equation (5) and shows that energy consumption affects emissions through two channels. The first concerns a direct scale effect through the term “[(Energy/capita)^β]” and an indirect efficiency effect through “[(Energy/capita)^{(γ × α × β)}]”. To interpret these parameters, we see that the parameter γ captures the strength of diminishing returns in energy use. It affects how quickly additional energy consumption becomes inefficient, plausibly through increasing entropies. The parameter α measures the sensitivity of emissions to entropy-related efficiency. A higher α means higher elasticity of emissions with respect to efficiency changes. Entropy under consideration is related to the process of generating production. It includes overall management aspects of units, plus direct or catalytic technical progress linking energy consumption and production. The parameter β represents the direct elasticity of emissions to energy use. All else remaining equal, the value of this parameter will depend on the nature of inputs consumed as energy. The quantity of CO₂ emissions generated by a one percent increase in coal should significantly differ from that of solar energy. A value of that parameter higher than unity can be interpreted as a need to continue the energy mix.

Before the end of this section, let us add an additional complexity referred to as the Jevons Paradox Connection [74]. It reveals a reversal case to the above rules. The improved efficiency can lead to increased resource consumption. The negative α captures this effect mathematically by modeling how efficiency gains might drive increased emissions through expanded production or usage. This paradox will not be considered in the remaining sections of this paper. It would require further research in an appropriate context.

Let us now discuss parameter interaction effects through µ (see Equation (8)) and β, where µ represents −γ × α. The interaction between µ and β parameters creates complex system dynamics. The µ parameter modulates the transmission of efficiency effects, with values greater than 1 leading to amplified emissions and values less than 1 resulting in mitigated emissions. Its negative values lower than −1 (which will be the case in the illustrative study provided in the next section) lead to amplified emission reduction for one percent of energy consumed. Meanwhile, the β parameter influences both the scale and efficiency components, creating intrinsic coupling in the system. This system, as one can deduce from the above comments, is governed by system feedback mechanisms. In fact, the model incorporates two primary feedback loops. The first loop connects efficiency improvements directly to emissions reductions. The second loop operates through changes in energy consumption patterns, affecting the scale component and feeding back into system efficiency. The strength and stability of these feedback loops depend critically on the parameter values and their temporal evolution.

Last but not least, the entire preceding presentation has considered Equation (5) as a production function. However, if we want to keep energy consumption at the center of emissions, the term entropy/MP could be seen as a “weight” factor of the per capita consumption variable. This weight is associated with the efficiency of productive systems. This way of understanding the presented model could help with the results in empirical research.

This section on the theoretical CO₂ metric has provided a foundation for analyzing complex relationships between energy efficiency, energy consumption, and their environmental impacts. For practical application, several key considerations emerge. Parameter estimation must account for regional variations and temporal stability. System monitoring should track both efficiency and scale components. Policy interventions need to balance short-term effects with long-term sustainability goals. Optimization strategies should consider the coupled nature of efficiency and scale effects.

II. Empirical Illustration: CO₂ Emissions Prediction

This section provides an illustrative case study consisting of CO₂ emissions prediction for 10 countries whose outputs have been presented in Table 2. We recall that the estimates were obtained via a power law related entropy. As explained earlier, outputs from this estimator are identical to those generated by most traditional econometric devices in the case of random errors exhibiting Gaussian law properties. In the following empirical illustration, we seek to generate the CO₂ emissions based on the model of Equation (5). Towards that purpose, we will use the values based on the results of the time series model presented in Table 2. These values concern entropy and marginal product. The ratio of these variables will constitute the first input of the model. The energy used per capita will constitute the second input. The best approach to gauge the pertinence of a new model is to implement it in a predictive simulation. Parameters from the time series model of Table 2 have been estimated based on the period 2001–2019. Now we will use estimates from Equation (5) to carry out prognostics of CO₂ emissions over the next few years from 2020 to 2022. Fortunately enough, emissions of the prediction period are already known from official statistics sources. Thus, a comparison between our prognostics and values published for the corresponding years will allow us to assess the performance of our new model. Nevertheless, we have noticed significant discrepancies between data on emissions prepared by different official sources. For instance, this is the case between World Bank and Eurostat data. The used data have been extracted from Eurostat sources. In this straight model, we first estimate the parameters of a cross-section model where a sample of 10 countries (see Table 2) is involved. Data on energy consumed from the year 2019, which is the last period of the series where estimates on components MP and entropy come from, should modulate the prediction having to start from 2020.

Table 3. Estimates of the cross-section model (Equation (5)) based on the outputs from Table 2.

Model: OLS, using observations 1–10 Dependent variable: CO2_capita Heteroskedasticity-robust standard errors, variant HC1
	Coefficient		Std. Error		z	p-Value
Energy-cons (β)	1.09547		0.106561		10.28	<0.0001
Entropy/beta (µ × β)	−1.04739		0.0385283		−27.19	<0.0001
Mean dependent var		2.210201		S.D. dependent var			0.240875
Sum squared resid		0.463154		S.E. of regression			0.240612
R-squared		0.990619		Adjusted R-squared			0.989446
F(2, 8)		435.3399		p-value (F)			6.87 × 10−9
Log-likelihood		1.172016		Akaike criterion			1.655969
Schwarz criterion		2.261139		Hannan-Quinn			0.992099

presents estimates of that model obtained from Gretl software (version 2022C) by the OLS method. We recall that these estimates are the exponents of a two-input production function as explained above.

We notice a high precision in the estimates. Before interpreting them, let us look at Figure 1, which describes the evolution of emissions of the 10 countries under analysis over the period 2001–2023. A general picture to retain is a downward shift of emissions, at different rates, of CO₂ emissions for all countries. Now we will comment on these estimates to understand what process seems to be hidden behind them. We start with the elasticity estimate of the parameter β, which values around 1.1%. A 1% increase in energy used per head leads to an average 1.1% increase in CO₂ emissions for the 10 countries under study, under the hypothesis that the factor efficiency in the production process remains unchanged. Since β > 1, emissions have increased more than proportionally with energy consumption. The next estimate is related to the parameter µ. Its value is deduced from the estimate µ × β equal to −1.04739, indicating µ is approximately equal to −0.96. This means µ is still negative but less negative than in the previous analysis. Entropy and MP will generally vary in opposite directions in the long run due to the second law of thermodynamics and the law of decreasing returns to scale. The parameter µ represents the elasticity of the system’s efficiency term (entropy/MP).

Its value becomes negative as a natural consequence of positive β to maintain µ = α × β = −1.05. Its magnitude depends inversely on β (always positive if and only if one observes a negative trend of CO₂ emissions (in the present analysis, see Figure 1).

To better understand the whole process leading to the negative sign of µ, let us show practical circumstances leading to that sign. Due to technical progress and innovations, MP may increase to a certain degree even in the long run, at least in the case of some countries. This increase may dampen the inexorable shift up of entropy but yet generate a positive µ. It is worthwhile to recall at this point that entropy could even increase as a result of higher MP, which generates higher production and further complexification of the economic system.

The biggest impact on CO₂ emissions is reached when β >> 1 while “Δ Entropy” tends to ∞ and “Δ MP” to 0+. Fortunately, there exist socio-physical mechanisms that will constrain µ to negativity. First of all, due to technological constraints, energy systems reach efficiency limits. This may be the case when entropy increases, forcing systems to hit physical or technological barriers (e.g., industrial processes involving high temperatures). This creates a natural ceiling effect, represented by a negative µ. Socio-economic pressures become unsustainable in the case of a high entropy/MP ratio (with a positive µ). In such circumstances, market forces push for efficiency improvements (thus leading entropy to marginally decrease) or companies are forced to innovate or lose competitiveness. Various other regulatory environments take place and become stricter with higher emissions. A good example could be carbon pricing mechanisms penalizing inefficient operations or different compliance requirements forcing efficiency improvements. The last important case concerns resource limitations, like the physical scarcity of energy resources or other related raw materials. This leads to cost increases with inefficient use, and finally, market pressures force process optimization. All the above cases may stabilize the system in front of efficiency barriers. This feedback mechanism is captured by the negative µ in the model.

Now we present an illustrative application of the above model and predict emissions. Based on the theoretical introduction of the model in the preceding section, prognostics based on it should concern a limited time frame. This is because the ratio entropy/MP explaining overall efficiency of the system may be subject to significant mutations in the medium and long run period. As alluded to above, this way of treating this component fits well with its alternative interpretation of being the weight of energy consumption as a main factor of emissions. Thus, in this illustrative example, we are going to carry out prognostics limited to 3 years ahead.

Table 4. Prediction of CO₂ emissions over the period 2019–2022.

Country	Year	Predicted	Actual	Error (%)
Belgium	2020	5636.59	7905.33	−28.70%
	2021	6101.36	8200.45	−25.60%
	2022	5440.06	7645.03	−28.84%
Bulgaria	2020	3151.34	5284.98	−40.37%
	2021	3457.56	6168.93	−43.95%
	2022	3354.21	6880.55	−51.25%
Czechia	2020	4977.23	8689.35	−42.72%
	2021	5503.07	9177.49	−40.04%
	2022	5091.47	8910.88	−42.86%
Denmark	2020	5390.49	4851.91	11.10%
	2021	5728.06	5051.84	13.39%
	2022	5286.29	4816.54	9.75%
Estonia	2020	5872.00	6917.67	−15.12%
	2021	6036.80	7802.32	−22.63%
	2022	5754.83	8708.43	−33.92%
France	2020	4807.96	4255.96	12.97%
	2021	5344.42	4624.56	15.57%
	2022	5034.34	4428.38	13.68%
Germany	2020	5411.34	7752.80	−30.20%
	2021	5523.96	8109.93	−31.89%
	2022	5204.12	7985.51	−34.83%
Ireland	2020	7379.12	7049.42	4.68%
	2021	7418.65	7466.40	−0.64%
	2022	7197.81	7184.21	0.19%
Netherlands	2020	7237.86	7749.17	−6.60%
	2021	7488.66	7878.13	−4.94%
	2022	6580.37	7106.68	−7.41%
Poland	2020	6413.02	7923.10	−19.06%
	2021	7082.41	8700.76	−18.60%
	2022	6750.30	8207.27	−17.75%

Source: Our own calculations.

presents the predicted values over the period 2020–2022. These prognostics are compared to the published estimates from Eurostat sources. The last column estimates the discrepancies between the Eurostat published values and the model prognostics. The mean absolute error amounts to 22.31% with very high variations among countries (from 0.19% to −51.25%). At first sight, this high error should indicate lower parameter estimator efficiency. Nevertheless, we should not neglect the fact that data published by different official statistical institutes were themselves estimated from methodologically limited approaches as seen in the first section of this paper.

Country-Specific Analysis

The relationship between MP and prediction accuracy: looking at the data, we can see a clear correlation between a country’s MP value and its prediction accuracy. Countries with higher MP values (like Denmark at 5.443 and France at 5.037) generally show better prediction stability, though not always better accuracy. This suggests that the MP value might be capturing some underlying structural characteristic of each country’s energy system that influences the stability of their emissions patterns. The role of entropy is well articulated. Its values show an interesting inverse relationship with MP values. Countries with lower entropy values (Denmark at 0.435, Belgium at 0.45) tend to have higher MP values. This relationship appears to affect prediction accuracy in complex ways. Low entropy/high MP combination: Denmark exemplifies this case with the lowest entropy (0.435) and highest MP (5.443). Its predictions consistently overestimate emissions by about 10–13%, but importantly, this error remains stable. This suggests that lower entropy values might indicate more predictable emission patterns, even if the absolute predictions are slightly off. Let us consider the reverse case of a high entropy/low MP combination. Bulgaria represents this case with the highest entropy (0.555) and lowest MP (2.01). Its predictions show the largest errors (−40% to −51%), and these errors grow over time. This combination appears to create a compound effect that reduces prediction accuracy significantly.

In spite of limited sample and official data coming themselves from methodologically questionable techniques, let us try to comment on error structure. Denmark and France show consistent overestimation, but with remarkably stable error margins. Denmark’s error stays within a 3.64% range (9.75% to 13.39%), while France’s stays within a 2.6% range (12.97% to 15.57%). This stability suggests that the model captures these countries’ emission dynamics better, even though it overshoots the actual values. The special case touches Ireland, which presents a fascinating case where moderate values for both MP (3.731) and entropy (0.525) result in exceptionally accurate predictions. The error decreases from 4.68% in 2020 to just 0.19% in 2022, suggesting that Ireland’s energy consumption and CO₂ emission patterns align particularly well with the model’s underlying assumptions. It is more likely that the estimates for Denmark, France, and Ireland remain more relevant than those of the other countries observed from the perspective of our model.

Before closing this section, let us make some comparative comments on the proposed metric compared to two others applied on a large scale: that of Eurostat and the Kaya identity.

Comparative Analysis: Entropy–Efficiency, Eurostat, and Yaka Identity Emissions Metrics

The analysis of carbon emissions requires robust methodological frameworks that can both explain historical patterns and inform future policy. This section examines three distinct approaches to emissions metrics: the established Kaya Identity (e.g., [75,76]), Eurostat’s standardized measurement system [66], and the proposed entropy-based metric. Each offers unique perspectives on the complex challenge of understanding and mitigating carbon emissions. The entropy-based metric introduces a novel thermodynamic perspective to emissions analysis. Unlike existing frameworks, the entropy-based approach grounds emissions analysis in thermodynamic efficiency. This offers new explanatory power by identifying physical limits to efficiency improvements and clarifying why similar technological interventions may yield different results across sectors or regions. The Kaya Identity, while widely used in climate policy discourse, takes a fundamentally different approach. It decomposes emissions into population, per capita GDP, energy intensity of economic output, and carbon intensity of energy. This accounting-style decomposition provides clarity and accessibility but lacks the physical foundation that connects emissions to underlying natural laws. Where the entropy-based metric highlights how physical transformations generate both useful output and unavoidable disorder, the Kaya Identity focuses on socioeconomic and technological factors without explicitly addressing their physical underpinnings. The entropy approach thus offers complementary insights that can explain phenomena that remain opaque within the Kaya framework. Eurostat’s approach represents yet another perspective, prioritizing standardized measurement protocols and sectoral categorization. While providing granular data essential for compliance monitoring and policy implementation, Eurostat’s methodology serves primarily as a measurement system rather than an explanatory framework. It excels at documenting what has occurred but offers limited theoretical insight into why emissions evolve as they do.

The introduction of the entropy-based metric into the literature advances emissions analysis in several ways. First, it establishes explicit connections between physical laws and carbon output, helping explain observed patterns that economic models alone struggle to capture. Second, its elasticity parameters offer flexibility to model non-linear relationships between efficiency improvements and emissions reductions. Finally, it provides a theoretical foundation for understanding why certain interventions reach diminishing returns.

When considering forecasting capabilities, the entropy-based approach offers particular promise. By modeling the physical constraints that govern energy transformations, it can potentially anticipate barriers to emissions reduction that might not be evident in trend-based projections derived from Kaya factors or Eurostat measurements.

For policymakers navigating complex climate challenges, each framework offers valuable but different insights. Besides the presented advantages of the entropy metric, the Kaya Identity provides an accessible factor decomposition that aligns with policy levers. Eurostat delivers standardized measurements essential for tracking progress and ensuring accountability.

As climate science evolves toward more integrated approaches, the entropy-based metric introduced in this article contributes an important physical perspective previously underrepresented in emissions analysis. By connecting carbon outputs directly to thermodynamic principles, it bridges a gap between physical science and policy applications, potentially enabling more effective interventions based on a deeper understanding of the fundamental processes driving emissions.

To conclude this subsection, each metric offers distinct advantages. The entropy–efficiency metric provides fundamental physical insights, Eurostat offers established implementation pathways, and Yaka delivers holistic sustainability assessment. The optimal approach likely involves strategic integration of these methodologies rather than exclusive adoption of any single framework. Moving forward, developing hybrid approaches that combine the thermodynamic rigor of entropy analysis with the practical implementation pathways of established frameworks represents a promising direction for emissions metrics evolution.

4. Concluding Remarks

The integration of entropy-based analysis with CO₂ emission prediction models represents a significant advancement in understanding the complex relationship between economic activities and environmental impacts. This research successfully establishes an additional framework connecting energy consumption, economic production, and CO₂ emissions through the lens of power law-based (Tsallis) entropy, offering unique insights into environmental impacts within economic system complexity. The model’s performance across EU countries demonstrates both its potential and areas for refinement. While achieving exceptional accuracy in some cases, such as Ireland, the model shows systematic underestimation patterns in others, highlighting the complexity of emission dynamics. The observed correlation between entropy values and prediction accuracy validates the metric’s ability to capture underlying systemic characteristics influencing emission patterns. From a practical perspective, the model offers stakeholders a novel tool for assessing environmental impacts of economic activities with the highest ratio of “information accuracy/information cost”. While current implementation shows some limitations in long-term predictions, it provides a foundation for more sophisticated analysis of emission patterns and their drivers. In fact, the introduction of the entropy-based metric into the literature advances emissions analysis in several ways. By combining the ratio entropy/MP of energy consumed, the model creates a new concept of efficiency that extends beyond conventional technical measures. This efficiency concept uniquely captures how systems convert energy inputs into productive outputs while accounting for inevitable disorder generation. The model’s elasticity parameters offer flexibility to represent non-linear relationships between efficiency improvements and emissions reductions, providing a theoretical foundation for understanding why certain interventions reach diminishing returns—an insight particularly valuable for policymakers evaluating long-term climate strategies.

Comparative studies with established frameworks such as the Kaya identity will be essential for positioning this approach within the broader scientific discourse. Future work will extend the analysis to longer time series and sector-specific breakdowns to further validate the model’s resilience across different economic contexts and structural shifts. Such expanded analysis will provide more robust evidence for the stability of model parameters under various conditions.

According to the authors, the integration of theoretical depth with practical applicability may position this work as a valuable contribution to both academic understanding and policy development in environmental economics. As the model evolves through suggested improvements, it has the potential to become an increasingly powerful tool for guiding sustainable development strategies and climate action initiatives, supporting more informed and responsible decision-making for a sustainable future.