Electrifying natural gas : a dynamic model of Québec's energy transition

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Electrifying Natural Gas: A Dynamic Model of Québec’s

Energy Transition

Mémoire

Julien Boudreau

Maîtrise en économique - avec mémoire

Maître ès arts (M.A.)

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Electrifying Natural Gas

A Dynamic Model of Québec’s Energy Transition

Mémoire

Julien Boudreau

Sous la direction de:

Markus Herrmann, directeur de recherche Luca Tiberti, codirecteur de recherche

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Résumé

Dans ce mémoire, j’évalue les implications économiques d’une électrification efficace des usages du gaz naturel. Pour ce faire, je développe un modèle dynamique en équilibre par-tiel des marchés du gaz naturel et de l’électricité dans lequel la demande est détaillée par service énergétique. Le modèle tient également compte des contraintes de capacité et des coûts d’ajustement du réseau électrique. L’implémentation numérique du modèle, laquelle est calibrée au contexte québécois, révèle que la transition énergétique est guidée par le prix des émissions de GES. Une transition optimale implique que les investissements dans les énergies renouvelables débutent tôt et soient répartis à travers le temps. Par ailleurs, le prix du carbone en place ne mène pas à une électrification qui permette l’atteinte des cibles du gouvernement du Québec.

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Abstract

In this thesis, I evaluate the economic implications of the efficient electrification of natural-gas usages. To do so, I develop a dynamic partial equilibrium model of the electricity and natural gas markets with multiple energy demands. The model accounts for the capacity constraint and adjustment cost of the electric grid. The numerical implementation, calibrated to Québec’s contexts, reveals that the electrification’s main driver is the cost of greenhouse gas emissions. An optimal transition implies that investments in renewables start early-on and are smoothed through time. Québec’s actual carbon price does not lead to an electrifica-tion pathway that satisfies the province’s emission target.

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List of Tables

3.1 Energy Services, Representative Equipment and Efficiencies (2017) . . . 29

3.2 Québec’s Energy Prices, Elasticities and Variable Costs (2017) . . . 32

3.3 Parameters of the Demand Functions . . . 33

3.4 Investment Costs of Electricity Production . . . 35

4.1 Carbon Price and Cumulative Emissions . . . 45

4.2 Additional Electricity Generation Capacity and Cumulative Investment Cost of Renewables . . . 48

4.3 Cumulative Carbon Revenues . . . 49

4.4 Costs and Benefits of Passing from a SPEDE to a CB Carbon Policy (G US$, 2017) 49 A.1 Representative Equipment . . . 58

B.1 Québec’s Energy Consumption and Energy Prices, 2017 . . . 60

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List of Figures

1.1 Energy transition modelling approaches . . . 5

1.2 Example of a simple Reference Energy System for home space heating . . . . 8

1.3 GEM-E3 economic circuit . . . 13

3.1 Energy Service Demand Functions by Sector . . . 34

3.2 Investment Cost Functions of Electricity Generation Capacity . . . 36

3.3 Growth and Carbon Constraint Scenarios . . . 36

3.4 Québec’s CO2Emission Targets . . . 38

3.5 Historical Price of Carbon Allowances Under the SPEDE . . . 39

4.1 Optimal Energy Consumption (G2S: SPEDE) . . . 42

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All models are wrong, but some are useful.

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Remerciements

J’aimerais tout d’abord chaudement remercier Markus Herrmann, mon mentor et directeur de mémoire. Peu de mots suffisent à exprimer ma gratitude pour tout le temps qu’il m’a si généreusement accordé pour discuter, relire et commenter mon mémoire. J’ai confiance que ce travail constitue le début d’une longue collaboration entre nous deux. Je remercie également Luca Tiberti, mon codirecteur, dont les commentaires m’ont permis de clarifier et élaborer ma démarche à bien des égards. J’aimerais aussi exprimer ma gratitude à mes évaluateurs, MM. Michel Roland et Carlos Ordás Criado, pour leur précieux commentaires et suggestions m’encourageant à poursuivre et peaufiner ma recherche.

Par ailleurs, je remercie MM. Oskar Lecuyer, Renaud Coulomb et Adrien Vogt-Schilb de m’avoir si généreusement fourni les codes de leur article intitulé Optimal Transition from Coal to Gas and Renewable Power Under Capacity Constraints and Adjustment Costs duquel je me suis largement inspiré tout au long de ma recherche. Ma démarche a aussi grandement bénéficié des très généreux commentaires de MM. Jean-Thomas Bernard et Bernard Decaluwé. Je remercie le Fonds de recherche du Québec - Société et culture (FRQSC), le Conseil de recherches en sciences humaines (CRSH), le Centre de recherche en économie de l’environnement, de l’agroalimentaire, des transports et de l’énergie (CREATE) et Énergir pour leur soutien fi-nancier.

Enfin, je remercie de tout mon coeur mes parents, ma copine et mes amis proches. Chacun à votre façon, vous m’avez offert le soutien, l’amour et l’inspiration sans lesquels ce projet n’aurait été envisageable.

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Introduction

Following the COP21 agreement, Québec’s government has set ambitious greenhouse gas (GHG) reduction goals (-20% by 2020, -37.5% by 2030 and -80 to 95% by 2050 as compared to 1990 emission levels, Gouvernement du Québec, 2020b). While provincial GHG emissions have declined since 2003, mitigation efforts are falling short of the 2020 objective. Nonethe-less, the 2030 and 2050 objectives are still achievable if sufficient investments are made to transition towards low-carbon energy resources (Dunsky, 2019). With natural gas being Québec’s second-biggest GHG source after petroleum (Whitmore and Pineau, 2020),1 sci-entists, analysts and politicians have called for the phaseout of natural gas, notably through the electrification of its usages (Bergeron, 2015; Breton et al., 2019; Croteau, 2019; Dunsky, 2019; Gagnon, 2019). With natural gas being mainly used for residential and commercial heating, most of its consumption could theoretically be converted to Québec’s low-carbon electricity.2 We know that the switchover from gas to low-carbon electricity is limited by Québec’s generation capacity (Paradis-Michaud, 2020; Michaud, 2020; Dunsky, 2019; CGA, 2019). There is, however, less knowledge regarding the characteristics and implications of optimal electrification of natural gas consumption in a dynamic setting.

I develop a stylized dynamic model of the energy market to characterize energy consump-tion, emission, and investment pathways under optimal natural-gas usage electrification. The energy demand is disaggregated into distinct energy service demands to account for the heterogeneous cost and efficiency between the energy sources, depending on their end-use. At each point in time, total electricity consumption is limited by the total installed genera-tion capacity. This capacity depreciates over time but can be increased through investments. The investments in new capacity incur adjustment costs, which are convex in investment size, reflecting the increasing opportunity cost of allocating scarce resources (e.g. labour) to speed up the renewable energy generation capacity (Amigues et al., 2015; Coulomb et al., 2019; Fischer et al., 2004). The model is calibrated to Québec data. Numerical simulations are run for three economic growth scenarios—temporary, slowing or constant growth—and

1_{Natural gas and petroleum accounted for 15.2% and 55.5% of Québec’s GHG emissions in 2017, respectively}

(Whitmore and Pineau, 2020).

2_{Hydro and wind account for 95% and 4.7% of Québec’s electricity, respectively (Whitmore and Pineau,}

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two alternative carbon constraints—a carbon budget (CB) based on Québec’s target and a backloaded price of emission based on Québec’s current cap-and-trade program (SPEDE). I find that the carbon price is the main driver of the energy transition. Under both carbon constraints (CB and SPEDE), the carbon price expressed in current-value terms increases over time. The total price of low-carbon electricity thus progressively becomes cheaper rel-ative to natural gas, and all convertible energy services (i.e. the services provided by both electricity and natural gas) are eventually electrified. In line with Fischer et al. (2004), it is optimal to start investing early in renewables and smooth investments over time to limit the cost of clean generation capacity.

Aggregate electricity consumption and generation capacity grow with the economy. In con-trast, service-specific electricity consumption reflects the tradeoff between allocating new capacity (i) to benefit energy services already provided by electricity or (ii) to avoid costly emissions from services provided by gas. Imposing a stricter carbon constraint favours (ii), resulting in a faster but more costly transition away from gas. Increasing economic growth affects the energy transition asymmetrically under the carbon budget (CB) and the exoge-nous emission price (SPEDE). In the first case, the scarcity (i.e. the price) of emissions “per-mits” increases and the switchover from gas to renewables occurs earlier and more abruptly. In the second case, the emission price is unaffected, and carbon emissions increase with gas consumption.

Regarding the policy implications, I find that, if the carbon price under Québec’s current cap-and-trade program does not depart from its floor value, the electrification of natural gas does suffice to meet Québec’s emission target. To reach Québec’s target, the carbon price needs to increase by 47%, 70% and 111%—depending on whether economic growth stops, slows to 1%, or stays to 2% per year beyond 2060. Though such an increase generates more carbon revenues for the government, it may not entirely cover the additional investment cost of renewable generation capacity. Following a back-of-the-envelope cost-benefit analysis, however, I find that increasing the carbon price might be profitable from a social perspective if energy demand keeps growing beyond 2060.

I identify two main contributions of my methodological approach. First, my results shed some light on the ongoing debate regarding the opportuneness of electrifying natural-gas usages to meet Québec’s emission targets (Breton et al., 2019; CGA, 2019; Croteau, 2019; Dunsky, 2019; Michaud, 2020; Paradis-Michaud, 2020). Paradis-Michaud (2020) identifies how much more power capacity would be needed to electrify all convertible gas usages. However, his approach supposes an instantaneous conversion, omitting the dynamic nature of the energy transition. Although CGA (2019) and Dunsky (2019) both account for this dynamic nature, the first study does not account for the specificity of Québec’s context, while the former does not focus on the electrification of gas. Moreover, no study on electrification

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depicts the economic implications of an “optimal” electrification pathway.

Second, the model adds to the economic literature on capacity constraints and adjustment cost of clean resources in which the models are either not calibrated (Fischer et al., 2004; Amigues et al., 2015) or does not account for the differing cost and efficiency between en-ergy sources depending on the enen-ergy service (Coulomb et al., 2019). In that regard, the model generalizes Coulomb et al. (2019) to a multiple-demand setup—similar to Chakra-vorty et al. (1997)—while neglecting natural gas scarcity. Moreover, the optimal investment and consumption pathways closely resemble the baseline simulation from Coulomb et al. (2019).

The rest of this document is structured as follows. In Chapter 1, I review the literature on energy system models to contextualize the theoretical model with respect to previous modelling efforts. In Chapter 2, I present the theoretical model and the efficiency conditions for an optimal solution. In Chapter 3, I calibrate the model to Québec data and present the alternative carbon constraints and growth scenarios. In Chapter 4, I present the numerical simulations and discuss some of their policy implications. In Chapter 5, I discuss the main limitations of the simulations before concluding.

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Chapter 1

Energy System Models in the

Literature

Most modelling efforts of the energy sector and the optimal energy transition belong to the broad category of energy system models. The first instances of these models are traced back to the late 1950s in a period of rising energy demand from developing OECD countries (Herbst et al., 2012). Models from that period were mostly national and international en-ergy balances aimed at forecasting enen-ergy demand to better plan supply and infrastructures development (Bhattacharyya and Timilsina, 2010; Hoffman and Wood, 1976). In the 1970s, and especially after the first oil crisis of 1973, modellers became increasingly interested in the interaction and interdependence between energy and the economy (Bhattacharyya and Timilsina, 2010). In particular, this period marks the schism between the top-down and the bottom-up modelling approaches. The former draws on macroeconomic mechanisms to evaluate the global effects of new policies and technologies; the latter describes tech-nologies in greater detail to identify optimal technological paths to meet energy demand under alternative policy scenarios (Bhattacharyya and Timilsina, 2010; Herbst et al., 2012). In the 1980s, the focus of energy system models broadened to include the interactions be-tween the environment and energy consumption; it shifted again in the 1990s to encompass climate-change-related issues (Bhattacharyya and Timilsina, 2010). Since then, significant resources have been invested in comparing and combining bottom-up and top-down ap-proaches to produce consistent estimates of optimal energy path while mitigating GES emis-sions (Böhringer and Rutherford, 2009; Fortes et al., 2013; Metz et al., 2007).

Before covering the modelling efforts of energy systems in more detail, I propose a distinc-tion between two approaches: Applied Energy System Models (AESMs) and Analytical Eco-nomic Models (AEMs). Though these approaches are not mutually exclusive per se, this categorization helps to situate the model presented in Chapter 2 concerning previous mod-elling efforts. The AESMs are generally subdivided between bottom-up (Section 1.1.1),

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top-Figure 1.1 – Energy transition modelling approaches

down (Section 1.1.2) and hybrid (Section 1.1.3) models. As for the AEMs, I concentrate on modelling efforts that focus on the energy transition resulting from technological change (Section 1.2.1), or on the switchover to renewables with adjustment costs and capacity con-straints (Section 1.2.2). Figure 1.1 portrays the main characteristics of these concurrent mod-elling approaches and how they relate to each other.

1.1 Applied Energy System Models (AESMs)

I define AESMs as numerical simulations of the production, transport and end-uses of a set of energy resources on a given territory and in a given time frame. In general, these models aim at providing a realistic forecast of either or both the supply and demand of (segments of) the energy market following alternative scenarios (e.g. business as usual, the introduction of a carbon tax or the advent of a technological breakthrough) to assist the policymakers and investors in the energy sector. Studies that present or rely on AESMs are typically published in energy-related and environment-related journals, focusing on economics and engineering, such as Energy Policy and Climate Policy. They are also often conducted by or for (inter-)governmental agencies such as the European Commission and published as reports. Since their introduction in the 1950s, AESMs have proliferated, with each model having its purpose and level of detail. Given this variety, different categorizations of AESMs ex-ist, notably differentiating AESMs following the modelling technique (Hoffman and Wood, 1976), the models’ attributes (Pandey, 2002), the spatial dimension and time horizon (Bhat-tacharyya and Timilsina, 2010; Nakata, 2004) and the level of abstraction and the complete-ness of the models (Pye and Bataille, 2016). Nonetheless, most authors classify the models based on their underlying conceptual framework, which distinguishes between the

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bottom-up (BU) and the top-down (TD) approach (Bhattacharyya and Timilsina, 2010; Böhringer and Rutherford, 2009; Herbst et al., 2012; Nakata, 2004; Fortes et al., 2013).

1.1.1 Bottom-up Models

Pye and Bataille (2016, S33) provides an eloquent definition of BU models.

[Bottom-up models] are focused on energy system technology stocks and how they can change through time. They have very detailed, often economy-wide, linked maps of energy use from supply through to end-use demand, and their operating paradigm is the minimization of the lifecycle costs for specific inter-mediate and end-use energy demands through technology competition, often in response to capital, labour, energy and emissions price changes.

BU models often take the form of optimization or simulation problems that identify the least-cost combination of energy technologies that meet specified energy service demands in a partial equilibrium setting (Fortes et al., 2013; Herbst et al., 2012). Because these mod-els depict technology in great detail, they are appropriate to simulate the market penetra-tion of new technologies and to assess the effects of sector and market-specific policies (i.e. command-and-control policies), e.g. emission standards on energy supply (Böhringer and Rutherford, 2009; Herbst et al., 2012; Pye and Bataille, 2016).

However, because they tend to be partial equilibrium models—i.e. only the energy sector is modelled—, BU models usually omit the effects of new technologies and climate policies on macro-level variables such as the employment rate, prices and overall economic activity (Böhringer and Rutherford, 2009; Fortes et al., 2013). They also tend to neglect the energy market’s second-best characteristics, such as market failures and tax distortions (Böhringer and Rutherford, 2009).

For these reasons, BU models often produce over-optimistic scenarios and underestimate climate policies’ societal costs (Herbst et al., 2012). Moreover, BU models usually depend on the availability of vast quantities of data (e.g. future cost of emerging technologies) and rely on simplistic depictions of economic behaviour (e.g. lifecycle cost-minimization of energy resources, exogenous energy demands) (Pye and Bataille, 2016). These models are generally designed and used by engineers, natural scientists and energy supply companies (Herbst et al., 2012). Of the various existing BU models, the most notable is probably the MARKAL/TIMES framework, as it has been adapted to more than 50 countries with over 150 teams working with it (IEA-ETSAP., 2019).

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MARKAL/TIMES Frameworks

TIMES, and its predecessor the MARKAL framework, have been applied to different regions and scope depending on the analysis (Loulou and Labriet, 2008; Pye and Bataille, 2016). De-veloped and maintained by the Energy Technology Systems Analysis Program, The Inte-grated MARKAL-EFOM System (TIMES) is described as follows:

[A] model generator for local, national or multi-regional energy systems, which provides a technology rich basis for estimating energy dynamics over a long-term, multiple period time horizon. It is usually applied to the analysis of the entire energy sector, but may also applied to study in detail single sectors (e.g., the electricity and district heat sector). (Loulou and Labriet, 2008, p.8)

The TIMES operates in a partial-equilibrium setting that assumes price-elastic energy de-mands, competitive markets and perfect foresight. Its inputs consist of alternative scenar-ios encompassing a set of one-period demand and supply curves and policies (Loulou and Labriet, 2008). The demand curves are exogenous in the TIMES; they can either be defined as fixed quantity per period or as functions of the demand drivers. In the latter case, the fore-cast of these drivers (e.g. population, GDP, sector outputs, price) must be provided by the analyst, which must also specify the elasticity of the energy service demands with respect to each driver (Loulou and Labriet, 2008). Regarding the supply curves, they are endoge-nous in the TIMES framework; they are built by linking available primary energy resources and material, for which supply curves must be specified, to a set of final energy services through a Reference Energy System (REF). As depicted in Figure 1.2 for the production of house heating, the REF is equivalent to a network diagram representing the production line of each energy service. It is composed of three entities: technologies (or processes), com-modities, and commodity flows. The commodities are the primary, intermediary and final goods used in the production of the energy services; the technologies are the nods where the commodities are refined (e.g. from wet gas to dry gas) or converted (e.g. from coal to electricity); these commodities and technologies are connected by commodity flows.

For each entity of the REF (i.e. technologies, commodities and commodity flows), the mod-eller has to define a set of technical parameters (efficiency, availability factor, commodity consumption per unit of activity), economic parameters (costs of investment, maintenance and operation) and policy parameters (upper or lower bounds). The TIMES uses the REF parameters to produce supply curves for each energy service included in the analysis. For each specified scenario, the TIMES translates the demand and supply curves into a linear optimization problem for which the objective function is the total market surplus over the horizon of the analysis. Depending on the scenario at hand, additional policy constraints can be added to the problem, such as emission targets or taxes and permit trading. The

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de-Figure 1.2 – Example of a simple Reference Energy System for home space heating

Source: Loulou and Labriet (2008, p.15)

cision variables of the problem are the investment level in equipment, equipment operation, primary energy supply and trade of each commodity (Loulou and Labriet, 2008, p.8). The solution of the TIMES corresponds to a partial equilibrium of the energy markets where the price of each commodity equals their marginal value Loulou and Labriet (2008, p.21). De-fined per period, the output of the TIMES correspond to the level of investments for each technology; the operation levels of each technology; the extraction levels of each primary energy form and material; the imports and exports of each tradeable commodity; the com-modity flows between each technology; and the emission levels of each substance that can be expressed by technology, sector and in total (Loulou and Labriet, 2008, p.11-12).

TIMES baseline framework can be extended or adjusted in various ways depending on the scope of analysis. The TIAM version of the model enables international and multi-regional analyses. It even includes a climate module, making it possible to evaluate alternative energy policy scenarios’ climate impact. The TIMES can also account for endogenous technologi-cal learning (or learning-by-doing) by defining the investment costs or efficiency parameters of specific technologies as functions of time, cumulative investment or cumulative produc-tion. Moreover, the perfect foresight assumption can be relaxed by adding uncertainty in

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the model by defining some (or all) parameters as random variables.3 (Loulou and Labriet, 2008)

Analysts have relied on MARKAL/TIMES models in various contexts. Börjesson et al. (2014) use it to assess the extent to which biofuels and electricity in the Swedish transportation sector can achieve a cost-efficient reduction in CO2 (Börjesson et al., 2014). Kannan and

Hirschberg (2016) implement it to analyze the interplay between the Sweden electricity and transport sectors and their trade-offs. Wright and Kanudia (2014) calibrate the TIMES to the U.S. in a multi-region, multi-sector setup to compare different energy technology options and policy scenarios while accounting for the interactions between state, regional and federal policies. Lastly, Fortes et al. (2013) use the TIMES to compare the optimal CO2 abatement

strategies between BU and TD approaches for Portugal. Other BU Models

Given the diversity of BU models, I concentrate on two models with distinctive features. The first one, the BENSIM (for Bioenergy Simulation Model), proposed by Millinger et al. (2017), is used to identify the most cost-competitive liquid or gaseous biofuel options from domestic biomass in Germany in the medium to long term. The BENSIM is described as a “recursive dynamic bottom-up least-cost simulation model with endogenous technological learning, seeking the least-cost mix of biofuel production options on a yearly basis for ful-filling a set demand” (Millinger et al., 2017, p.395). The model first maps the existing biofuel plants in the region of focus (i.e. Germany) after removing the plants that have reached the end of their lifetime. It assumes that all biofuels (e.g. synthetic natural gas, biomethane, biomethanol) to be equivalent for the end-users. For a given year, the model calculates the “marginal costs” (MC), the “investment costs” (IC) and the “total costs” (TC) of produc-tion for each technology, all of which are expressed in AC/GJf uel. The TC are the sum of

the MC and the IC.4 _{All technologies are then sorted in the merit order of their marginal}

costs, enabling the estimation of a supply curve that maps the marginal cost of the biofuels onto cumulative biofuel production capacity. The market price of biofuels is the point on that curve where the production capacity equals a given (i.e. exogenous) time-increasing demand. The technologies for which the TC is lower than the market price are subject to ca-pacity ramp-up through investments until their TC reach the market price. In the following year, the technologies that experienced an expansion are subject to a reduction in investment costs because of learning effects by the learning rate for each doubling of capacity. The other technologies are subject to lower, exogenous yearly learning effects. The steps above are

re-3_{With learning-by-doing, the optimization problem is no longer linear and must be solved using}

mixed-integer programming (see Loulou (2008) for more details). With uncertainty, the objective function must be adjusted (e.g. to expected utility) and can even account for risk aversion (Loulou and Labriet, 2008, p.36).

4_{The MC accounts for the costs of the operation and maintenance and the different inputs used in the}

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peated until the model reaches the end of its time horizon. The BENSIM thus produces a projection of the production structure and costs of biofuels.

Spatial techno-economic models offer another interesting approach to BU modelling as the spatial dimension is core to the analysis. Parker et al. (2017) use such a model to estimate a potential supply curve for renewable natural gas (RNG) in California. In short, the au-thors map the sources of RNG (landfills, dairy herds, municipal wastewater and municipal wastes) and estimate each source’s annual RNG generation potential their levelized costs.5 These levelized costs—which include transportation costs and the cost of injecting the RNG into the network—are minimized using a linear programming model that allocates each RNG source to a biogas facility. Thus, the solution identifies the optimal allocation of RNG sources to biogas plants and these plant’s optimal location and size. The potential supply curve that results from this analysis can be disaggregated by RNG sources. It can also be adjusted following different policy scenarios (source’s specific subsidies, feed-in tariffs, etc.). Similar spatial techno-economic models have been used to estimate potential supply curves of biofuels in California (Tittmann et al., 2010), of RNG in Chile (Bidart et al., 2013) and of carbon sequestration in the US (Dooley et al., 2005).

1.1.2 Top-down (TD) Models

Drawing on the Arrow-Debreu paradigm, top-down (or macroeconomic) models adopt a na-tional or regional view of the economy by modelling all producers and end-users through a set of equations (Bhattacharyya and Timilsina, 2010; Böhringer and Rutherford, 2009; Herbst et al., 2012). TD models mostly take the form of computable general equilibrium models (CGE), which compare an initial macroeconomic equilibrium to the new equilibrium reached after the introduction of energy or climate policies.6 Through this comparison, the analyst can determine the effect of these policies on key macroeconomic variables such as employ-ment, growth and net balance (Herbst et al., 2012; Pye and Bataille, 2016).

In typical CGE models, the equilibrium results from the maximization of consumers’ welfare and producers’ profits subject to a set of operational constraints (e.g. a market clearing con-dition and zero windfall profits) (Pye and Bataille, 2016; Herbst et al., 2012). The formulation of the optimization problem varies in complexity. It ranges from a static setup in which the analyst compares the equilibria before and after an exogenous shock, such as a new policy, to a dynamic setting in which time-dependent equations endogenously determine savings and investments. The model can also be dynamic recursive, meaning that savings are passed on through a series of forward-looking one-period static models that determine the investment

5_{Levelized costs of energy are defined as the present value of lifetime costs of an energy source divided by}

its total energy production.

6_{Herbst et al. (2012) distinguish between four types of TD models: input-output models, econometric}

mod-els, system dynamics and computable general equilibrium (CGE) models. Only the latter is covered in this section since it is the most common in the literature.

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levels in each (Pye and Bataille, 2016). In TD models, the consumers are generally charac-terized by a representative quasi-concave utility function increasing in consumption or final goods produced from a mix of capital, labour, energy and materials (or KLEM). Besides la-bor, the inputs are themselves produced from a similar KLEM mix. For each factor and final good, a specified production function determines the mix of inputs used to produce one unit; at the equilibrium, the mix is usually set such that the marginal productivity of each factor equals its marginal cost.

As a result, TD models can account for feedback loops between each market that result from a policy-induced change in relative prices and income: depending on the elasticity of substi-tution between each factor, a change in relative prices will translate into a change in relative usage (Böhringer and Rutherford, 2009; Fortes et al., 2013; Herbst et al., 2012). Accordingly, TD models generally provide consistent estimations of the economy-wide impacts of en-ergy policies, such as emission taxes and trading schemes and feed-in tariffs, regarding the macroeconomic theory (Böhringer and Rutherford, 2009; Herbst et al., 2012). While the inclu-sion of feedback loops is considered a key strength of TD models, they also cause TD models to produce more conservative cost estimations of environmental and energy policies than BU models; the effects of changes in the relative prices having repercussions throughout the economy (Fortes et al., 2013).

One of the TD models’ limitations comes from their broad focus, which prevents them from providing a detailed portrait of technology and its evolution. As a result, TD models are con-sidered unsuited to characterize the evolution of discrete technologies or evaluate the impact of technology-specific policies such as emission standards (Böhringer and Rutherford, 2009; Herbst et al., 2012). Moreover, because they usually rely on the assumptions of perfect infor-mation and efficient market allocation, TD models tend to underestimate the non-monetary barriers to energy efficiency and sector-specific policies (Herbst et al., 2012). In contrast to BU models, TD models are usually designed by economists and employed by economists and public administrators (Herbst et al., 2012). Of the various instances of TD models, the GEM-E3 is one of the most commonly used. Other examples include the GTAP-E, the PACE and the Intertemporal General Equilibrium Model (IGEM).

The GEM-E3 Model

The GEM-E3 has been developed as an international collaboration project partly funded by the European Commission. Coordinated by the National Technical University of Athens, its development involves more than ten research teams.7

7_{These teams include the National Technical University of Athens (NTUA/E3M-LAB), Katholieke}

Univer-siteit of Leuven (KUL), University of Manheim and the centre for European Economic Research (ZEW), Ecole Centrale de Paris (ERASME), PSI, University of Toulouse (IDEI), Stockholm School of Economics, CORE, CEA and University of Strathclyde (Capros et al., 2013).

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The GEM-E3 is defined as a “multi-regional, multi-sectoral, recursive dynamic computable general equilibrium model which provides details on the macro-economy and its interaction with the environment and the energy system” (Capros et al., 2013, p.13). The model includes all European and major non-European countries (e.g. USA, China) separately; the remaining countries are aggregated into regions. Thus, it is particularly suited to evaluate the impact of environmental policies in Europe or specific European countries (Capros et al., 2014). More specifically, the GEM-E3 enables assessing the distributional effects of policies on national accounts, investment, consumption, public finance, foreign trade and employment per eco-nomic sectors and agents across countries or regions (Capros et al., 2014). The model pro-ceeds by first calibrating Social Accounting Matrices (SAM) for each country/region to base year data (Capros et al., 2013; Herbst et al., 2012). Following the Walras law, the GEM-E3 then computes the prices of goods, services, labour and capital that simultaneously clear all markets through demand and supply interactions in a perfectly competitive setup (Capros et al., 2014). Similar to other recursive dynamic models, the GEM-E3 is sequentially solved based on the current capital stock, the endogenous motion of the capital linking each succes-sive equilibria.

To identify each period’s equilibrium, the GEM-E3 formulates the behaviour of the house-holds, the firms, and the governments separately. The households are modelled by country-specific myopic representative consumers that seek to maximize their inter-temporal utility subject to an inter-temporal budget constraint. Each period, the household first decides its current consumption in goods and leisure (which determines its labour supply) and its sav-ings (which determine the amount of capital available in the next period) based on its avail-able disposavail-able income. The household’s income sources are its wage from labour, interest payments from assets, share in the firm’s profits, and cash transfers. Once the household has chosen its current consumption, it chooses its joint consumption of durable goods (e.g. cars, heating systems, electric appliances) and non-durable goods (composed of different energy resources and consumables such as food and clothing).

The firms are assumed to maximize profits, i.e. the difference between their revenues and the cost of the production factors and intermediary goods, subject to their production tech-nology. The production is defined by branch, each branch producing a differentiated good (e.g. agriculture, electric goods). For each good, the production function has a typical CES functional form with learning-by-doing effects and taking capital, labour, energy and mate-rial as inputs. The branch-specific elasticities of substitution between each input are constant through time.8 With the zero-profit condition, the optimal firm behaviour is such that each period the firm decides its supply of goods given the inputs’ prices and its selling price (Capros et al., 2013, p.41). The firms can only increase their capital stock for the next pe-riod through current investment, the demand of which depends on the rental price of capital

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Figure 1.3 – GEM-E3 economic circuit

Source: Capros et al. (2013, p.16)

and its replacement cost (Capros et al., 2013). That is, investments at t increase capital and production capacity at t+1.

As for the government, its behaviour is exogenous to the model. It can consume goods, make public investments, do cash transfers and levy taxes, the level of which are all deter-mined exogenously.9 In each country/region, the demand for products by the households, firms (for investment and intermediary goods) and the government are aggregated into the total domestic demand. The latter is allocated between domestic and imported products fol-lowing the Armington specification. Thus, the mix of imported and domestically produced goods is chosen by the buyers in each country, making the bilateral trade flows between each country/region endogenous to the model (Capros et al., 2013, p.58). For illustration, Figure 1.3 depicts a diagram of GEM-E3’s structure.

For policy evaluation, the analyst must first define the reference case scenario based on current policy parameters and solve the model over the analysis horizon. This reference scenario can then be compared with “counterfactual simulations” by altering the values of selected exogenous policy parameters and solving the model for these new values (Capros et al., 2013). The different scenarios can then be compared with the following model’s output:

9_{The GEM-3E differentiates between 9 categories of receipts: indirect taxes, environmental taxes, direct taxes,}

value-added taxes, production subsidies, social security contributions, import duties, foreign transfers and gov-ernment firms (Capros et al., 2013, p.56).

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dynamic annual projections in volume, value and deflators of national accounts by coun-try; full Input-Output tables for each country/region identified in the model; distribution of income and transfers in the form of a social accounting matrix by country; employment, capital, investment by country and sector; greenhouse gasses, atmospheric emissions, pollu-tion abatement capital, purchase of pollupollu-tion permits and damages10; consumption matrix by product and investment matrix by ownership branch; public finance, tax incidence and revenues by country; full bilateral trade matrices (Capros et al., 2013, p.17-18).

Since its development in 1985, the GEM-E3 has been used frequently for analyses carried out for the Directorate General of the European Commission or national authorities (Capros et al., 2013). In the recent academic literature, Karkatsoulis et al. (2017) use the GEM-E3 to simulate the transition towards low-carbon transport in the EU from 2020 to 2050 and as-sess the macroeconomic and sectorial impacts of such a transition. Fragkos et al. (2017) rely on it to assess the macroeconomic implications of the EU Intended Nationally Determined Contribution (EU INDC) for 2030 as submitted to the UNFCCC and EU’s decarbonization targets for 2050. Ciscar et al. (2013) use it to analyze the outcomes of various allocations of GHG emissions across countries following different low-carbon scenarios. Recently, the base module of the GEM-E3 has been complemented to include a detailed bottom-up rep-resentation of the power generation technologies, making it a hybrid energy system model. Moreover, addressing Herbst et al. (2012), the latest version of the GEM-E3 has been ex-tended to include endogenous energy efficiency investments undertaken by the firm of each branch and the representative household (Capros et al., 2013, p.137).

1.1.3 Hybrid Models

To overcome the BU and TD approaches’ limitations mentioned above, significant efforts have been invested in developing hybrid energy system models in the last years. These models integrate at varying degrees elements from both BU and TD approaches. The first degree of integration consists of linking a BU model with a TD model through transfers of data, parameters and estimates. Such transfers can be either done in a manual procedure called a “soft-link”, or through an automatic process called a “hard-link”. “Soft-linking” is notably done in Fortes et al. (2013) where the authors calibrate the Portuguese version of the TIMES and the GEM-E3 to a common baseline scenario to compare models’ GHG abatement strategies under alternative low-carbon scenarios. The second degree of integration takes a model from one approach and complements it with a reduced form of a model from the other approach. With its BU representation of power generation technologies, the latest version of the GEM-E3 achieves such second-degree integration (Capros et al., 2013). A third and more complete form of integration is to combine both BU and TD approaches directly through the specification of market equilibrium models as mixed complementarity problems (Bohringer

10_{The emissions are linked to the activity level of each polluting sector. For more detail, see Capros et al.}

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and Rutherford, 2008; Böhringer and Rutherford, 2009; Fortes et al., 2013; Herbst et al., 2012). This more recent approach is developed in Bohringer and Rutherford (2008) and Böhringer and Rutherford (2009); however, it remains to be applied to specific policy analyses.11

1.2 Analytical Economic Models (AEMs)

In this review, I define the AEMs as stylized mathematical models that seek to characterize the economy’s evolution under an energy transition, be it in a partial or general equilibrium setup. Compared to AESMs, AEMs focus less on yielding precise forecasts, but instead con-centrate on identifying theoretic insights regarding the energy transition dynamics—in some instances (e.g. Amigues et al. (2015) and Fischer et al. (2004)), the model is not calibrated with data, but only solved analytically.12

Studies drawing on AEMs are typically published as working papers or as articles in aca-demic journals specialized in economics (e.g. American Economic Review, Journal of Political Economy, Environmental and Resource Economics, Energy Economics). As defined above, AEMs have not been the subject of a systematic review nor a comprehensive categorization effort, making it hard to appreciate existing modelling efforts’ depth and width. However, given this review’s purpose is to contextualize my methodological approach concerning previous modelling efforts, I focus on two branches of AEMs: endogenous substitution among energy resources resulting from technological change and energy transition with adjustment costs and capacity constraint.

1.2.1 Endogenous Substitution, Technological Change and the Environment “Endogenous substitution” refers to the switchover of energy consumption from one source to another—generally from a polluting resource to renewables—resulting from changes in energy costs. Accordingly, the models falling in this category aim to characterize how tech-nological change (TC) could induce the energy transition away from fossil fuels, ultimately impacting the climate through lower emissions.

In this line of models, Chakravorty et al. (1997) bears the most similarities to my model. In brief, this deterministic, partial equilibrium model of the global energy market analyzes the

11_{Some authors, such as Pye and Bataille (2016), consider integrated assessment models (IAMs) as variants of}

the hybrid models. However, these models generally have a global focus, making them unsuited for analyzing the energy sector at a national level. Instances of IAMs include Nordhaus’ 2013 DICE/RICE models, Tol’s 2020 FUND model and Hope’s 2011 PAGE model, all of which have been notably used to evaluate the social cost of carbon (Metcalf and Stock, 2017).

12_{Note that other categorizations of energy-economic models have been proposed. Notably, Popp et al. (2010,}

p.39) denote under Aggregate Energy-Environment Models (AEEMs) models which “assess the role of technological change on long-term well-being”. While AEMs and AEEMs are not mutually exclusive model categories per se, the AEEMs presented in Popp et al. (2010) exclude the models that are not implemented numerically or focus on other characteristics of the energy transition (e.g. adjustment costs and capacity constraints).

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fossil fuel extraction path, renewables’ rate of adoption, and global warming under alter-native technological change and carbon tax scenarios. It uses a multiple-energy-demands and multiple-energy-resources framework. The energy resources are distinguished between cheap and exhaustible fossil fuels (i.e. oil, coal and gas), which emit carbon emissions, and more expensive but unlimited renewable energy (i.e. solar energy). The economic model is complemented with a simplified model of the climate, making it possible to estimate fossil fuel consumption’s effect on climate change and world GDP.

A particularity of the model is that it accounts for the cost and efficiency of converting a spe-cific resource into a spespe-cific demand (e.g. solar energy into transportation); the energy de-mands and consumption prices are thus expressed in “delivered” energy unit. The model’s solution is given by the energy consumption path that maximizes the total consumer surplus of energy net of the extraction and conversion costs over the simulation’s horizon. Calibrat-ing their model to world data, the authors analyze how different carbon tax regimes and exogenous technological change pathways impact the energy transition.

The economic literature on technology and climate change has evolved considerably since Chakravorty et al. (1997). Notably, climate-economic models with endogenous technologi-cal change (ETC) have gained in popularity and elaboration. In this branch of the literature, it is acknowledged that alternative energy resources’ cost and efficiency depend not only on time, but also on past, present and future expected prices and policy. Generally, TC un-der ETC models is induced by either of the following mechanisms: relative prices, learning and research and development (R&D) (Gillingham et al., 2008). The intuition behind price-induced TC is that a change in relative prices stimulates the innovation towards cost reduc-tion of the more expensive producreduc-tion inputs (e.g. renewable energy). Learning-induced TC instead implies that the cost of a specific input/output depends on prior experience with it; for example, its cumulative adoption/production in the case of learning-by-doing. Finally, TC is R&D-induced when the R&D investments in each technology determine its speed and direction. ETC in climate-economic has been extensively reviewed in Gillingham et al. (2008) and in Popp et al. (2010).

In the line of AEMs with ETC, work by Acemoglu et al. (2012, 2016) has attracted substantial attention. In these two endogenous growth models, Acemoglu et al. represent the econ-omy as a two-sector model in which clean and dirty technologies—which are either com-plements (2012) or substitutes (2016)—are used to produce a final good. The cost of each technology depends on their accumulated “stock” of knowledge, which can be increased by profit-maximizing researchers. Using a simple quantitative example (Acemoglu et al., 2012) and an elaborate micro-founded calibration (Acemoglu et al., 2016), the authors show that under laissez-faire the dirty technology remains dominant in the production of the final good, resulting in a considerable loss of well-being, either from an environmental disaster (2012) or the adverse effect of atmospheric pollution on production (2016). The optimal governmental

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strategy is to implement early-on subsidies for R& D in clean technology and a back-loaded carbon tax to avoid such loss. Subsequently, the models from Acemoglu et al. (2012, 2016) have been extended in Lemoine (2017), who shows that the orientation of R&D and the sub-sequent adoption of renewable energy depends to a large extent on whether machines and energy resources are complements or substitutes. Through the calibrated numerical imple-mentation of his model, and contrarily to Acemoglu et al. (2012, 2016), Lemoine (2017) finds that a transition to renewable energy resources is conceivable under laissez-faire. However, and echoing Acemoglu et al. (2012, 2016), he finds that it is optimal for the government to im-plement early-on R&D subsidies to permanently redirect innovation towards low-emission resources, to accelerate the energy transition and to avoid costly environmental degradation. 1.2.2 Adjustment Costs and Capacity Constraints

While TC undoubtedly determines the extent to which low-emission energy resources will be adopted in the future, such adoption will also largely depend on the adjustment of capital stocks upon which the economic processes depend. Even if TC made renewables substan-tially cheaper than fossil fuels, renewable production capacity first needs to be increased substantially before the energy transition can happen.

Fischer et al. (2004) sets the theoretical basis of this branch of the literature. In their article, the authors develop a dynamic, general equilibrium model. A uniform good can be produced by either a polluting or clean technology, with production from the clean technology being more expensive. The initial clean technology stock is limited but can be increased through investments in labour. The labour cost of these investments is given by a positive, strictly increasing convex function. Production using dirty technology generates emissions that add to a pollution stock, a portion of which is dissipated at each moment. This pollution stock generates environmental damages. The labour cost of investment and the environmental damage are both given by convex functions, which are positive and strictly increasing with their derivative around zero set to zero.13 _{The solution of the model is given by the clean}

and dirty consumption and the investment trajectories that maximize over an infinite time horizon the discounted utility of consumption, net of production, investment and pollution costs. Using this simple analytic model, Fischer et al. (2004) find that the optimal transition largely depends on whether the environment is initially clean or dirty. In the first case, the marginal cost of pollution is initially low and clean technology is not instantaneously adopted. Nevertheless, there will always be current investments in clean capacity to take advantage of the negligible costs at low capacity installation levels.14 In that scenario, if the depreciation rate is sufficiently low, there could be an over-shooting in the shadow price

13_{Fischer et al. (2004) shows that this implies that installation cost of the first unit of new capacity is negligible.}

14_{Fischer et al. (2004) show that “the existence of any positive level of investment at any time in the future}

along the optimal path will trigger some current investment” (p.336-7), resulting from the derivative of the investment cost function being set to zero when the investments are null (see note 13).

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of carbon (i.e. its value may momentarily surpass its steady-state level) resulting from the smoothing of investment costs over time. In the second case, clean technology is directly adopted and increases rapidly, possibly over-shooting its long-term level.

Using a similar dynamic model, but within a partial equilibrium setting where dirty re-sources are exhaustible and without accounting for pollution, Amigues et al. (2015) find that it is optimal to invest in renewables before exhausting dirty resources. Also, if dirty resources are sufficiently abundant, the equilibrium path toward a steady state is composed of three stages: the exclusive usage of dirty resources followed by their progressive substitution until they are depleted and only renewables are used. Contrary to Fischer et al. (2004), it may be optimal to wait before investing in new capacity depending on the marginal cost of the first unit of clean capacity.

In the same vein, Coulomb et al. (2019) develop a partial equilibrium representation of the electricity market in which electricity can be produced using either fossil fuels or renewables. However, Coulomb et al. (2019) expand previous models by incorporating three energy re-sources: coal, which is dirty but cheap and abundant; natural gas, which is cleaner but scarce; and a renewable but more expensive clean resource (solar or wind). Coal and gas consump-tion emits CO2that accumulates in the environment until cumulative emissions reach an

ex-ogenous carbon budget. The authors suppose that coal capacity is over-abundant (i.e. coal capacity exceeds allowed cumulative production under the carbon budget), that initial gas capacity is positive but finite, and that initial clean capacity is null. As in Fischer et al. (2004) and Amigues et al. (2015), the investment cost functions are convex in investment size. This model’s solution is given by the investment and consumption paths that maximize, over an infinite time horizon, the actualized electricity consumption surplus net of investment costs. Coulomb et al. (2019) identify two types of energy transition: sequential and an overlapping transition. In a sequential transition, it is optimal to abandon coal completely before in-vesting in renewables. In the overlapping transition, investments in renewables start before coal’s phasing-out, either before or after investments in the gas capacity started. In any case, the specification of a carbon budget makes the transition towards a renewables-only phase inevitable. However, to reach that phase, it may be profitable to invest in gas to re-duce short-term emissions temporarily, thus smoothing costly investments in renewables over time. With these results at hand, the authors implement a numerical simulation of their model calibrated to European data. The resulting investment path corresponds to the overlapping type, with coal consumption being instantaneously replaced by gas consumption while clean capacity rapidly increases.

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Chapter 2

The Theoretical Energy Transition

Model

The following model relates more closely to AEMs modelling efforts presented in the pre-vious chapter than to AESMs. It aims at characterizing the switchover from gas to electric-ity rather than predicting with precision the evolution of the energy sector under alterna-tive electrification policies—in fact, the formulation and notation are mostly adapted from Chakravorty et al. (1997), and Coulomb et al. (2019).

The model is purposely kept simple. This simplicity enables the identification and interpre-tation of the efficiency conditions guiding the “optimal” electrification pathway, but it also facilitates the calibration and numerical solving of the model. Hence, the analysis of the op-timal solutions (presented in Chapter 4) is relatively straightforward. The energy demand is nevertheless calibrated with some degree of technological detail for greater accuracy as in bottom-up AESMs.

2.1 Baseline Formulation

I assume that I energy sources can be used within J energy services. Each energy service is associated with a specific demand. Expressing the demands in terms of energy services presents many advantages. First, it reflects the fact that energy is not consumed for itself, but for the services it produces (Hunt and Ryan, 2015). Second, it provides a more rep-resentative depiction of energy demand since it explicitly accounts for the effect of energy efficiency on price elasticity. In response to a surge in energy prices, consumers may invest in energy efficiency in addition to adopting energy-saving behaviours (e.g. taking shorter showers). Moreover, some energy sources are better substitutes than others for some end-uses, depending on the state of the technology (e.g. natural gas and electricity are better sub-stitutes for residential heating than household appliances). Using energy service demands

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instead of aggregate energy demands allows for an explicit depiction of the evolution of the cost and efficiency of converting specific energy sources into energy services. Similar to Chakravorty and Krulce (1994), I assume the demand for energy service j to be given by a positive, bounded, continuous, strictly decreasing function of effective energy prices, Dj(·),

with R₀∞Dj(p)dp < ∞.15 The net quantity of energy source i delivered for the provision of

service j is given by dij(t) =qij(t) ×vij(t), where qij(t)denotes the quantity of energy i used

for service j at time t with qij(t) ≥0, ∀i, j, t, and vij(t)is the efficiency of converting energy i

into service j given exogenously.

The total consumption of energy source i at t given by Qi(t) = ∑jqij(t)cannot exceed

in-stalled distribution capacity ki(t):

Qi(t) ≤ki(t), ∀i, t. (2.1)

The distribution capacities account for the physical limitations of the distribution networks. For example, the distribution of gas is limited by the maximum pressure a pipeline can withstand. Likewise, power generation facilities and transmission lines are limited in terms of their power capacity.

For each energy source, the distribution capacity ki(t) can be increased through costly

in-vestments in the network given by xi(t) ≥0. The positivity constraint on xi(t)implies that

investment are irreversible; i.e. it is not possible dismantle installed capacity to retrieve the capital costs (Coulomb et al., 2019).16 The investments xi(t) add to the installed capacity

which depreciates at a rate δ. For simplicity, I assume the depreciation rate to be constant and similar for all networks. Thus, the instantaneous variation of the distribution capacity is given by:

˙ki(t) =xi(t) −δki(t). (2.2)

We define three types of costs: investment costs, variable distribution costs and conversion costs. As in Fischer et al. (2004) and Coulomb et al. (2019), I assume the investment costs to be positive, increasing and convex functions of investment flows:

Cn

i =Cin(xi(t)) >0, Cin0(xi(t)) ≥0, Cin00(xi(t)) ≥0 and

C_in(0) =0, Cn_i0(0) >0. (2.3) These conditions reflect the increasing opportunity cost of using scarce resources—e.g. labour, capital and material—to increase the distribution capacities faster.

The variable costs account for the cost of distributing more energy through each network, regardless their distribution capacities. In Quebec’s context, these costs notably include the

15_{These restrictions on the functional form of D}

j(·) ensure that the surplus accruing from energy service

consumption is finite and, therefore, a maximum exists (Chakravorty and Krulce, 1994).

16_{As stated by (Coulomb et al., 2019), this correspond to a putty-clay capital formation, as opposed to a}

putty-putty process. For further readings on putty-putty vs putty-clay capital formation in the context of energy consumption, see Atkeson and Kehoe (1999).

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procurement cost of the natural gas produced from western Canada and northeastern United States or the opportunity cost of consuming the electricity produced locally instead of ex-porting it. These variable costs are given by continuous, positive and increasing functions of the quantities of energy distributed by each network i and time t:

C_iv =Cv_i(Qi(t)) >0, Civ0(Qi(t)) ≥0, and

C_iv(0) =0, Cv_i0(0) >0. (2.4)

The third type of costs is the conversion costs. These costs account for the capital, mainte-nance, and replacement costs of the consumers’ equipment to convert energy source i into service j at t (e.g. residential gas-fired furnaces and water boilers). These energy-use specific costs are given by zij(t).

Finally, each energy source emits fi tons of CO2 equivalent (tCO2eq) per unit distributed

where the emission factor fi is assumed exogenous.17 Let m(t)be the cumulative emissions

over the horizon of the analysis and ˙m(t) be the emission flow at t. With an exogenous carbon budget ¯m, we have the following emission constraint:18

˙

m(t) =

∑

i

fiQi(t), (2.5)

m(t) ≤m.¯ ∀t. (2.6)

2.2 The Social Planner’s Problem

The social planner seeks to maximise the discounted sum of the net surplus associated with each energy service:

max {qij(t),xi(t)} Z ∞ 0 e −ρt "

∑

j Z ∑_iq_ij(t)v_ij(t) 0 D −1 j (θ)dθ −

∑

i Cn_i(xi(t)) +Civ(Qi(t)) +

∑

j qij(t)zij(t) # dt (2.7)

17_{It could be more realistic to define each f}

ias endogenous variables. For instance, the natural gas utility

could invest in carbon capture and storage technologies resulting in a lower fi.

18_{In order to represent the year-specific emission targets set by public deciders in most countries, a more}

representative emission constraint would be to limit emission levels following a “descending” staircase pattern. For instance, from 2030 onward Quebec’s annual emissions must be 37.5% lower than their were in 1990; after 2050, that target is set to 80–90%. With ¯M(t)the emission cap at t, the emission constraint would be:

m(t) ≤M¯(t) with M¯(t) =      ¯ M1 for t<T1 ¯ M2 for T1≤t<T2 ¯ M3 for t≥T2 and M¯1≥M¯2≥M¯3.

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subject to ˙ki(t) =xi(t) −δki(t), (νi(t)) Qi(t) ≤ki(t), (γi(t)) qij(t) ≥0, (λij(t)) xi(t) ≥0, (ξi(t)) ˙ m(t) =

∑

i fi(t)Qi(t), (µ(t)) m(t) ≤m,¯ (η(t))

where ρ is a constant social discount rate and the Greek letters are the Lagrangian multipliers associated with each constraint. The Lagrangian multipliers are given a similar interpreta-tion as in Coulomb et al. (2019); notably, νi(t)is the marginal value of distribution capacity i

at t, which is equal to the sum of the future social costs of the capacity constraint evaluated in present value at t (as shown in Section 2.3.3); γi(t)is the social cost of distribution capacity

constraint i at t; η(t)is the social cost of the carbon budget at t; and µ(t)is the shadow price of carbon at t, which is equal to the social cost of the emission budget evaluated in present value at t (as shown in Section 2.3.1).

Adapted from Chakravorty et al. (1997), the formulation of the objective function does not exclude cases where some energy services can only be provided by some energy sources— for instance, high-temperature ovens used in some industrial processes which can only work with natural gas. Let j0be an energy service that can only be provided by a set of compatible energy sources defined by I0 where I0 ⊂ I. I account for the incompatibility between other energy sources and service j0by imposing vij0(t) =0, ∀i /∈ I0.

2.3 Efficiency Conditions

From (2.7), I derive the following current value Hamiltonian (we drop the time argument to simplify notation): H(q_ij, xi, ki, νi, µ) =

∑

j Z ∑_iq_ijv_ij 0 D −1 j (θ)dθ−

∑

i h C_in xi +C_iv Qi +

∑

j qijzij i +

∑

i νi xi−δki +µ

∑

i fiQi. (2.8)

Accounting for the inequality constraints given in (2.7), the optimization problem yields the following Lagrangian:

L(qij, xi, ki, m, λij, νi, γi, ξi, η, µ) = H +γi ki−Qi

+λijqij+ξixi+η m¯ −m. (2.9)

We define the social value of energy i for service j as pij = D−j 1 qijvij. The following

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conditions for static efficiency are: ∂L ∂qij =0 ⇐⇒ pij = Cv_i0(qij) +zij−µ fi+γi−λij vij , (2.10) ∂L ∂xi =0 ⇐⇒ C_in0 =νi+ξi. (2.11)

The necessary conditions for dynamic efficiency are:

− ∂L

∂ki(t)

=δνi(t) −γi(t) = ˙νi(t) −ρνi(t) ⇐⇒ −γi(t) = ˙νi(t) − (ρ+δ)νi(t), (2.12)

− ∂L

∂m(t) =η(t) = ˙µ(t) −ρµ(t), (2.13)

where ˙νi(t)and ˙µ(t)are the instantaneous variations of νi and µ at t, respectively.

The solution must also satisfy the laws of motion:

∂L ∂νi

= ˙k_i =xi−δki, (2.14)

∂L

∂µ =m˙ =

∑

_i fiQi, (2.15)

and the slackness conditions:

∂L ∂γi =ki−Qi ≥0, ∂L ∂γi γi =0, γi ≥0, (2.16) ∂L ∂λij =qij ≥0, ∂L ∂λij λij =0, λij ≥0, (2.17) ∂L ∂ξi = xi ≥0, ∂L ∂ξi ξi =0, ξi ≥0, (2.18) ∂L ∂η =m¯ −m≥0, ∂L ∂ηη=0, η≥0. (2.19)

Finally, we have the following transversality conditions: lim t→∞e −ρt ν_i(t) ≤0, lim t→∞e −ρt ν_i(t)k_i(t) =0, (2.20) lim t→∞e −ρt µi(t) ≥0, lim t→∞e −ρt µi(t) m¯ −m(t) =0. (2.21)

Transversality condition (2.20) ensures that the marginal value of remaining distribution ca-pacity at the end of time evaluated in present value is finite.19

The transversality condition (2.21) ensures that if the carbon budget has not been depleted at the end of time, its social value in present terms is equal to zero. Alternatively, if the carbon budget is depleted when t tends to infinity (limt→∞m¯ −m(t) = 0), society may associate a positive value to an increase in the carbon budget.

19_{For k}

i(t) to be positive as t tends to infinity, (2.14) implies that xi(t)must also be positive. As demonstrated in Section 2.3.2, the marginal value of capacity i must then be equal to its marginal cost.

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2.3.1 Energy Prices

From (2.10) and (2.17), we have: pij ≤

Cv

i0+zij−µ fi+γi

vij

, (2.22)

with a strict equality sign when qij > 0. This means that for all positive values of qij, the

marginal value of energy i for service j must be equal to its effective marginal cost which corresponds to the full marginal cost adjusted for vij. The latter is composed of the marginal

cost of distribution C_iv0; a conversion cost zij; the shadow price of pollution µ fi, which is set

to zero for non-emitting energy sources; and the capacity rent γi, which is also set to zero if

current capacity i is non-binding. Because higher emissions decrease the objective function by increasing the stringency of the carbon budget, we have µ ≤ 0. If the marginal value of energy i for service j is lower than its effective marginal cost, then static efficiency requires that qij =0.

2.3.2 Investments

From (2.11) and (2.18), we have:

νi ≤ Cin0(xi), (2.23)

with a strict equality sign when xi > 0. Hence, it is profitable to invest in network i as long

as the marginal value of capacity i equals its marginal cost. If νi ≤ Cn_i0(0), the initial cost of

investment is greater that its marginal value; because Cn_i(xi)is convex, any positive amount

of investment into network i is inefficient. Alternatively, if νi > Cin0(0), it will be profitable

to increase xi until νi =C_in0(xi).

2.3.3 Current Value of Distribution Capacity

Differential equation (2.12) links the marginal cost of the capacity constraint γi(t) to the

dynamics of the marginal value of capacity νi(t). From (2.12), we can express νi(t)in terms

of γi(t). To do so, I first multiply both sides of the equation by the integration factor e−(ρ+δ)t:

˙νi(t)e−(ρ+δ)t−νi(t)(ρ+δ)e−(ρ+δ)t = −γi(t)e−(ρ+δ)t. (2.24) Integrating (2.24), we have: νi(τ)e−(ρ+δ)τ ∞ t = lim T→∞νi(t)e −(ρ+δ)T₋ νi(t)e−(ρ+δ)t = − Z ∞ t γi (τ)e−(ρ+δ)τdτ.

Isolating νi(t)and from (2.20) we get: νi(t) =

Z ∞

t γi

(35)

Hence, for the capital accumulation path to be optimal, the current marginal value of capac-ity νi(t)must always be equal to the sum of the future social costs of the capacity constraint

in present value. Here, δ is added to the discount rate ρ. Thus, the capacity constraint’s social cost will be discounted at a greater rate, including the depreciation rate, which allows accounting for the finite lifetime of capacity.

2.3.4 Carbon Price

Solving in the same way for the differential equation as in Section 2.3.3, we can express the shadow cost of carbon µ(t)in terms of the social cost of the carbon budget η(t). I first multiply both sides of (2.13) by the integration factor e−ρt_:

˙µ(t)e−ρt₋

ρµ(t)e−ρt =η(t)e−ρt. (2.26)

Integrating both sides of (2.26), we get:

µ(τ)e−ρτ ∞ t = lim T→∞µ(T)e −ρT₋ µ(t)e−ρt = Z ∞ t η (τ)e−ρτdτ. (2.27)

Isolating µ(t)and from (2.21), we get the following expression for µ(t):

µ(t) = −

Z ∞

t

η(τ)e−ρ(τ−t)dτ. (2.28)

Thus, on the optimal emission path, the current shadow price of carbon is always equal to the social cost of the remaining carbon budget in present value. Moreover, in conjunction with (2.19), (2.21) implies that as long as the carbon budget has not been depleted, the carbon price in current-value terms grows at the social discount rate ρ such that: