# An equation for global energy efficiency gains in the long-run

Hervé Bercegol

► To cite this version:

Hervé Bercegol. An equation for global energy efficiency gains in the long-run. Ecological Economics, 2025, 236, pp.108666. 10.1016/j.ecolecon.2025.108666. cea-05086843

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Analysis

# An equation for global energy efficiency gains in the long-run

Hervé Bercegol

SPEC – Service de Physique de l'État Condensé, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

## ARTICLE INFO

### Keywords:

Energy-GDP

Energy consumption

Energy efficiency

Climate energy scenarios

History of energy consumption

## ABSTRACT

This work focuses on the global economic efficiency of energy use, defined as the ratio at world scale of Gross Domestic Product to Final Energy Consumption, including food for humans and feed for draft animals. With a simple hypothesis of energy efficiency gains being proportional to economic activity, it evidences that for the last two centuries energy efficiency grew on average exponentially with the cumulative energy consumption. By extrapolating this relationship, I estimate that the global economic efficiency of energy doubled or so from the Neolithic transition up to 1820, whereas it roughly tripled since then. Concerning the present energy transition, the International Energy Agency's scenario for "Net zero emission" by 2050 would reverse a recent slowdown in efficiency gains and retrieve the trend of the last two centuries, eventually overpassing it. The equation thus provides a historical reference for analyzing the past and calibrating future energy consumption scenarios.

## 1. Introduction

Contrasting with most other living species, human beings evolve culturally in their use of energy resources both qualitatively and quantitatively. From their ancient status as foragers (Haberl, 2001b), they have added many new ways of producing food energy, using it to make tools, taming and feeding wild animals to put them to work. They have harnessed water and wind streams, used available stocks of biomass or fossil fuels to warm themselves, to reduce ores into metals and to manufacture materials, forming them into buildings, machines, appliances, and so on². These energy usages and conversion processes are the basis of economic activity (Cleveland et al., 1984; Kander et al., 2013), and they also fuel the historical, social, and cultural development of civilizations (Debeir et al., 2013; Morris, 2013).

Economic activity is usually measured in terms of annual Gross Domestic Product [GDP], denoted by Y, and its per capita value, $Y$. The anthropologist Leslie White (White, 1949) proposed an analogue to $Y$, which he called cultural development, and denoted with C. He defined C as "the magnitude of human need-serving goods and services produced" and expressed it as the product of energy consumption per capita, denoted by $\mathcal{E}$, times a "technology" function T describing "the quality or efficiency of the tools employed in the expenditure of the energy":

$$C = \mathcal{E}T = Y \qquad (1)$$

Multiplying $Y$ and $\mathcal{E}$ of Eq. (1) by the population number P, one obtains the annual GDP Y and the annual energy consumption E. White's technology function T can be identified with the economic efficiency of

energy A, defined as the ratio:

$$A = \frac{Y}{E} = \frac{Y}{\mathcal{E}} \qquad (2)$$

A (Tripathy et al., 2024), and more generally the relationship between energy and GDP (Kraft and Kraft, 1978), have been widely studied for the last 50 years, mostly at a national level (Popkova and Sergi, 2021; Ayres and Warr, 2010), especially in the context of the climate-energy transition. Contrary to most of these studies, I focus on the global economy and avoid either a search for causality (Kraft and Kraft, 1978) between GDP and energy, which has been found in general inconclusive (Bruns et al., 2014; Szustak et al., 2021), or an analysis with explicit economic factors (capital, labor, research expenditures, etc.) contributing to production and innovation (Popkova and Sergi, 2021; Kümmel et al., 1985). As White (White, 1949) and most macroeconomists (Romer, 2012) did, I will consider average human beings only and average economic or cultural levels associated with the global economy, avoiding the consideration of inequality between humans or unequal terms of trade between regions.

The purpose of this paper is to derive conceptually, test empirically and apply a quantitative formula for the time evolution of the global economic efficiency of energy use A. From here on, A will be simply called "energy efficiency". I will present simple considerations on economic activity and inventive processes in §2.1, leading to Eq. (3), which links E and the growth rate of A, and Eq. (5), its integral form. In §2.2, this law is demonstrated in Fig. 1 for the last two hundred years, through a reappraisal of a recent co-authored work (Bercegol and Benisty, 2022)
<figure><img src="image_3.png" /><figcaption>Fig. 1. Historical values of A plotted vs the cumulative FEC, starting in 1820. The yearly GDP/FEC ratio, expressed as 2011 US$ at Purchasing Power Parity [PPP] per MegaJoule [MJ], is a green continuous line in this semi-log graph as a function of the time integral of FEC in abscissa, expressed in ExaJoule [EJ]. To provide simultaneous historical reading, decennial values from 1900 on are marked as green circles, and 50-year periods as black ones. A ± 5% error bar is indicated on these decennial marks. A fit following Eq. (5), black dashed line, approximates rather well the data except for spikes around the years 1920 and 1940, and a bend in the 2010s, beginning in the late 2000s. See §2.2 for comments, and Materials & Methods for more details; series represented is #10 in Table 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

in which the relation of world level primary energy consumption [PEC] to global GDP was studied. Due to inconsistencies in the historical series used, corrections were needed as explained in Materials & Methods (section 4), with as a major consequence the study of final energy consumption [FEC]. In §2.3 I will argue for the relevance of Eq. (3) in the very long run, and apply it to derive quantitative estimates of efficiency gains in the last 12,000 years. In §2.4, turning the focus to our future, I will consider published mainstream energy transition scenarios through the prism of Eq. (3), which constrains the resulting {GDP; FEC} trajectories. I will discuss the limitations and prospects of the present analysis in section 3, and finally in section 4 provide the important details of the reasoning and data treatment that led to the results.

## 2. Results

### 2.1. Final energy consumption, the fuel of activity and invention

Economic services are rendered by energy converters that deliver goods or actions by consuming energy. Primary energy consumption [PEC] covers all the energy extracted from the environment (Blesl and Kessler, 2021; Way et al., 2022) previously embodied in natural streams (sun, rivers, etc.) or stocks. As for final energy, it is consumed, i.e. dissipated, in the production process by converters and in the life cycle process to build, maintain, organize, and eventually dismantle those converters as well. Final Energy Consumption [FEC] is also essential for managing the whole system, structuring and organizing social life and reproductive needs, producing and storing knowledge, and many other crucial activities. Thus defined, FEC, also noted E, is the quantity directly provided and totally consumed in both processes, taking into account the energy needs of investment in agreement with present-day net energy analysis (Sahin et al., 2024). In line with a recent practice (Kander et al., 2013; Haberl, 2001a; O'Connor and Cleveland, 2014), I aggregate together traditional and modern energy sources. Final energy sources are food, fodder, fuelwood, fossil fuels, electricity, etc. needed to generate, nurture, feed and maintain the converters. All converters are put in one ensemble: humans, draft animals, fireplaces and stoves, machines, etc. This accounting choice contains a strong assumption: that

the fuel consumed by converters of one type, coal by machines for instance, is commensurable in terms of final energy with another fuel consumed by of another type of converters, say fodder by draft animals.

In a stationary economy, with a fixed number of converters of fixed types, and immutable processes, the activity would be proportional to the energy stream, and A would be a constant. Without any change in the quality of converters, type of goods, or social organization, final energy consumption is a measure of the activity that may change with the number of converters or goods. Any innovation, in material, process, or organization, results in a change of either the nature or usage of converters or their life cycle or all at once. To survive after an adaptation period, those innovations that deliver already available goods have to save energy; those providing new goods produce an economic value at least worth their energy consumption.

In Eq. (2) rewritten as $Y = AE$, the accumulation of knowledge is accounted for in A, while E takes into account the quantity of energy, from as many sources as are made available by the progress of knowledge. A crucial hypothesis is that the rate of change of A (thus due to innovation) is proportional to the economic activity Y with a proportionality factor $\chi$. This is written in two forms:

$$ 
\begin{aligned}
\frac{dA}{dt} &= \chi Y = \chi AE \\
\frac{dA}{Adt} &= \chi E
\end{aligned}
\qquad (3) $$

The coefficient $\chi$ articulating FEC (E) and energy efficiency growth rate $\frac{dA}{Adt}$ is later determined empirically. As E is dimensionally energy divided by time, and $\frac{dA}{Adt}$ is an inverse time, $\chi$ is an inverse energy, expressed later in $EJ^{-1}$. Eq. (3) could be viewed as a simplified version of endogenous growth theory (Romer, 1990; Kremer, 1993) extending the observations of Ester Boserup (Boserup, 1965). It stresses that the rate of growth due to energy efficiency gains is itself proportional to total energy consumption. Consequently, if one assumes a stable consumption per capita, a bigger population would induce a higher A growth as noticed by Boserup (Boserup, 1965). The growth of GDP is thus a sum of two contributions:

$$ \frac{dY}{Ydt} = \frac{dA}{Adt} + \frac{dE}{Edt} = \chi E + \frac{dE}{Edt} \qquad (4) $$

respectively due to energy-saving innovations for the first term, and to energy consumption growth for the second. In this simple model, all humans are supposed to share knowledge at the same time. A refined framework would treat local inventions and their geographical diffusion. To further deal with this hypothesis, a localized version of the model, briefly discussed below, remains to be developed. Actually, for a simple and global Eq. (3) to hold, invention diffusion and sharing do not have to be instantaneous, although modern information techniques in a global open market could provide such a case. One only has to suppose that the communication technologies and the marketing speed both evolve with the average innovation rate.

### 2.2. Global energy efficiency growth in the last 200 years

Fig. 1 illustrates the empirical validity of Eq. (3) using its integral form:

$$ A[t] = A[0] \text{Exp} \left[ \chi \int_{0}^{t} E[u] du \right] \qquad (5) $$

well suited for comparison with data. In Eq. (5) and Fig. 1, FEC plays the role usually devolved to a time parameter: cumulative FEC is like years adding to years on a time axis. To establish the data for Fig. 1, several published series of energy consumption, GDP (per capita or total), and population were used: all graphs of A vs cumulative FEC showed a similar trend to that in Fig. 1, as discussed in Materials and Methods (section 4). I obtain $\chi = 5.5 \times 10^{-5} EJ^{-1}$ with an error bar of about 5% if
calibrated on the whole 1820–2014 period (see section 4). However, in Fig. 1, a noticeable bend occurs towards the end of the 2000s. When calibrating Eq. (5) from 2000 on, one obtains a somewhat smaller $\chi$ near $4 \times 10^{-5}$ EJ$^{-1}$. This feature is further discussed below when considering future scenarios, as well as in section 4.

A general word of caution is in order when it comes to data. Prior to the 20th century, global data on energy, population and GDP are often simple estimates (see section 4). In particular, GDP data (Bolt and Van Zanden, 2024), here expressed in Purchasing Power Parity, have limitations and sources of error that could affect the trend in Fig. 1. The underestimation of unpaid household work, the undervaluation of qualitative service improvements or radical innovations could create biases in either direction, artificially increasing or decreasing the value of $\chi$. Bearing in mind these data problems and the difficulty in establishing long term measurements of well-being, as described by Bolt & Van Zanden (Bolt and Van Zanden, 2024), I analyze here the consequences of the model in the current state of knowledge.

Looking at Eq. (4), the efficiency growth coefficient, $\frac{dA}{Adt} = \chi E$ with $E \approx 16$ EJ was a mere 0.1 % year$^{-1}$ in 1820, nearly 10 times smaller than FEC growth $\frac{dE}{Edt}$. However, efficiency growth was nearly 25 times bigger in 2014, when $E \approx 400$ EJ. In 200 years, efficiency growth went from negligible to providing almost two-thirds of global GDP growth, as given by Eq. (4). With the hypotheses discussed above, this happened not because humans would be more innovative now than in the past, but simply from Eq. (3) due to a larger economy, with more people consuming more energy, thus innovating more today than yesterday, and inventing ever more efficient processes able to spread over the geosphere.

In Eq. (5), energy efficiency $A[t]$ is explicitly a function of the sum of all consumed energy up to time $t$. This dependence of $A$ on cumulative consumption illustrates the irreversible accumulation of knowledge and inventions. Once progress has been made through a better tool, process, or theory, it is never lost, and its positive effect on GDP remains forever, since cumulative FEC can only grow. In Fig. 1, annual FEC linearly commands the rate of global efficiency improvement: while the logarithmic slope of $A$ vs cumulative FEC appears fixed, very diverse choices can still be made in energy types and consumption levels.

Affecting aggregated quantities, FEC or GDP, Eqs. (3) and (5) mean that any human activity in any place and time does contribute to the growth of $A$, if not directly at least probabilistically by making innovation more probable. $A$ was multiplied by about 3 in 200 years; this low rate of growth compares unfavorably with more considerable progress made in specific devices or generic services. For example, the efficiency of engines grew by about a factor of 50 from 1700 to 2000 (Smil, 2010; Meyer and Vallee, 1975), and information and communication technologies [ICT] saw their performance increase millionfold in the last century (Meyer and Vallee, 1975; Nagy et al., 2011). There is no mistake, nor is there any paradox in this discrepancy for two main reasons. First of all, $A$ measures an overall, systemic efficiency gain, which integrates by construction all the activities necessary to social life, including those least amenable to progress. Second, $A$ also integrates all investments, energy investment to be specific, indispensable to obtain a gain, and often not accounted for in technology surveys. New improved machines generally need a new production line, and better materials, not forgetting research efforts that enabled the progress. They also require general infrastructure, such as roads, networks, and education facilities.

## 2.3. Global energy efficiency gains in the very long run: 10,000 BCE to 1820 CE

A striking fact is the empirical validity of a simple relation as Eq. (3), or its integral counterpart Eq. (5), in such diverse periods as the exponential deployment of rail transport in the 1840–1910 (Tostes et al., 2024), the oil-fueled “trente glorieuses” – postwar economic boom in

1945–1973 – or the ICT era from the 1970–1980s to our time (Kander et al., 2013). This rather wide applicability of Eq. (3) suggests that its validity could extend much farther in the past and the future than the 200 years on which we have data. It is fair to mention that these 200 years were marked by the transition from an agrarian to a fossil-fueled economy, drawing on huge geological reserves. However, the reasoning that brought Eq. (3) is not limited to energy provision from stocks of fossil energy carriers, and does not exclude its application to an agrarian economy or one based on modern renewables, both based on energy flow. Since this is the first attempt (that I know of) to estimate economic progress in the very long run from considerations of energy consumption, I calculate here first approximations of efficiency gains in the very long run according to Eq. (5), with a constant $\chi$ value obtained in Fig. 1.

To determine efficiency gains from 10,000 BCE, I only need the integral of FEC on the last 12,000 years. Consistently with the model discussed above, I consider the population of a geographical zone across which exchanges could have taken place between any two groups, whatever the speed of social vehicles such as trade or oral transmission. This “connected world” comprises Africa and Eurasia from 10,000 BCE to 1500 CE, then the Americas, then finally Oceania from 1800 CE. Table 1 summarizes the application of Eq. (5) to 10,000 BCE-1820 CE, divided into two periods. Energy efficiency $A$ grew by about 25 % from the Neolithic revolution to the beginning of the Common Era, then by a factor $\approx 1.6$ between 0 CE and 1820, thus $A$ grew by a total factor of $\approx 2$ when combining the two periods. In section 4 (Table 3), I present various alternative hypotheses on FEC per capita before 1820 CE: results that differ by 10 % at the most are averaged in Table 1. From this estimate one can deduce that the average human being around 0 CE was 2.5 wealthier than her/his hunter-gatherer ancestor in 10,000 BCE, with 25 % due to efficiency gains, and a factor of 2 coming from FEC per capita growth. From 10,000 BCE to 1820 CE, the total improvement in per capita living standards was a factor of 4, i.e. a doubling of both efficiency gains and FEC per capita growth. This slow progress in nearly 12,000 years contrasts with the factor $\approx 3$ for efficiency, and $\approx 4$ for FEC per capita, gained between 1820 and the present. In the present framework, this is correlated to an accelerated FEC, with a cumulative FEC nearly twice as big between 1820 and the present than in the previous 12,000 years.

## 2.4. Energy efficiency in energy transition scenarios

Turning to the future, I now illustrate how Eq. (5) could be used to gauge proposed world-scale scenarios of clean energy transition in the forthcoming decades. I focus on recent International Energy Agency [IEA] energy transition scenarios (IEA, 2023), while in-depth analysis of the many published scenarios (IPCC, 2022) should be the subject of future work. STEPS (Stated Energy Policy Scenario) follows policies already set, APS (Announced Pledges Scenario) supposes that all climate and energy governmental targets will be reached in due time, while NZE (Net Zero Emissions) scenario limits global warming to 1.5 °C (IEA, 2023). Global FEC projections differ appreciably between the three scenarios, but the corresponding global GDP trajectory turns out to be the same for all of them. This outlook can be usefully analyzed with the present work. In Fig. 2, $A$ is plotted as a function of cumulative FEC, with both quantities calculated from IEA data, with FEC of food for humans and feed for draft animals obtained from Malanima (Malanima, 2020; Malanima, 2022); details are provided in section 4.

As already commented above, $\chi$ takes a lower value of about $4 \times 10^{-5}$ EJ$^{-1}$ after the year 2000. One notices in Fig. 2 that STEPS lies near, slightly below the extrapolation of 2010–2022 trend. STEPS is a plausible projection on the same energy efficiency tracks observed in the last 15 years. On the contrary, APS appears as a path aiming at retrieving the longer-term trend in efficiency gain observed in 1820–2014, without reaching it though. As for NZE, it is an ambitious target, based on annual gains double those seen over the last two centuries (IEA, 2023), setting a
<caption>**Table 1**</caption>

Estimates of A growth from 10,000 BCE to 1820 CE. Hypotheses on population numbers (Klein Goldewijk et al., 2017) and FEC per capita value yield FEC and permit to estimate energy efficiency gains by applying Eq. (5) with $\chi = 5.5 \times 10^{-5} \text{ EJ}^{-1}$. The last column presents values in 2014, relevant to the series represented in Fig. 1. See section 4 for further details, in particular Table 3.

<table><thead><tr><th></th><th>Quantities</th><th>Value in 10,000 BCE</th><th>Period from 10,000 BCE to 0 CE</th><th>Value in 0 CE</th><th>Period from 0 CE to 1820 CE</th><th>Value in 1820</th><th>Value in 2014</th></tr></thead><tbody><tr><td rowspan="2">Inputs of the calculation</td><td>Hypotheses on FEC per capita (GJ/cap/y)</td><td>7.0</td><td>Linear with time</td><td>14.4</td><td>stable</td><td>14.4</td><td>60.1</td></tr><tr><td>Published population data (million) of connected world</td><td>1.9</td><td>Near exponential growth</td><td>213</td><td>Accelerated exponential growth with bumps</td><td>1066</td><td>7205</td></tr><tr><td rowspan="2">Resulting energy consumption</td><td>FEC (EJ/y)</td><td>0.013</td><td>Slightly quicker growth than the population</td><td>3.1</td><td>Following population</td><td>15.4</td><td>433</td></tr><tr><td>Cumulative FEC from 10,000 BCE (EJ)</td><td>0 (as per convention)</td><td>/</td><td>≈ 4400</td><td>/</td><td>≈ 13,500</td><td>≈ 34,700</td></tr><tr><td>Deduced A values</td><td>A = GDP to FEC ratio (normalized to 1 in 10,000 BCE)</td><td>1</td><td>most of the gain in the last 2000 years</td><td>≈ 1.25</td><td>continuous, accelerating gain</td><td>≈ 2</td><td>≈ 6.4</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Historical and projected values of A plotted vs the cumulative FEC, starting in 2010. Comparison of NZE (Net Zero Emission), APS (Announced Pledges Scenario) and STEPS (Stated Energy Policy Scenario) energy transition scenarios from the International Energy Agency [IEA] (semi-log scale). Also shown, extrapolations with Eq. (5) of 2010–2022 IEA data, black solid line with $\chi = 4.0 \times 10^{-5} \text{ EJ}^{-1}$, and of 1820–2014 data from Fig. 1, black dashed line with $\chi = 5.3 \times 10^{-5} \text{ EJ}^{-1}$, as well as a fit of NZE with Eq. (5), blue double line with $\chi = 7.9 \times 10^{-5} \text{ EJ}^{-1}$. IEA provides GDP, Population, and FEC (excluding food and fodder), for 2010–2022, and projected values for the three scenarios in 2030, 2035, 2040, and 2050 (IEA, 2023). Malanima (Malanima, 2020; Malanima, 2022) provides FEC for food for humans and feed for draft animals in 2010, and the 2010–2020 trend of per capita consumption, extrapolated in 2020–2050 with IEA population figures (IEA, 2023). See details in section 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

radical trajectory with a $\chi$ value 50% above that of 1820–2014.

At this exploratory stage of knowledge and understanding of using Eqs. (3) or (5), interpreting them in a normative way is questionable. The recent period of smaller $\chi$ could be a sign of larger final energy investments, to develop green energy (Sahin et al., 2024; Slameršak et al., 2022) or other technologies, with a delayed economic benefit. In particular, subsidies and investments in infrastructure for the energy transition were part of “green stimulus” policies following the global financial crisis in 2008, which could explain the bend around that date. Conversely, a future bigger $\chi$ could be associated with the generalization of electricity usage, implying a much higher final to useful energy conversion ratio (Way et al., 2022; Eyre, 2021) and a quicker growth of A. Such an oscillation around the long-term trend would be similar to what happened in 1900–1950, during which two successive oscillations

of A values around the long-term trend can be seen in Fig. 1. This was a time of a major energy shift towards oil and gas, away from coal dominance. The resumption of a strong growth in coal consumption in the 2000s could also be contributing to the recent slowdown of efficiency gains.

It could be argued that the approach of physical limits is contributing to the slowdown of A growth. For instance, progress in the efficiency of thermal machines is now hindered by the thermodynamic limits of heat to work conversion (Smil, 2010). However, economic efficiency, as defined here, can benefit from qualitative jumps from one technology to the other, with long term trends showing superexponential growth (Meyer and Vallee, 1975; Nagy et al., 2011). Moreover, the energy efficiency of computing, data processing and storage, irreplaceable building blocks of future technologies or social innovations (car sharing, second-hand markets, circular economy practices and strategies in general...), can still gain orders of magnitude before reaching any physical limit (Koomey et al., 2010; Prieto et al., 2025). ICT plays a pivotal role in the clean energy industry, where economies of scale and automation are crucial factors in the sustained decrease of the cumulative energy demand, for photovoltaics in particular (Louwen et al., 2016).

What major changes in technology, economics or energy use policy (Rosenow and Eyre, 2022) could help us move from the current trend towards the more ambitious scenarios?

### 3. Discussion

This last question illustrates the unexplored area opened by extrapolating the present model. Eq. (3), and especially its integral form Eq. (5), are global and “mean field”, relating the inextinguishable, and irreversible innovation trend to dissipated energy in the global economic process. Due to its neglect of relations between economic actors and its temporal smoothing of the innovation process itself, the model cannot explain nor exclude crises and fluctuations, as seen in Fig. 1, especially in the 1900–1950 sequence. Nor can it rule out the possibility of an upturn in energy efficiency trends, as implied by IEA’s ambitious NZE transition scenario in Fig. 2. A rapid transition to a low carbon society appears to be economically viable (Way et al., 2022). However, this industrial revolution would require a significant portion of global energy consumption to be allocated to investing in clean energy infrastructures (Slameršak et al., 2022). Concurrently, there is a risk of financial turmoil due to the rapid emergence of new industries and potential stranded assets (Semieniuk et al., 2021).

Going deeper into this new realm of knowledge requires not only applying Eq. (3) to more specific issues, like the rebound effect
(Brockway et al., 2021), but also deciphering the inner social and economic dynamics of individual actors and bigger entities, firms, nations, etc. Using the present validated model of energy efficiency A, one can try to link energy consumption growth to A growth, either theoretically with a set of differential equations or empirically using available historical series. Both are work in progress, motivated by the relevance of the rebound effect for our understanding of future policy needs (Brockway et al., 2021).

We should also explore the dynamical economic outcome of basic hypotheses of the model, studying for example how innovation, economic growth and FEC have been linked at the level of individual nations. To achieve this goal, we need to go beyond the current simplification that neglects any differentiations and inequalities between people and people groups, sub-regions, etc. For that, we should think of projecting the model on a spatial grid, and probably also introducing independent economic agents (Amendola et al., 2024). A future application of Eq. (3) at local levels, with hypotheses of unequal distribution of energy sources, and social processes to structure power and diffuse knowledge between human groups, could convene with recent research endeavors (Freeman et al., 2021; Freeman et al., 2024) in describing the long-run history of the human economy in the framework of the model. It should also benefit to the understanding of

regularities and peculiarities of energy efficiency at national level (Tripathy et al., 2024; Popkova and Sergi, 2021).

In the 18th and 19th century, the low value of $\chi^2$, smaller than 0.1 % year$^{-1}$, made innovation growth practically invisible to economic thinkers (Smith, Malthus, Ricardo, among others) (Klitgaard, 2022), who then mostly saw diminishing returns when more investment would induce a lesser production growth. On the contrary, increasing returns were studied and popularized by endogenous growth theorists in the late 20th century (Romer, 1986), in a period when $\chi^2$ was worth about 1 % year$^{-1}$, when more energy consumption induces efficiency gains. This work not only sheds new light on previous thinking and theories, it also materializes the reality of increasing returns, evidencing a continuous improvement of the economic efficiency of energy, while anchoring growth into energy consumption.

## 4. Materials and methods

I present and discuss the data, and explain the work done to reach results presented above. I begin with population, and Gross Domestic Product [GDP]. Then, I go deeper into energy consumption data. When the idea of Eq. (3) was first conceived (Bercegol and Benisty, 2022), energy consumption was taken from primary energy series (Malanima,

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Original population and GDP data used for Figs. 1 and 5.</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

(A) Global population series from five different sources, from 1800. Symbols in (A), diamonds (Maddison Project Database (Maddison Project Database, 2023)) and circles (decadal series from HYDE 3.3 until 1950), represent by themselves a ± 3 % uncertainty on population in this semi log plot. Diamonds also bear a vertical error bar of fixed 10⁸ value; also shown Gapminder V5 (Gapminder Population Series V5, 2025), World bank (World Bank, 2024) and Malanima's (Malanima, 2020). (B) Global average of GDP per capita, in unit of 2011 US$ at Purchasing Power Parity (2011 PPP$). Krausmann (Krausmann et al., 2018) and de Stercke (de Stercke, 2014) followed Maddison (Maddison, 2010) (squares), who gave annual data post 1950 (not represented for clarity). The World Bank provides data beginning in 1990. Maddison Project Database 2023 is a main reference (diamonds with 10 % error bar). Discrepancies of up to 10 % exist between series. (C) Comparison of Gapminder V25 series (Gapminder V25 GDP per capita Series, 2025), used in this paper, with V27 and V30 (Gapminder V27 and V30 Global GDP per capita Series, 2024), two series released more recently, and very near V25 as evidenced here. (D) Global GDP, in unit of 2011 US$ at Purchasing Power Parity (2011 PPP$). After correction with HYDE 3.3 population series (Klein Goldewijk et al., 2017), both in GDP per capita determination and in GDP calculation, GDP from Gapminder V25 series changes by up to 5 %.
2020). Data were problematic since the energy series utilized were a mix of primary and final energy. Here, various sources had to be combined to obtain annual global final energy consumption [FEC] series spanning 1820–2022. Explanations are given on the choice of FEC as an accounting quantity, and on how the resulting series were built. Then, I describe extensively the empirical test of Eqs. (3) and (5) using various series of population, GDP per capita, and FEC. Finally, some important elements relevant to the elaboration of the estimates of energy efficiency in the very long-run (Table 1) are detailed and discussed.

## 4.1. Population data

Population data for Fig. 1 and Table 1 come from the recent HYDE 3.3 series (Klein Goldewijk et al., 2017). The data is at millenary intervals (10,000 BCE-0 CE), then at century (0 CE-1700 CE), and then at decadal (1700-1950) intervals. These were interpolated with a Spline function to produce an annual series that matches the annual HYDE 3.3 data after 1950. UN data are available from 1960, after which most data sources are very similar. Fig. 3A reports the world population series from those and other sources. After 1950, relative differences between them are very small. However, before that date, and especially in the 19th century, HYDE 3.3 series (Klein Goldewijk et al., 2017), also used as reference by “our world in data” web site, is up to 10 % higher than all other series, enough to modulate the value of $\chi$ by a few %. HYDE 3.3 was used to rescale, with a unique population standard, FEC and GDP data based on per capita basis, as explained below in 4.2.

## 4.2. GDP data 1800–2022

As for global GDP per capita at Purchasing Power Parity [PPP], the main sources remain the seminal, millenia-spanning work of Maddison (Maddison, 2007; Maddison, 2010) and its continuation by the Maddison Project Database [MPD] (Bolt and Van Zanden, 2024). Their recent MPD2023 release (Maddison Project Database, 2023) constitutes a reference here for the 200-year period, expressed in 2011 PPP$. MPD (Maddison Project Database, 2023) gives global GDP per capita for discrete years between 1820 and the late 20th century, and yearly data after 2010. MPD provides more detailed figures for individual countries, often year by year down to 1820 or earlier. A 2018 release by Gapminder (Gapminder V25 GDP per capita Series, 2025), version 25, was also used here as previously (Bercegol and Benisty, 2022) since it provides an annual series from 1800 to 2016. De Stercke (IIASA) (de Stercke, 2014) and Krausmann (Krausmann et al., 2018) proceed from Maddison (Malanima, 2020; Malanima, 2022). Those four sets were rescaled to $2011 with a 1.59 factor (taken from Fig. 3 in Ref. (Bolt and Van Zanden, 2024)). After 1990, GDP data are available from the World Bank (World Bank, 2024), which provides GDP in US$ 2017 PPP, here converted to 2011 PPP$ using a 0.95 factor. In Fig. 3B, one notices that a discrepancy exists between series: it is up to 10 % and reduces progressively after 1990, being less than 1 % after 2010.

Gapminder V25 (Gapminder V25 GDP per capita Series, 2025) data provides an annual series very near MPD2023 in 1820–1940 and intermediate between MPD2023 and the World Bank data after 1990, progressively reaching the latter after 2000. I kept V25 for Fig. 1 (see §4.4), because it follows closely the original academic work (Maddison Project Database, 2023), provides an annual series and was used previously (Bercegol and Benisty, 2022). New releases (Gapminder V27 and V30 Global GDP per capita Series, 2024) of GDP per capita by Gapminder, V27 and V30, are very near V25, as shown in Fig. 3C. See also in Table 2 determinations of $\chi$ values obtained with them.

In order to use HYDE 3.3 as a population reference in a consistent way, it was desirable to use it at every step of the building process of both FEC and GDP data. FEC data generally come with an associated population series, which could be as much as 10 % different from the reference: then, I simply considered that food and fodder series had to be rescaled proportionally to the population ratios. Global GDP per capita

is logically determined by averaging a set of local values of GDP per capita characterizing separate regions. The average thus rests on the relative values of population in the regions considered. Different population series (Fig. 3A) also come with different local distributions. I adjusted the global GDP per capita when GDP sources had regional details (Maddison, 2007; Gapminder V25 GDP per capita Series, 2025), by applying a correction factor of the regional population ratio (see Table 2 and supplementary spreadsheets).

## 4.3. Energy data 1820–2022

A first version of Eqs. (3) and (5) was published previously (Bercegol and Benisty, 2022), based on primary energy consumption [PEC] data made available by Paolo Malanima (Malanima, 2020). Global energy consumption data were given as the sum of eight different types: coal, oil, and gas; electricity (excluding fossil-based and nuclear); nuclear; fuelwood, food, and fodder. This neglects traditional hydro and wind energy, i.e. not converted to electricity: sailing, water and windmills had a significant importance to enable development prior to and during the 19th century, but are quantitatively negligible (O'Connor and Cleveland, 2014; Malanima, 2020). In ref. ³², food – as an energy resource for human converters – and fodder – energy for animal converters – were both quantified by the amount of energy ingested, thus were not primary. Following the trophic-dynamic approach (Haberl, 2001a), I count as Final Energy Consumption [FEC] food for humans and feed for draft animals, but not feed for other livestock. Primary energy from the biomass would include all appropriation from organic matter, such as feed for livestock, or forest burned to clear and fertilize land, etc.... To further illustrate the difference between primary and final energy, here are two specific examples. First, for charcoal, the total combustion energy of the wood fuel is primary, and the energy content of the charcoal is final. Secondly, in the case of coal used for electricity, coke or town gas: the total combustion energy of the coal is primary; the total energy delivered in the form of electricity, coke, or gas is final. After explaining why FEC was chosen for energy accounting, I will detail how the global FEC series were constructed.

### 4.3.1. About alternative choices of energy accounting

I discuss here the alternatives of energy accounting, first Primary Energy Consumption (PEC), then Useful Energy Consumption (UEC), and Final Exergy.

Malanima's series can provide PEC in the 19th century, excluding food and fodder that are given as FEC. This confusion is not alone in the field since many historical accounts of energy consumption (e.g. ref. (Debeir et al., 2013; Morris, 2013) do not explicitly distinguish primary from final stages of traditional energies, especially food, whenever data are provided. PEC for food should obviously include food waste, during storage, transport, trade, or after its purchase. More importantly, PEC should include all the vegetal energy ingested by animals kept for food. FEC to PEC ratio is low, due mainly to a notoriously low ratio of vegetal to animal energy conversion (around 10 % for cattle meat (Alexander et al., 2016)). Other primary consumption comes also with agricultural practices such as fire-clearing (burn-beating), or plowing of covered field, which results in vegetal cover nurturing the future crops. In a recent appraisal focusing on exergy accounting, Carmona et al (Carmona et al., 2021). found that primary exergy was about 10 times bigger than useful exergy for food throughout the 20th century, going from more than 16 times bigger in 1900 to less than 9 times in 2010. Knowing that primary energy and primary exergy are in general very close to each other, and that useful exergy of food considered in that work is very similar in value to food energy given by Malanima (Malanima, 2020; Malanima, 2022), I retain that PEC for food would be an order of magnitude greater than FEC. I also retain that PEC for food, and for draft animal feed, is rarely studied in the modern period and mostly unknown in the past.

The definition of PEC is also unclear in important cases of renewable
<caption>**Table 2**</caption>

Characteristics of the GDP and FEC series with $\chi$ values obtained. Numbers in the last column refers to figures where data are plotted: Fig. 1, 2 and 5. All data are gathered in Dataset S3. For #8 and #10, a second slightly lower $\chi$ value is given in italic; they both correspond to the exponential functions shown in Fig. 2 (number also in italic in the last column), adjusted for a slightly bigger FEC counts in the IEA reference (IEA, 2023). FEC data have been built from several references, named in the 3rd column, as explained in Materials & Methods.

<table><thead><tr><th>Id. #</th><th>Time span</th><th>Ref. # for FEC data</th><th>Ref. # for Population data for FEC</th><th>Ref. # for GDP per capita data</th><th>Nature of GDP data</th><th>Ref. # for Population data for GDP</th><th>χ (in 10<sup>-5</sup> EJ<sup>-1</sup>)</th><th>Where to find plots of A vs cumulative FEC</th></tr></thead><tbody><tr><td rowspan="2">1</td><td>1820–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(MPD, 2023)</td><td>Discrete</td><td>(MPD, 2023)</td><td>5.47</td><td>Fig. 5A</td></tr><tr><td>1900–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(MPD, 2023)</td><td>Discrete</td><td>(MPD, 2023)</td><td>5.50</td><td>Fig. 5B</td></tr><tr><td rowspan="2">2</td><td>1820–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Gapminder V5)</td><td>5.74</td><td>Fig. 5A</td></tr><tr><td>1900–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Gapminder V5)</td><td>5.66</td><td>Fig. 5B</td></tr><tr><td>3</td><td>1820–2003</td><td>*</td><td>(Malanima, 2020)</td><td>(Maddison, 2007)</td><td>Discrete</td><td>(Maddison, 2007)</td><td>5.99</td><td>Dataset S3</td></tr><tr><td>3 bis</td><td>1820–2008</td><td>*</td><td>(Malanima, 2020)</td><td>(Maddison, 2010)</td><td>Discrete + yearly post<br>1950</td><td>(Maddison, 2010)</td><td>5.66</td><td>Dataset S3</td></tr><tr><td>4</td><td>1900–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(de Stercke, 2014)</td><td>Yearly</td><td>(de Stercke, 2014)</td><td>5.65</td><td>Dataset S3</td></tr><tr><td rowspan="2">5</td><td>1900–2014</td><td>*</td><td>(Malanima, 2020)</td><td>(Krausmann, 2018)</td><td>Yearly</td><td>(Krausmann, 2018)</td><td>5.50</td><td>Dataset S3</td></tr><tr><td>1970–2022</td><td>†</td><td>(Malanima, 2022)</td><td>(MPD, 2023)</td><td>Discrete</td><td>(MPD, 2023)</td><td>5.37</td><td>Fig. 5A</td></tr><tr><td rowspan="2">6</td><td>1970–2010</td><td>†</td><td>(Malanima, 2022)</td><td>(MPD, 2023)</td><td>Discrete</td><td>(MPD, 2023)</td><td>5.84</td><td>Fig. 5C</td></tr><tr><td>2010–2022</td><td>†</td><td>(Malanima, 2022)</td><td>(MPD, 2023)</td><td>Discrete + yearly post<br>2015</td><td>(MPD, 2023)</td><td>4.16</td><td>Fig. 5c</td></tr><tr><td rowspan="2">7</td><td>1970–2008</td><td>†</td><td>(Malanima, 2022)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Gapminder V5)</td><td>4.90</td><td>Dataset S3</td></tr><tr><td>2008–2016</td><td>†</td><td>(Malanima, 2022)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Gapminder V5)</td><td>4.24</td><td>Dataset S3</td></tr><tr><td rowspan="2">8</td><td>1990–2008</td><td>†</td><td>(Malanima, 2022)</td><td>(World Bank, 2024)</td><td>Yearly</td><td>(World Bank, 2024)</td><td>5.10</td><td>Dataset S3</td></tr><tr><td>2008–2022</td><td>†</td><td>(Malanima, 2022)</td><td>(World Bank, 2024)</td><td>Yearly</td><td>(World Bank, 2024)</td><td>4.94</td><td>Fig. 2</td></tr><tr><td>9</td><td>1820–2003</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Maddison, 2007)</td><td>Discrete</td><td>(Klein Goldewijk et al., 2017)</td><td>5.98</td><td>Dataset S3</td></tr><tr><td>9 bis</td><td>1820–2008</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Maddison, 2010)</td><td>Discrete + yearly post<br>1950</td><td>(Klein Goldewijk et al., 2017)</td><td>5.66</td><td>Dataset S3</td></tr><tr><td rowspan="2">10</td><td>1820–2014</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Klein Goldewijk et al., 2017)</td><td>5.48</td><td>Fig. 1 &amp; 5A</td></tr><tr><td>1900–2014</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Klein Goldewijk et al., 2017)</td><td>5.28</td><td>Fig. 2</td></tr><tr><td rowspan="2">11</td><td>1970–2008</td><td>†</td><td>(Malanima, 2022)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Klein Goldewijk et al., 2017)</td><td>5.34</td><td>Fig. 5B</td></tr><tr><td>2008–2016</td><td>†</td><td>(Malanima, 2022)</td><td>(Gapminder V25)</td><td>Yearly</td><td>(Klein Goldewijk et al., 2017)</td><td>4.93</td><td>Dataset S3</td></tr><tr><td>12</td><td>1820–2014</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Gapminder V27)</td><td>Yearly</td><td>(Gapminder V27)</td><td>5.41</td><td>Dataset S3</td></tr><tr><td>13</td><td>1820–2014</td><td>‡</td><td>(Klein Goldewijk et al., 2017)</td><td>(Gapminder V30)</td><td>Yearly</td><td>(Gapminder V30)</td><td>5.68</td><td>Dataset S3</td></tr></tbody></table>

<footnote>* (Malanima, 2020; de Stercke, 2014); Red double line in Fig. 4B.</footnote>

<footnote>† (IEA, 2021; Malanima, 2022; IEA, 2023); Black solid line in Fig. 4B.</footnote>

<footnote>‡ (Malanima, 2020; de Stercke, 2014); Within a few percent of red double line in Fig. 4B (effect of change in population reference).</footnote>
energy sources. For photovoltaic [PV] energy, PEC is often defined as the equivalent fossil fuel energy that would have been consumed to produce the same amount of final electricity with a standard thermal plant (BP Statistical Review of World Energy, 2022). Alternatively, it could be obtained as the solar energy intercepted by PV panels (Malanima, 2020), or even from the total solar flow on the facility, including the non-productive areas. If we choose PEC as the total energy intercepted by a PV farm, shouldn't we also apply the same convention to fuelwood, fodder, and food? Since solar energy conversion by biomass is known to be of the order of a few tenths of 1 %, PEC in that convention would be worth hundreds of times FEC.

Primary energy is thus at the moment ill-defined for renewable flows. This subject must not be neglected for sustainability studies, since primary flows are probably faithful indicators of the burden imposed on the environment (Haberl et al., 2007; Krausmann et al., 2013; Fischer-Kowalski et al., 2014; Rodríguez et al., 2023). However, for what concerns the present work, final energy is a better choice, because it is more easily defined and evaluated, and because it fits the model described here: FEC is a reliable indicator of the activity of energy converters, including their fabrication, maintenance, etc.

One could also attempt to consider useful energy consumption [UEC] instead of FEC. Indeed, it has been proposed (Ayres and Warr, 2010; Aramendia et al., 2021) that UEC is the most relevant energy parameter to analyze the economic activity. However, such a choice would also bump into major obstacles. The main one would be the inexistence of data prior to 1900. A second one would be the uncertain definition of useful energy for humans (Sousa et al., 2017), whereas final energy is easily identified as the ingested energy. Should we choose to consider muscle work only (Meyer and Vallee, 1975; Steenwyk et al., 2022) or consider, as Carmona et al (Carmona et al., 2021), that the whole ingested energy is useful? Another important roadblock is the difficulty to define what would be UEC in some sectors like information and communication technologies. Retaining FEC was the only practical choice so far, indeed. Moreover, the present model does certainly capture the relevant and major case in which improvement of GDP to FEC ratio is due to an increase of the FEC to UEC conversion ratio.

To complete this discussion, one must mention final exergy as a way of accounting energy consumption. By its ability to take into account thermodynamic limits, exergy is of prime importance in technological studies and economic considerations on specific sectors. However, at the final stage, exergy and energy provide rather similar values (de Stercke, 2014). Since final exergy is not available for the 19th century, and is more difficult to calculate or infer, requiring details of the specific processes involved, it has not been considered here.

### 4.3.2. Constitution of a global FEC series 1820–2014

The lack of data consistency (mixing PEC with FEC) remained unnoticed by the authors of previous studies (Bercegol and Benisty, 2022; Malanima, 2020; Malanima, 2022; Court, 2016). To correct this mistake, I had to build a new series of FEC. No complete source of global FEC exists in the literature for the whole period from 1820; moreover, for food and fodder, the only long series are from Malanima (Malanima, 2020; Malanima, 2022) and Court (Court, 2016). I will conform to the previous conclusion (Bercegol and Benisty, 2022) and prefer Malanima's after comparing the two databases. The same choice was made by Steenwyk et al (Steenwyk et al., 2022), who used Malanima's data (Malanima, 2020) as original source for human final energy. The two series from Malanima are very similar in the food and fodder consumption numbers, differing mainly by their final year: 2014 (Malanima, 2020), or 2020 (Malanima, 2022). The longest available FEC series is obtained from the Primary, Final, Useful database from IIASA made by de Stercke (de Stercke, 2014). The data give annual values for 1900–2014, detailing many different energy types, sectors, and uses, at the level of individual countries (France, UK, etc.), regions (OECD, ...), or the whole world. Since these data do not include food and fodder, the global FEC series for 1900–2014 was made by adding IIASA

FEC data to Malanima's food and fodder (Malanima, 2020).

For the 19th century period, no ready-made final energy estimate was available. Thus, I had to elaborate such a series from Malanima's database (Malanima, 2020). I used the same source as in ref. (Bercegol and Benisty, 2022), knowing that 19th century data are very similar in Malanima's more recent web release (Malanima, 2022). As already mentioned, Malanima's data were PEC for most types of energy, but FEC for food and fodder. De Stercke (de Stercke, 2014) provides the primary to final energy conversion ratio (ratio of FEC over PEC) from 1900 on (see Fig. 4A). These ratios decreased steadily on average during the period 1900–2024, mainly due to a growing conversion of fossil fuels to electricity: burned fuels are PEC while the energy content of electricity is FEC, with typically low conversion ratios near 0.33. By extrapolating these ratios towards the pre-1900 period, it was possible to amend Malanima's data to produce a complete and annual FEC series in 1820–1900. Fig. 4 explains and illustrates this construction; the whole set of data is available in supplementary spreadsheets. While the ratios obtained by extrapolation cannot be definitely validated, their slow, then accelerating decrease throughout 19th century is consistent with the development of secondary energy carriers, counted as final, like

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

**Fig. 4.** Building a Final Energy Consumption series from 1820. (A) Ratios of Final Energy Consumption [FEC] to Primary Energy Consumption [PEC] obtained from de Stercke (de Stercke, 2014) for Biomass, Petroleum, Natural Gas, and Coal; these four main components of Malanima's series (Malanima, 2020) were corrected with the extrapolated ratios to yield FEC without food and fodder in 1820–1900. (B) Global FEC (red double line), including food and fodder, reconstructed for the whole 1820–2014 period, from de Stercke (de Stercke, 2014) in 1900–2014 adding food and fodder from Malanima (Malanima, 2020); in 1820–1900 using the FEC to PEC ratios of (A) and a correction coefficient near 0.9 to match de Stercke's value in 1900. Also shown, is the original series from ref. (Malanima, 2020), and another Global FEC series spanning 1970–2022, black solid line, obtained by adding IEA FEC data (IEA, 2023; IEA, 2021) to Malanima's food and fodder (Malanima, 2022).
coke, town gas and electricity at the end of the century.

### 4.3.3. Another global FEC series 1970–2022

Since IIASA data terminate in 2014, another FEC series was obtained from 1971 to 2019 using data from the International Energy Agency [IEA] (IEA, 2021). I complemented once again FEC from IEA, which include only technical energy, with food and fodder from Malanima (Malanima, 2022). This series was extrapolated two years to complete historical FEC data until 2022, using yet other IEA FEC figures (IEA, 2023) and an extrapolation of the food and fodder series. The year 1970, absent from IEA FEC data (IEA, 2021) and useful to combine with decadal GDP data, was also extrapolated back by proportionality to the longer series of IIASA (de Stercke, 2014). This 1970–2022 series is plotted in Fig. 4B. It merges with the previous 1820–2014 series within less than 1 %.

I also used ref. (IEA, 2023) in Fig. 2, combined with food and fodder series from ref. (Malanima, 2022), in this case extrapolated to 2050. Since ref. (IEA, 2023) gives FEC data 3 % higher than the other sources (de Stercke, 2014; IEA, 2021), I made slight adjustments to FEC data or $\chi$ values. In the 1970–2022 series (Fig. 4B), the year 2010, given by both ref. (), permitted to rescale 2021 and 2022 of ref. (IEA, 2023) in line with ref. (IEA, 2021). In Fig. 2, the $\chi$ values for the application of Eq. (5) are thus 3 % lower; they had to be changed accordingly when going from Fig. 1 (or Fig. 5, below) to Fig. 2.

When directly comparing Fig. 1 and 2, it should be noted that different data sets had to be used: that described in 4.3.2 in Fig. 1, that of 4.3.3 in Fig. 2. 3 % higher FEC data in Fig. 2 compared to Fig. 1, acting on cumulative FEC, induce 3 % lower $\chi$ values ($5.3 \times 10^{-5}$ in Fig. 2 compared to $5.5 \times 10^{-5}$ in Fig. 1). Differences in FEC data and GDP unit (2022 PPP$ compared to 2011 PPP$ in Fig. 1) affect the Y-axis proportionally, by a factor of 1.25, without affecting the growth rate.

### 4.4. Obtaining $A = GDP/FEC$ relationship to cumulative FEC

The main result of this paper (Fig. 1) is obtained by calculating A (ratio of GDP over FEC) and plotting it as a function of cumulative FEC. All the {GDP, FEC} pairs studied are presented in Table 2. The reference data are plotted in Fig. 5; all data can be found in Dataset S3. Plots of A very generally follow Eq. (5). Fig. 5A shows four A series, from which a mean value of $\chi = 5.5 \times 10^{-5} \text{EJ}^{-1}$ is determined with about a 5 % error bar ($\pm 0.25 \times 10^{-5} \text{EJ}^{-1}$). The approximation by Eq. (5) is rather good on the whole range, from 1820 to 2014, with $R^2$ values of 0.99; most other fits, in Figs. 1 or 5, show $R^2$ values of at least 0.98. Fig. 5B reproduces three series of 4a with the exponential fit performed on a limited period, 1900–2014, which corresponds to the complete IIASA FEC series (de Stercke, 2014), complemented with food and fodder from Malanima (Malanima, 2020). The value of $\chi$ is almost the same as in 4a for 1820–2014, with the same average $5.5 \times 10^{-5} \text{EJ}^{-1}$. First, this shows that the construction of the 1820–1900 FEC series had no impact on the fit, and secondly that the same exponential trend is found for the 19th and 20th centuries.

However, if food and fodder were not included in the energy series, the GDP to energy ratio would not be monotonous, showing a minimum around the turn of the 19th to 20th century, as already discussed (Bercegol and Benisty, 2022), and demonstrated also in the case of the USA (Haberl, 2001a). In ref. (Bercegol and Benisty, 2022) with mixed PEC-FEC series as explained above, the fit by Eq. (5) was possible in the period 1930–2016 only, whereas turning here to complete FEC series reveals that the law has been verified on average for the last 200 years.

In Figs. 1, 2, and 5, a lower slope is evidenced in the later period, after 2000. Fig. 5C illustrates this distinctive bend in the slope of Log[A] vs cumulative FEC occurring around 2008–2010, with a $\chi$ value decaying from $5.84 \times 10^{-5}$ to $4.16 \times 10^{-5} \text{EJ}^{-1}$ for #6, and from $5.10 \times 10^{-5}$ to $3.93 \times 10^{-5} \text{EJ}^{-1}$ for #8 (see Table 2 for the series #). One may notice that a linear law of A vs cumulative FEC would fit Fig. 5C results as well as two successive exponentials from 1970 on. Without

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 5. Various plots of A vs the cumulative FEC, starting in 1820 (semi-log scale). Series #1, 2, 6, 8 and 10, described in Table 2, are plotted in log-linear graphs and fitted according to Eq. (5). (A) #1, 2 and 6 are plotted and fitted on 1820–2014, and #6 on 1970–2022. To enhance historical reading, 1850, 1900, 1950, and 2000 are marked as black diamonds of series #1. 5 % error bars are indicated on these decennial marks, as well as at decennial points of curve #10. (B) #1, 2, and 10 are plotted and fitted on 1900–2014. (C) #6 and 8 are plotted on 1970–2022 and fitted in two periods, before and after 2008–2010. See §4.4 for comments, and Table 2 for main characteristics of the series and fitted $\chi$ coefficients.</figcaption></figure>

precluding any rationale for a change of behavior from exponential to linear, which remains to be analyzed and explained, I will stick here to the exponential fit by Eq. (5). In any case, a linear approximation is worse than Eq. (5) in the earlier period (1820–1970) and simply wrong when looking at extrapolation towards the past, for mathematical and economic reasons. A linear law of A vs cumulative FEC would project A to impossible negative values.

Table 2 of the paper provides information on the 13 {GDP,FEC}
couples of series used for calculating A (ratio of GDP on FEC) and plotting it as a function of cumulative FEC in Figs. 1, 2 and 5. These data and calculations provide on 1820–2014 a determination of $\chi = 5.5 \times 10^{-5}$ EJ$^{-1}$ with about a 5 % error bar (±0.25 $10^{-5}$ EJ$^{-1}$).

### 4.5. FEC per capita and population data, 10,000 BCE to 1820 CE, used for Table 1

A key figure is the level of energy consumption in the distant past, in the Paleolithic period of hunter-gatherers. Haberl (Haberl, 2001b) proposed 7 GJ per person and per year for FEC, including food and fuel-wood. FEC per capita $\mathcal{E}$ of 7 GJ/cap/y is a lower bound for our calculations. From 10,000 BCE, region after region, human societies have shifted to agriculture as a way of subsistence. $\mathcal{E}$ grew, reaching 14.4 GJ/cap/y in 1820, at the end of the period discussed here.

Numerous historical accounts of energy consumption report that at the turn of the Common Era, in year 0 CE, energy per person was probably as high as at the beginning of the Industrial Revolution, or even higher (Morris, 2013). Consequently, I consider two periods in Table 1, the first 10,000 years when $\mathcal{E}$ changes from 7 to 14.4 GJ/cap/y, and a second period from 0 CE to 1820 CE with a constant $\mathcal{E}$ at 14.4 GJ/cap/y. Four alternative consumption scenarios, detailed in Table 3, yield results averaged in Table 1.

HYDE 3.3 recent population database (Klein Goldewijk et al., 2017) is the reference used in calculating the results of Table 1. This source gives data down to 10,000 BCE, specified by geographical areas relevant to the problem. This database also provided the global population from 1800, as said above.

Eqs. (3) or (5) should be applied to a global system, within which the sharing of inventions, new practices and organizations is possible. Kremer (Kremer, 1993) used this concept when comparing the evolution of separated sets of population in the very long run. He convincingly showed that a low initial population induced a very slow economic and population growth in the case of Australia, even a history of zero growth and decline for the smaller Tasmania, whereas the main part of the human species experienced positive and increasing growth rates.

I consider a geographical set, denominated “connected world”, which comprises Eurasia and Africa until 1500 CE. These two continents are connected by a land bridge and have been always so for the last

12,000 years. During this period, the diffusion of innovations occurred over most of the land mass. For example, the invention of the chariot in the early second-millennium BCEE (Lindner, 2020), or before that the domestication of horses (Hosek et al., 2024), both originating in the steppes of central Eurasia, took hundreds of years but finally reached Western and Eastern parts of the Eurasian continent, as well as Northern Africa. The same is true for many inventions, from the horse collar to paper (Smil, 2017; Debeir et al., 2013) and Indo-Arabic numerals, all of which crossed Eurasia in a timescale of centuries to millennia, in the last 2000 years. One can challenge the inclusion of Sub-Saharan Africa in this set, or even the connectivity of the West and the East before 2000 BCE. However, one can see from precise data that contributions to population, thus to cumulative FEC, of the period 10,000 to 2000 BCE in Eurasia, or of Africa during the ancient history are very small, totaling a few tens of EJ.

The Americas, isolated in the pre-Columbian era, join the “connected world” after 1500 CE, with a similar $\mathcal{E}$. Oceania joins after 1800 CE, with a minute incidence on the final calculation. Using HYDE 3.3 regional data, I calculated the population of this “connected world” as a discrete series, which was interpolated with a Spline function to produce an annual series from 10,000 BCE until 1820 CE (see Dataset S4). Multiplying population by $\mathcal{E}$ yields a total FEC series. Table 1 recalls the hypothesis on FEC per capita, gives the determined total cumulative FEC of the “connected world”, and provides the estimate of efficiency gains by applying Eq. (5) (averaged from 4 different scenarios, see Table 3).

Kremer's argument (Kremer, 1993), recalled above, on population divergence between regions also applies to the present model, population P being the major evolving factor in $E = P \times \mathcal{E}$. The smaller cumulative FEC of continents isolated from the “connected world” induces smaller subsequent innovation change $\Delta A$ at the later time of civilization encounters. Within the model, using similar assumptions on $\mathcal{E}$ to those in Table 1 and population data available in Dataset S4, efficiency gains in the Americas are estimated to have been around 10–13 % between 10,000 BCE and 1500 CE, depending on scenarios of Table 3 (assuming Neolithic transition scenarios in Americas similar might be discussed). Calculations in Table 1 neglect numerically this contribution. Nevertheless, it is certainly non-negligible culturally, with e.g. energy-rich plants such as corn and potatoes, brought from the New to the Old World.

<caption>**Table 3**</caption>

Four different scenarios concerning FEC per capita trajectories before the Common Era. The later part from 0 CE to 1820 CE is left unchanged. The value of A is calculated with Eq. (5) using $\chi = 5.5 \times 10^{-5}$ EJ$^{-1}$, with an initial value =1 in 10,000 BCE. Another parameter change consists in changing the value of $\chi$. In the case of scenario 1, using a lower value of $\chi = 4 \times 10^{-5}$ EJ$^{-1}$ yields A = 1.19 in 0 CE (instead of 1.28) and 1.71 in 1820 CE (instead of 2.1); whereas using a higher value of $\chi = 6 \times 10^{-5}$ EJ$^{-1}$ yields A = 1.30 in 0 CE and 2.24 in 1820 CE.

<table><thead><tr><th>Scenarios</th><th>state in 10,000 BCE</th><th>period from 10,000 BCE to 0 CE</th><th>state in 0 CE</th><th>period from 0 CE to 1820 CE</th><th>state in 1820 CE</th><th>A in 10,000 BCE (norm)</th><th>A in 0 CE</th><th>A in 1820 CE</th></tr></thead><tbody><tr><td>1 hypothesis on FEC per capita (GJ/cap/y)</td><td>7.0</td><td>linear increase with time</td><td>14.4</td><td>stable</td><td>14.4</td><td rowspan="2">1</td><td rowspan="2">1.28</td><td rowspan="2">2.10</td></tr><tr><td>Resulting Cumulative FEC from 10,000 BCE (EJ)</td><td>0</td><td>/</td><td>4422</td><td>/</td><td>13,470</td></tr><tr><td>2 hypothesis on FEC per capita (GJ/cap/y)</td><td>7.0</td><td>linear increase with population</td><td></td><td>Same as scenario 1</td><td></td><td rowspan="2">1</td><td rowspan="2">1.21</td><td rowspan="2">1.99</td></tr><tr><td>Resulting Cumulative FEC from 10,000 BCE (EJ)</td><td>0</td><td>/</td><td>3474</td><td>/</td><td>12,522</td></tr><tr><td>3 hypothesis on FEC per capita (GJ/cap/y)</td><td>7.0</td><td>new population has fixed FEC per capita (14.5 GJ/cap)</td><td></td><td>Same as scenario 1</td><td></td><td rowspan="2">1</td><td rowspan="2">1.30</td><td rowspan="2">2.14</td></tr><tr><td>Resulting Cumulative FEC from 10,000 BCE (EJ)</td><td>0</td><td>/</td><td>4772</td><td>/</td><td>13,820</td></tr><tr><td>4 hypothesis on FEC per capita (GJ/cap/y)</td><td>7.0</td><td>FEC per capita grows proportionally to A (path dependent)</td><td></td><td>Same as scenario 1</td><td></td><td rowspan="2">1</td><td rowspan="2">1.20</td><td rowspan="2">1.95</td></tr><tr><td>Resulting Cumulative FEC from 10,000 BCE (EJ)</td><td>0</td><td>/</td><td>3300</td><td>/</td><td>12,348</td></tr></tbody></table>
Alternatively, instead of neglecting the effect of pre-Columbian efficiency growth after the discovery of the Americas, one could calculate an upper bound of A, by supposing that the growth of A in the Americas is incorporated instantaneously in the global efficiency. A would then experience a sudden jump around 1500 CE when New World inventions combine with Old World ones. Assuming that hunter-gatherers efficiencies in 10,000 BCE are similar in both worlds, the mathematics of Eq. (5) imply that after 1500 CE A figures would be 10 to 13% bigger, as if continents had always been connected. Similar reasoning with Oceania around 1800 CE would cause an effect around 100 times smaller, utterly negligible. In this hypothesis of a perfect combination of New and Old World technologies, the overall growth in A from 10,000 BCE to 1820 CE would then be about a factor of 2.25.

## 5. Conclusions

Data at world level for the last 200 years have been gathered and analyzed in order to compute global energy efficiency defined as the ratio GDP/FEC (gross domestic product to final energy consumption). I evidenced a steady trend and modelled it by assuming that the increase of energy efficiency depends linearly on the economic activity. The model is summarized in a linear relationship between energy efficiency growth and FEC (Eq. (3)). Its integral form (Eq. (5)) expresses that energy efficiency grows exponentially with cumulative FEC. This was used to estimate the course of energy efficiency from the far past and to perform comparisons to existing energy transition scenarios up to 2050. The three periods – 10,000 BCE to 1820; 1820 to 2016; 2016 to 2050 – show comparable levels of cumulative FEC, about 13,000 to 25,000 ExaJoule (EJ), and consequently levels of energy efficiency growth of the same order of magnitude.

The model provides a first estimate of energy efficiency growth from the Neolithic to the Industrial Revolutions, by a factor of around 2. The assumptions made to obtain this result could and should be challenged; moreover, there is plenty of room for data improvement. Completing historical series, comparing results with other research programs and perspectives, compiling and consolidating data should remain high priority efforts.

Following a recent downturn in energy efficiency growth, it comes as no surprise that an ambitious energy transition scenario like NZE of the IEA hypothesizes a radical upturn, and envisages decades of hitherto unseen progress of energy efficiency. Whether or not such an upturn could occur probably depends on a still unclear set of political, economic and technological decisions. However, to aid and monitor the transition, the present model could now be used comprehensively and systematically to grade energy efficiency hypotheses, and outcomes, of energy-economy modelling. This should improve the debate on growth and degrowth, of GDP and of consumption of resources, energy or matter.

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ecolecon.2025.108666.

## Author contribution

He conceived this research, realized it, and wrote the paper.

## Funding

No other funding than the author's salary was used in this investigation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This work benefitted from many interesting discussions with Henri Benisty, in its beginning, and with Paul Brockway, in its final writing stage. The author warmly thanks both of them. He also acknowledges fruitful debates when co-organizing and hosting interdisciplinary seminars at Laboratoire Interdisciplinaire des Énergies de Demain (LIED) of Université Paris Cité, notably with Mathieu Arnoux and Petros Chatzimpiros. Finally, the author would like to thank two anonymous reviewers for their careful reading and helpful suggestions.

## Data availability

All data, code, and computations used in this research are available as supplementary data files.

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