List of Figures

3.1 The economy, as a metabolism embedded in the society and in the environment, or our "Earth Spaceship". Own elaboration. . . . . . . . . . . . . . . . . . . . . . 17 3.2 The feedack driven economic growth model in the absence of natural resource constraints, an ecological approach. Inspired from [START_REF] Warr | Useful work and information as drivers of economic growth[END_REF]. . . . . 18 5.1 On the left, the Spanish real GDP (blue) and the Spanish useful exergy (red) growth rates. On the right, the Spanish real GDP (blue) growth rate in function of the Spanish useful exergy (red) growth rate -each data point representing the couple (Useful exergy growth, GDP growth) for a specific year. Own elaboration of useful exergy data, and GDP data directly from the Penn World Table -"rgdpe" variable used [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF]. Data for the years 1960-2013. . . . 30 6.1 On the left, actual GDP (dark circles), as well as fitted GDP with a two factors of production Cobb-Douglas APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 On the left, actual GDP (dark circles), as well as fitted GDP with a two factors of production Constant Elasticity of Substitution APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.3 On the left, actual GDP (dark circles), as well as fitted GDP with a three factors of production Cobb-Douglas APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period

Summary

This thesis explores the energy-economy interactions with a focus on the particular Spanish case. It is argued that the current economic paradigm promotes economic growth, which is expected to bring about greater wealth and is thought to be compatible with sustainability. However, it is evidenced that economic growth has been historically tightly correlated with energy consumption. Besides, two main constraints to energy consumption are presented: the climate change issue, which is entailed to a great extent by energy consumption, and the perhaps less widely known impending fossil fuel -particularly oil -depletion. As such, these energy constraints appear to jeopardise future economic growth and justify further work in order to shed light on the energy-economy relationship.

The main limitations of the mainstream neoclassical theory of growth are thereafter presented, and it is stressed that this theory does neither explicitly includes physical flows nor natural resources consumption, and relies considerably on exogenous and unexplained technological progress. In opposition, an ecological theory of growth which explicitly considers physical flows is presented. In this view, natural resources consumption and efficiency gains in natural resources use are main drivers of economic growth. Conversely to the mainstream theory of growth, the ecological theory of growth presented therefore makes explicit the role of natural resources consumption in the economy. This thesis takes the ecological standpoint in order to explore the energy-economy nexus through three different approaches.

Firstly, a Granger causality test between energy and GDP growth rates is carried out in order to determine whether a statistical causality can be identified. Secondly, this thesis explores whether including energy as an additional factor of production in Aggregate Production Functions (APFs) enhances the fit and the predictive capacity of such APFs. Lastly, a conceptual energy-economy model that makes explicit the underlying mechanisms of the energy-economy interactions is presented. All these approaches rely on the exergy concept, which accounts for both energy quantity and quality. It is suggested that useful exergy, namely exergy at the enduse stage, is the best proxy for energy services and therefore the energy metric most appropriated for studying energy-economy interactions. This thesis includes notable novelties. Firstly, it provides a new causality test with useful exergy as a quality adjusted metric for energy consumption, as only one previous study has been identified in the literature. Besides, the parameters of the test -causality test performed, number of lags, time span, energy and GDP metrics -are considerably varied and a metaanalysis is carried out on the results of the causality tests. Secondly, and conversely to the conventional approach, the Total Factor Productivity (TFP) is excluded of the fitted APFs, as it is argued a priori that the justification for such a TFP is weak in an ecological perspective. Then, the predictive capacity of the tested APFs is also assessed as best fits are extrapolated for the post-economic downturn period (2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016), which is an unusual approach. Finally, the conceptual energy-economy model presented includes parameters from different literatures, including energy prices and affordability, Energy Return On Investment (EROI), stocks, flows, demand, investments in energy industries... Relevant findings are drawn from the results. Firstly, from the causality tests conducted, it can be said that no salient evidence of causality could be reliably identified as the results of the tests depend widely on the parameters, which is consistent with the literature. When specific parameters -useful exergy and output-side GDP -are used, energy consumption is found to Granger cause GDP in most cases, although this is reckoned to be limited evidence. Limitations vii of causality tests are pointed, and the ability of such tests to identify causality in the energyeconomy context is seriously questioned, both because of the underlying mathematical model and on a conceptual basis. It is suggested that exploring and modelling the energy-economy relationships is likely to be a more fruitful activity than expanding the inconclusive causality literature.

Secondly, from the APF analysis conducted, it is found that the inclusion of energy in APFs improves the fit in most cases, although some acceptable fits are also obtained without energy. The predictive capacity during the downturn is improved in most cases when using energy, although some functions reject energy as a factor of production, which is thought to be a consequence of a strong correlation between the factors of production before 2009. Besides, it is found that using quality adjusted metrics does often not provide a better fit than using unadjusted quality metrics, which raises the issue of double counting when adjusting all factors of production, particularly for energy and capital. Finally, caveats to APF modelling are identified. Indeed, large changes in fitted parameters can have a low impact on the fitted GDP, and fitted parameters may be economically meaningful but unrealistic. It is therefore suggested that other modelling approaches which capture the underlying mechanisms of the energy-economy nexus, such as system dynamics, can provide more useful and reliable insights than APF modelling.

Thirdly, a conceptual energy-economy model which successfully integrates concepts from different literatures -such as dynamics of fossil fuels stocks, declining EROI of energy sources, Renewable Energy Systems (RES) development, investments in energy industries, and price dynamics -is presented. It therefore provides a helpful synthesis of how different parameters interact. It stresses the importance of supply constraints, both in terms of physical availability and economic affordability. Therefore, this conceptual model seems an appropriate framework for energy-economy modelling in a context of finite resources and supply constraints, i.e. in the current context. Besides, the suggested model rejects standard economic theory, according to which energy and oil prices will rise as resources are depleted and consequently entail efficiency improvements and the development alternatives that will solve the shortage. Conversely, an affordability limit is presented as a limiting factor to the rise of energy prices, and therefore as an obstacle to the development of costly alternatives and extraction of more costly resources.

Chapter 1

Introduction

Economic growth is currently the dominant economic and political paradigm. The endeavour of governments to obtain economic growth is hardly questioned. Meanwhile, environmental impacts and resource consumption are rising alongside economic growth and become a major concern [Steffen et al., 2015a,b, Ripple et al., 2017]. But the dominant paradigm assumes that environmental impacts can evidently be decoupled from economic output and economic growth through eco-efficiency, or resource productivity [START_REF] Von Weizsacker | Factor Five: Transforming the Global Economy through 80% Improvements in Resource Productivity[END_REF][START_REF] Hawken | Natural Capitalism: The next Industrial Revolution[END_REF]. In this chapter, the economic growth paradigm is questioned, and evidence of linkages between economic output and energy consumption are underlined. Then, two physical constraints to energy consumption, namely climate change and fossil fuel depletion, are evidenced. Due to these constraints on energy consumption, it is argued that a better understanding of the relationship between energy consumption and economic output is needed. Finally, the research questions and the study case are presented.

Economic growth, a controversial paradigm

As mentioned, economic growth is currently the unquestioned paradigm. At the country level, economic growth shapes national policies, as showcased with the British "Clean Growth Strategy" [UK Government, 2017] and the French "law for energy transition and green growth" [French Parliament, 2015], as growth is said to create wealth and employment. International organisations such as the OECD are equally keen on green growth [OECD, 2011]. Economic growth is also seen as the solution to world poverty and regarded as an urgent need in the so-called developing and non-industrialised countries [OECD, 2012]. As such, the Sustainable Development Goals (SDGs), released by the United Nations, regard economic growth as a key development target, as stated in Goal number 8, "Decent Work and Economic Growth" [UN, 2017]:

• "Sustain per capita economic growth in accordance with national circumstances and, in particular, at least 7 per cent gross domestic product growth per annum in the least developed countries."

• "Improve progressively, through 2030, global resource efficiency in consumption and production and endeavour to decouple economic growth from environmental degradation."

Even though such a dominant concept, there are still strong and diverse criticisms to economic growth. These criticisms include the idea that more wealth does not make people happier above certain standards of living [START_REF] Jackson | Prosperity without Growth: Economics for a Finite Planet[END_REF], and even that too much material wealth alienates human beings [START_REF] Illich | Tools for conviviality[END_REF] 1 . Other critics state that economic growth is a substitute for equality, as stated by Henry Wallich, former governor at the Federal Reserve [START_REF] Wilkinson | The Spirit Level: Why Greater Equality Makes Societies Stronger[END_REF]: "Growth is a substitute for equality of income. So long as there is growth there is hope, and that makes large income differentials tolerable."

According to these criticisms, what is needed is greater equality instead of further economic growth [Dietz and O'Neill, 2013, chapter 7]. Others argue the impossibility of decoupling environmental impacts from economic growth [START_REF] Ward | Is Decoupling GDP Growth from Environmental Impact Possible[END_REF], or the fact that such decoupling is not happening when looking at the global scale [Bithas andKalimeris, 2013, 2018]. Besides, physical limits to growth are also often regarded as a key barrier to infinite economic growth. This last group of critics takes roots on the famous "Limits to Growth" report [START_REF] Meadows Donella | The limits to growth. A Report for The Club of Rome's Project on the Predicament of Mankind[END_REF], although [START_REF] Forrester | World Dynamics[END_REF] described earlier the physical limits to growth approach. Criticisms based on limits and finiteness of resources are not new, and [START_REF] Malthus | An Essay on the Principle of Population[END_REF] is often referred as being the first publication depicting a forthcoming end of growth due to resources scarcity. Such ideas have been discussed quite before the "Limits to Growth" report, as Weil [1934] underlines: "[...], it is clear that capitalism means principally economic expansion, and that capitalist expansion is not far from the moment where it will hit the Earth's limits."2

In the present study, the focus is on the relationship between energy consumption and economic growth. This is directly related to two of the criticisms mentions above; firstly, to the increasing environmental impacts and particularly climate change, and secondly, to the natural resources finiteness, and especially those of fossil fuels. Indeed, if energy consumption is a key element in economic growth, then these criticisms are legitimate, as growth would be directly related to greenhouse gases emissions and threatened by fossil fuel depletion. Section 1.2 underlines how global economic growth is statistically correlated to global energy consumption, notwithstanding eco-efficiency advocates. Section 1.3 explains shortly the climate change issue and how it is directly relates to energy consumption. Then, Section 1.4 elucidates the impending fossil fuel depletion, which is likely to bring about energy supply constraints.

Economic growth and energy consumption

It has already been mentioned that eco-efficiency and resource productivity are the mainstream solutions to environmental impacts and natural resources depletion. Notwithstanding these pervasive ideas, the truth is that at the global scale, there has not been any decoupling3 between energy consumption and economic growth4 [START_REF] Bithas | Re-estimating the decoupling effect: Is there an actual transition towards a less energy-intensive economy[END_REF]. Figures 1.1 and 1.2 showcases the evident link between economic output and energy consumption at the global scale.

The linear regression of global real GDP in function of energy consumption -both primary and final -gives a very good fit (R 2 of 0.9940 for primary energy, and of 0.9935 for final energy). Besides, the growth rates of GDP and energy consumption seem to evolve simultaneously, supporting the evidence for a strong link between the two values. It is clear from these figures that there is presently no global decoupling between energy consumption and economic growth.

At the national scale, very few countries have achieved absolute decoupling between energy and GDP [START_REF] Csereklyei | Theoretical efficiency limits for energy conversion devices[END_REF]. And the few countries that have achieved absolute decoupling between energy consumption and economic growth, such as the UK, have relied heavily on 1.2. Economic growth and energy consumption q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1.0 1.5 2.0 2.5 Global primary energy consumption -index 71 Global real GDP -index 71 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1.0 offshoring production to developing countries [Hardt et al., 2018, Moreau and[START_REF] Moreau | Decoupling energy use and economic growth: Counter evidence from structural effects and embodied energy in trade[END_REF]. It can therefore been said that economic growth and energy consumption have been, so far, tightly correlated, and that there is no evidence of decoupling happening.

Besides, a new field of research, exergy economics, has been providing different evidence for the role of energy consumption in economic growth for the last 15 years. The main findings of this field are shortly presented in Section 2.1. These different elements bring us to the following statement: there is a tight relationship between economic output and energy consumption, although this relationship is not fully understood. Consequently, it is extremely likely that more economic growth will lead to an increase in energy consumption. However, there are physical constraints to energy consumption, and therefore to economic growth.

Energy consumption and climate change

The greenhouse effect and its consequences on the global climate were first described by [START_REF] Arrhenius | On the influence of carbonic acid in the air upon the temperature of the ground[END_REF], who identified roughly the increase in the global average temperature one could expect from burning fossil fuels. Since then, our knowledge of the climate system has considerably increased through decades of research, and it seems clear that climate change is human made [IPCC, 2014a, page 47]. There is considerable research on the likely physical impacts of climate change depending on the increase in temperature [IPCC, 2014b]. Economists have also assessed the economic impacts of climate change in order to recommend specific policies [Nordhaus, 1992, Stern and[START_REF] Stern | The Economics of Climate Change: The Stern Review[END_REF]. Although there is significant debate on how to assess the impacts of climate change from an economic standpoint [START_REF] Nordhaus | A Review of the "Stern Review on the Economics of Climate Change[END_REF][START_REF] Weitzman | On Modeling and Interpreting the Economics of Catastrophic Climate Change[END_REF] and significant uncertainty on the feedback loops of the Earth's climate system (e.g. permafrost thaw, increase in water vapour in the atmosphere...) [IPCC, 2018, Section 2.2.1], it is clear that climate change is about to trigger considerable impacts on human societies.

Yearly climate summits hold at the Conference of the Parties (COP) have led to the Paris Agreement in 2015, that should enable to keep the rise in global temperatures below 2°C and potentially under 1.5°C. Despite such an agreement, greenhouse gases emissions are still increasing [START_REF] Jackson | Warning signs for stabilizing global CO2 emissions[END_REF][START_REF] Quéré | Global Carbon Budget[END_REF].

The new IPCC Special Report on Global Warming of 1.5°C explores the scientific literature on climate scenarios that would enable to remain below both 1.5 and 2°C. This report underlines that 1.5°C consistent pathways "tend to cluster on the lower end for energy and food demand", although the scenarios cover a wide range of developments from decreasing to increasing energy demand [IPCC, 2018, Section 2.3]. But still, 1.5°C pathways rely on lower energy consumption than higher global temperature increase pathways. This is directly due to the fact that most greenhouse gases emissions are related to energy consumption processes [IPCC, 2014a, Section 1.2], which underlines the strong link between energy consumption and climate change. Based on these elements, it can be said that if we are to realistically limit climate change under a critical threshold and with a high probability, energy consumption needs to be capped, or even lowered.

Fossil fuel depletion and looming energy constraints

Besides climate change, a second physical limit is directly linked to the finiteness of non renewable natural resources, and particularly fossil fuels. The finiteness of a non renewable resource constrains its extraction profile. This one must, mathematically, go through a peak, and decrease until eventually insignificant levels5 , when one considers a time period long enough. This phenomenon is known as an extraction peak. Such a peak in the extraction of a non renewable resource does not mean that this resource becomes unavailable after the peak, but that it will become increasingly scarce, unless the demand changes accordingly.

These peaks have been forecasted for different fossil fuels [Capellán-Pérez et al., 2017a, Section 2.3] and different countries [START_REF] Mohr | Projection of world fossil fuels by country[END_REF]. Similar works predicting extraction curves for diverse minerals have also been built [START_REF] Calvo | Assessing maximum production peak and resource availability of non-fuel mineral resources: Analyzing the influence of extractable global resources[END_REF]. Although this concept exists and has been discussed for natural gas, coal, uranium, and numerous non energetic minerals, the focus in this section is on peak oil, as it is (i) the most widely known and debated peak, (ii) the most imminent one, and (iii) likely to have massive impacts on the world economy, as oil provided 34% of global primary energy consumption in 2017 [British Petroleum, 2018].

The concerns for a peak in the extraction of fossil fuel were firstly raised by [START_REF] Hubbert | Nuclear energy and the fossil fuel[END_REF], who proposed a new model for fossil fuel depletion (known as the Hubbert model or the bell shaped curve) and forecasted a peak of conventional oil for the US around 1970, and for the world, around 2000. The forecast was found to be accurate for the US, that indeed passed its peak oil around 1970. However, as the oil prices plummeted in the second half of the 80s, the interest in oil depletion and in Hubbert predictions dropped significantly. However, Hubbert's work was updated in 1998 [START_REF] Campbell | The End of Cheap Oil[END_REF], who obtained results consistent with Hubbert, with a peak in global conventional oil production forecasted before 2010. This landmark paper brought about the Association for the Study of Peak Oil (ASPO), which largely contributed to popularising this concept. Nevertheless, a couple of decades after this warning, a scarcity in oil can hardly be noticed, and the global peak does not seem to have occurred. As a result, the interest for the peak oil concept has dropped anew.

Nevertheless, and as explained in [START_REF] Bardi | Peak oil, 20 years later: Failed prediction or useful insight[END_REF], there is no valid reason for such a dismissal of the peak oil argument. Although the bell shaped curve that Hubbert proposed is not valid in each case [START_REF] Brandt | Testing Hubbert[END_REF], a broad understanding of Hubbert's methodology would shed light on the fact that "depletion of conventional oil is inevitable and becoming rapidly more important" [START_REF] Brandt | Testing Hubbert[END_REF]. Indeed, the shape of the extraction curve depends on numerous political and economic factors that cannot be taken into consideration in geological models. As such, the bell shaped curve can be seen as the idealised extraction profile when there are neither political nor economic constraints [Bardi, 2019, Kerschner andCapellán-Pérez, 2017]. Besides, numerous recent studies are deeming peak oil as an imminent phenomena that is to be expected shortly. Figure 1.3, extracted from Capellán-Pérez et al. [2017a], summarises the oil depletion curves projected for different studies. The general trend showcased by the figure is that peak oil is to be expected imminently. The International Energy Agency (IEA) seems overly optimistic in its World Energy Outlook 2012 [IEA, 2012] when compared to the other studies. One main caveat to this figure, however, is the definition of oil, which is largely inconsistent between studies (e.g. inclusion or not of natural gas liquids, unconventional oil...). However, the IEA seems to have recently changed its mind about oil projections and is currently releasing less optimistic analysis. The IEA currently recognises that the peak of global conventional oil production was attained in 2008 [IEA, 2018a, page 45]. This gives legitimacy to the previous works of Hubbert, Campbell and Laherrère [Hubbert, 1956, Campbell and[START_REF] Campbell | The End of Cheap Oil[END_REF], that projected respectively a peak of conventional oil around 2000 and before 2010. Despite fierce criticisms, these forecasts have therefore proved to be relatively accurate. Besides, the IEA's new WEO in 2018 releases worrisome forecasts of the future of oil, as displayed in Figure 1.4. The natural decline observed in Figure 1.4 represents the decline in oil production if oil producing companies were not to invest anything in maintaining current production, which is extremely unlikely to happen. The observed declined represents the decline in oil production if companies were to invest in order to maintain production, but if new exploitation permits were not granted, which is also unlikely to happen. In order to avoid a mismatch between oil demand and supply, the IEA reckons that the annual approval of new resources needs to double. The current rate of new resources approved would lead to a mismatch of 13 mb/d in the forthcoming 7 years, which represents 13% of the forecasted oil demand in 2025. This constitutes an implicit assumption that the global peak in oil production is imminent and that the supply in 2025 is likely to be lower than the present one, and is a worrisome statement by the IEA.

Besides this graph from the IEA, two more reasons that supports the imminence of a global peak oil are presented. Firstly, there is a general decrease in upstream investment in fossil fuels industries. Indeed, investments have dropped since 2013 in most producing countries [IEA, 2018b, figure 1.5], which raises concerns about the ability of the oil industry to provide sufficient oil over the long term. Secondly, it has been shown that the currently increasing supply of oil has been exclusively supplied by an increase in the US oil production [Tverberg, 2018a]. And this increase is due to the steep development of the tight oil (or shale oil) industry, as displayed in Figure 1.5. This industry currently provides 7.5 bbl/d (around 10% of the current global oil consumption) of unconventional oil, and has increased by more than 6 bbl/d since 2010. However the tight oil industry has been shown to be financially inefficient as even with a barrel price of $100 these businesses are cash flow negative [St Angelo, 2017a,b]. As such, the US tight oil industry is presented as a "Ponzi scheme", namely a deceptive investment that leads to a financial bubble that will burst when investors stop financing it [START_REF] Angelo | The Shale Oil Ponzi Scheme Explained: How Lousy Shale Economics Will Pull Down The U.S. Economy[END_REF]. And this may be already happening, as stated by [START_REF] Olson | Frackers Face Harsh Reality as Wall Street Backs Away[END_REF]: "The once-powerful partnership between fracking companies and Wall Street is fraying as the industry struggles to attract investors after nearly a decade of losing money. Frequent infusions of Wall Street capital have sustained the US shale boom. But that largess is running out. New bond and equity deals have dwindled to the lowest level since 2007. Companies raised [...] less than half the total in 2016 and almost one-third of what they raised in 2012."

What will happen with these daily 7.5 bbl that are provided by the US tight oil industry when this financial bubble bursts and investors withdraw their investments? The financial instability of this industry is very likely to make peak oil quicker and steeper. And the numerous environmental concerns raised by these industries that could lead to a massive political opposition and that are underlined in a new report for the British Columbian government (US) are not raised here [START_REF] Shfrp | Scientific Review of Hydraulic Fracturing in British Columbia[END_REF].

Research questions

One can draw the following conclusions from the previous sections:

(i) Economic growth is the current dominant paradigm in economics and policy making, although it is fiercely criticised on different grounds.

(ii) Economic growth seems to be empirically tightly linked and correlated to energy consumption, which raises the question about whether energy consumption is a key driver of economic growth. If this is the case, then decoupling these two processes seems overly optimistic, and economic growth may be constrained by the two following dynamics.

(iii) The first limiting dynamic is global climate change. Greenhouse gases emissions are mainly a result of burning fossil fuels in order to provide the society with energy. If destructive climatic changes are to be avoided, then fossil fuel consumption needs to dramatically decreased, which will lower energy availability, and, consequently, economic growth.

(iv) The second limiting dynamic is fossil fuel depletion. There are some worries around the issue of the peak of production of fossil fuels and about the impact that these may have on the different national economies. Particularly, there is presently a deep concern on the question of peak oil, as the peak of production of conventional oil already passed and as the IEA seems to announce an imminent peak in global oil production.

This analysis brings us to the question about whether there is evidence that energy consumption is a main driver of economic growth or not. If there is such evidence, then economic growth seems constrained by the two aforementioned physical limits, and it is likely that global economic growth will come shortly to an end. This question will be explored based on the specific Spanish case. The reasons for choosing this specific case are twofold. Firstly, there is presently no detailed study about energyeconomy interactions in the Spanish case. Secondly, Spain has been particularly struck by the economic crisis in 2008 and the economic downturn has lasted several years. This makes it a particularly interesting case when studying the interactions between energy consumption and economic growth. The following research questions are explored in this study:

To which extent can energy consumption explain the recent Spanish economic history (1960-2016)?

• Is there statistical evidence of causality between GDP and energy consumption?

• To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)?

• What conceptual model can explain the interactions between the economy and energy consumption?

One key note on the formulation of the research questions is related to the loose use of the term "energy". As Chapter 2 underlines, this report relies considerably on the exergy concept, which is deemed to be a better measure when studying the interplay between energy consumption and the economy. However, exergy can be regarded as a particular measure of energy consumption, which justifies the simple use of the term "energy" in the research questions. Likewise, when the term "energy" is used in this report, exergy is encompassed by the term as a particular energy metric.

The first research question is explored by studying the growth rates of energy consumption and economic growth, as well as conducting a Granger causality test between these. The second research question is explored through the use of Aggregate Production Functions (APFs) that describe the economic output (real GDP) in function of capital, labour, and energy. The third research question is discussed through literature review, and a conceptual energy-economy model is proposed. The methodology for each research question is further developed in Chapter 4.

Chapter 2

Societal Exergy Analysis

This section introduces the exergy concept, and gives an overview of the different uses of this concept. The field of exergy economics, in which this study is rooted, is subsequently presented, and the advances of this field are shortly presented. Then, the societal exergy analysis approach, which is the foundation of exergy economics, is detailed. The two main characteristics of societal exergy analysis, namely the extension of the analysis to the useful stage, and the focus on exergy, are highlighted. Finally, it is underlined how the societal exergy analysis approach directly relates to the research questions.

The exergy concept

The exergy concept was firstly defined in Rant [1956], although the physical concepts were already introduced 70 years earlier by [START_REF] Anderson | On the Conversion of Heat into Work[END_REF]. Exergy is a physical value that completes the traditional energy concept, and explicitly includes the second law of thermodynamics in its definition. As such, exergy does not obey the conservation principle that applies to energy. While energy is conserved in each process, exergy is partly destroyed as entropy increases. The exergy content of a system is a measure of its ability to perform work, and is expressed in traditional energy units (e.g. Joules). In thermodynamics, the exergy content of a system is defined as the maximum amount of work that a system can provide to its surroundings as it reversibly approaches thermodynamic equilibrium.

Two things can be stated from this definition. Firstly, exergy is defined according to the reference environment [START_REF] Romero | Exergy as a global energy sustainability indicator. A review of the state of the art[END_REF]. For instance, the ocean is plentiful of thermal energy that originates in the temperature of its water. However, the exergy content of this water is extremely low as the temperature of the atmosphere is similar to the one of the ocean. Conversely, if the atmosphere was at a significantly lower temperature, one could harness a large amount of the ocean water energy. The Carnot efficiency1 defines the available energy -i.e. exergy -of a thermal source, depending on the temperature of its surrounding environment. Secondly, as exergy measures the capacity of a system to provide work, it is both a measure of energy quantity and quality. It accounts for the fact that not all the forms of energy are as valuable. The same amount of energy in electrical, mechanical and thermal forms have different exergy contents. While electrical and mechanical energy can be regarded as "pure exergy", the exergy content of thermal energy is defined by the aforementioned Carnot efficiency.

Exergy economics, a new branch of research, hinges upon the exergy concept, and seeks to apply it to macroeconomics. The state of the art and diverse achievement of this field of research is developed in [START_REF] Brockway | Exergy Economics: New insights into energy consumption and economic growth[END_REF]. Among other things, studies have been looking at national exergy efficiencies and evidencing that efficiency gains are slowing down in industrialised countries [START_REF] Williams | Efficiency Dilution: Long-Term Exergy Conversion Trends in Japan[END_REF][START_REF] Brockway | Divergence of Trends in US and UK Aggregate Exergy Efficiencies[END_REF], due to inefficient final uses of energy (for instance, air conditioning) taking a more important share of final energy consumption. In regard to the energy intensities of national economies, it has been shown that conversely to primary and final energy intensity metrics, useful exergy intensities have been staying remarkably constant [START_REF] Serrenho | Decomposition of useful work intensity: The EU (European Union)-15 countries from[END_REF][START_REF] Serrenho | Structure and dynamics of useful work along the agriculture-industry-services transition: Portugal from 1856 to[END_REF], therefore contrasting with the idea that our societies are becoming less dependent on energy (exergy) consumption. Neoclassical economic growth theory has been discussed and it has been showcased that the inclusion of exergy as a factor of production is a key element in understanding recent economic growth [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF]Warr, 2005, 2010]. Exergy-based methods for assessing future energy demand have also been developed, and exergy-based projections generally find a future energy demand that is significantly higher mainstream projections [START_REF] Brockway | Understanding China?s past and future energy demand: An exergy efficiency and decomposition analysis[END_REF], thereby questioning the possibility of meeting both economic and energy goals simultaneously [START_REF] Heun | Meeting 2030 primary energy and economic growth goals: Mission impossible?[END_REF]. The rebound effect has also been studied through exergy lens [START_REF] Brockway | Energy Rebound as a Potential Threat to a Low-Carbon Future: Findings from a New Exergy-Based National-Level Rebound Approach[END_REF], and exergy-based macroeconomic models are currently being developed, the findings of which support the idea of economic growth driven by energy consumption and efficiency [START_REF] Sakai | Thermodynamic Efficiency Gains and their Role as a Key 'Engine of Economic Growth[END_REF].

Although the applications of exergy economics are very diverse, all these exergy economic research rely upon societal exergy analysis studies, which constitute the starting point of further work linking exergy and the macroeconomy. The next section therefore presents the societal exergy analysis approach.

Societal exergy analysis

Exergy analysis is the study of exergy flows throughout a system. Although this can be done at different levels, for instance at the industrial or sectorial level [START_REF] Seckin | Extended exergy analysis of Turkish transportation sector[END_REF], here we focus on the societal level, hence the name societal exergy analysis. The usual scale for such studies is the national scale, even though it is also be possible to carry out such studies at a regional or municipal level [START_REF] Causone | An Exergy Analysis for Milano Smart City[END_REF]. The first national exergy analysis was carried out by [START_REF] Reistad | Available Energy Conversion and Utilization in the United States[END_REF] for the US economy. A few exergy analysis studies were published in the 70s as an answer to energy crisis stemming from the oil shocks [START_REF] Carnahan | Technical aspects of the more efficient utilization of energy: Chapter 2-Second law efficiency: The role of the second law of thermodynamics in assessing the efficiency of energy use[END_REF], Ford et al., 1975a,b]. At this time, energy studies where moving from energy accounting to energy analysis, due to the growing evidence that "the world has many natural resources in finite supply" [START_REF] Webb | The economics of energy analysis[END_REF]. Such studies focused on the energy requirements of products and services [START_REF] Slesser | Accounting for energy[END_REF]. However, mainstream economists fended off energy analysis as it impinged on their realm, and brought about a heated debate on the usefulness of energy analysis [START_REF] Webb | The economics of energy analysis[END_REF]Pearce, 1975, Common, 1976]. Eventually, the interest in energy and exergy analysis decreased, in part due to the decline in energy prices which lowered energy constraints on the economy. However, there has been a renewal of interest since the 2000s as the new field of exergy economics developed, led by Ayres and Warr [START_REF] Ayres | The Economic Growth Engine: How Energy and Work Drive Material Prosperity[END_REF].

Two kinds of societal exergy analysis can be distinguished as underlined by [START_REF] Ertesvåg | Society exergy analysis: A comparison of different societies[END_REF]. The first approach originates in exergy studies for Sweden [START_REF] Wall | Exergy -A Useful Concept Within Resource Accounting[END_REF], and is known as the exergy extended analysis. It aggregates in terms of exergy flows both energy and material flows, using the exergy content of materials [START_REF] Szargut | Exergy Analysis of Thermal[END_REF], Szargut, 1989]. Conversely, the second approach only takes into consideration exergy flows embedded in energy carriers and their use within an economy, and therefore only focuses on tracking down exergy flows due to energy uses. Consequently, this latter approach is adopted, as it is the most relevant one in order to study the relationship between energy and the economy, and therefore to answer the present research questions. However, it is expected that this methodological choice would only make a minor difference as exergy flows included in materials are typically significantly lower than those embedded in energy carriers [START_REF] Ayres | The Economic Growth Engine: How Energy and Work Drive Material Prosperity[END_REF].

The energy conversion chain considered in societal exergy studies in considered in Figure 2.1. Energy flows from the primary to final to useful stage in order to provide energy services.

Chapter 2 -Societal Exergy Analysis

The different stages of energy are defined as:

• Primary energy refers to energy as found is the environment. This form of energy includes wind energy, solar radiation, fossil fuels as extracted from the ground...

• Final energy refers to energy as used by the final user. A conversion process in order to make the primary energy ready to use by the final user is therefore needed. Final energy includes electricity, gasoline as available at the pump, coal as used in coal fired power plants... As Figure 2.1 highlights, one can look at both energy and exergy values for each of the stages included in the system boundaries. The first and second law efficiencies (η, ε)2 enable to calculate the flows of energy and exergy at each stage. Finally, energy is consumed in order to provide energy services (e.g. transportation, illumination...). Energy services are excluded of the boundaries for two main reasons. Indeed, energy services are obtained when useful energy and exergy are dissipated in passive systems (e.g. building envelope, vehicle) as defined by [START_REF] Cullen | Reducing Energy Demand: What Are the Practical Limits[END_REF]. It is not possible to define the efficiency of a passive system as no energy conversion takes place in such systems; the energy is merely dissipated [Cullen and Allwood, 2010]. Then, one could think about defining the efficiency with which useful energy provides energy services as the ratio of useful energy to the minimum amount of energy needed for providing the energy service. But energy services are loosely defined. If we consider transportation from A to B, the service could be defined as going from A to B (i) by gas guzzler, (ii) by public transportation, (iii) by bike, etc... Each choice would give a different minimum energetic requirement. Energy services are therefore left outside the scope of societal exergy analysis studies.

Here, it is worth highlighting two main characteristics of societal exergy analysis studies. Firstly, the focus in on exergy instead of energy. As explained in Section 2.1, exergy is an energy measure that accounts for both energy quantity and quality. It is therefore considered that exergy is closer to the energy service than energy, since it accounts for the work that a particular energy form can provide. Secondly, societal exergy analysis extends the energy conversion chain to the useful stage, instead of stopping at the final stage, as is often done. This is crucial, as the useful stage corresponds better to the satisfied needs and to the physical and economic output. [START_REF] Percebois | Is the concept of energy intensity meaningful[END_REF] points out that energy analysis needs to be carried out at the "output end of energy-using devices", as this enables to capture "changes in the structure of production" and "substitution of different energy forms having different efficiencies or yields". The focus on the useful stage enables to take into consideration the efficiency with which final energy is used in order to provide energy services.

The Spanish societal exergy analysis carried out in [START_REF] Aramendia | Energy Intensity of the Spanish Economy: A Useful Work Study[END_REF] constitutes the starting point of this master thesis, as it provides the energy and exergy data (for each of the three primary, final and useful stages) that are used as inputs for answering the research questions.

Chapter 3

Theories of growth

It is recognised here that every analytical work is embedded in a wider picture, namely the analyst's perception of the world, which biases the subsequent analysis. This has been described as a "pre-analytic vision" [START_REF] Schumpeter | History of Economic Analysis[END_REF][Schumpeter, , first published in 1954]], or as a "frame" [START_REF] Goffman | Frame Analysis: An Essay on the Organization of Experience[END_REF]. Two paradigms and the associated theories of growth are presented and opposed here. Neoclassical economics is presented as the mainstream paradigm, while ecological economics is a subversive paradigm that presents a totally different picture of the relationship between the economy and the environment. After presenting shortly each of these paradigms, the theories of growth associated to each one are presented. It is finally presented how this thesis takes the ecological economics approach in order to answer the research questions.

Neoclassical paradigm

Neoclassical economics

In the neoclassical paradigm, agents are supposed fully rational and be driven by utility maximisation, i.e. satisfaction maximisation1 [START_REF] Dequech | Neoclassical, mainstream, orthodox, and heterodox economics[END_REF]. Hence the notion of economic man, or "Homo Oeconomicus", which behaves according to its own interests. This perception of man can be traced back to Adam Smith's seminal work [START_REF] Smith | An Inquiry into the Nature and Causes of the Wealth of Nations[END_REF], although Smith also emphasized other qualities. Besides, these rational agents are often endowed with perfect knowledge and foresight.

Agents are connected through the market, which is seen as an efficient, or even optimal, way to allocate and distribute resources [START_REF] Agboola | Neoclassical economics and new institutional economics: An assessment of their methodological implication for property market analysis[END_REF]. As a result, a liberal ideology is often praised, as public interventions might hamper the market's efficient allocation, and place the economy on a non-optimal path. This market is supposed to be in equilibrium, namely in a situation where supply equals demand. Prices are supposed to be the element keeping the market in equilibrium and working properly, as they are supposed to be fully representative of the items traded. In particular, prices are supposed to be able to account for the relative scarcity of an item. This trust in prices makes the neoclassical school particularly techno-optimistic. Indeed, a shortage in a commodity would result in a rise in prices, which would stimulate investments in efficiency measures and development of alternatives such as a "backstop technology" in order to solve the situation [START_REF] Bamett | Scarcity and growth: The economics of natural resource availability[END_REF]Morse, 1963, Nordhaus et al., 1973].

These assumptions have all been challenged by other economic schools2 . From an ecological perspective, the sharpest criticism is probably the lack of consideration of physical flows. This non consideration of physical flows leads mainstream economists to present an economic theory, and particularly a theory of growth that is disconnected with the use of natural resources, as underlined in the next section.

Neoclassical theory of growth

The standard theory of growth is based on the Solow-Swan model that was presented simultaneously by [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF][START_REF] Solow | Technical Change and the Aggregate Production Function[END_REF] and [START_REF] Swan | Economic Growth and Capital Accumulation[END_REF]. In this early formulation of the neoclassical theory of growth, there is neither place for energy nor material flows. On the contrary, what appears to drive economic growth, is labour force growth, capital accumulation, and exogenous technological progress (also known as Solow's residual or Total Factor Productivity -TFP). At the core of Solow-Swan based models lie Aggregate Production Functions (APFs), that express the economic output (GDP) in terms of factors of production. The factors of production usually considered in the neoclassical paradigm are labour and capital, while technological progress is an exogenous time multiplier. In his early formulation, Solow found that these factors of production were insufficient in order to describe economic growth from 1909 to 1949 in the US, and that technological progress was responsible for 87.5% of economic growth over this period.

Since then, the Solow-Swan model of economic growth has been considerably elaborated, but the role of energy and natural resource has remained disregarded, or considered as being marginal. This disregard of energy and natural resources is based on two main rationales. Firstly, the neoclassical theory distinguishes between primary factors of production, namely those that are not used up during consumption, and intermediary factors of production, namely those that are used up, or transformed, in the process. In this approach, it is considered that natural resources and energy are just intermediary goods, that can be made out of capital and labour through extractive activities [START_REF] Denison | Explanations of declining productivity growth[END_REF][START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF]. As such, these are not considered as factors of production in the APF approach. Secondly, neoclassical theory postulates the cost share theorem, according to which the output elasticity3 of each factor of production is equal to its cost share in the national accounts. However, cost shares associated to natural resources and energy can hardly be defined, and when they are considered, they are equated to energy industries revenues (as this has been done in the case of energy), which are typically lower than 10%, thereby giving energy a marginal role [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF]. From this, three main criticisms of the neoclassical theory of growth appear:

• Firstly, neoclassical models rely to a large extent to an exogenous TFP in order to explain for economic growth. Although there are currently attempts to reduce the TFP by accounting for capital services, that account for capital productivity, instead of capital stocks, and quality adjusted labour (accounting for skills and working hours instead of only employment), this reliance on exogenous TFP is a shared feature by recent implementations of Solow-Swan based growth models [START_REF] Van Ark | Total factor productivity: Lessons from the past and directions for the future[END_REF]. But ascribing the unexplained economic growth to technological improvements seems hardly a satisfactory explanation of economic growth.

• Neoclassical theory justifies the non inclusion of natural resources claiming that these are intermediary inputs that can be made out of capital and labour and are used up in the production process. However, it is only the availability of natural resources, e.g. through extraction, that is made out of capital and labour, not the resource itself. And this statement could also be reverted, as one can create capital when associating e.g. labour and natural resources. Hence capital could also be regarded as intermediary factors of production [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF]. Besides, it is unclear why an input that is used up in the production process should be of lower importance than a long lasting input such as capital or labour. This argument seems consequently invalid.

• Lastly, neoclassical theory states that the cost share theorem has to be obeyed, and that factors of production output elasticities need to equate their cost share in the national accounts. On an empirical basis, [START_REF] Kümmel | Cost shares, output elasticities, and substitutability constraints[END_REF] showcases how this approach leads to hardships in explaining economic recessions associated to energy crisis4 . On a theoretical basis, this theorem can only be demonstrated with very specific assumptions [Ayres and Warr, 2010, Appendix A], such as the much debated equilibrium assumption [START_REF] Kornai | [END_REF]. Such theorem has been shown to be erroneous when considering more than two factors of production [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF], and does not hold in a disequilibrium economy. Following Ayres, it is argued here that only payments to capital (interests, dividends, rents, royalties...) and to labour (wages) are included in the national accounts. Payments to nature for natural resources are not included, as no one pays for it. Thus, if natural resources are to be considered as a factor of production, the cost share theorem is to be regarded as not applicable from the onset. Eventually, it seems curious to equate a priori the cost share of a factor of production to its output elasticity, while it can only be demonstrated under questionable assumptions. Why should a priori a factor of production be paid in the same proportion as it contributes to economic output? This appears as a normative approach, and output elasticities should rather be empirically determined.

The reasons for not including physical flows in terms of natural resources and energy seem therefore far from satisfactory, and besides, the neoclassical theory of growth leaves a major part of economic growth unexplained. This incomplete description of economic growth calls for a biophysical stance.

Ecological paradigm

Ecological economics

The classical essay of Kenneth Boulding set up the foundations for ecological economics 5 [Boulding, 1966]. In this seminal essay, the economic system is viewed as an open system that relies on inputs -natural resources -and on its capacity to get rid of outputs -wastes. This economic system processes inputs such as fossil fuels and ores, and rejects effluents to the environment. Therefore the essay describes an embedded economy, vulnerable both in terms of supply of physical materials, and in terms of waste disposal capacity. Both the threats of both the exhaustion of natural resources and pollution accumulation are explicitly described. The Earth Spaceship is the closed system that encompasses this open economy. A common strand of this early essay and of the ecological economics field in general, is the perspective of the economy as a system embedded in the environment. Ecological economics appears therefore as the science that describes the economic system from a physical perspective, namely looking at physical flows (energy, materials, wastes) within the economy, but also in relation to the environment, as natural resources are extracted and wastes emitted.

The role of the laws of thermodynamics have proved crucial in order to carry out such physical descriptions. The principle of mass and energy conservation are drawn from the first law of thermodynamics. For each physical process, mass outputs are equal to mass inputs, and energy outputs are equal to energy inputs6 . As the mass of useful products is generally lower than the mass of inputs, wastes are created, and are rejected to the environment in terms of emissions. The implication of such a law is that wastes should be regarded as intrinsic to economic activity, instead of isolated failures of the market [START_REF] Ayres | Production, consumption, and externalities[END_REF]. The view of the economy as a metabolism that transforms low entropy valuable inputs into high entropy wastes stems from the second law of thermodynamics. Other direct implications of this second law for the economy is the difficulty to obtain a perfect recycling process without massive energy inputs, which underlines the hurdles to a perfectly circular economy. The concept of energy quality and exergy, which is of great interest in this study, equally stems from this second law.

Stemming from the ecological economics perspective, the framework proposed in this study is presented in Figure 3.1. The economy is embedded in the society, which is embedded in the environment. The economic process needs societal inputs, namely labour services and capital services, which are provided by the societal "stocks" of labour and capital. It also needs natural resources inputs, that are extracted from the environment, thereby lowering stocks. It provides useful products and services that flow into the society in order to fulfil societal needs and wants. However, it also emits wastes to the environment, that are absorbed by natural sinks. The economy, as a metabolism embedded in the society and in the environment, or our "Earth Spaceship". Own elaboration.

Ecological theory of growth

According to the framework presented in Figure 3.1, economic growth should be explained by the evolution of the following factors of production: capital, labour and natural resources. In this section, we present an ecological theory of growth that stems from the work of [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF]Warr [2005, 2010], [START_REF] Warr | Useful work and information as drivers of economic growth[END_REF]. This theory relies on positive feedback mechanisms that are presented in Figure 3.2.

In this view, efficiency gains, either led by technological progress, new knowledge or economies of scale tend to lower production costs of products and services. This happens both by improving the production process in itself, and by improving the extraction processes that underpin the whole economic activity (e.g. oil drilling, ores mining...), thereby providing the rest of the economy with cheaper natural resources. The decrease in production costs tend to trigger a decline in the prices of goods and services as a way to maximise profits in a competitive economy, which in turns increases the aggregate demand for products and services. Whenever natural resources are available in sufficient quantities, this brings about an increase in the aggregate production of products and services that fulfils the increasing demand. Consequently, the use of natural resources also rises, as more products and services are produced, and as energy7 and capital are substituted for labour in order to increase production8 . This rise in the consumption of natural resources stimulates in turn further efficiency gains in order to diminish costs of production and particularly the costs of natural resources. The described feedback mechanism, with efficiency gains and increase in aggregate production as its core, can be regarded as an economy wide description of the rebound effect, or Jevon's paradox, firstly introduced in Jevons [1866]. Following [START_REF] Ayres | The Economic Growth Engine: How Energy and Work Drive Material Prosperity[END_REF], it is suggested that at the aggregate level, a main driver for the increase in production, i.e. economic growth, is efficiency gains.

The interplay between efficiency gains, prices and increase in demand is backed up by empirical evidence. In the particular case of energy consumption, there is evidence that the price of energy services has decreased over the years. [START_REF] Fouquet | Seven Centuries of Energy Services: The Price and Use of Light in the United Kingdom (1300-2000)[END_REF] gives the particular example of lighting in the UK over the last seven centuries, and [START_REF] Fouquet | Long-Run Demand for Energy Services: Income and Price Elasticities over Two Hundred Years[END_REF] showcases how the prices of energy services for lighting, transportation, and heating have decreased dramatically in the two last centuries. This work also estimates the price elasticities 9 for each of these

dQ Q 0 dP P 0
, where Q refers to the demand, and P to the price. Q0 and P0 can either energy services. These are found to be negative, which implies that an increase in the demand occurs whenever the price decreases, thereby supporting the model presented in Figure 3.2.

As energy is a main input in every process, and particularly in transformation processes where materials are transformed into products, it is argued that at the aggregate level, energy consumption can be used as a proxy for natural resource consumption. As argued in Chapter 2, the relevant energy metric is deemed to be useful exergy, as it is closer to the energy service delivered and accounts for energy quality. Therefore, it is assumed that the total consumption of natural resources can be approximated by the useful exergy consumed, as it is directly used in order to transform natural resources into goods and services. The rest of the work therefore focuses on capital, labour, and useful exergy as factors of production. By no means this means that the role of other natural resource inputs is disregarded. It is merely considered that an accurate construction of useful exergy time series will serve as a acceptable proxy for total natural resource consumption.

As discussed previously, technological progress is seen as exogenous in the mainstream theory of growth. Conversely, technological progress, modelled as efficiency gains, are at the core of the presented ecological theory of growth. These gains are regarded as triggered by the rise in resource consumption and goods and services production, and reciprocally drive the decrease in costs and subsequently the increase in aggregate demand. This leads to main differences in the modelling process. While the standard aggregate production function approach includes technological progress as a crucial time dependent multiplier, usually in the form e λ.t in APFs, technological progress is not directly included as a factor of production in the ecological approach. The reasons, inspired by [START_REF] Kümmel | Technical progress and energy dependent production functions[END_REF], are twofold:

• Firstly, it is argued that technological progress, instead of being homogeneous across the different sectors of the economy, is heterogeneous, namely different for each factor of production. As such, it is argued that it is possible to capture independently the technological progress related to each one of these factors by using quality adjusted measures of these. Technological improvements related to capital are accounted through capital services, which consider for the productivity of capital, while labour can be measured using the human capital index, which accounts for the skills and education of workers [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF]. And energy is measured as useful exergy, which enables to account for technological progress related to the energy conversion chain -see Chapter 2.

• Secondly, and as stated by [START_REF] Kümmel | Technical progress and energy dependent production functions[END_REF], "Technological progress has often meant more energy-intensive production processes in the form of more powerful machines handled by labour". More generally, it is argued here that technological progress needs useful exergy in order to be activated. The development of computers, airplanes, cars, high speed trains, obviously rely on crucial technological progress. However, each of these technologies, in order to become operational, needs a crucial input in useful exergy flows, or it would just stay at the pilot level.

Two main advances are therefore presented by the ecological theory of growth. Firstly, it includes explicitly physical flows in the production process. Particularly, energy is accounted for by using useful exergy, and it is argued that this variable represents an acceptable proxy for natural resource consumption. Secondly, technological progress is regarded as endogenous, as it is described in terms of quality adjusted factors of production (e.g. capital services, human capital, useful exergy), which account for the specific technological progress related to capital, skills, and energy. Moreover, it is argued that technological progress related to other aspects (e.g. transportation, Information and Communication Technologies (ICTs)...) are also represented by refer to the initial values, or to the mean of the initial and final values.

the evolution of the useful exergy variable. Indeed, useful exergy flows are needed to activate technological progress: more computers stemming from technological improvements need inputs of energy, the increase of airfreight due to advances in aviation also require additional inputs of energy compared to road or rail based freight transportation... Thus, useful exergy is deemed as a decent proxy for the technological progress that is not directly captured by the factors of production.

Although this ecological theory of growth comes with main advances, there are also caveats and limitations associated. Criticisms include the incomplete description of the economy provided in this theory, as key aspects of the economy (i.e. financial sector, wages, debt...) are not included. The ability of useful exergy as a proxy for natural resources consumption can also be questioned, as not all the energy is used to transform natural resources (for instance energy use in transportation and ICTs). Declining marginal returns in extractive industries (e.g. declining EROI and decrease in ore grades) and limits in supply are not considered, which questions the validity of the theory in a context of natural resources depletion.

Application of the theoretical framework

The theoretical framework described in Section 3.2 constitutes the "pre-analytic vision" of this thesis and is therefore embedded in different aspects of this research. The very research subject and question, that seek to underline and understand the role of energy in the Spanish economy, are rooted in an ecological vision of the economy, where energy inputs are seen as a key input to the economic process. Then, each research question relates in a different way to the presented framework:

Is there statistical evidence of causality between GDP and energy consumption?

This research question seeks evidence of correlation and causality between GDP and energy consumption. A high correlation indicates that it is plausible to consider GDP and energy consumption as dependent and would support the ecological vision. A causality relationship running from energy to GDP, or both ways around, would support the ecological theory of growth. Conversely, a non-causality or a causality relationship running from GDP to energy consumption would support a vision of growth as essentially independent from energy consumption.

To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)? This research question explores whether including energy consumption in APFs enhances the fit of the APFs to real GDP, and thereby helps to account for economic output. If the inclusion of energy consumption reduces the residual (i.e. the difference between the actual and fitted GDP), then this supports the ecological approach. However, this research question does not explain economic output, it merely accounts for it by using statistical correlations and mathematical formulas. Saying that one can account for economic output using capital, labour and energy time series does not explain the interplay between these factors of production.

What conceptual model can explain the interactions between the economy and energy consumption?

Finally, this research question seeks to provide a conceptual model that makes explicit the cause-effect relationships and the interplay between energy consumption and economic growth. Thus, it seeks to identify the relevant parameters in order to describe the relationship between energy and the economy and how these interact.

Chapter 4

Methodology

In this Chapter, the methodology applied for answering the research questions is described. An overview of the different analysis methods can be found in Table 4

Testing for causality

Granger causality test was firstly introduced by [START_REF] Granger | Investigating Causal Relations by Econometric Models and Cross-spectral Methods[END_REF]. It is crucial to distinguish between causality as usually understood, namely as an event X entailing another event Y, and causality in the Granger sense, as tested with a causality test. Saying that a variable X Granger causes a variable Y means that knowing both the past values of X and Y provides more information on the evolution of Y than knowing only the past values of Y. Namely, that the past values of X provide additional information when forecasting Y [START_REF] Granger | Investigating Causal Relations by Econometric Models and Cross-spectral Methods[END_REF][START_REF] Granger | Testing for causality: A personal viewpoint[END_REF].

In this thesis, causality tests between GDP and energy consumption growth rates are carried out for the time period 1960-2013. Time constraints impeded extending the analysis until 2016, but the conclusion is unlikely to be sensitive to this fact. R [R Core Team, 2019] language is used as the programming environment for carrying such tests. Numerous tests have been carried out as subsequently explained, where the following parameters have been modified: the Granger test carried out, the variables tested, the numbers of lags when specifying the Vector Autoregression (VAR) model (see Appendix A for more details on VAR models), and the time period covered. The following sections elucidate how these parameters were changed.

Selected Granger tests

Three different Granger tests have been carried out for each situation. Firstly, the grangertest function from the lmtest package [START_REF] Zeileis | Dynamic interacting relationships among international oil prices, macroeconomic variables and precious metal prices[END_REF] has been used. Secondly, the causality function from the vars package [START_REF] Pfaff | SVAR and SVEC Models: Implementation Within R Package vars[END_REF] has been used. These two functions are different R implementations of the Granger causality test. Thirdly, the Toda-Yamamoto (T-Y) version of the Granger causality test [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF] is carried out according to the R implementation described in [START_REF] Pfeiffer | Toda-Yamamoto implementation in 'R'[END_REF].

It is worth underlining the difference when using these tests. Testing for causality using a standard Granger causality test requires the considered time series to be stationary1 , otherwise causality results may be spurious [START_REF] He | On spurious Granger causality[END_REF]. Conversely, the Toda-Yamamoto implementation of the Granger causality does not require time series to be stationary in order to provide reliable results [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF]. Time series of energy and GDP growth rates must therefore be transformed until they are stationary in order to use the grangertest and the causality functions, while the Toda-Yamamoto procedure can be directly used.

Testing for stationarity

Stationarity can be tested with different statistical tests. In this study, three tests are carried out: the Augmented Dickey-Fuller (ADF), the Phillips-Perron (PP), and the Kwiatkowski Phillips Schmidt Shin (KPSS) tests 2 . The ADF and PP procedures test the H 0 hypothesis 3 that the time series is not stationary, while the KPSS procedure tests the H 0 hypothesis that the time series is stationary. Using tests with different H 0 hypothesis enable a greater reliability of results.

The energy and GDP growth time series are found to be non-stationary, but all the tests find them stationary with 95% confidence once they are differentiated 4 . As such, the differentiated time series of energy and GDP growth rates are used as inputs to the grangertest and causality functions.

Choice of variables

Different energy and GDP variables are used in order to test for differences in the results when changing variables.

Energy variables:

The tests were carried out with the growth rates of three different energy variables: final energy, useful energy, and useful exergy.

GDP variables:

The tests were carried out with the growth rates of three different GDP variables: rgdpe, rgdpo, and rgdpna, which are all extracted from the Penn World Table 9.1 (PWT) [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF]. This data is described in Section 4.4.

This gives a total of nine couples (energy variable, GDP variable) that are tested for each of the three presented tests.

Considered time period

As mentioned, the time period covered is 1960-2013, which gives 53 data points once the growth rates are calculated. A minimum time period of 30 years has been deemed necessary in order to obtain reliable results of the Granger causality test. The time period for the test has been modified in order to cover every possible time period of length superior or equal to 30 years. Let us define i as the length of the time span considered (therefore varying from 30 to 53 in the T-Y case) and j as the number of possible time periods for a specific length (therefore varying from 1 to (53-i+1), in the T-Y case). The number of time periods where causality tests can be carried out is presented in Equations 4.1 (Toda-Yamamoto) and 4.2 (direct Granger implementations) are obtained. Note that these numbers are different as one data point is lost when differentiating the data in order to use the grangertest and causality functions. The three tests, for each of the nine couples of energy and GDP variables, are carried out for each possible time period.

Specification and validity of the VAR model

Once the test, the variables, and the time period have been selected, one needs to specify the VAR model used for the causality test. The only additional parameter needed in order to characterise fully the VAR model is the number of lags that are considered. Different criteria can be used for selecting the most appropriate number of lags. The VARselect function from the vars library has been used in order to identify the preferred number of lags to be included according to each of the following criteria: the Akaike Information Criterion (AIC), the Hannan-Quinn Criterion (HQ), the Schwarz-Bayes Criterion (SBC), and the Akaike's Final Prediction Error (FPE)5 . Each criterion returns a particular optimal number of lags6 . For each test, couple of variables, and selected time period, the approach has been to carry out each causality test with each of the lags returned by each criterion. So for each causality test, couple of variables, and selected time period, between one and four tests have been carried out depending on whether criteria did return the same number of lags. Often this number differed from one criterion to another.

Finally, the VAR model defined was tested for misspecification and stability. Following [START_REF] Pfeiffer | Toda-Yamamoto implementation in 'R'[END_REF], this was done through two tests. Firstly, residual serial correlation was tested using the serial.test function7 from the vars package and was deemed successful if the p-value returned was lower than 0,1 (corresponding to a situation of 90% confidence that the VAR model is not serially correlated). Secondly, a stability test was carried out. Unit roots of the VAR model are tested using the roots8 function of the vars library. Whenever all the units roots of the matrix associated to the VAR model have modulus inferior to unity, the test is successful and the model is regarded as stable [START_REF] Lütkepohl | New Introduction to Multiple Time Series Analysis[END_REF].

When these two tests (residual serial correlation and stability) were successful, the causality test carried out with the defined VAR model was deemed valid. Conversely, if one of these tests was unsuccessful, the causality test associated was regarded as void.

Summary of the tests conducted

Analysis of the results

In order to analyse the results, only the VAR models which were stable -according to the tests presented in Section 4.1.5 -are kept and deemed reliable. Results are subsequently analysed based on the total number of tests supporting each of the four following outcomes: no causality, causality running from energy to GDP, from GDP to energy, or both ways around.

Once general results are analysed, the rest of the study focuses on the influence of particular parameters on the results of the tests. The influence of the test carried out, of the GDP and energy variable considered, and of the number of lags specified in the VAR model is discussed.

Energy as a factor of production

The methodology used for including energy as a factor of production is presented thereafter following the theory to policy 6 steps process introduced in Heun et al. [2017].

Formulating theoretical framework

The theoretical framework, which stems from an ecological economics perspective, has already been introduced in Section 3.2. Concretely, this has three main implications for the downstream modelling. Firstly, the usual cost share theorem is a priori rejected. Secondly, the TFP usually included in studies is equally excluded. And thirdly, energy is included as a factor of production as a proxy for natural resources consumption. These modelling choices have all been defended in Chapter 3.

Formulating modelling approaches

The first assumption made in the present modelling is the constant returns to scale assumption (see Appendix B for more details), which is an usual economic assumption and accounts for the fact that the productivity of countries does not depend on their size. This seems a reasonable assumption once an economy reaches a certain size.

Then, different APFs are fitted to the historical data. A description of the fitted APFs as well as a discussion of their respective properties and limitations can be found in Appendix B. Firstly, and as a preliminary test, two APFs (Cobb-Douglas (CD) and Constant Elasticity of Substitution (CES) are tested without including energy as a factor of production. Secondly, functional forms including energy are tested (Cobb-Douglas, Constant Elasticity of Substitution following the three possible nesting choices described in Appendix B, and Linex). For each of these APFs, the data is fitted using both quality adjusted and quality unadjusted data.

Finally, the fits are conducted for two different time periods: the whole time period 1960-2016, and the time period 1960-2008. The second fit is performed in order to compare the predictive capacities of the different APFs, which is further presented in Section 4.2.4.

Fitting to historical data

The fitting technique used in this thesis relies on an Ordinary Least Squares (OLS) approach, as described in [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF]. The ordinary least squares approach seeks to minimise the sum of squared errors (SSE), defined as9 :

SSE = i ln(y i ) -ln(ỹ i ) 2 (4.3)
where ỹi is the fitted economic output for the year i. Concretely, these fits are conducted with R by using the micEconCES package [START_REF] Henningsen | Econometric Estimation of the 'Constant Elasticity of Substitution' Function in R: Package micEconCES[END_REF] for the CES APFs, and the MacroGrowth package [START_REF] Pruim | MacroGrowth: Economics Models[END_REF] for the Linex and CD APFs. For the CD APF, the MacroGrowth package was amended in order to fit CD functions without TFP. Due to time constraints, this was not possible for the CES APFs, hence the choice of the micEconCES package. It is however noteworthy that the MacroGrowth package offers significant added features, including a better assessment of boundary models, which may not be properly handled when the micEconCES package is used, for the PORT and L-BFGS algorithms employed have difficulties dealing with boundaries [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF]. The micEconCES package is therefore chosen as a reasonable trade-off between time investment, and needed precision.

Comparing and evaluating modelling approaches

The different modelling approaches (e.g. the use of a specific APF with particular factors of production data) are evaluated based on the quality of the fit obtained in two situations. Firstly, the quality of the fit over the whole period covered (i.e. 1960-2016) is assessed and discussed by comparing the SSE obtained for each APF.

Secondly, the capacity of an APF to account for the structural break and economic downturn starting in 2008 is equally assessed and discussed. The rationale behind this is as follow: APFs are usually used in order to describe the economy and to guide policy making, for instance in energy or climate policy. Therefore, a decent APF model should be able not only to fit the past economic output, but also to reproduce structural breaks when it is fitted on years predating these breaks. Otherwise, the predictive power of an APF is questionable and its policy implications and recommendations would seem spurious. In order to do so, all APFs are fitted for the time period 1960-2008, and then the fitted GDP is projected for the period 2009-2016 by using the available time series for the factors of production and the fitted parameters of each specific APF. Then, the SSE obtained for the period 2009-2016, designated as SSE ext , is used as an indicator of the predictive capacity of an APF -the lower, the better.

SSE ext = 2016 i=2009 ln(y i ) -ln(ỹ i ) 2 (4.4)
Then, the main parameters obtained for each fitted APF are presented, whether these are realistic or not, and their implications are shortly discussed.

Interpreting the economy

While one could draw numerous different interpretations of the economy from this APF study, the main output of this step for this thesis will be to answer the research question "To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)?". Hence, the main interpretation of the economy will be to analyse whether including energy as a factor of production provides a fit of actual economic output that seems closer to reality than the conventional approach, which includes only capital and labour.

Formulating policy

This final step is not relevant in the present case as formulating policy recommendations is outside the scope of the thesis. However, significant caveats when modelling the energy-economy interactions through APFs are drawn from the present study.

Modelling energy-economy linkages

Based on a literature review, a conceptual model that makes explicit the cause-effect relationships and the interplay between energy consumption and economic growth is suggested. The geographical scope of the model is flexible, and this one could be implemented at different scales (regional, national, global). The conceptual model provides a thorough description of the energy industry, and describes how it interacts with the rest of the economy, by focusing on how the energy industry and key economic parameters feedback. However, the model does not describe an economic model, as this is out of the scope of this thesis. Likewise, numerous other aspects, such as materials and land constraints, or pollution and climate change, are left outside the model, as they are also out of scope.

Literature review

The methodology for this literature review was to firstly identify a few key parameters in order to model the energy-economy interactions from an ecological approach. For each key parameter identified, some key articles and literature have been identified from the ecological economics and biophysical literature. Then, the literature review has been widened by cross referencing these articles as relevant references were found and by searching directly articles on a specific subject when the cross referencing did not provide sufficient literature.

Following other biophysical energy-economy models that have been of important inspiration, such as MEDEAS [Capellán-Pérez et al., 2017a] or GEMBA [START_REF] Dale | Global energy modelling -A biophysical approach (GEMBA) Part 2: Methodology[END_REF], a system dynamics modelling framework is adopted. Although the presented model is not formally a system dynamics model, it is indeed based on feedback relationships that stem from system dynamics. This is an intentional choice, as system dynamics is regarded as the most appropriate methodological framework in order to deal with complex systems that feedback in non-linear ways and include tipping points [START_REF] Meadows | Thinking in Systems: A Primer[END_REF]. It is worth noting that the core of the model, namely the idea of the energy-economy feedback whereby the energy demand is fulfilled or not depending on the physical availability of energy stems directly from previous biophysical models, particularly MEDEAS [Capellán-Pérez et al., 2017a] and GEMBA [START_REF] Dale | Global energy modelling -A biophysical approach (GEMBA) Part 2: Methodology[END_REF]. Both models include key parameters such as the EROI and the Maximum Extraction Rate and essential feedbacks loops. The presented model therefore presents numerous similarities with these two noteworthy models, which have been of considerable inspiration, as Table 4.4 evidences. The added value of the suggested model is to integrate other important parameters, such as the price of energy, the affordability concept, and the efficiency with which energy is used in the economy.

Sources of inspiration for the main elements

Data

All the data described thereafter has been indexed according to the first year of study -1960 -before being used as input for the causality tests and for the APF modelling.

Energy and exergy data

All the energy and exergy data, for each of the three primary, final and useful stages, have been calculated following the methodology presented in [START_REF] Aramendia | Energy Intensity of the Spanish Economy: A Useful Work Study[END_REF], that draws considerably from the Physical Supply Use Table (PSUT) framework described in [START_REF] Heun | A physical supply-use table framework for energy analysis on the energy conversion chain[END_REF]. The time series, that were firstly constructed until 2013, have been extended until 2016 following the same methodology. Final energy is adopted as the unadjusted quality measure for energy consumption, while useful exergy is adopted as the quality adjusted measure, and useful energy lies in between.

Economic data

GDP data

All GDP data is extracted from the Penn World Tables 9.1 (PWT) [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF]. Three different GDP measures from the PWT are used in this study:

• rgdpna: this measure corresponds to real GDP as provided in the national accounts. [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF] suggests that it is appropriate when studying economic output of a specific country over time: "If the sole object is to compare the growth performance of economies, we would recommend using the rgdpna series".

• rgdpe: this measure is the expenditure-side real GDP, which uses final goods prices that are constant over time and countries [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF].

• rgdpo: this measure is the output-side real GDP, which uses final goods exports and imports prices that are constant over time and countries. [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF].

These are all real GDP10 measures, namely suitable for studies over time, as is done in this thesis. However, only the rgdpe and rgdpo measures are suitable for cross countries comparisons: "The variables with the R-prefix are best suited for comparisons over time, though only rgdpe and rgdpo are simultaneously suitable for over time and cross country comparisons" [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF]. As such, the three variables presented above are suitable for the present study, 4.4. Data although the rgdpna seems to be more appropriate for studying the economic performance of a single country over time. Indeed, the Spanish GDP, expressed in PPP, may be affected by the performance of the rest of the world economy, as it is defined in comparison to all the other countries' economies. Thus, the three GDP measures are used in order to conduct the causality tests, but only the rgdpna measure is used as economic output in APFs. The sensitivity of this choice for APF modelling is quickly discussed in Appendix D.

Capital data

The conventional approach for quantifying capital is to use the perpetual inventory method (PIM), which calculates capital stocks year by year by adding new stocks and subtracting out of date stocks. These subtracted stocks can be defined as the retired existing assets, in which case a gross capital stock (GCS) metric is adopted. From the GCS, one can define the net capital stock (NCS), as equal to GCS minus the depreciation of existing assets [Brockway et al., 2017a]. In this study, the NCS is adopted as the unadjusted quality measure for capital.

However this conventional PIM approach does not account the heterogeneity of capital, for its productivity, and for the services it provides. Conversely, capital services, defined as the "flow of productive services from the cumulative stock of past investments" [Schreyer et al., 2003, page 7], account for the productivity of capital. As such, the capital services measure is regarded as the quality adjusted measure for capital. Both measures are taken directly from the PWT 9.1, where the NCS is equated to the rnna variable and capital services are equated to the rkna variable.

Labour data

Likewise, quality adjusted and non-quality adjusted measures are introduced for labour. The unadjusted measures accounts for the total number of hours worked by labourers in one year, and is calculated as:

L = emp.avh (4.5)
where emp is the number of persons engaged and avh is the average annual hours worked by engaged individuals. Both measures are taken from the PWT 9.1. The quality adjusted value for labour is calculated by multiplying the unadjusted value by the human capital index, which is equally available in the PWT 9.1.

Chapter 5

Testing for causality

In this chapter, preliminary results that showcase a strong correlation between energy consumption and GDP in the Spanish case are firstly presented, thereby underlining the relevance of undertaking a causality test. Then, a literature review on the state of the art of causality studies between energy consumption and the economy is presented. The results of the Granger causality tests performed are subsequently presented, and finally conclusions on causality for this specific Spanish case are drawn from the results.

Preliminary results

Figure 5.1 displays on the left the temporal evolution of Spanish growth rates of real GDP and useful exergy growth rates. As these evolve simultaneously, an obvious correlation between energy consumption and economic growth can be inferred 1 . 1960 1970 1980 1990 2000 2010 -0.10 0.00 0.10 0.20 Year Growth rates q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.10 0.00 0.10 0.20 -0.05 0.00 0.05 0.10

Useful exergy growth rate Real GDP (RGDPE) growth rate Figure 5.1: On the left, the Spanish real GDP (blue) and the Spanish useful exergy (red) growth rates. On the right, the Spanish real GDP (blue) growth rate in function of the Spanish useful exergy (red) growth rate -each data point representing the couple (Useful exergy growth, GDP growth) for a specific year. Own elaboration of useful exergy data, and GDP data directly from the Penn World Table -"rgdpe" variable used [START_REF] Feenstra | The Next Generation of the Penn World Table[END_REF]. Data for the years 1960-2013. This graphs are similar to the ones presented in Figure 1.2, with the crucial difference that it corresponds to national data. On the right, Figure 5.1 displays real GDP growth rates in function of useful exergy growth rates for each year. As only four data points are in the top left corner, it can be said that decoupling between useful exergy consumption and economic growth has only happened for four specific years in the Spanish case2 . This positive correlation is consistent with earlier findings pointing that very few countries have achieved to decouple energy consumption from economic growth [START_REF] Csereklyei | Theoretical efficiency limits for energy conversion devices[END_REF]. From these preliminary results, a strong correlation between energy consumption and economic growth can be inferred in the Spanish case. As such, performing a Granger causality test in order to unpick the directionality of the relationship seems more than justified.

Literature review

An inconclusive literature

The causality between energy consumption and economic growth has been widely studied since the seminal work of [START_REF] Kraft | On the Relationship Between Energy and GNP[END_REF]. These applied Sim's methodology3 , and found an unidirectional causality link running from economic growth to energy consumption in the US. However, this early finding was quickly discussed, as [START_REF] Akarca | On the Relationship Between Energy and GNP: A Reexamination[END_REF] pointed that results were different when changing the considered time period, using the exact same methodology and datasets. Numerous papers have been published on the subject since then, and the literature is inconclusive, as shown in four recent literature reviews [START_REF] Ozturk | A literature survey on energy-growth nexus[END_REF][START_REF] Omri | An international literature survey on energy-economic growth nexus: Evidence from country-specific studies[END_REF][START_REF] Chen | Energy consumption and economic growth-New evidence from meta analysis[END_REF][START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: Implications for the growth-degrowth dialogue[END_REF]. Four different hypothesis are generally confronted in this field of study, each with specific policy implications [START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: Implications for the growth-degrowth dialogue[END_REF]]:

• the neutrality, or no causality hypothesis: in this view, there is no causality between energy consumption and economic growth. The direct policy implication of this view is that energy conservation policies will not hamper economic growth and should be conducted.

• the conservation hypothesis: this view assumes that there is a unidirectional relationship running from economic growth to energy consumption. Hence, economic growth causes an increase in energy consumption. According to this view as well, energy conservation policies will not hamper economic growth and are desirable.

• the growth hypothesis: in this hypothesis, there is a unidirectional causality running from energy consumption to economic growth. Thus, increasing energy consumption causes economic growth. In this view, a reduction in energy consumption will have adverse effects on the economic activity, and energy conservation measures are likely to hamper economic growth.

• the feedback, or bi-directional hypothesis: in this view, there is a bi-directional causality between energy consumption and economic growth, which trigger one another. This feedback hypothesis also implies that energy conservation measures are likely to hinder economic growth.

The literature has been inconclusive, as results have been found to vary significantly according to a few criteria and parameters of the tests carried out. The main criteria that have been found to greatly influence the results of the test are the following [START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: Implications for the growth-degrowth dialogue[END_REF]]:

• the time span considered: as showcased by the first studies on the US economy of [START_REF] Kraft | On the Relationship Between Energy and GNP[END_REF] and [START_REF] Akarca | On the Relationship Between Energy and GNP: A Reexamination[END_REF], the time span on which the test is carried out has a crucial importance.

• the country studied: depending on the characteristics and the history of the studied country, results also differ. An example of such differences can be found in [START_REF] Mutascu | A bootstrap panel Granger causality analysis of energy consumption and economic growth in the G7 countries[END_REF] and [START_REF] Bozoklu | Energy consumption and economic growth for selected OECD countries: Further evidence from the Granger causality test in the frequency domain[END_REF], who apply the same methodology on different countries, using the same time span, and nonetheless obtain results that differ depending on the country.

Both studies showcase that even on countries with similar characteristics (e.g. European and industrialized) results can differ.

• the methodology applied: a wide range of econometric techniques exist in order to test for causality. Each method is likely to provide a different result.

• the energy source considered: different energy values can also be used when running a causality test. One can distinguish between studies that consider total energy consumption and energy consumption from a specific fuel (e.g. oil, coal...) or energy carrier (e.g. electricity). This may also have a decisive impact. [START_REF] Omri | An international literature survey on energy-economic growth nexus: Evidence from country-specific studies[END_REF] provides a more precise literature review on these aspects.

Intended contribution to the literature

From the 158 studies presented in the literature review carried out in [START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: Implications for the growth-degrowth dialogue[END_REF], it seems complex to bring additional useful material to the energy-economy causality inconclusive literature. As stated by [START_REF] Ozturk | A literature survey on energy-growth nexus[END_REF], "research papers using the same methods with the same variables, just by changing the time period examined, have no further potential to make a contribution to the existing energy-growth literature". However, most of the studies considering causality between quantity of energy consumed and economic output are using primary or final energy quantities, and besides, the energy measures considered are not quality adjusted, which assumes that all energy inputs have the same economic value. [START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: Implications for the growth-degrowth dialogue[END_REF] shows that only a few studies (5 out of 158) use quality adjusted values of energy. However, studies that account for quality adjusted values of energy, either using a divisia index adjustment method4 [START_REF] Stern | Energy and economic growth in the USA[END_REF][START_REF] Stern | A multivariate cointegration analysis of the role of energy in the US macroeconomy[END_REF][START_REF] Zarnikau | A Reexamination of the Causal Relationship between Energy Consumption and Gross National Product[END_REF][START_REF] Cleveland | Aggregation and the role of energy in the economy[END_REF], or a useful exergy based metric [START_REF] Warr | Evidence of causality between the quantity and quality of energy consumption and economic growth[END_REF], tend to find results supporting either the growth or the feedback hypothesis.

As described in the Societal Exergy Analysis framework, it is here argued that the energy that actually provides energy services and therefore economic output is the useful energy instead of primary or final energy. Besides, it is argued that useful exergy gives a better proxy for provided energy services as it accounts for energy quality. According to the literature review, only one causality tests based on useful exergy measures have been published so far, and it points to unidirectional causality running from energy consumption to economic growth [START_REF] Warr | Evidence of causality between the quantity and quality of energy consumption and economic growth[END_REF]. However, this study did not include a significant sensitivity test.

The contribution to the literature of this section is therefore to provide a new useful exergy based causality test, and to carry out the associated sensitivity test by varying numerous parameters. If the results do not significantly vary when changing the parameters of the test, it will be argued that energy quality adjusted tests may be able to contribute to this literature and seem a reasonable way to investigate further the energy-economy nexus. Conversely, if results vary broadly depending on the parameters of the test, it will be suggested that the Granger causality test may not be a appropriate test when studying the energy-economy nexus.

Results

Results

A total of 13 610 Granger tests have been carried out following the methodology presented in Section 4.1. Of these tests, 12 170 are deemed valid once they are tested for misspecification and stability. The following section presents the main results. In this section, e → y stands for the growth hypothesis, y → e stands for the conservation hypothesis, and e y stands for the feedback hypothesis.

General overview

An overview of the results of all the valid tests is presented in Table 5.1. One can draw two main conclusions from this table. Firstly, most of tests, almost 80%, seem to support a situation of no causality between energy consumption and economic growth. Secondly, the conservation hypothesis, according to which economic growth drives energy consumption, is backed up by more tests than the growth and feedback hypothesis. However there is still a substantial amount of tests supporting the growth hypothesis.

Another key remark is needed here. For each of the three different tests described in Section 4.1 (i.e. the Toda Yamamoto, the causality and the grangertest functions), the H 0 hypothesis is the non causality hypothesis. Hence, results supporting the growth, conservation and feedback hypothesis should be interpreted as a situation in which one Granger causality is detected with 95% certainty. However, the interpretation of the tests supporting the non dependency situation is quite different. Such test should not be interpreted as a situation where non causality is detected with 95% certainty, but rather, as a situation where there is not enough evidence in order to affirm Granger causality with 95% certainty. Consequently, the number of tests supporting the non causality hypothesis is overestimated, for all the tests for which causality cannot be detected with 95% certainty are included5 . As this large number can bias the analysis, these tests are omitted in the following sections.

Influence of the test carried out

The test carried out also has a significant influence on the results, as displayed in Table 5

Influence of the energy variable

Table 5.3 showcases the influence of the selected energy variable on the results. While tests using final energy are more supportive of the conservation hypothesis than of the growth and feedback hypothesis, this changes when energy is quality adjusted. Firstly, when useful energy is used this trend exists but is less pronounced. Then, when the variable used is useful exergy, there are almost as many tests supporting the growth than the conservation hypothesis. This is consistent with the fact that useful exergy is reckoned to be a more accurate measure in order to describe the relationship between energy and the economy. Indeed, when useful exergy is employed, the results seem to give a larger importance to energy consumption than with other units. However, even when using useful exergy, no salient conclusion can be drawn, for the different hypothesis are still backed up by a considerable number of results.

Influence of the GDP variable

Likewise, the influence of the GDP variable used for the test is also considerable, as showcased in Table 5 While tests using rgdpe and rgdpna mostly support the conservation hypothesis, tests carried ou with the rgdpo variable support equally the growth hypothesis. Considering that the rgdpe and rgdpna variables are expenditure-side GDP measures, i.e. consumption based measures, while the rgdpo is an output-side GDP measure, i.e. a measure that accounts for the productive capacity of a country, results are consistent. Indeed, testing for causality with an output-side GDP measure finds more causality from energy to GDP than doing so with an expenditure-side GDP measure. This is consistent with the fact that energy consumption drives the productive processes, rather than consumption, which also depends on imports and on the consumption of embodied energy.

Influence of the number of lags

The influence of the number of lags included in the VAR model is also found to be significant. Table 5.5 showcases the results for each number of lags. However, even though the number of lags seems to be a critical parameter, there is no intuitive tendency of the influence on the results. For instance, tests with a very low number of lags (i.e. one and two) support the conservation hypothesis, while tests with three, five and six lags are more supportive of the growth hypothesis, and finally tests with seven and eight lags seem to support more the growth hypothesis as well. It does not seem that the number of lags tend to flaw the test in a specific direction.

Specific test: useful exergy and rgdpo

Finally, the specific test that was deemed to represent better the relationship between energy and the economy was carried out, using the useful exergy and rgdpo measures. The choice of useful exergy has already been extensively argued in Chapter 2. Then, it is considered that energy consumption impacts directly the production process within a country, more than the consumption process, as this one is partly relies on energy consumption elsewhere, i.e. on the consumption of embodied energy. Thus, in order to study the impact of energy on GDP, the rgdpo measure seems to be the most appropriate amongst the ones that were available for this thesis, as it is the only one that accounts for GDP from a production perspective. To some extent, it could be argued that rgdpo is the quality adjusted variable for GDP, alike useful exergy for energy consumption. With these specific variables, the tests carried out seem to be more supportive of the growth hypothesis. This is very limited evidence for claiming that energy consumption is driving economic growth, but it underlines that the metrics used need to be carefully and soundly chosen when choosing the energy-economy interplay. It also supports the idea that when the metrics are adjusted to be more representative of the relationship between energy and the economy, here using useful exergy and rgdpo, the observed relationship shifts towards the growth hypothesis, namely energy consumption driving economic growth.

Conclusion on causality

Let us consider the first research question of the present thesis; "Is there statistical evidence of causality between GDP and energy consumption?". Although strong evidence of correlation between these energy consumption and GDP has been identified, evidence of causality has not been found, and therefore, no conclusion can be drawn about causality. Results of the Granger causality test vary significantly depending on the test carried out, the time period covered, the variables selected, and on the number of lags selected when defining the VAR models. This is consistent with the findings of the literature.

Although the causality tests carried out with useful exergy showed higher cases supporting the growth hypothesis, no salient conclusion could be drawn, conversely to the results in [START_REF] Warr | Evidence of causality between the quantity and quality of energy consumption and economic growth[END_REF]. However, when the GDP unit was also adjusted to better represent the outputside of an economy, namely the productive capacity, most of the results pointed to the growth hypothesis, but this remains limited evidence in order to claim causality. However, one can easily take a critical approach to the present results and their inconclusiveness. Firstly, from a conceptual perspective, Fizaine and Court's criticism is reminded as it is fully representative of the present point: "Suppose we try to determine the effect of energy consumption on the average speed of a car traveling between a series of equidistant refueling points. If we make a Granger causality test between the fuel bills obtained at each gasoline station (representing energy consumption) and the recorded average speed of the car (representing GDP growth), it would probably indicate a causal relation running from the latter to the former. Indeed, the higher the speed of the car, the higher the energy consumption (and the higher the gasoline bill). But no one can reasonably assume that energy does not play the primary role in propelling the car at some speed or other, and that we can cut energy consumption without affecting the car's motion." [START_REF] Fizaine | Energy expenditure, economic growth, and the minimum EROI of society[END_REF].

Secondly, from an analytical perspective, it seems that the Granger causality test suffers from crucial drawbacks, which seriously question the validity of these tests when used in order to determine the relationship between energy consumption and the economy. From the description of the VAR models provided in Appendix A, one can draw the following remarks 6 : (i) Firstly, the possibility of having instantaneous interactions between energy consumption and the economy is not considered, as the VAR models only allow a variable to influence another one through its past values (or lags), thereby neglecting the possible influence of the variable's present value. In the case of energy-economy modelling, this may be a crucial limitation, as a constraint in energy at a specific time will constrain the economy at the same specific time. Conversely, sufficient energy supply at a specific time allows economic activity at the same specific time. This is further discussed in Chapter 7.

(ii) Secondly, VAR models are linear models. Hence, the influence of the lag of a variable in the VAR model increases linearly with its value. However, relationships between energy and the economy are likely to be non-linear. Indeed, while a mild energy constraint may not entail substantial economic impacts, important shortages or reductions in energy consumptions may trigger salient economic crisis. The impacts of such constraints are likely to evolve in a strongly non-linear way.

(iii) Thirdly, a VAR model is, to some extent, a static model. If one considers that the coefficients of the VAR model describe the relationship between energy and the economy in a specific country, then one can argue that these coefficients should evolve over time as the economy evolves. Considering for instance Spain and most Western countries, important changes have happened in the economy that modify the energy-economy interplay over the years. Structural changes and offshoring of industrial processes have reduced the amount of energy intensive processes within the economy, while efficiency gains have allowed the economy to use energy more efficiently. A such, when carrying out a causality test over such long periods (more than 30 years), it seems that a static relationship between the variables is highly questionable.

In a nutshell, the following points about causality tests for studying the relationship between energy consumption and GDP have been made in this thesis:

(i) Despite a large literature on the causality between energy consumption and GDP, there is no consensus on the directionality of the relationship. This literature can be said to be inconclusive so far.

(ii) Even when considering a particular case (here the Spanish case) and using particular GDP and energy data, the time period and the number of lags assumed are critical in determining the direction of the causality. Varying these parameters changes the results of the test.

(iii) Lastly, it has been discussed that the inherent characteristics of the Granger causality test, which assumes a linear and static relationship in order to model the feedback of the variables, is probably not appropriate to the energy-economy interactions. This a crucial caveat and arguably invalidates results of causality tests.

Hence, such statistical tests, despite their sophistication, are likely to be misleading. From a biophysical perspective, all processes need energy inputs, notwithstanding causality test results, as reminded by the car's metaphor. Considering this, as well as the linear and static assumptions of causality tests, such tests seem superfluous and unsuitable. As such, if biophysical researchers reckon that energy is a key input to the economic process, exploring and modelling the relationship is likely to be a more fruitful activity than expanding the causality literature.

Chapter 6

Energy as a factor of production

In this chapter, a literature review on APFs is firstly presented, which includes the limits of the conventional APF approach, a review of the studies including energy as an additional factor of production, and the main criticisms addressed to the APF approach. Then, the fitted GDP with each different APF as defined in Section 4.2 are presented. Fitted parameters, fully available in Appendix C, are equally discussed. The second research question is subsequently answered according to the findings. A few caveats to APF modelling are finally presented based on the results.

Literature review

Limits of the conventional APF approach

As mentioned in Section 3.1.2, neoclassical growth models, that respect the cost share theorem, rely heavily on technological progress (represented by the TFP) in order to account for economic growth. As such, [START_REF] Abramovitz | Resource and Output Trends in the United States Since 1870[END_REF] states that productivity increase "may be taken to be some sort of measure of our ignorance about the causes of economic growth". Since then, numerous studies have been undertaken in order to explain this technological progress. [START_REF] Jorgenson | The Explanation of Productivity Change[END_REF] firstly introduced the concept of quality adjusted measures of the factors of production and underlined its importance. Recent efforts have focussed on quality adjusted metrics for the conventional factors of production, labour and capital. Labour has been adjusted through the use of a "human capital index", which accounts for the skills of workers [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF][START_REF] Whalley | The contribution of human capital to China's economic growth[END_REF][START_REF] Manuelli | Human Capital and the Wealth of Nations[END_REF], while capital has been adjusted by measuring "capital services", which accounts for the productivity of capital [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF][START_REF] Schreyer | OECD capital services estimates[END_REF][START_REF] Oulton | Capital stocks and capital services: Integrated and consistent estimates for the United Kingdom, 1950-2013[END_REF]. However, although such adjustments for the factors of production reduce the reliance on technological progress, "there is still a TFP [...] that cannot be "explained away"" [Van Ark, 2014].

Energy as a factor of production: main outcomes

Other researchers have emphasized the role of energy in driving economic growth and considered energy as a key factors of production using an APF approach, with mixed results [Tintner et al., 1977, Hannon and[START_REF] Hannon | Energy and technical progress[END_REF]. Amongst the first of them, Kümmel et al. introduced the Linex production function and showcased that it could account for economic growth for the US industrial sector and for Western Germany's economy [START_REF] Kümmel | The Impact of Energy on Industrial Growth[END_REF][START_REF] Kümmel | Technical progress and energy dependent production functions[END_REF]. More recently, Ayres and Warr have pursued the inclusion of energy in APFs through Linex functions. One of their key contributions was to introduce useful exergy (also designated as useful work) as the key metric in order to asses the contribution of energy to economic growth, instead of primary or final energy. These authors have been successful in accounting for recent economic growth in the US and in Japan without relying to a technological progress factor [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF]Warr, 2005, 2010]. However, their results also point to an increasing discrepancy between the fitted and the actual GDP in recent years. As such, Ayres and Warr considered that this might be due to the effect of ICTs as a factor of production, that are becoming increasingly important since roughly 1975. In a subsequent work, they extended their analysis to ICTs and find that the fit was improved [START_REF] Warr | Useful work and information as drivers of economic growth[END_REF]. Finally, in a recent work, Santos et al. explores the identification of economically plausible Cobb-Douglas APFs using a cointegration and causality framework, and finds that such APFs are only identified when quality adjusted metrics are employed [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF].

Criticisms to the APF approach

Last but not least, it is crucial to keep in mind that the APF approach has faced criticisms for the last 70 years, although these are often neglected. The main criticisms are summarised thereafter:

(i) Firstly, and as already underlined, APFs usually rely to a great extent on the use of a TFP, and therefore often leave most of economic growth unexplained.

(ii) A second criticism is related to the aggregation principle [START_REF] Felipe | Aggregation in Production Functions: What Applied Economists should Know[END_REF]. Aggregating the whole economy in a simple APF, as well as aggregating heterogeneous capital in one single measure, is not straightforward. This was extensively discussed during the so-called "Cambridge Controversy" (50s-70s) [Harcourt, 1969, Cohen and[START_REF] Cohen | Retrospectives Whatever Happened to the Cambridge Capital Theory Controversies[END_REF], which opposed detractors of the use of APFs [START_REF] Robinson | The production function and the theory of capital[END_REF], and proponents of their use [START_REF] Samuelson | Parable and Realism in Capital Theory: The Surrogate Production Function[END_REF].

(iii) Thirdly, other criticisms have focused on the existence of an underlying identity that explains the statistically very good fit observed when using APFs. This criticism was firstly introduced by [START_REF] Shaikh | Laws of Production and Laws of Algebra: The Humbug Production Function[END_REF], and then extensively developed both for CD and CES APFs [Felipe and[START_REF] Felipe | Why do Aggregate Production Functions Work? Fisher's simulations, Shaikh's identity and some new results[END_REF][START_REF] Felipe | The CES Production Function, the accounting identity, and Occam's razor[END_REF]. It has led these authors to consider that the APF approach is "not even wrong" [START_REF] Felipe | The Aggregate Production Function: 'Not Even Wrong[END_REF], as it is merely a particular formulation of an underlying mathematical identity, which impedes any meaningful economic interpretation.

Following [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF], the present approach is to recognise that despite these solid criticisms, APFs are widely used in the academic and policy making fields [START_REF] Van Der Werf | Production functions for climate policy modeling: An empirical analysis[END_REF], Brockway et al., 2017a]. Besides, there are great differences in the modelling process of APFs, and these can lead to significantly different outcomes and policy recommendations [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF]. Considering this, this chapter adopts an APF approach notwithstanding relevant criticisms.

Intended contributions to the literature

Finally, this thesis intends to contribute to the outlined literature in different ways:

(i) Most conventional APFs used in energy-economy models include a TFP factor for modelling technological progress. This thesis will discuss the quality of the fit of different functional APFs (CD, CES, Linex), when removing the TFP factor e λ.t . As explained in Section 3.2.2, this TFP is deemed unsuitable in an ecological approach. Considering the important implications of modelling assumptions for APFs [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF], if the TFP can be removed when using quality adjusted metrics, it is recommended that the policy implications of using a TFP instead of relying exclusively on the factors of production are explored in subsequent studies, as these may differ from conventional recommendations.

(ii) Lastly, it has been argued in this thesis, particularly in Chapter 1, that there are limits to global energy consumption, due to two main constraints: climate change and fossil fuel depletion. Thus, it is deemed crucial to understand the likely impacts on the economy and on society of energy consumption constraints. As such, while most of the studies using APFs so far have focused on the causes of economic growth (i.e. factors of production), one main emphasis of this chapter will be on the capacity of such APF models to account for recessions as well as to foresee recessions.

Preliminary results: APFs without energy

Firstly, the CD and CES APFs are fitted without including energy as a factor of production, which gives a baseline.

Cobb-Douglas

Figure 6.1 showcases the CD performance when fitting GDP without considering energy consumption. It can be said that the quality of the fit seems acceptable until 2008, but then the APF seems unable to account for the economic downturn, both for quality adjusted and unadjusted factors of production. This APF can therefore be said to be insufficient in order to account for economic output in Spain over the considered time period. A first caveat can however be identified here. Had this study be carried out until 2008, or a few years earlier, one could have drawn the conclusion that CD APFs were robust enough in order to model the economy, as it could fit closely historical GDP. It is only by including the post crisis years that the limitations of this APF are identified.

q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 An additional noteworthy remark is that in all the cases, fitting with an OLS approach gives functional forms that depart largely from the usual cost share theorem. Indeed, as presented in Table C.1, the fitted output elasticities are larger for capital than for labour in each single case, while the cost share theorem would support output elasticities (α k , α l ) ≈ (0.3, 0.7). This mismatch is observed in all the following models, and dismisses the cost share theorem on an empirical basis.

Constant Elasticity of Substitution

When using a CES APF with capital and labour as factors of production, Figure 6.2 showcases that one can obtain a very good fit for the whole time period. However, when the fit for the period 1960-2008 is projected until 2016, the extrapolation departs from the observed GDP. The CES using unadjusted factors of production is unable to account for the downturn, while the CES using quality adjusted factors of production is somewhat able to account for the downturn, although there is a notable mismatch. q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 A direct implication from the very good fit obtained for the whole time period is that including energy as a factor of production cannot improve considerably the fit. However, including energy may be able to improve the forecasting ability of the APFs by improving the fit of the extrapolated GDP for the period 2009-2016 to the observed GDP.

Results: inclusion of energy

In this section, the fits obtained for APFs with three factors of production are presentedfirstly CD, then CES with each of the three possible nesting structures, and finally Linex.

Cobb-Douglas

Figure 6.3 showcases that the fits obtained with a three factor of production CD APF 1 are similar to those obtained with a two factors of production (see Figure 6.1). The fits are decent until 2008, and then the function is unable to account for the economic downturn, with either quality adjusted or unadjusted factors of production. q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 The failure of the CD APF can be related to its simple modelling assumptions as described in Appendix B, for it assumes that factors of production are perfectly substitutable over time, which is arguably an unrealistic assumption. Indeed, one can hardly imagine that capital, labour and energy are freely and indefinitely exchangeable for one another.

Constant Elasticity of Substitution

Figure 6.4 showcases the fits for all the three possible CES nesting structures. As already pointed, the quality of the fit for the whole time period was already satisfactory, thus the addition of energy as a factor of production could only marginally enhance this fit -the SSE values for each model are available in Appendix C. These are found in general lower when energy is included in the fitting period. Likewise, the projection is found to be enhanced when energy is included in the CES function with quality unadjusted factors of production 2 . However, the inclusion of energy for APFs with adjusted factors of production does not enhance the projection as the fitted APFs collapses to the two factor of production CES functional 1 For the period 1960-2008, the quality adjusted APF was fitted with a TFP. This was due to resources constrains; the amended code did not work as intended in this specific case, and it was not possible to fix this issue within the time limits of this thesis. It is acknowledged that this brings inconsistencies when comparing the different functional forms, but it does not affect the conclusions of this thesis.

2 SSEext is equal to 2.4187, 1.2532 and 2.0842 for respectively the CES with (KL,E) (LE,K) and (KE,L) nesting structures, while the SSEext is found equal to 6.0819 for a CES with two factors of production -see Appendix C. q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980

KL,E

Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980

KL,E

Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980

LE,K

Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980

LE,K

Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980

KE,L

Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 A possible interpretation of this is the fact that the factors of production are rather correlated for the time period 1960-2008, while this correlation disappears in the 2009-2016, as showcased in Figure 6.5, particularly for the couples (labour, capital) and (capital, energy consumption). The consequence is that in the quality adjusted model adding energy as a factor of production (useful exergy) fails to add any additional information that was not already captured by the two other factors of production, and the model obtained with energy does not provide a better fit. Conversely, when the fit is conducted until 2016, the correlation disappears and energy consumption provides some additional information to the model in the last years. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 4 6 8 10 12 0.8 0.9

1.0

1.1

1.2 1.3 capital labour q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 4 6 8 10 12 1 2 3 4 5 6 7 capital final energy q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.8 0.9 1.0 1.1 1.2 1.3 1 2 3 4 5 6 7 labour final energy q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 4 6 8 10 12 14

1.0 1.2 1.4

1.6

1.8 capital services adjusted labour q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 4 6 8 10 12 14 2 4 6 8 10 12 capital services useful exergy q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q 1.0 1.2 1.4 1.6 1.8 2 4 6 8 10 12 adjusted labour useful exergy q q q q q q q q Although the fits displayed in Figure 6.4 are relatively close in numerous cases, the underlying APFs differ greatly in the fitted parameters. This issue was already pointed in [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF], where it is stated that "It is quite possible for substantial changes in a parameter to have a relatively modest effect on the objective function that is determining the parameters estimates (in our case, SSE)". These authors apply bootstrap resampling in order to estimate the precision of the fitted parameters. Conducting a bootstrap resampling is out of the scope of this thesis, however Table 6 The first example showcases that when the the APF is fitted for the time period 1960-2008 with unadjusted factors of production, the role of energy is relatively important: its output elasticity varies between 0 and 0.4, and the elasticities of substitution between capital and energy (σ k,e ) and between the couple (capital, energy) and labour (σ ke,l ) are relatively low. In this model, capital and energy can difficultly be substituted for one another, and the couple (capital, energy) and labour can be substituted, although with constraints. Therefore, energy consumption can be regarded as a key factor of production alongside capital and labour. Conversely, when the same APF is fitted with adjusted factors of production for the same time period, the value fitted for δ 1 ,and therefore for α e , rejects energy as a factor of production.

Likewise, when the APF with adjusted factors of production is fitted for a slightly longer time period, here 1960-2016, energy consumption recovers a key role as a factor of production. The elasticities of substitution obtained are even lower (particularly σ ke,l ) which points that in this model, it is very complex to substitute a factor of production for energy, which is a key and essential factor of production.

It can also be said that the best fit may not return reasonable values. In many cases the elasticities of substitution obtained reach infinity, which would mean that a factor of production can fully substitute for another. Hence, the view supported by these models is that a constraint on a factor of production for which the elasticity of substitution is infinity is not worrisome, as increasing inputs in other factors of production will easily bridge the gap. This is hardly realistic.

Linex

Figure 6.6 showcases that the Linex APF provides a decent fit of real GDP for both quality adjusted and unadjusted factors of production, although a significant mismatch is observed before the economic crisis in the case of quality adjusted factors of production.

However, the Linex APF manages to account for the economic downturn both when it is fitted for the whole time period and when the projection is compared to real GDP, and it is the APF that provides the projection closest to the real GDP; the SSE ext is reduced to respectively 0.95165 (unadjusted) and 0.51698 (adjusted) -see Table C.7. It is also worth noting that this Linex approach does not reproduce the growing mismatch observed in [START_REF] Ayres | The minimum complexity of endogenous growth models: The role of physical resource flows[END_REF]Warr [2005, 2010] between fitted and real GDP in the Spanish case. This may be simply due to the specific country data, or to the considered time period, as these studies covered economic growth since 1900. q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 .6: On the left, actual GDP (dark circles), as well as fitted GDP with a Linex APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

An additional remark that can be drawn from this best fits is that contrarily to what was expected, quality adjusted APFs may return worse fits than quality unadjusted APFs. This is particularly evident for the Linex function, but is actually observed for most of the studied APFs (see Appendix C). This raises the possibility that quality adjusting all the factors of production may lead to substantial double counting (for instance between capital services and useful exergy).

One main difference when using the Linex APF, is that contrarily to the CD and CES APFs, that may support a marginal role of energy within the economy depending on the best fit obtained -namely, on the statistical correlation between the factors of production and the economic output -the Linex APF, due to its mathematical formulation (see Equation B.15), compels energy to be a key factor of production. It may therefore be an APF that fits better an ecological economics approach and modelling.

Conclusion

Two different methods have been followed in order to answer the second research question: "To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)?". Different APFs have been fitted for different time periods; firstly, for the whole time period considered , and secondly, only for the time period 1960-2008, thereby excluding the post crisis period. In this second approach, the fitted APF has subsequently been extrapolated until 2016 by using the fitted coefficients and the observed values of the factors of production. The underlying argument is that a decent and reliable APF should be able to account for both economic growth and economic downturns, no matter on which data it is fitted. Otherwise one would be using economic models for policy making that are unable to account for economic downturns which would be worrisome.

When the APF was fitted for the whole time period, including energy as a factor of production generally improved the fit4 , as evidenced by the SSE values in Appendix C, which accounts for the fact that energy adds supplementary and useful information in order to fit GDP time series. However, the fit obtained with a two factors of production CES was already good, so the improvement of adding energy was marginal.

Conclusions changed when the APF was fitted for the period 1960-2008 and then extrapolated until 2016. The inclusion of energy always improved the fit for CES functions when using quality unadjusted factors of production, but this was not observed when using quality adjusted CES, as the fitted CES collapsed to a two factors CES. It has been discussed that this is probably due to the strong correlation of the factors of production for the period 1960-2008. An additional remark is that the APFs that managed to minimise the residual when extrapolated until 2016 -as showcased by the SSE ext in Appendix C -are the Linex based APFs, although the quality adjusted departed from GDP before the crisis.

As such, it can be answered that the inclusion of energy in APFs can successfully account for recent economic output in Spain, with some reservations. Indeed, when adding energy, most of the APFs successfully account for both economic growth and the economic downturn, although not very accurately for the downturn when the fitted APFs are extrapolated. However, the fit obtained for the whole time period was already good with a two factor CES so the inclusion of energy does not substantially improve the whole time period fit. Besides, when the fit is conducted until 2008 and then extrapolated, adding energy only betters the fit with a quality unadjusted CES, and with a Linex, as the quality adjusted CES converges towards the previous two factors CES. Hence, it has been pointed that Linex APFs, as they ensure that energy consumption will be a key factor of production, may be a modelling choice that fits better an ecological economics framework.

Crucial caveats to APF modelling have also been identified in this thesis. It has been shown that the fitted parameters and economic interpretations can be very sensitive to modelling choices, including the APF choice, the time period for which the data is fitted, and whether the factors of production are quality adjusted or not. Besides, Appendix D showcases that results and best fits are heavily dependent on the GDP metric adopted. An additional caveat is that these important variations in the fitted parameters can occur with only very light modifications of the fitted GDP -which can give a false sense of confidence. Bootstrap resampling can be used in order to assess the precision with which parameters are fitted and can be a way to obtain more reliable results. [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF] performs such resampling and obtains parameters highly uncertain in some cases. APF modelling with best fits may also return fitted parameters that seem hardly realistic, such as infinite elasticities of substitution between factors of production. These caveats are aligned with the conclusion in [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF] as they "raise[s] the possibility that energy-economy modelling with APFs may tell us more about theory and modeling approaches than about the economy". Therefore it seems that other approaches such as input-output economic modelling, stock flow consistent models or system dynamics modelling can provide more useful and reliable insights than APF modelling. It seems hazardous to base policy recommendations on APF modelling, which relies on such arbitrary and sensitive modelling choices.

Chapter 7

Modelling energy-economy linkages

As showcased in the previous section, energy can be used as a factor of production in an APF framework in order to account for economic output. This enables to account for growth as well as recessions without resorting to a technological progress (TFP) factor. However, accounting for growth is quite different than explaining the interactions between energy consumption and economic output. In this section, a simplified conceptual model using a few key concepts is presented in order to describe these interactions. Section 7.1 provides a general overview of the model and of its main modules. Then, Sections 7.2 and 7.3 provide respectively a description of the economy and energy modules. Section 7.4 presents thereafter the different feedbacks of the model. Then, Section 7.5 elucidates the implications of the model described.

General overview

The purpose of this model is to provide a conceptual framework in order to describe the interactions between energy consumption and economic output which uses the main concepts identified in the literature. The model consists of the interaction of two main modules, the economic and the energy module. Thus, some important parameters, such as materials extraction and availability, or land requirements, are omitted (see Section 8.3.3 for a further discussion), although they should be taken into consideration in comprehensive energy-economy modelling. This is however beyond the scope of the present thesis. Figure 7.1 gives an overview of the proposed model, although most feedbacks are omitted from this first description. As explained subsequently, a key differentiation is drawn between the physical availability and the economic affordability of energy. The economic module determines the desired economic activity, and therefore also the ag-

Economic module

gregate demand for goods and services. Providing such goods and services requires energy consumption. As argued in Chapter 2, useful exergy is reckoned to be closer to the goods and services delivered, thus the demand in goods and services is translated into a useful exergy demand. This useful exergy demand is subdivided in demand per energy carrier category: electricity, heat, liquids, gas, solids. Indeed, energy carriers are not perfectly substitutable, given a particular energy system and infrastructure. For instance, gasoline could be somewhat substituted for electricity as oil becomes scarcer, but this could not be instantaneous given the massive infrastructure changes such a shift would demand.

Then, this demand in useful exergy per energy carrier category is compared to the available useful exergy, which is computed by the energy module, which is presented further in Section 7.2. If the energy module is not able to provide the needed useful exergy for each energy carrier, then the demand needs to be adjusted, and the economy module is solicited until the useful exergy demand matches energy availability. If conversely, the required useful exergy is available, the availability test is successful, and an affordability test is triggered. Indeed, the physical availability of enough useful exergy does not mean that this one is available at an economically acceptable price. The economic module also generates an affordable price for useful exergy from each energy carrier, which is associated to the useful exergy demand. The actual price of useful exergy for each energy carrier category is defined dynamically by other parameters as defined in Section 7.4. If the actual price is higher than the affordable price for one of the energy carriers, then the test is not successful, and the economic module is solicited again in order to generate a new couple of useful exergy demand and affordable price, until the affordability test is validated. Conversely, if the affordability test is successful, the aggregate useful exergy demand is fulfilled.

Here, a short but key note needs to be made on the affordability notion. Although it seems clear that the physical availability represents a crucial constraint on energy consumption, the existence of an affordability limit may not be straightforward. As already mentioned, payments to energy are low in national accounts (tipically lower than 10%), when even considered [START_REF] Santos | Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth[END_REF]. One could therefore argue that energy is only a limited share of consumers' expenses, and that energy prices cannot be a key constraint to the economy. However, it is defended in this thesis that energy consumption underpins the whole economic activity. Consequently, one pays indirectly for energy, and the price of goods and services depend heavily on energy prices, since these significantly impact production costs. It has for instance been shown that prices of precious metals, as well as agricultural commodities, tend to evolve in tandem with energy prices [START_REF] Abbott | What's driving food prices?[END_REF][START_REF] Imai | Food and oil prices[END_REF], Zhu et al., 2015]. As such, the affordable price and affordability limit are not concepts that apply strictly to energy prices. Rather, there is an affordability limit on the price of energy because high energy prices means high general prices, and low consumers' affordability for commodities, goods and services.

Economic module

The economic module is not described in depth as this is beyond the scope of this thesis. Rather, the economic module can be regarded as a "black box" which provides as outputs relevant macroeconomic indicators which feedback and are influenced by the rest of the model. The key economic module outputs indicators identified are presented subsequently:

• the aggregate demand: as already presented, the aggregate demand for goods and services is generated by the economic module and is a key element, since it defines the desired level of economic activity, and the demand for useful exergy by energy carrier category. Aggregate stands for the fact that this demand is neither broken down by sector of the economy nor geographical area.

• the affordable price: the aggregate demand in useful exergy, whenever physically available, can only be fulfilled if it can be provided at a reasonable price, i.e. a price lower than the affordable price. For the sake of simplicity, the price concept is used here in terms of price of an energy carrier category, e.g. price of liquids, electricity, solids... instead of price of useful exergy, which would depend upon the exergy efficiency of the economy. For each energy carrier category, the economic module generates an affordable price associated with the aggregate demand.

• the aggregate debt: debt is a key macroeconomic element as it enables to afford higher prices by purchasing against future economic activity and income. Therefore, the higher the debt is, the higher the affordable prices of energy are.

• the interest rates: interest rates control the evolution of debt. Low interest rates tend to support an increase of the debt, while high interest rates tend to prevent debt from increasing. Interest rates therefore have an indirect effect on the affordable prices of energy, through their influence on debt.

• the investments in fossil fuel industries: these investments are crucial as they define the future availability of fossil fuels. High investments translates in exploration, development of new projects, and renovation of infrastructure, maintaining the producing capacity high.

Conversely, low investments endanger future fossil fuel supply as the productive capacity decreases.

• the investments in RES: likewise, investments in RES increase future availability of RE, while a deficiency of investments reduce future production capacity.

The way this parameters interact with the rest of the model is presented in the next sections.

Energy module

In this module, the useful exergy physically available for the economy is computed. This calculation involves some economic parameters introduced in the previous section. The conceptual model is different for fossil fuels and renewable energy availability, as showcased in Figure 7.2.

Energy sources breakdown

As this is a conceptual model, the fossil fuels and renewable energy categories can be defined as one deems relevant. However, a pertinent breakdown of energy sources adapted from Capellán-Pérez et al. [2017a] is suggested in Table 7.1. These tables also include the type of energy demand that each energy source can fulfil. For each of these energy sources, be they fossil fuels or renewables, the conceptual model presented in Figure 7.2 applies in order to assess the physically available useful exergy.

Fossil fuels availability

Main steps

In order to assess fossil fuels availability, a four step process is followed:

(i) For each fossil fuel, the maximum extraction, or maximum flow, is determined, depending on the Remaining Ultimately Recoverable Resources (RURR) and on the physical infrastructure. (ii) A fraction of this maximum flow is subtracted from the actual flow, as it represents the energy consumed by the energy industry in the process of extracting the fossil fuel. This fraction depends on the Energy Return On Investment (EROI).

(iii) A fraction of the remaining flow is subtracted from the actual flow as it represents the energy investment for building new Renewable Energy Systems (RES) infrastructure. This depends on the investments in new RES.

(iv) The available useful exergy flow is calculated from the actual fossil fuel availability by taking into consideration the aggregate exergy efficiency of the economy demanding energy.

(i) Maximum extraction rate

In the case of fossil fuels, the maximum extraction is constrained by two main parameters. Firstly, the RURR plays a crucial role. The RURR is defined according to the Ultimately Recoverable Resources (URR), which accounts for the total amount of the fossil fuel that will eventually be extracted, and to the Cumulative Extraction (CE) of the considered fossil fuel [Capellán-Pérez et al., 2017a]. This is described in Equation 7.1.

RURR t = URR -CE t (7.1)
Obviously, the RURR cannot be negative, which constrains the value of the maximum extraction value. But most importantly, and following Hubbert's peak oil concept, resource extraction profiles tend to go through a maximum and then follow a decline similar to the previous increase, leading to a bell shaped curve, as illustrated in Figure 7.3. Although it has been also argued that the extraction profile, at the global scale, may be asymmetric and follow a steep decline [START_REF] Bardi | The mineral economy: A model for the shape of oil production curves[END_REF], it does not change the present point. The idea is that the extraction rate cannot increase indefinitely until the resource is exhausted. It must go through a peak and then decline. This means that there is a certain moment where the RURR starts to constrain the extraction and sets a maximum extraction rate. One can describe this relationship in terms of feedback mechanism, as displayed in Figure 7.4, where the "Maximum Extraction Rate" only starts constraining the consumption once the peak of extraction has been reached [START_REF] Mediavilla | The transition towards renewable energies: Physical limits and temporal conditions[END_REF]. The value of the URR is therefore a crucial input to the model, although no values are provided here. A literature review of different studies proposing different URR for each fossil fuel can be found for instance in Capellán-Pérez et al. [2017a].

Flow -Extraction Rate

The second limiting factor is the physical infrastructure for fossil fuel extraction. Extractive activities are capital intensive, and a lack of investment or capital can considerably hamper production. One can think of the current tight oil boom in the US, which requires heavy continuous investments in order to balance the steep and rapid decrease in extraction rates [START_REF] Angelo | The Shale Oil Ponzi Scheme Explained: How Lousy Shale Economics Will Pull Down The U.S. Economy[END_REF]. More generally, installing new capacity for extracting fuels takes time, and therefore an inadequate infrastructure may impede to fulfil the demand for a significant length of time. As described in Figure 7.2, the available infrastructure is affected by investments and depreciation, the effects of which compete. Investments higher than depreciation will increase the maximum extraction rate, whereas a high depreciation rate will decrease the maximum extraction rate.

Then, the lower of the two maximum extraction rates, defined with the RURR and the infrastructure methods, is kept as actual maximum extraction rate for the next steps.

(ii) Energy industry own use

All processes consume energy, and the extraction, processing and refining of fossil fuels is not an exception. The concept of net energy can be defined from this observation. This idea was firstly emphasized in energy studies by [START_REF] Cottrell | Energy and Society: The Relationship between Energy, Social Change, and Economic Development[END_REF][START_REF] Cottrell | Technology, Man, and Progress[END_REF], who defined the concept of surplus energy as the difference between the energy obtained from a process and the energy invested in that process, thereby emphasising the need of investing some of the harvested energy in order to obtain additional energy. He subsequently based his analysis of the industrial revolution and economic growth on the considerable increase of surplus energy delivered by the new harnessed energy sources: "It will only be when we get another response from nature, in the form of greatly diminished return in the form of surplus energy, that we can expect the present (industrial) revolution markedly to slow down... There is knowledge that leads to the conclusion that the time this will take place is not so far away as we would think" [Cottrell, 1972, page 142] The same concept was described as net energy and energy costs of getting and concentrating that energy by [START_REF] Odum | Energy, Ecology, and Economics[END_REF], who also underlines the key role of net energy. Although Cottrell and Odum were probably the first in linking this concept to economic and societal activity, it has to be said that biologists introduced the idea of net energy yields as a main criterion for natural selection of organisms considerably earlier [Lotka, 1922a,b]. Since then, the net energy concept has become a key concept in ecological economics. The Energy Return On Investment (EROI), firstly formally introduced in [START_REF] Cleveland | Energy and the U.S. Economy: A Biophysical Perspective[END_REF], is a different formalisation of the surplus energy or net energy concepts that is often preferred. The EROI of a system can be defined according to Equation 7.2: EROI = Energy delivered by the system Energy invested in the system (7.2)

Since then, the EROI concept has proved useful for different purposes [START_REF] Murphy | Year in review-EROI or energy return on (energy) invested: Review: Energy return on investment[END_REF], including the study of whether and to which extent specific energy sources are net energy yielders [START_REF] Farrell | Ethanol Can Contribute to Energy and Environmental Goals[END_REF], the study of EROI evolution for specific fuels [Hall et al., 2014, Court and[START_REF] Court | Long-Term Estimates of the Energy-Return-on-Investment (EROI) of Coal, Oil, and Gas Global Productions[END_REF], which usually decrease over time as the most accessible resources are depleted first, the comparison of different energy sources in terms of net energy and EROI [START_REF] Hall | EROI of different fuels and the implications for society[END_REF], the relationship between energy and the economy [START_REF] Cleveland | Energy and the U.S. Economy: A Biophysical Perspective[END_REF], Capellán-Pérez et al., 2017a], and the minimum societal EROI in order to have an industrial or sustainable society [Hall et al., 2009, Fizaine and[START_REF] Fizaine | Energy expenditure, economic growth, and the minimum EROI of society[END_REF].

In the present case, the EROI concept is useful in describing the share of extracted fossil fuels that are directly consumed by the fossil fuel industry and the share that is made available for society. For each of the fossil fuel presented in Table 7.1, one can define the Energy Industry Own Use (EIOU) and the Net Energy (NE) shares directly with the EROI value:

EIOU share = 1 EROI (7.3) NE share = EROI -1 EROI (7.4)
which are the coefficients used to describe EIOU and NE flows of energy in Figure 7.2. The demonstration of Equations 7.3 and 7.4 is presented in Appendix E.

(iii) Energy investment in RES

Once the energy used by the energy industry has been subtracted, the conceptual model also subtracts the energy invested in developing new RE infrastructure and RES. Following [START_REF] Sgouridis | The sower's way: Quantifying the narrowing net-energy pathways to a global energy transition[END_REF] and [START_REF] Bardi | In Support of a Physics-Based Energy Transition Planning: Sowing Our Future Energy Needs[END_REF], it is argued that the development of RES needed for reaching Paris Agreement's targets or EU targets [European Parliament, 2009] will have massive energy costs, which will reduce the availability of energy for other producer sectors of the economy. The flow of fossil fuel energy invested in new RES is therefore directly linked to the level of financial investments in developing this new infrastructure. The higher financial investments are, the more energy is diverted from its other societal uses.

One could argue that taking into consideration both this energy investment and the EROI of RES is double counting, which is indeed true, as the EROI can be defined as presented in Equation 7.5 [Sgouridis et al., 2016, Supplementary Information]:

EROI = E g E c + E op + E d (7.5)
where E g refers to the energy generated by a system, E c to the energy invested in developing the system, E op to the energy invested in operating the system, and E d to the energy invested in decommissioning the system. The present suggestion is to take into consideration the E c of new RES by subtracting it from the flow of available energy as new RES are installed instead of including it in the EROI of RES, as presented in Section 7.3.3. This is suggested in order to have a more accurate modelling of temporal aspects and constraints, which are crucial when modelling energy-economy linkages, as getting available energy one specific year differs significantly from getting the same amount of energy 10 years later. Indeed, modelling energy flows with the EROI implies that the net energy obtained from a system is constant over the years (leaving temporal variations of the EROI aside), which, at the macro scale, makes sense once a production capacity is installed. In this situation, old plants are maintained, retrofitted and dismantled while new plants are constructed, each year, leaving the aggregate net energy available from a specific RES roughly constant. Conversely, if a massive investment in new RES has to be made, then the net energy obtained will depend heavily on temporal aspects. This investment will be an effort at the beginning and constitute a trough in energy availability at first, until the desired production capacity is reached. Then, net energy returns will be considerably higher, as only the energy expenses of operating the RES will be made, until the infrastructure becomes obsolete and needs to be gradually renewed and retrofitted. The flow of available net energy in this final situation can be modelled using a constant EROI as a suitable proxy.

(iv) Exergy efficiency of the energy demanding society

Once the available fossil fuel primary energy has been determined according to the previous steps, the actual useful exergy that this primary energy can provide to the economy needs to be calculated. This depends directly on the efficiency with which the society converts primary energy to useful exergy, and can be directly calculated by using the second law efficiency as defined in Section 2.2. Considering i as being the energy demand category (e.g. solids, liquids, gases, electricity, heat), and j as being the energy source (here one of the fossil fuel of Table 7.1), it can be written: Useful exergy i,j = Primary energy i,j .φ i .ε i,j (7.6) where φ i corresponds to the exergy factor for the energy source j, namely the factor that gives primary exergy when multiplied per the primary energy1 , and ε i,j corresponds to the second law efficiency with which the energy source i provides useful exergy for the specific energy demand category j.

Renewable energy availability

A three step process is followed in order to assess the renewable energy availability:

(i) For each renewable energy source, the maximum extraction, or maximum flow, is determined, depending on the available infrastructure.

(ii) A fraction of this maximum flow is subtracted from the actual flow, as it represents the energy consumed by the energy industry in the process. This fraction depends on the Energy Return On Investment (EROI). This step is subtle as double counting with the fossil fuel energy invested in new infrastructures has to be avoided.

(iii) The available useful exergy flow is calculated from the actual maximum flow of renewable energy by taking into consideration the aggregate exergy efficiency of the economy demanding energy.

Step 1 and 3 are similar to what has already been described for the fossil fuel industry and as such, are not described in the following. However a particular attention is given to Step 2 as double counting with the fossil fuel energy invested in new RES needs to be avoided.

Energy industry own use

Similarly to the fossil fuel industry, the RE industry also consumes energy in order to deliver additional energy to the economy. However, and as described in Section 7.3.2, the energy investments required from the fossil fuel industry in order to develop new RES is already taken into consideration by subtracting it from the available fossil fuel energy. Therefore, for each specific type of RES i (e.g. hydro, windmill, solar PV...), the average lifetime T i is introduced. Then, the EROI of the installed capacity is calculated differently depending on the age of the considered RES:

• whenever the capacity of the considered RES is newer than T i , the EROI used for modelling the flow is similar to the one presented in Equation 7.5 without including the E c energy costs as these energy costs are covered by the contribution of the fossil fuel industry.

• whenever the capacity of the considered RES is older than T i , the complete EROI as presented in Equation 7.5 is used in order to model the net energy flow.

This differentiation enables to take into consideration both the EROI of RES and the fact that a massive and punctual energy investment is needed in order to develop the required RE infrastructure, which will suppose high energy costs in a short timespan.

Intermodule feedbacks

Decrease in EROI and depletion

The first feedback mechanisms presented are the ones relative to the decrease of the EROI of fossil fuels entailed by gradual depletion of the resources. As the easiest resources to extract are depleted first, the direct effect of depleting resources is to decrease the EROI of fossil fuels. This is consistent with findings in the EROI literature pointing to a decrease in the EROI of fossil fuels [Court andFizaine, 2017, Hall et al., 2014]. In turn, the decrease of this EROI decreases the physical availability of energy, as an increasing share of the energy produced has to be consumed by the energy industry. However, three mechanisms oppose this decrease in energy availability. The first one, that has been historically widely most employed, is to increase extraction of fossil fuels. But this method somewhat backlashes as it engenders a faster depletion of fossil fuels stocks and consequently a quicker decrease in the EROI.

The second method to counter the decrease in energy availability, not as promptly and widely implemented -a glance at the historical figures of fossil fuel versus renewable energy use will convince the sceptic reader -is to develop new RES. However, RES have currently lower EROIs than fossil fuels (leaving unconventional fossil fuels apart) [START_REF] Hall | EROI of different fuels and the implications for society[END_REF], which means that the development of RES also entails a decrease in the average EROI. Nevertheless, RES development does not imply an ever decreasing EROI as it relies on renewable sources, hence non-exhaustible. This negative effect will therefore fade off once fossil fuels' EROI become lower than RES EROIs as extraction turns to very costly resources.

Thirdly, one possible answer is to increase efficiency. As displayed in Figure 7.5 increasing efficiency opposes the declining EROI trend, although considering historical data, its influence has remained marginal. However, one key impact of the efficiency improvements is to increase the efficiency with which energy is used within society, thereby enabling to obtain the same energy service with less energy inputs. Therefore, efficiency improvements can be regarded as a way of buffering reduction in energy availability due to physical constraints. However, efficiency improvements tend to be associated with higher energy demand due to macroeconomic rebound effects, as described in Figures 3.2 and 7.7, which questions its buffer capacity.

Demand fulfilment, availability and affordability

Once the demand in goods and services is translated into a demand for useful exergy, this one is fulfilled or not depending on the physical availability of useful exergy (Section 7.3) and on the economic affordability of energy. As showcased in Figure 7.6, whenever this demand is fulfilled, it entails a positive feedback on the economic activity, which tends to move forward, and to demand greater amount of useful exergy for the following time period. Conversely, whenever this demand is not fulfilled, the economy has to adapt and to reduce the demand. Hence, a rejection of the demand entails a negative feedback on the economic activity, which tends to slow down.

Besides these feedbacks, Figure 7.6 presents factors affecting availability and affordability of energy. As already described in Section 7.3, the physical availability of useful exergy is supported by the investments in energy infrastructure and by the efficiency with which the economy converts primary energy into useful exergy. The decline of the EROI of energy sources however reduces physical availability.

The economic affordability notion represents whether the economy is able to pay in order R&D. Furthermore, the rise in prices also makes the extraction of resources harder to extract profitable, thereby propelling new investments in the extraction of resources that were not profitable with lower prices. In the case of RES, the increase in prices of energy makes them increasingly profitable, as RE can be sold at higher prices. This is for instance obvious in the case of biofuels, which have been considerably developed during the 2004-2008 period, when oil prices considerably increased, making biofuels production increasingly profitable [START_REF] Abbott | What's driving food prices?[END_REF]. The investments triggered by high energy prices turn into a higher physical availability of energy, which in turns supports economic growth as the physical demand for useful exergy can be more easily fulfilled, and as higher availability supports affordability, as described in the previous section. However, if prices increase above affordable levels (Figure 7.8), this positive retroaction is broken. Indeed, unaffordable prices will entail a reduction of the demand for goods and services and consequently for useful exergy, which in turn will decrease energy prices. In other words, too high energy prices will trigger an economic recession, during which energy prices are low. These low prices will have two effects; firstly rising the demand as goods and services become more affordable, and secondly reducing investments in energy industries as the benefits of energy companies shrink and as extracting costly resources is not profitable with low energy prices. This diminution of investments will curtail future energy production capacity. The combination of these two effects will entail a situation where the demand is higher than the supply, namely a situation of low physical availability of energy, which will in turn rise the prices of energy.

The combination of these two feedback loops, when prices are affordable and prices unaffordable, suggests that energy prices will remain bounded below an affordable price -which can however vary. Prices will then fluctuate up and down as periods of high and low affordability alternate.

Implications and conclusion

The conceptual model is consistent with the ecological theory of growth presented in Chapter 3 to the extent that it includes useful exergy as a main driver of growth, since the economy needs to consume sufficient useful exergy in order to move forward. Likewise, efficiency is also presented as a key element in economic growth as it increases availability (Figure 7.6) and increases economic growth (Figure 7.7).

However, the presented model significantly expands the ecological theory firstly described. Particularly, it explicitly includes and stresses the importance of supply constraints to economic growth, both in terms of physical availability and economic affordability. These constraints are described through the dynamics of fossil fuels stocks, declining EROI of energy sources, RES development, investments in the energy industry, and price dynamics. The implications of these constraints is that the ecological model of growth presented in Chapter 3 may be valid in a world of unconstrained energy supply, but that it does not hold when confronted to a world of finite resources and supply constraints. It can be argued that an unconstrained world is an appropriate description of the economy for the past historical economic growth. Conversely, there is now increasing evidence, as already mentioned, that we enter a period of finiteness and resources constraints, in which supply constraints will play a key role. As such, the theory presented in Figure 3.2 may be valid in order to describe historical economic growth, but is not suitable in order to model present and future energy-economy interactions. Energy-economy models need to explicitly and consistently include supply constraints.

Another direct implication that can be drawn from this model, is that contrarily to standard economic theory, which assumes that energy and oil prices will rise as resources are depleted and consequently entail efficiency improvements and the development alternatives that will solve the shortage, energy prices are likely to remain bounded below an affordability limit and to fluctuate up and down as cycles of increasing and shrinking demand alternate. This affordability limit is likely to be a key constraint to the extraction of costly fossil fuels, which will probably remain in the ground because of affordability limits, thereby constraining future energy availability [Murphy andHall, 2011a,b, Tverberg, 2018b].

Finally, this chapter has attempted to answer the third research question of this thesis; "What conceptual model can explain the interactions between the economy and energy consumption?". This research question has been directly answered by presenting one conceptual model that describes energy-economy interactions. It is however not claimed that the model is a comprehensive, and some considerable limitations appear, as underlined in Section 8.3.3.

Chapter 8

Conclusion

In this conclusion chapter, the main steps of this master's thesis argumentation are firstly summarised. Then, the answers of the research questions are synthesised. The main implications of the findings are thereafter discussed, and directions for further work are suggested. Finally, the main limitations of the present thesis are finally underlined, and the validity of the findings is discussed.

Conclusion

In this thesis, the tight historical correlation between energy consumption and economic growth was firstly introduced. It was thereafter argued that two critical and imminent constraints impede to keep energy consumption growing indefinitely. The first one is environmental damage and particularly climate change, that is to a great extent caused by greenhouse gases emissions related to energy consumption. The second impending constraint is fossil fuel depletion, and particularly oil depletion. Indeed, the global production of oil is thought to be likely to reach a peak shortly, and while its impacts on the economy and society are poorly understood, they are likely to be significant and unpleasant. The dominant economic paradigm, however, praises further economic growth, as if it was the solution to all evils. It was therefore argued that a better understanding of the underlying mechanisms of the energy-economy interactions is needed. In order to contribute to this inquiry, the following research questions were investigated:

To which extent can energy consumption explain the recent Spanish economic history (1960-2016)?

• Is there statistical evidence of causality between GDP and energy consumption?

• To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)?

• What conceptual model can explain the interactions between the economy and energy consumption?

It was subsequently discussed that the neoclassical theory of growth does not offer satisfying explanations for the causes and drivers of economic growth. Indeed, the most salient criticism from a biophysical perspective is the lack of inclusion of natural resources consumption (i.e. both energy resources and other natural resources) in the theory, according to which a combination of capital, labour and technological improvements can be hold responsible for the evolution of economic output over the years. The arguments of neoclassical theory for the neglect of natural resources -i.e. the cost share theorem and the fact that natural resources are regarded as intermediary inputs -were deemed unpersuasive. An ecological theory of growth, according to which natural resources consumption and efficiency gains are key drivers of economic growth, was presented. Although this theory also suffers from flaws, it successfully includes physical flows and natural resource consumption in the mechanisms explaining growth. Finally, it was argued that useful exergy, as the metric closest to the energy service gained by the final user, can be used as a decent proxy for natural resources consumption.

After this theoretical discussion, each research question was investigated, and the main findings are summarised thereafter.

Is there statistical evidence of causality between GDP and energy consumption?

This first question was answered by conducting numerous Granger causality tests between energy consumption and GDP growth rates. Many parameters, including the length of the time period considered, the kind of causality test conducted, the energy and GDP metric used, were changed in each test. The main finding of this analysis was that a conclusion on causality can hardly be drawn in the Spanish case, as results depended heavily on the parameters of the tests, which is consistent with the literature. The influence of some parameters was assessed, and particularly it was found that when using energy metrics closer to the energy service (i.e. useful exergy) and GDP metrics adapted for measuring the output-side of the economy (i.e. rgdpo), more tests tended to support a causality from energy consumption to GDP, although this remained limited evidence.

To which extent can the inclusion of energy as a factor of production in aggregate production functions (APF) account for Spain's economic output (GDP)? This second research question was explored by fitting different APFs to the Spanish historical GDP data. The quality of the fit as well as the predictive power of the APF (when APFs were fitted on the data predating the economic crisis) were tested. It was found that in general adding energy as a factor of production improved the quality of the fit and the predictive capacity of such APFs, although this was not salient in all cases, as the fit was in some cases already satisfactory without energy. Therefore, it can be said that the inclusion of energy in APFs improves moderately the capacity of an APF to account for Spain's economic output, as well as the predictive power of an APF.

What conceptual model can explain the interactions between the economy and energy consumption?

In this third research question, a conceptual energy-economy model was suggested in order to make explicit the underlying mechanisms of the energy-economy interactions. This model was presented building on different models and concepts of the energyeconomy literature. Although the model could not be tested, the different important parameters identified in the literature were all meaningfully included, and it can therefore be said that a conceptual model explaining the interactions between the economy and energy consumption was successfully presented.

Discussion

In this section, the main implications of the findings are discussed, and suggestions for further research are provided based on these implications.

APF modelling and TFP

It was showcased in Chapter 6 that fitting GDP through APFs can provide a satisfactory fit without including a TFP -at least for this specific case study. Mainstream APF modelling however systematically include technological progress through the use of a TFP, and thereby ascribe to "technological progress" a major role in the production process. Therefore, policy recommendations that stem from these models may support investment in innovation, R&D, new technologies... However the use of a TFP seems questionable both on conceptual grounds (section 3.2) and on empirical grounds (decent fits are obtained without TFP). As such, the policy implications of APF modelling with or without TFP would be an interesting area for further research.

Limitations of mainstream approaches

The findings of this thesis also point to serious limitations in the mainstream methods -i.e. Granger causality testing and APF modelling -through which the energy-economy interactions are usually studied. Firstly, the literature on Granger causality testing was found particularly extensive, but nonetheless inconclusive. Besides, it was discussed that the mathematical VAR model upon which causality tests are based seemed inappropriate in order to describe the relationship between energy and the economy, as it relies on linear and static relationships. Secondly, it has been showcased that the parameters fitted through APFs and the downstream economic interpretations depend heavily on arbitrary modelling choices, including the time span considered for the fit. Besides, the fitted parameters obtained may not be realistic, which rises the question of the reliability of the results and insights gained through APF modelling.

As a consequence of these serious limitations, it is argued that other approaches, such as system dynamics modelling -which enables to deal with crucial elements for energy-economy modelling, such as flows, stocks, and tipping points -may be able to provide more useful and reliable insights than these two mainstream approaches.

Neoclassical theory and unlimited growth

The conceptual energy-economy model presented contradicts important statements of the neoclassical economics school, including the fact that prices mechanisms will solve any energy shortage situation. Indeed, neoclassical theory states that prices will soar in response to a shortage, which will in turn stimulate the development of alternatives and remedy the shortfall.

Conversely, the presented model suggest that a rise in prices will be constrained by an affordability limit, above which end users will not be able to purchase goods and services. Indeed, if this affordability limit is exceeded, the aggregate demand will diminish, thereby entailing a recession and a decrease in prices. The direct consequence of such an affordability limit is that costly energy sources -such as shale oil and tar sands -are not likely to be harnessed in a large scale, and that a considerable amount of fossil fuels will remain unexploited, as prices will not rise enough to make the extraction of such resources profitable. This is likely to intensify energy supply constraints, which will adversely affect the economy. As such, it seems unlikely that the economy can grow indefinitely, as suggested by neoclassical economic theory.

Limitations

This study also presents a few limitations, which are subsequently underlined. Taking into consideration these limitations, the validity of the findings is equally discussed.

Data limitations

The first limitations can be related to the data used in this thesis. Firstly, one can question the value of useful exergy as an accurate valuation of the economic usefulness of energy. Other valuations based on economic considerations, such as the Divisia index method, exist, and have been reckoned superior by some analysts [START_REF] Cleveland | Aggregation and the role of energy in the economy[END_REF]. According to Chapter 2, useful exergy is still regarded as a superior metric, but controversy exists, and it could be of interest to explore to which extent the results would change if a different quality adjusted metric for energy, based for instance in the Divisia index method, was used. Likewise, the GDP metric used can also be questioned, particularly in Chapter 6. It is showcased in Appendix D that results can differ importantly depending on the GDP metric used. Conclusions with different data could therefore have differed to some extent. Other caveats on the data include uncertainty on the data used, for instance in the useful energy and useful exergy data.

However, as the main outcome of the causality and APF analysis conducted was to point the weaknesses and insufficiencies of such approaches, it is believed that this main conclusion is robust to changes in the data used.

Aggregate Production Function approach

There are also caveats related to the use of APFs in this thesis. It is worth noting that the time span covered can change importantly conclusions. For instance, [START_REF] Ayres | Accounting for growth: The role of physical work[END_REF] account for growth in the US over a century, and they obtain an increasing residual in the fitted APF that they ascribe to ICTs. In this study, no increasing residual is identified, which may be a consequence of the shorter time period considered. Findings might therefore have differed, had a longer time period been studied.

An additional limitation is related to the use of the micEconCES package for fitting CES APFs. As already underlined, this package does not have a proper handling of model boundaries. This means that the best fits might have somewhat differed for CES APFs with a different fitting algorithm, and the conclusions might have to some extent differed as well.

It is also believed that these limitations only have marginal implications for the findings and conclusions. The target was not to conduct a thorough economic analysis, but rather to investigate the consequences of including energy consumption as a factor of production in APFs. The limits and caveats to APF modelling that have been identified seem robust to the mentioned limitations.

Conceptual energy-economy model

The presented energy-economy conceptual model has serious limitations as important parameters when describing the energy module have been omitted. First of all, material constraints have been neglected, while there is evidence that energy systems, and particularly RES, are highly demanding in specific minerals and metals. The supply of particular minerals and metals will consequently be playing a key role in the energy transition [START_REF] Moss | Critical metals in strategic energy technologies. Assessing rare metals as supply-chain bottlenecks in low-carbon energy technologies[END_REF][START_REF] Arrobas | The growing role of minerals and metals for a low carbon future[END_REF]]. An overview of some material requirements for different energy systems can be found in [START_REF] Moss | Critical metals in strategic energy technologies. Assessing rare metals as supply-chain bottlenecks in low-carbon energy technologies[END_REF], who identifies 14 key metals for energy technologies, 5 of which may be at high supply risks. As such, [START_REF] Vidal | Metals for a low-carbon society[END_REF] states that "a shift to renewable energy will replace one non-renewable resource (fossil fuel) with another (metals and minerals)". Besides the issue of material requirements for energy systems, materials interact in another crucial way with energy supply. Like for fossil fuels, the energy requirements of minerals and metals increase as extraction moves towards lower grade resources, due to exhaustion of high quality resources [START_REF] Bardi | The Mineral Question: How Energy and Technology Will Determine the Future of Mining[END_REF][START_REF] Bardi | Extracted: How the Quest for Mineral Wealth Is Plundering the Planet[END_REF]. This means that rising energy costs of mineral and metal extraction will divert an increasing amount of energy from the satisfaction of societal needs. Currently, the share of primary energy utilised in extracting and refining minerals and metals is already around 10% of the global primary energy supply [START_REF] Bihouix | Quel Futur Pour Les Métaux?: Raréfaction Des Métaux: Un Nouveau Défi Pour La Société[END_REF]De Guillebon, 2012, Vidal et al., 2013]; which indicates that this effect may have significant consequences.

Secondly, land constraints have also been neglected in the present model, although these may be crucial for particular RES, such as solar and biomass based systems [START_REF] Dijkman | Comparison of renewable fuels based on their land use using energy densities[END_REF]. Biofuel production, for instance, has been shown to be highly demanding in land, and as such, competes for food production [START_REF] Abbott | What's driving food prices?[END_REF][START_REF] Tyner | Biofuels and food prices: Separating wheat from chaff[END_REF]] and brings about deforestation and indirect land use change (ILUC), which emits considerable GHGs [START_REF] Searchinger | Use of U.S. Croplands for Biofuels Increases Greenhouse Gases Through Emissions from Land-Use Change[END_REF][START_REF] Fargione | Land Clearing and the Biofuel Carbon Debt[END_REF]. Concerns about the sufficient availability for land have been raised in 100% renewable energy scenarios [START_REF] Capellán-Pérez | Assessing vulnerabilities and limits in the transition to renewable energies: Land requirements under 100% solar energy scenarios[END_REF], although this is currently an ongoing debate, as other studies defend that land requirements are not compelling [START_REF] Jacobson | Providing all global energy with wind, water, and solar power, Part I: Technologies, energy resources, quantities and areas of infrastructure, and materials[END_REF]. Taking into consideration that land is globally already subject to an intense and fierce competition driven by numerous factors [START_REF] Smith | Competition for land[END_REF], it seems reasonable and cautious to include it as a possible constraint in energy-economy models.

A serious modelling of energy-economy interactions aiming at policy recommendations needs to include these two decisive constraints, or else, results may point to erroneous recommendations. Thus, the conceptual model drawn here would need further elaboration if it was to be used in order to fully represent the energy-economy nexus.

B.1.3 Elasticity of substitution

The elasticity of substitution is a measure of the substitutability between factors of productions, namely of how well these can substitute for one another. The higher the elasticity of substitution is, the better factors of production substitute for one another. There are different definitions for the elasticity of substitution, which can be employed for different purposes, but this diversity can be somewhat confusing and cause inconsistencies in the literature [Brockway et al., 2017a]. A review of different definitions can be found in [START_REF] Stern | Elasticities of substitution and complementarity[END_REF].

The Hicks Elasticity of Substitution (HES) is usually preferred when modelling the economy through APFs, for instance in CGE models [START_REF] Sorrell | Energy Substitution, Technical Change and Rebound Effects[END_REF]. The HES between two factors of production x i and x j is defined as: The HES therefore "measures the ease with which a decrease in one input i can be compensated by an increase in another j while holding output fixed" [START_REF] Sorrell | Energy Substitution, Technical Change and Rebound Effects[END_REF]. An additional note for what follows is that the HES is symmetric, namely that σ i,j = σ j,i .

∀(i, j) ∈ [1 : n] 2 , i = j, σ i,j = ∂ln x i

B.2 Functional forms employed in this thesis

This section directly describes the functional forms by employing the same three factors of production that are considered in the thesis, nameely capital, labour, and energy. Therefore, y stands for economic output, k for capital, l for labour, and e for energy. This section does however not provide a literature review of the use of these functional forms and of the modelling consequences for each parameter. Further discussions can be found in Brockway et al. [2017a], [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF].

B.2.1 Cobb-Douglas

Formulation

The CD production function is defined as: y = θ.e λ.t .k α k .l α l .e αe (B.5) or, in its formulation without TFP:

y = θ.k α k .l α l .e αe (B.6)
where θ is a scale parameter, λ is a term representing the rate of technological progress and e λ.t is the TFP.

Output elasticities

From the definition of output elasticities provided in Equation B.3, one can conclude that each α i stands for the output elasticity of the associated factor of production. Output elasticities are therefore constant over time. Likewise, a direct application of Equation B.1 gives that: ν = α k + α l + α e (B.7) thereby imposing α k + α l + α e = 1 in order to fulfil the constant returns to scale condition.

Elasticities of substitution

The HES for any of the factors of production is constant and equal to unity, which is given directly by the application of Equation B.4.

Limits of the CD APF

Serious criticisms can be addressed to the CD production function. Firstly, the elasticities of substitution between the factors of productions is equal to unity, thereby assuming perfect and constant over time substitution. This is arguably unrealistic, or only valid in very particular cases. One can hardly imagine that capital, labour and energy are freely and indefinitely exchangeable for one another. The CES production function offers more flexibility when modelling the substitutability of factors of production. Besides, output elasticities are assumed to be constant, while it can be argued that these are likely to change depending on the specific situation of the considered economy.

B.2.2 Constant Elasticity of Substitution

Formulation

The mathematical formulation of a three factors of production CES production function depends on the nesting modelling choice, as two factors of production are usually tied together in the equation. Equations B.8 and B.9 prodives the example of nesting capital and energy against labour The three possible nesting structures are therefore {k,e),l}, {(k,l),e}, and {(e,l),k}. The CES mathematical formulation will remain the same, but factors of production need to be arranged according to the nesting structure chosen. Further discussion about the choice of a nesting structure is provided in [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF].

Output elasticities

Conversely to the CD production function, the output elasticities of the factors of production are not constant over time in a CES configuration. Equations B.10, B.11 and B.12, directly adapted from [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF], provide the output elasticities for each of the three factors of production according to the nesting structure {(k,e),l}. These depend directly upon the evolution of the factors of production.

Elasticities of substitution

Conversely, the elasticities of substitution are constant over time and can be determined directly by using the parameters of the CES:

σ (ke),l = 1 1 + ρ (B.13) σ k,e = 1 1 + ρ 1 (B.14)
As such, it can be said that the CES functional form does enable to depart from the unity elasticity of substitution of the CD functional form. However, elasticities of substitution remain constant over time, which is a questionable assumption. This assumption has been a main criticism addressed to CES type function. The Linex production function enables to depart from this limitation.

B.2.3 Linex

Formulation

The Linex function was firstly formulated in [START_REF] Kümmel | Technical progress and energy dependent production functions[END_REF]. This first formulation was equivalent to the one proposed thereafter: Note that in the Linex formulation the TFP and the time dependency are usually removed. Conversely to the CD and CES APFs, the role of the factors of production cannot be swapped in this formulation, which is further justified in [START_REF] Kümmel | Technical progress and energy dependent production functions[END_REF].

y = θ.e.

Output elasticities

In the Linex case, output elasticities are also dependent on the evolution of the factors of production and are therefore not constant over time. These are somewhat simpler than in the CES case, as calculations give the following expressions: The constant returns to scale condition is equally fulfilled by these output elasticities.

Elasticities of substitution

The elasticities of substitution are not calculated here as these may not be definable by analytical means. However, these are variable over time, thereby providing an additional flexibility to the Linex APF, which is part of the broader family of the Variable Elasticity of Substitution APFs.

C.2.2 Constant Elasticity of Substitution

As explained in Section 4.2.3, the package micEconCES that was used in order to conduct the fits does not have a precise handling of meaningful boundaries, which implies that when a boundary model is returned as best fit, non relevant parameters will be returned as well. This issue is further discussed in [START_REF] Heun | From Theory to Econometrics to Energy Policy: Cautionary Tales for Policymaking Using Aggregate Production Functions[END_REF], which underlines the different boundary models and the relevant parameters for each. [Heun et al., 2017, Table 2] description of boundary models is used in order to adjust the parameters returned by the cesEst function, which explains the NA values for σ and δ when found in the following tables.

KL,E nesting structure

Sensitivity -GDP metric

The influence of the GDP metric used for fitting APFs is quickly discussed in this appendix. Best fits for the whole time period are compared when using the rgdpna and the rgdpe metrics with the CES and Linex functional forms. The best fits seem to depend heavily on the GDP metric used, which is one additional limitation to the results presented in this thesis. Indeed, it rather supports the statement according to which APF modelling does not seem reliable enough for energy-economy modelling.

D.1 Constant Elasticity of Substitution

Figure D.1 and D.2 showcase that the fit is considerably poorer when the data is fitted to the rgdpe GDP metric, both when quality adjusted and unadjusted factors of production are used. q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q q qq q q q qqq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q 1960 1970 1980 1990 2000 2010 Chapter D -Sensitivity -GDP metric q q q q q q q q q q q q q q qq q q qq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q q qq q q q qqq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q 1960 1970 1980 1990 2000 2010

D.2 Linex

Figure D.3 showcases a poorer fit when a Linex APF is used in order to fit the rgdpe metric. q q q q q q q q q q q q q q qq q qqq qqq q q q q q q q q qqq q q q q q q q q q q q q q q q qq q q q q q q 1960 1970 1980 1990 2000 2010 Year GDP (rgdpna) -index 1960 q q q q q q q q q q q q q q qq q q qq q q q qqq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q 1960 1970 1980 1990 2000 2010 These sensitivity tests are not supportive of the use of APFs for energy-economy modelling as results seem highly dependent on the GDP metric used, and therefore question the robustness of APF modelling.

  ment. In dashed lines, the different possible consequences of the decrease in the physical availability of fossil fuels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The demand is fulfilled or not depending on the physical availability and economic affordability of energy, which are influenced by several other factors. . . . . . . . 7.7 Interplay between demand, prices, investments, and availability, when prices remain affordable and the economy grows. . . . . . . . . . . . . . . . . . . . . . . . 7.8 Interplay between demand, prices, investments, and availability, when prices are unaffordable and the economy slows down. . . . . . . . . . . . . . . . . . . . . . . C.1 Output elasticities for each CES model, with unadjusted factors of production on the left and adjusted on the right. Dashed lines correspond to fits for the period 1960-2008, while solid lines correspond to fits for the whole time period. . . . . . D.1 On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a two factors of production Constant Elasticity of Substitution APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe. . . . . . . . . . . . . . . D.2 On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a three factors of production Constant Elasticity of Substitution APF -(KE,L) nesting -using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a Linex APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 .

 1 Figure 1.3: Depletion curves for oil in different studies, where oil can stand for different liquid fuel categories [Capellán-Pérez et al., 2017a, page 64].

Figure 1 .

 1 Figure 1.4: Projections of oil production and oil demand in the New Policies and Sustainable Development scenarios. Source: [IEA, 2018a, World Energy Outlook, page 74. All rights reserved.]

Figure 1 . 5 :

 15 Figure 1.5: Dramatic increase in US tight oil production, data from the US Energy Information Administration (EIA).

  Figure3.1: The economy, as a metabolism embedded in the society and in the environment, or our "Earth Spaceship". Own elaboration.

Figure 3

 3 Figure 3.2: The feedack driven economic growth model in the absence of natural resource constraints, an ecological approach. Inspired from Warr and Ayres [2012].

  Figure 5.6 showcases results in this specific setting for the test. 6: Results of the causality tests when using useful exergy and rgdpo as variables.

Figure 6 . 1 :

 61 Figure 6.1: On the left, actual GDP (dark circles), as well as fitted GDP with a two factors of production Cobb-Douglas APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

Figure 6

 6 Figure6.2: On the left, actual GDP (dark circles), as well as fitted GDP with a two factors of production Constant Elasticity of Substitution APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

Figure 6

 6 Figure 6.3: On the left, actual GDP (dark circles), as well as fitted GDP with a three factors of production Cobb-Douglas APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

Figure 6 . 4 :

 64 Figure6.4: On the left, actual GDP (dark circles), as well as fitted GDP with Constant Elasticity of Substitution APF for each possible nesting structure using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

Figure 6 . 5 :

 65 Figure 6.5: Factors of production plotted against one another. The first row displays quality unadjusted factors, the second row displays quality adjusted factors. Dark dots represent values for the period 1960-2008, while blue dots correspond to values for the period 2009-2016.

  Figure6.6: On the left, actual GDP (dark circles), as well as fitted GDP with a Linex APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines).

Figure 7 . 1 :

 71 Figure 7.1: Overview of the conceptual model presented.

Figure 7

 7 Figure 7.3: Bell shaped curve as introduced by Hubbert, own elaboration, inspired from Kerschner and Capellán-Pérez [2017].

  Figure7.4: Feedback mechanism: Remaining Ultimately Recoverable Resources (RURR) constraining the Maximum Extraction Rate, which in turns may constrain the fossil fuel consumption if the demand is higher than this rate.

  Figure 7.5: Feedback interactions with depletion and decrease in Energy Return On Investment. In dashed lines, the different possible consequences of the decrease in the physical availability of fossil fuels.

  Figure7.7: Interplay between demand, prices, investments, and availability, when prices remain affordable and the economy grows.

Figure 7

 7 Figure7.8: Interplay between demand, prices, investments, and availability, when prices are unaffordable and the economy slows down.

1 :y 1 ρ 1 ρ

 111 = θ.e λ.t . δ. δ 1 .k -ρ 1 + (1 -δ 1).e -ρ the formulation without TFP:y = θ. δ. δ 1 .k -ρ 1 + (1 -δ 1 ).e -ρ

1 - 1 δ 1 ρ 1 - 1 δ 1 ρ 1 ρ

 1111111 1 .k -ρ 1 . δ 1 .k -ρ 1 + (1 -δ 1 ).e -ρ 1 ρ ρ . δ 1 .k -ρ 1 + (1 -δ).e -ρ 1 -δ 1 ).e -ρ 1 . δ 1 .k -ρ 1 + (1 -δ 1 ).e -ρ 1 ρ ρ . δ 1 .k -ρ 1 + (1 -δ).e -ρ ρ . δ 1 .k -ρ 1 + (1 -δ 1 ).e -ρ

Figure C. 1 :

 1 Figure C.1: Output elasticities for each CES model, with unadjusted factors of production on the left and adjusted on the right. Dashed lines correspond to fits for the period 1960-2008, while solid lines correspond to fits for the whole time period.

Figure D. 1 :

 1 Figure D.1: On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a two factors of production Constant Elasticity of Substitution APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe.

Figure

  Figure D.2: On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a three factors of production Constant Elasticity of Substitution APF -(KE,L) nesting -using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe.

Figure D. 3 :

 3 Figure D.3: On the left, actual GDP (rgdpna) (dark circles), as well as fitted rgdpna with a Linex APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, the fitted GDP is rgdpe.

  1.1 Economic output (real GDP) in function of energy consumption, both indexed in 1971. Economic data from the World Bank, and energy data from the International Energy Agency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 On the left, global real GDP (blue) and global primary energy (red) growth rates. On the right, global real GDP (blue) and global final energy (red) growth rates. Economic data from the World Bank, and energy data from the International Energy Agency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Depletion curves for oil in different studies, where oil can stand for different liquid fuel categories [Capellán-Pérez et al., 2017a, page 64]. . . . . . . . . . . . . . . . 5 1.4 Projections of oil production and oil demand in the New Policies and Sustainable Development scenarios. Source: [IEA, 2018a, World Energy Outlook, page 74. All rights reserved.] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Dramatic increase in US tight oil production, data from the US Energy Informa-

tion Administration (EIA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Energy Conversion Chain as considered in this article. Inspired from Brockway et al. [2015]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

  On the left, actual GDP (dark circles), as well as fitted GDP with Constant Elasticity of Substitution APF for each possible nesting structure using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period and thereafter extrapolated (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Factors of production plotted against one another. The first row displays quality unadjusted factors, the second row displays quality adjusted factors. Dark dots represent values for the period 1960-2008, while blue dots correspond to values for the period 2009-2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 On the left, actual GDP (dark circles), as well as fitted GDP with a Linex APF using quality unadjusted (red) and quality adjusted factors of production (blue) for the whole period 1960-2016. On the right, GDP is fitted for the time period 1960-2008 and thereafter extrapolated (dashed lines). . . . . . . . . . . . . . . . 7.1 Overview of the conceptual model presented. . . . . . . . . . . . . . . . . . . . . 7.2 Overview of the energy module, which computed the useful exergy physically available for each energy carrier category. . . . . . . . . . . . . . . . . . . . . . . 7.3 Bell shaped curve as introduced by Hubbert, own elaboration, inspired from Kerschner and Capellán-Pérez [2017]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Feedback mechanism: Remaining Ultimately Recoverable Resources (RURR) constraining the Maximum Extraction Rate, which in turns may constrain the fossil fuel consumption if the demand is higher than this rate. . . . . . . . . . . . 7.5 Feedback interactions with depletion and decrease in Energy Return On Invest-

and thereafter extrapolated (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.4

  Figure 1.1: Economic output (real GDP) in function of energy consumption, both indexed in 1971. Economic data from the World Bank, and energy data from the International Energy Agency.

							4.0
						Global real GDP -index 71	1.5 2.0 2.5 3.0 3.5
							1.0
								1.2	1.4	1.6	1.8	2.0	2.2
							Global final energy consumption -index 71
	Growth rates	0.01 0.03 0.05				Growth rates	0.01 0.02 0.03 0.04 0.05
		-0.01					-0.01
		1970	1980	1990	2000	2010	1970	1980		1990	2000	2010
				Year						Year

Figure 1.2: On the left, global real GDP (blue) and global primary energy (red) growth rates. On the right, global real GDP (blue) and global final energy (red) growth rates. Economic data from the World Bank, and energy data from the International Energy Agency.

Table 4 .

 4 .1. Section 4.1 describes the methodology for testing causality, Section 4.2 elucidates how energy has been included in APFs, and Section 4.3 presents how the conceptual model has been constructed.

	Research questions	Analysis methods
	(Q1) Statistical evidence?	Granger causality test Sensitivity test
	(Q2) Energy in APFs?	Use of different APFs Commparison of actual and fitted GDP
	(Q3) Conceptual model?	Compiling literature review Proposal of a conceptual model

1: Used methods in this study.

Table 4 .

 4 2 provides a short summary of the number of parameters tested for each causality test. In total, 13610 tests have been conducted, 12170 of which are deemed valid.

	Causality test Couples of variables Time periods Different lags Total Valid tests
	T-Y	9	300	1 ≤ n ≤ 4	4264	3664
	causality	9	276	1 ≤ n ≤ 4	4673	4253
	grangertest	9	276	1 ≤ n ≤ 4	4673	4253
	Table 4.2: Summary of the test conducted (T-Y: Toda-Yamamoto) and of the different param-
	eters modified.

Table 4 .

 4 3 summarises the different fits. The data used for these variables is presented in Section 4.4.

	w/ or w/o energy Functional form Factors of production	Time periods
	Without	CD CES	Quality unadjusted Quality adjusted Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2008 & 1960-2008 1960-2016 & 1960-2008 1960-2016 & 1960-2008
		CD	Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2016 & 1960-2008
		CES -(KL,E)	Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2016 & 1960-2008
	With	CES -(LE,K)	Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2016 & 1960-2008
		CES -(KE,L)	Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2016 & 1960-2008
		Linex	Quality unadjusted Quality adjusted	1960-2016 & 1960-2008 1960-2016 & 1960-2008

Table 4

 4

.3: Summary of the different fits conducted. The factors of production (K,L,E) stand respectively for capital, labour and energy, and the APFs CD and CES for Cobb-Douglas and Constant Elasticity of Substitution.

Table 4 .

 4 4 presents the main elements of the suggested model and the main sources of inspiration.

	Element of the model	Sources of inspiration
	Energy-economy feedback	[Capellán-Pérez et al., 2017a] [Dale et al., 2012]
	URR & RURR	[Capellán-Pérez et al., 2017a] [Mediavilla et al., 2013]
		[Percebois, 1979]
	Energy, exergy efficiency	Exergy economics literature, e.g. [Ayres and Warr, 2010]
		[Sakai et al., 2018]
		[Capellán-Pérez et al., 2017a]
	Maximum Extraction Rate	[Mediavilla et al., 2013]
		[Dale et al., 2012]
	EROI	[Hall et al., 2014] [Dale et al., 2012]
		[Tverberg, 2018b]
	Price of energy and affordability	[Tverberg, 2012] [Murphy and Hall, 2011a]
		[Murphy and Hall, 2011b]
		[Sgouridis et al., 2016]
	Investments in energy industries	[Bardi and Sgouridis, 2017]
		[Dale et al., 2012]
		[Sgouridis et al., 2016]
	Transition to RES	[Bardi and Sgouridis, 2017]
		[Dale et al., 2012]
	Debt & Interest rates	[Tverberg, 2018b]

Table 4

 4

.4: Inspiration sources for each element of the energy-economy model presented in Chapter 7. EROI: Energy Return On Investment; (R)URR: (Remaining) Ultimately Recoverable Resources.

Table 5 .

 5 These categories are exclusive, therefore a test presenting a e y relationship is neither included in the e → y nor in the y → e columns.

		e → y y → e e	y Non causality Total
	Number of tests	785	1 526		150	9 709	12 170
	Percentage	6,45	12,54	1,23	79,78	100

1: Overview of the results. Whenever Granger causality is detected, it is done with 95% confidence. e → y and y → e causality is directly obtained from the causality tests conducted, while e y causality is identified when the two previous causalities are simultaneously identified.

  .2. The T-Y test supports more the conservation hypothesis than the two other tests. Conversely, the tests carried out with the causality and grangertest functions provide more neutral results, as the conservation and growth hypothesis are backed up in similar amounts of cases. This influence of the chosen test is consistent with the literature reviewed.

		e → y y → e e		y Total
	T-Y	197	950	66	3664
	causality	350	337	54	4 253
	grangertest	238	239	30	4 253
	Table 5.2: Results depending on the type of test carried out. T-Y: Toda-Yamamoto.

Table 5

 5

		e → y y → e e		y Total
	Final energy	122	569	19	3 851
	Useful energy	127	393	90	3 977
	Useful exergy	536	564	41	4 342

.3: Results depending on the energy variable considered.

  .4. The description of the different GDP variables has been presented in Section 4.4.

		e → y y → e e	y Total
	rgdpe	380	736	29	4 110
	rgdpna	194	552	115	3 155
	rgdpo	211	238	6	3 589
	Table 5.4: Results depending on the GDP variable.

Table 5

 5

		e → y y → e e		y Total
	1	6	750	0	6536
	2	95	246	24	1847
	3	65	6	0	195
	4	9	12	0	111
	5	187	86	39	1081
	6	143	48	3	499
	7	187	232	30	1053
	8	93	146	54	848

.5: Influence of the number of lags included in the VAR model.

  .1 presents two examples of very different economic implications of APFs which present very similar fits. The two examples are drawn from the (KE,L) nesting structure as it may be the case for which contrasts between models are the most evident, but Appendix C provides data for investigating other contrasts.

	Nesting	Factors	Time period SSE SSE ext	Key parameters
	KE,L	Unadjusted	60-08	3.05	2.08	(σ k,e ; σ ke,l ) ≈ (0.07; 0.89) α e ∈ [0; 0.4]
		Adjusted	60-08	4.84	1.99	(δ 1 ; α e ) = (1; 0)
	KE,L	Adjusted	60-16	5.72	NA	(σ k,e ; σ ke,l ) ≈ (0.07; 0.39) α e ∈ [0; 0.4]
		Adjusted	60-08	4.84	1.99	(δ 1 ; α e ) = (1; 0)

Table 6 .

 6 1: Different economic implications. SSE and SSE ext are in a 10 -2 scale. More precise values are available in Appendix C.

  exp 2.a 0 . 1 -

	l + e 2.k	+ a 1 .	l e	-1	(B.15)

Table C .

 C 4: Fitted parameters for the CES APF with 3 factors of production and KL,E nesting.

	Quality	Time period	log(θ)	σ k,l	σ kl,e	δ	δ 1	SSE 10 -2 SSE ext 10 -2
	Unadjusted	1960-2016 1960-2008	1.0812 0.72034 1.1087 0.84832	∞ ∞	0.72948 0.78554 0.86140 0.75953		3.6803 3.0542	NA 2.4187
	Adjusted	1960-2016 1960-2008	0.98250 0.32520 1.0261 0.40283 NA ∞	0.95666 0.94532 1.0000 0.90409		8.1150 4.8432	NA 1.9941
	LE,K nesting structure
	Quality	Time period	log(θ)	σ e,l	σ el,k	δ	δ 1		SSE 10 -2 SSE ext 10 -2
	Unadjusted	1960-2016 1960-2008	1.0850 1.1026	∞ ∞	0.45947 0.19152 0.81236 0.58292 0.21759 0.86903	2.9876 2.8197	NA 1.2532
	Adjusted	1960-2016 1960-2008	0.99404 0.73436 0.30337 0.086399 0.77189 1.0261 NA 0.40283 0.095906 1.0000	7.2769 4.8432	NA 1.9941

Table C .

 C 5: Fitted parameters for the CES APF with 3 factors of production and LE,K nesting.

	KE,L nesting structure
	Quality	Time period log(θ)	σ k,e	σ ke,l	δ	δ 1	SSE 10 -2 SSE ext 10 -2
	Unadjusted	1960-2016 1960-2008	1.1029 0.23214 0.84423 0.80319 0.83959 1.1166 0.068745 0.88828 0.77375 0.99240	3.2231 3.0501	NA 2.0842
	Adjusted	1960-2016 1960-2008	1.0018 0.070090 0.39182 0.91292 0.66295 1.0261 NA 0.40283 0.90410 1.0000	5.7175 4.8432	NA 1.9941

Table C .

 C 6: Fitted parameters for the CES APF with 3 factors of production and KE,L nesting.

	C.2.3 Linex
	Quality	Time period	θ	a 0	a 1	SSE 10 -2 SSE ext 10 -2
	Unadjusted	1960-2016 1960-2008	1.1558 0.40608 0.84985 1.1531 0.33270 0.75384	6.8448 6.4129	NA 0.95165
	Adjusted	1960-2016 1960-2008	1.2603 0.065832 1.0231 1.2574 0.043238 1.0038	29.374 28.941	NA 0.51698

Table C.7: Fitted parameters for the Linex APF.

This thesis therefore usefully contributes to the energy-economy literature as it integrates three different and important approaches of the literature, and points their respective strengths and limitations. The suitability of these approaches is discussed, and it is argued that further research should focus on energy-economy models that make explicit the underlying mechanisms of the energy-economy interactions, instead of focusing on mere statistical correlations, which can lead to misleading findings and conclusions.

Note that Illich did not focus on economic growth, but his ideas are consistent with the thesis that there is a critical level of material standards above the one growth is not anymore desirable.

Direct translation from French from the author.

In this study, the word decoupling stands for absolute decoupling, in opposition to relative decoupling, as defined in UNEP[2011]. Otherwise the full denomination relative decoupling will be used.

Note that this has neither been the case for material use nor greenhouse gases emissions, as showcased in[START_REF] Krausmann | From resource extraction to outflows of wastes and emissions: The socioeconomic metabolism of the global economy[END_REF] and[START_REF] Quéré | Global Carbon Budget[END_REF].

The extraction curve can also have a more complex profile, e.g. reaching different peaks in different years, but the main trends over a period long enough remains a profile with a maximal peak and then a decline until insignificant extraction values.

The Carnot efficiency defines the maximum efficiency of a machine converting thermal energy at the temperature T1 in to work, considering a reference environment at the temperature T2. It is defined as η Carnot = 1 -

T 2 T 1

The first law efficiency η is defined as the ratio of energy output to energy input, and the second law efficiency ε is defined as the ratio of exergy output to exergy input.

Utility maximisation is often equated to total consumption.

The rationality of agents has been particularly questioned by the behavioural economics school[START_REF] Dequech | Neoclassical, mainstream, orthodox, and heterodox economics[END_REF], while the very concept of free market has been questioned by institutional economics, that argue that "a market looks free only because we so unconditionally accept its underlying restrictions that we fail to see them"[START_REF] Chang | 23 Things They Don't Tell You about Capitalism[END_REF].

The output elasticity, also called marginal productivity, of a factor of production xi is defined as αi =∂y ∂x i y x i, where y stands for the economic output (often GDP)

While the economic recession stemming from the 1973 oil shock should have lowered the US GDP by 0.26% according to the cost share theorem, the US GDP decreased four times more. This underlines that this cost share theorem, besides being disconnected form biophysical grounds, does not provide an accurate or valid view of the economy.

Note that this discipline has also been called biophysical economics[START_REF] Cleveland | Biophysical economics: Historical perspective and current research trends[END_REF] or bioeconomics[START_REF] Georgescu-Roegen | The Entropy Law and the Economic Process[END_REF]].

However, in the case of energy, one needs to take into consideration that energy is dissipated in the form of heat, and that entropy created. Consequently, the first law is particularly useful for mass balances and mass flow accounting.

In this argumentation, energy is a subset of natural resources. Therefore, an increase in energy consumption also implies an increase in natural resources consumption.

Note that this step is particularly true for industrialising periods, where labour still has a significant importance in the production process. Nevertheless, the underlying reasoning also holds in an industrialised economy. Instead of substituting for labour, industries may substitute larger and more efficient machineries for old and out of date equipment in order to produce more. This is also likely to increase the energy consumption of the industry.

Roughly, one can consider that a time series is stationary when its main characteristics (e.g. mean, variance...) do not change over time. A mathematical of stationarity can be found in[Lütkepohl,

2005, page 24]2 The R functions used are respectively the adf.test, pp.test, and kpss.test, from the tseries package[START_REF] Trapletti | Tseries: Time Series Analysis and Computational Finance[END_REF].3 In statistics, the H0 hypothesis stands for the "null hypothesis", namely the hypothesis that the test will either reject or be unable to reject -note that the inability to reject is different from the validation the hypothesis.4 A time series is differentiated when the difference between two successive terms is taken. In this case, it means that the new time series is constituted by the difference of the successive growth rates.

A discussion on these criteria can be found in[Lütkepohl, 2005, Chapter 4]

For each criterion, the optimal number of lags is the one that minimises such criterion

This function carried out the multivariate Portmanteau and Breusch-Godfrey tests, which test for residual serial correlation.

This function returns the eigenvalues of the matrix equivalent to the VAR model defined.

Here the logarithmic scale is chosen for the residuals, as the R packages used for fitting models use the logarithmic scale.

Real GDP measures GDP in constant prices, namely adjusting prices for inflation. Conversely, current GDP does not take into consideration inflation.

Note that this graph is only marginally modified when changing the energy metric (e.g. using primary, final, useful energy) or the GDP metric. Conclusions are therefore robust to changes in these metrics.

The case in which useful exergy consumption grows while GDP degrows, and which is relevant for four data points, is here not considered as decoupling.

See[START_REF] Sims | The Role of Approximate Prior Restrictions in Distributed Lag Estimation[END_REF]

The divisia index method accounts for energy quality by weighting each energy quantity based on the price of fuels or energy carriers.

Note that the same can be said for the tests supporting either the conservation or growth hypothesis. Indeed, these do not reject the feedback hypothesis, they barely correspond to a situation in which one the causality could be detected with 95% probability in one direction, while causality could not be detected in the other direction at 95% confidence.

It is worth noting that there are adaptations of the Granger causality test which enable to depart from some of these limitations. However, these limitations are relevant when using the standard version of the test, as done here.

The SSE decreases from 10.275 to respectively 8.1150, 7.2769 and 5.7175 for the (KL,E), (LE,K) and (KE,L) nesting structures.

The CD APF with energy was still not found able to provide a better fit than the two factors CES, therefore evidencing the limits of such a function. Besides, the Linex APF with quality adjusted factors of production also shown a worse fit than the two quality adjusted factors CES, as the Linex fit departed considerably of the real GDP observed before the crisis.

These factors are standard for different fossil fuels and can be found for instance in[START_REF] Ayres | The Economic Growth Engine: How Energy and Work Drive Material Prosperity[END_REF].

Indeed, for each Ai matrix, the coefficient am,n (placed on row m and column n) corresponds to the weight of the lag number i of the variable yn when determining the present value of the modelled variable yn.

Note that different CES formulations can be found in the literature, although equivalent in mathematical terms. A typical case is two define the couple (ρ * , ρ * 1 ) as being (-ρ, -ρ1) according to the convention of this section. In this case, the subsequent formulae need of course to be adapted.

Acknowledgements

I am greatly thankful to the different people who supported this work and provided invaluable advice and support. Massimo Pizzol from Aalborg University (Denmark), provided very helpful advice and supervision. Paul Brockway, from the University of Leeds (UK), also provided valuable recommendations, readings, and ideas. I am equally grateful to Matthew Heun, from Calvin College (Grand Rapids, USA), for providing the R code upon which parts of the work are based, as well as providing technical support with the code.

Availability

Fossil fuels Availability

Renewables Availability Infrastructure Availability Affordability

Contents

to fulfil the energy demand. The affordability is directly assessed from the comparison between the actual price of energy and its affordable price. An increase in prices will therefore decrease affordability. Conversely, the creation of additional debt, which enables to pay for additional goods and services, at higher prices, enables to delay affordability issues by raising the affordable price of energy [Tverberg, 2018b]. In short, more debt increases affordability for goods and products, which enables commodity prices, particularly energy prices, to rise. Although interest rates are not displayed in Figure 7.6, they play a key role in monitoring the debt. The lower they are, the more creation of debt is entailed. Conversely, high interest rates are likely to stop increasing the aggregate debt [Tverberg, 2018b].

Finally, the physical availability and economic affordability also interact. A high physical availability tends to lead to a sufficient energy supply in comparison to energy demand, and therefore to affordable prices. Likewise, a situation of high affordability, namely a high affordable price, increases physical availability through investments in energy infrastructures -fossil fuel exploration and RES development. The interplay between prices, demand and investments is further discussed in the following section.

Prices, demand, and investments

This section is considerably inspired from the thesis developed by Tverberg (see for instance [START_REF] Tverberg | Oil supply limits and the continuing financial crisis[END_REF][Tverberg, , 2018b]]), and is also consistent with the work of other authors [Murphy and Hall, 2011a,b]. Figure 7.7 presents the dynamics of prices, demand and investments when energy prices remain affordable, and Figure 7.8 when energy prices reach unaffordable levels.

In the first case, a context of economic growth leads to an increase in aggregate demand for goods and services and therefore for energy, which consequently increases energy prices. This increase in prices triggers firstly efficiency improvements, which on the one hand reduce the demand by making the use of energy more efficient, although on another hand it drives further economic growth. Secondly, the increase in energy prices leads to a situation with higher investments, both in fossil fuels industries and RES development. Indeed, as prices rise, the profits of fossil fuels industries rise, enabling such industries to invest in new projects and Appendix A

On Granger causality testing A.1 VAR models

Let us consider define a VAR model of k different variables and of order p (namely including p lags), which is designated as a VAR(p) model. The variable y t = (y 1,t , y 2,t , ..., y k,t ) is a vector of k elements representing each of the considered k variables at the time t. One can define the VAR(p) model as follows [START_REF] Lütkepohl | New Introduction to Multiple Time Series Analysis[END_REF]:

where each A i is a fixed matrix of (k, k) dimensions which contain the coefficients defining the VAR(p) model 1 The vector ν = (ν 1 , ν 2 , ..., ν k ) represents the means of each of the k variables considered. Finally, the vector u t = (u 1,t , u 2,t , ..., u k,t ) is a white noise vector representing the part left unexplained by the model, namely the difference between each modelled variable and its actual value. Further conditions on u t can be found in [Lütkepohl, 2005, Chapter 1].

A VAR(p) model linking only two variables x and y may be defined with a couple of equations:

2)

A.2 The Granger causality test

Basically, when a Granger causality test is carried out in order to determine whether a variable x causes a variable y, the idea is to check whether a VAR(p) model of y including the past values of x (Equation A.4) or excluding these values (Equation A.5) provides a better description of y.

This is tested by using a Wald test on the VAR(p) models defined. A description of the Wald test can be found in [START_REF] Lütkepohl | New Introduction to Multiple Time Series Analysis[END_REF]. Although the underlying test is not presented here, it is helpful to elucidate the VAR(p) models upon which the Granger causality test rely. Indeed, the limitations of the test as pointed in Section 5.4 stem directly from the VAR model structure.

Functional APF forms

In this chapter, the main parameters of APFs are introduced. Then, the three functional forms used as APFs in this study (i.e. the Cobb-Douglas, the Constant Elasticity of Substitution, and the Linex APFs) are presented. Their main characteristics and limitations are described. Other relevant APFs used in economic studies, such as Leontief, Translog, and Variable Elasticities of Substitution, are out of the scope.

B.1 Important parameters of APFs

In this section, economic output is designated by the letter y, while the factors of productions are designated by (x 1 , x 2 , ..., x n ).

B.1.1 Returns to scale

In mathematical terms, the returns to scale factor ν can be defined as the number verifying the following equation:

This factor is often assumed to be equal to unity. From an economic perspective, this equates to assuming that the productivity of economies are not dependent on their size, e.g. Russia is not more productive (in terms of economic output per unit of factor of production) than Denmark, just because it is a bigger country [START_REF] Warr | Useful work and information as drivers of economic growth[END_REF]. This seems to be a reasonable assumption when a certain economic size is reached, although it may not be quite true for small economies. The second crucial impact of such an assumption is to considerably reduce the number of eligible functional forms for an APFs.

B.1.2 Output elasticities or marginal productivities

For each factor of production, the output elasticity, or marginal productivity, is defined as:

or, in its normalised formulation:

In the following, the normalised formulation will be prefered as it enables to refer to the returns to scale. According to these definitions, the output elasticity associated with a factor of production therefore accounts for its capacity to increase economic output when the availability of this factor of production in the economy increases. It is therefore a key indicator: the higher this output elasticity is, the more increasing this factor of production will increase economic output.

Appendix C

Fitted parameters

This chapter presents the parameters that were fitted for each APF in Chapter 6. All fitted parameters are provided with five significant digits. Note that for the period 1960-2008, the quality adjusted APF was fitted with a TFP. This was due to resources constrains; the amended code did not work as intended in this specific case, and it was not possible to fix this issue within the time limits of this thesis. It is acknowledged that this brings inconsistencies when comparing the different functional forms, but it does not affect the conclusions of this thesis. The fitted value for λ in this fit is -5, 5599 -2 .

C.1 APFs without energy

Appendix E

Shares of energy

From the definition of the EROI as presented in Equation 7.2, one can directly write: