WPS6985 Policy Research Working Paper 6985 Optimal Transition from Coal to Gas and Renewable Power under Capacity Constraints and Adjustment Costs Oskar Lecuyer Adrien Vogt-Schilb Climate Change Group Office of the Chief Economist July 2014 Policy Research Working Paper 6985 Abstract This paper studies the optimal transition from existing costs. Gas plants may be used to reduce short-term invest- coal power plants to gas and renewable power under a ment in renewable power and associated costs, but must carbon budget. It solves a model of polluting, exhaustible eventually be phased out to allow room for carbon-free resources with capacity constraints and adjustment costs power. One risk for myopic agents comparing gas and (to build coal, gas, and renewable power plants). It finds renewable investment is thus to overestimate the lifetime that optimal investment in renewable energy may start of gas plants—e.g., when computing the levelized cost of before coal power has been phased out and even before electricity—and be biased against renewable power. These investment in gas has started, because doing so allows for analytical results are quantified with numerical simula- smoothing investment over time and reduces adjustment tions of the European Commission’s 2050 energy roadmap. This paper is a product of the Office of the Chief Economist, Climate Change Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at oskar.lecuyer@vwi.unibe.ch and avogtschilb@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Optimal Transition from Coal to Gas and Renewable Power under Capacity Constraints and Adjustment Costs Oskar Lecuyer 1 , Adrien Vogt-Schilb 2, 3 1 Department of Economics and Oeschger Centre for Climate Change Research, University of Bern, Bern, Switzerland 2 The World Bank, Climate Change Group, Washington D.C., USA 3 CIRED, Nogent-sur-Marne, France Keywords: climate change mitigation; path dependence; optimal timing; investment; resource extraction; dynamic efficiency; early-scrapping; JEL classification: Q54, Q58 Many governments aim at stabilizing climate change to avoid important cli- mate damages, which requires reaching near-zero greenhouse gas (GHG) emis- sions in the long term (Collins et al., 2013; Steinacher et al., 2013). Abating GHG emissions from power generation is key to reach this goal, as the power sector is currently responsible for nearly 40% of carbon emissions worldwide, and fuel switching to clean electricity is a major technical option to reduce emissions from other sectors (Williams et al., 2012; Kriegler et al., 2014). Two important features of the electricity sector are that (i) it depends on long-lived capital that in general is tied to a specific fuel, and (ii) today’s power production greatly relies on polluting fossil fuels such as coal (IEA, 2014). Sev- eral alternatives are thus available to abate GHG emissions from electricity production. Emissions may be reduced by replacing coal plants with new gas power plants,1 or with more-expensive but almost-carbon-free options such as renewable power. Also, decision makers can either wait for existing plants to reach their natural lifetime, or decide to decommission them earlier in order to switch faster to cleaner energy sources. Email addresses: oskar.lecuyer@vwi.unibe.ch (Oskar Lecuyer), avogtschilb@worldbank.org (Adrien Vogt-Schilb) We thank Mook Bangalore, C´ eline Guivarch, Louis-Ga¨ etan Giraudet, St´ ephane Halle- gatte, Jean-Charles Hourcade, Guy Meunier, Jean-Pierre Ponssard, Antonin Pottier, Philippe Quirion, Julie Rozenberg, Fran¸ e, Ralph Winkler and one anonymous referee from cois Salani´ the FAERE Working Paper series for useful comments and suggestions on earlier versions of this paper. We are responsible for all remaining errors. We are grateful to Patrice Dumas for technical support, and to Institut pour la Mobilit´ ´ e Durable (Renault and ParisTech), Ecole des Ponts ParisTech and EDF R&D for financial support. The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reflect the views of the World Bank, its executive directors, or the countries they represent. 1 This paper assumes gas is a low-carbon substitute for coal. The relative carbon content of gas and coal may actually depend on the type of gas and coal and on the particular processes used for extracting and transporting the fuels (e.g., Alvarez et al., 2012). The relative merits of coal and gas also depend on factors disregarded here, such as impact on energy security or impact on local pollution. This paper analyzes the optimal timing of investment made to transition from coal power to gas and renewable power under a carbon budget. To our knowledge, it proposes the first theoretical model capable of investigating this question, as related issues have been treated in separate strands of the analytical literature. On the one hand, a strand of the literature studies the optimal usage of dif- ferent fossil fuels through the lens of nonrenewable resources theory (Hotelling, 1931; Herfindahl, 1967), with little attention to the dynamics of capital accu- mulation (Chakravorty et al., 2008; Smulders and van der Werf, 2008; van der Ploeg and Withagen, 2012). For instance, Chakravorty et al. (2008) find that a carbon price may accelerate the burning of the most polluting resource: coal would be used first, followed only later by natural gas. In this literature, re- newable energy (modeled as a clean backstop) should never be used early, that is before fossil fuel production stops. More recently, Amigues et al. (2013) study the extraction of a renewable and a non-renewable energy source, taking into account that the extraction of the renewable source requires to first invest in appropriate capital (renewable power plants) and pay for adjustment costs. They prove that optimal investment in renewable is in this case independent from existing fossil resources; the best strategy is to start to build renewable power early and spread investment over time.2 Amigues et al. (2013) let climate policy for further research (the authors do mention that climate policies is an obvious application), consider only one type of fossil fuel, and disregard capacity constraints limiting the extraction of fossil fuels. On the other hand, there is a nascent literature on the optimal investment in abatement capital (Fischer et al., 2004; Williams, 2010; Vogt-Schilb et al., 2014b). This literature disentangles optimal emission reduction pathways from optimal investment pathways and from carbon prices. In particular, Vogt-Schilb et al. (2014b) study optimal investment in different economic sectors, and find that the same carbon price translates in more short-term investment in sectors with larger abatement potential and sectors where abatement capital is more expensive. Also, Rozenberg et al. (2014) analyze the trade-off between (i) early- scrapping existing dirty capital to reduce emissions quickly and (ii) using dirty capital at full capacity to increase short-term production. They find that early- scraping part of the dirty capital built before climate policies were implemented is optimal for achieving stringent climate targets.3 These two last studies are close to our question, but they focus on investment in perfectly clean capital (e.g. renewable power), letting aside any intermediate option (e.g., switching from coal to gas). This paper attempts to bridge the gap between these two veins of the lit- 2 A whole literature started by Kemp and Long (1980) has established that if extraction of an expensive energy source is exogenously constrained, it may be optimal to use that source simultaneously with cheaper alternatives (e.g., Amigues et al., 1998; Holland, 2003). Amigues et al. (2013) confirm this result for endogenous constraints. Other reasons for not extracting resources according to a strict Herfindahl sequence include heterogeneity of both producers and consumers – Gaudet (2014) provides an extensive review. 3 We focus on the analytical literature. Early decommissioning of existing coal power plants has also been studied using numerical models (e.g., Rogelj et al., 2011; IEA, 2014). 2 erature.4 We model different types of nonrenewable resources (cheap coal and more expensive gas) and an expensive renewable source, all subject to capacity constraints and adjustment costs (to make irreversible investment in coal, gas, and renewable power plants). We study the optimal transition from existing coal plants to gas and to renewable power under the constraint of a carbon budget. The analysis brings three findings. First, we confirm and extend the result by Amigues et al. (2013): under capacity constraints and adjustment costs, different energy sources may be used simultaneously. In particular, investment in gas can start before coal is phased out, and most importantly investment in renewable can start before gas or coal are phased out. The reason is that smoothing investment over time reduces adjustment costs. In other words, the availability of appropriate resources (skilled workers, production lines) and the stringency of the carbon budget set an optimal speed at which to deploy renewable power. We also extend the finding from Rozenberg et al. (2014): it may be optimal to build gas power plants and to decommission them before their natural end of life, even in the absence of a policy shock. Doing so allows for reducing investment in expensive renewable power plants in the short term, moving some efforts to the medium term, when gas power is under-used and replaced by renewable power. Finally, we extend the result from Vogt-Schilb et al. (2014b), as we find that more investment should go to the expensive zero-carbon renewable power than to the intermediate low-carbon gas power. This is not only because renewable power plants save fossil fuels and carbon taxes, but also because they keep a value for ever, while gas plants eventually become obsolete. These results stress the importance of correctly anticipating the effective lifetime of different types of power plants when choosing among them. Such effective lifetime is not only a technical parameter, but also an endogenous choice depending on future investment and production decisions. For instance, the levelized cost of electricity (LCOE), in theory a good metric for comparing investment in different power plants, may be biased against renewable power in practice. Indeed, different types of power plants are usually compared using the LCOE (e.g. Kost et al., 2012; EIA, 2013; IPCC, 2014), that is the ratio of discounted costs of building and using a power plant, including fuel and GHG emission 4 The interaction of investment and natural resources extraction was also the subject of the theory of the mine (e.g., Campbell, 1980; Gaudet, 1983; Lasserre, 1985), in which installed capital similarly limits the extraction rate of (a single type of) minerals. After reviewing this literature, Cairns (1998) notes that “there can be three phases in the exploitation of the mine, namely (1) a period of positive investment after time t = 0, in which production is at full capacity, then (2) a period in which investment is zero and production is at full capacity, and finally (3) a period of declining production ”. This paper is different as we model several resources and a climate constraint, we find however a similar trajectory for exploitation of gas resources. In addition, Dasgupta and Heal (1974), Solow (1974) and Stiglitz (1974) have started a literature that studies the impact of resource finitude on growth, in green Ramsey models that feature both capital accumulation and resource extraction — van der Ploeg and Withagen (2014) is a recent example. This literature also focuses on a single type of capital and a single fossil resource, while in this paper we study the transition from coal to gas and renewable power plants. 3 costs, over the discounted expected electricity production. We find that the LCOE computed against future production, investment and prices should be equal to the average future electricity price. But in practice, fossil fuel, carbon and electricity prices and the resale value of power plants of any given technology may be difficult to anticipate. They depend on future investment and production from all technologies, which depend in turn on future prices and regulations. We thus investigate whether LCOEs computed against current signals can provide an operational rule of thumb to guide investment decisions.5 In a numerical calibration using data from the European electricity sector, we find that such operational LCOE should be several times higher for new renewable power plants than for new gas power plants. Indeed, the operational LCOE does not account for future increases in fuel and carbon prices, and does not consider that gas plants will be under-used in the medium term once replaced by renewable power. As a result, if an agent was to assess investment in new capacity for the European electricity market by computing LCOEs from current prices, she would recommend to invest too much in gas capacity, and not enough in renewable capacity. The bad performance of such imperfectly computed LCOEs is neither sur- prising nor a theoretical challenge under the rational expectation hypothesis. It stresses that low-carbon capital cannot be valued from current price sig- nals only, such valuation requires to anticipate entirely a future decarbonization strategy.6 The novelty here is in determining the direction and magnitude of the error induced by myopic decision rules in the transition to clean electricity. The remainder of the paper is structured as follows. Section 1 details the model. Section 2 derives the first-order conditions, optimal production decisions at each point in time, and the electricity price. It then studies the different phases through which the power sector may go during its transition to renewable power. Section 3 tackles optimal investment decisions and the levelized cost of electricity. Section 4 provides numerical simulations calibrated with data from the European electricity sector. Section 5 concludes. 1. Model A social planner controls the supply of electricity, using and investing in three different technologies: an existing high-carbon technology (h, coal power), a fossil-fueled low-carbon technology ( , gas), and an inexhaustible zero-carbon technology (z , renewable power). 5 Such usage of the LCOE would be consistent with the view, popular in the early literature on investment, that firms may take optimal investment decisions based on current prices and current returns only (e.g., Arrow, 1964; Jorgenson, 1967, p. 145). It has already been signaled that the efficiency of such myopic investment decisions critically depends on investment prices being exogenous: under endogenous investment costs, optimal investment decisions require to perfectly anticipate all future prices (Gould, 1968; Mussa, 1977). 6 In practice, such pathways are usually provided by numerical models of the electricity sector (e.g., Seebregts et al., 2002; Loulou, 2008) or wider energy-environment-economy models (e.g., Weyant, 2004; Waisman et al., 2012; Kriegler et al., 2014). These models do not always model adjustment costs explicitly, but they take into account the limited ability to switch quickly from high- to low-carbon capital by building on maximum investment speeds (Wilson et al., 2013). Several authors have stressed that in this case optimal investment pathways are in part driven by such exogenous bounds, therefore calling for more explicit discussion of their calibration (Wilson et al., 2013; Iyer et al., 2014; Vogt-Schilb et al., 2014a). 4 At each time t, the social planner chooses the physical investment xi,t in technology i. The investment adds to the installed capacity ki,t , which depreci- ates at the constant rate δ (dotted variables denote temporal derivatives):7 ∀i, ˙ i,t = xi,t − δki,t k (1) Physical investment is made at a positive, increasing and convex cost ci (xi,t ): i > 0, ci > 0 ci > 0, c′ ′′ (2) This captures the increasing opportunity cost to use scarce resources (skilled workers and appropriate capital) in order to build and deploy capacities faster. We assume that low-carbon capacity is cheaper than zero-carbon capacity in the sense that: ∀x z ( x) c′ (x) < c′ (3) Without loss of generality, we assume low-carbon and zero-carbon capacities to be nil at the beginning (k ,t0 = kz,t0 = 0).8 The social planner then chooses how much output to produce from each technology. We assume production exhibits constant returns to scale: two gas plants can produce twice as much electricity as one gas plant.9 The positive production qi,t with technology i cannot exceed the installed capacity ki,t : ∀i, 0 ≤ qi,t qi,t ≤ ki,t (4) For simplicity, we assume the existing carbon-intensive capital is overabundant (such that (4) is not binding for coal); this assumption is relaxed and confirmed in the numerical application.10 Let Fi be the carbon intensity (or emission factor) of technology i. The high- carbon technology is more carbon-intensive than the low-carbon technology: Fh > F > Fz = 0 (5) The social planner is subject to an exogenous carbon budget (or emission ceil- ¯: ing), cumulative emissions cannot exceed a given ceiling M ∀t, mt ≤ M ¯ (6) 7 Through this paper, capacity is to be understood as equivalent capacity, e.g., in kWh/yr, unless otherwise specified. For instance if 2 kW of windmills are required to provide as much output per year as 1 kW of coal, then 2 kW of windmills are accounted as 1 kWeq . 8 The reason why this assumption does not result in a loss of generality is that our model studies the transition from the existing situation to cleaner and to clean power. 9 We thus neglect the imperfect substitution between gas and renewable plants that comes from their respective cost structures (in terms of fixed versus variable costs), for instance making gas more suitable for peak power than coal. One argument is that such considerations have more impact on the optimal electricity mix at the equilibrium than on the transition from the existing state to that equilibrium, which is the focus of the paper (see also Vogt-Schilb et al., 2014b). 10 Rozenberg et al. (2014) offer a proof that carbon-intensive capital becomes overabundant at the moment when a carbon price is implemented. 5 Where cumulative emissions mt grow with emissions Fi qi,t : ˙t= m Fi qi,t (7) i Cumulative emissions have been found to be a good proxy for global warming (Allen et al., 2009; Matthews et al., 2009). Some policy instruments, such as an emission trading scheme with unlimited banking and borrowing, set a similar constraint on firms (Slechten, 2013). Using fossil fuel (gas or coal) requires to extract exhaustible resource from an initial stock, such that the current stock Si,t classically satisfies: Si,t0 given ˙ i,t = −qi,t S (8) Si,t ≥ 0 While it is convenient to use the above general notations (indexed by i), we will follow van der Ploeg and Withagen (2012) and focus on the case where the zero carbon technology is renewable and coal is overabundant (Sz,t0 = Sh,t0 = ∞). This is equivalent to recognize the carbon budget is more stringent than the scarcity of coal resources — as it will be discussed later, this assumption is likely to hold for gas resources as well, especially with the recent developments of non-conventional extraction methods. Consumers derive utility u (∑i qi,t ) from electricity consumption, where u satisfies the Inada conditions and is smooth enough. The program of the so- cial planner consists in determining the trajectories of investment xi,t and pro- duction qi,t that maximize discounted utility net from investment costs while complying with the carbon budget M ¯ and the various constraints: ∞ max e−rt u qi,t − ci (xi,t ) dt (9) xi,t ,qi,t 0 i i ˙ i,t = xi,t − δki,t s.t. k (νi,t ) qi,t ≤ ki,t (γi,t ) qi,t ≥ 0 (λi,t ) ˙t= m Fi qi,t (µt ) i mt ≤ M ¯ (ηt ) ˙ i,t = −qi,t S (αi,t ) Si,t ≥ 0 (βi,t ) Where r is the constant discount rate and the Greek letters in parentheses are the costate variables and Lagrange multipliers (all chosen such that they are positive): among them, νi,t is the shadow cost of new power plants; γi,t , the social cost of the capacity constraint, can also be interpreted as the shadow rental cost of capital or the marginal productivity of capital; µt is the shadow carbon price; and αi,t the shadow price of resource i (all notations are gathered in Table A.5). 6 (a) Production and capacity (b) Output price Figure 1: Output price during the three types of phases. t = (δ + r ) cz (xz,t ) − dt cz (xz,t ) d ′ (1): u′ ′ 2. Sequencing production from and investment in competing tech- nologies 2.1. Optimal production decisions We start the resolution with the first order conditions, which translate the equilibrium between production decisions and electricity prices at each point in time. Indeed, during a time interval (τi+ , τi- ) when production and investment with technology i are strictly positive, the first-order conditions simplify to the following (see Appendix A for the complete set of equations): d ′ ∀t ∈ (τi+ , τi- ), i (xi,t ) − ( δ + r ) c′ t − αi,t − µt Fi c (xi,t ) = u′ (10) dt i On the right hand side of (10), u′ t is the competitive electricity price (ut ′ stands for u (∑i qi,t )), αi,t is the fuel cost, Fi the carbon intensity of fuel i, and ′ µt is the carbon price. The left hand side is what Vogt-Schilb et al. (2014b) call the marginal im- plicit rental cost of capital (MIRCC), extending the concept of implicit rental cost of capital proposed by Jorgenson (1967) to the case of endogenous capacity prices. It is the rental price that ensures agents are indifferent between buying and renting capacity. More specifically, the relation means there is no prof- itable tradeoff between the two following strategies: (i) buy capital at t at a cost c′i (xi,t ), rent it out during one period dt at the rental price, then sell the depreciated (δ ) capacities at t + dt at a price c′ i (xi,t ) + dt ci (xi,t )dt or (ii) simply d ′ lend money at the interest rate r. Equation (10) can thus be seen as an application of the equimarginal princi- ple: it provides a simple rule to arbitrate production decisions at each moment It shows that at each time step, production decisions may be taken relying only on current prices — the rental price of capacities and the price of electricity. 2.2. Electricity price and types of phases Equation (10) can also be used to determine the price of electricity, depend- ing on which one is the (short-term) marginal power plant: Lemma 1. The output price u′ t is given by the variable costs of the marginal technology, which includes a strictly positive rental cost only if capacities are fully used. Three types of situations are possible: 7 1. When coal capacities are used at less than full capacity, electricity price is given by variable costs from coal generation: 0 < qh,t < kh,t t = αh,t + Fh µt ⇒ u′ (11) 2. When gas capacities are used at less than full capacity, electricity price is given by variable costs from gas generation: 0 cz (0) by assumption (2) and cz (0) > c (0) by assumption (3). c′ ′ ′ ′ 11 3.3. Levelized cost of electricity (LCOE) Here we investigate whether the Levelized cost of electricity (LCOE) can be used as an operational rule of thumb to value different power plants. Let us con- sider an incremental unit of capacity built during an interval when investment is positive (at t ∈ (τi+ , τi- )). Dividing its optimal investment costs (15) with its discounted production over the same time interval yields: τ- τ- ∫t e i uθ dθ ∫t i e−(r+δ)(θ−t) (µθ Fi + αi,θ )dθ −(r +δ )(θ −t) ′ ∀t ∈ (τi+ , τi- ), τi- = - τi (17) ∫t e−(δ+r)(θ−t) dθ ∫t e−(δ+r)(θ−t) dθ - i (xi,t ) − e ci (xi,τi- ) −(r +δ )(τi −t) ′ c′ + - τi ∫t e−(δ+r)(θ−t) dθ Equation (17) defines the levelized cost of electricity (LCOE) produced with technology i. The first term in the right hand side is the discounted variable cost (from carbon and resource prices) over the discounted production; the second term corresponds to the investment costs of the marginal unit of capacity at t minus its residual value at τi- , over the discounted future production (the levelized cost of the capacity net of its residual value). The left-hand side is the average output price weighted by discounted production during (t, τi- ). Equation (17) states that at the optimum, the LCOE is equal to the weighted average future price of the output. The relevant weight is the discounted pro- duction from the deprecating capacity. In other terms, electricity priced at the optimal LCOE covers for the complete cost of building and operating a power plant.13 In practice, using this relationship to assess investment requires to know in advance the future prices of additional capacity, resources, emis- sions and output. This may be challenging for actual agents willing to schedule or monitor investment in different types of power plants. For instance, it is recognized that governments have little capacity to commit in general, and in particular to future carbon prices (Helm et al., 2003; Golombek et al., 2010; Ulph and Ulph, 2013).14 The question we address here is whether LCOEs computed against current price signals may provide an operational rule of thumb to value different types of power plants.15 Such an operational LCOE reads: i (xi,t ) + µt Fi + αi,t Li,t = (r + δ ) c′ (18) Li,t is the sum of the annualized investment cost, the carbon cost and the fuel cost. Proposition 2. Equalizing the operational LCOE across technologies would lead to invest relatively too much in gas power and too little in renewable power 13 We ignore here all intermittency issues, which add a cost to renewable sources, and may be a rationale to use some gas during the final phase (Joskow, 2011; Ambec and Crampes, 2012; Bazilian et al., 2013; Ueckerdt et al., 2013). 14 See also Spiro (2012) on the limited capacity of investors and regulators to anticipate prices over long time horizons. 15 As discussed in footnote 5, such operational LCOE is consistent with the idea that agents can take optimal investment decisions based on current price signals only. 12 Table 1: Technology sets considered in the numerical model Set Abbreviation Composition Average thermal production mix in High carbon technology Legacy 2008: 40% gas , 50% coal , 10% oil (EN- ERDATA, 2012) Low carbon technology Gas Efficient gas Zero carbon technology Wind Onshore wind Table 2: Fuel price trajectories of the fossil technology sets in $/MWh (EU, 2011) 2008 2025 2035 2050 Legacy 42.5 46 48 39 Gas 65 76 75 60 ˜z,t and x Proof. Let x ˜ ,t be two investment schedules that equalize the opera- tional LCOE accross technologies, such that: z (x (r + δ )c′ ˜ ,t ) + µF + α ˜z,t ) + µFz + αz = (r + δ )c′ (x (19) µF + α ⇒ c′ ˜z,t ) − c′ (x z (x ˜ ,t ) = (20) r+δ Comparing the allocation implied by (20) with the optimal rule stated in Corol- lary 1 shows that equalizing the operational LCOE across technologies misses the future increase for carbon and fossil resources prices, and misses the term correcting for future changes in the value of capacities, leading to relatively too much investment in gas power and not enough investment in renewable power. In the next section we use numerical simulations calibrated on the European power sector and find that during the optimal transition to clean power, renew- able power plants should be built at an operational LCOE several times higher than new gas power plants. 4. Numerical application to the European electricity sector 4.1. Functional forms, data and calibration We calibrate the model with data from the European power sector as de- scribed in the European 2050 Energy Roadmap (EU, 2011). In this numerical application, efficient gas power plants (the low-carbon technology) and onshore wind (the zero-carbon technology) are used to phase out the existing polluting capacities represented as the average legacy thermal production mix (Table 1). To better fit the data, we express installed capacity ki,t in peak capacity (GW), and production qi,t in GWh/yr. Production is constrained by a maxi- mum number of operating hours Hi . For instance, a given windmill will produce 13 Table 3: Technology-specific data used in the numerical application Description Unit Legacy Gas Wind Source m Ci Nominal $/kW IEA (2010) investment costs 1 800 1 200 2 000 Average annual Xi new capacity in GW/yr 4.2 11 10 ENERDATA (2012) Europe Hi Average annual h/yr IEA (2010) operating hours 7 500 7 500 2 000 ENERDATA (2012); Fi Carbon intensity gCO2 /kWh 0 Trotignon and Delbosc 530 330 (2008) Table 4: General parameter values used in the numerical application Description Unit Value Source r Discount rate %/yr 5 ¯ M Carbon budget (central value) GtCO2 22 UE (2011) D0 TWh/y 1 940 ENERDATA (2012); Electricity demand in 2008 EU (2011) G Annual growth of demand TWh/y 16.5 EU (2011) δ Depreciation rate %/yr 3.33 IEA (2010) A Convexity parameter (central value) ⋅ 0 electricity only at the moments where it is windy, which expectedly happens a given number of hours per year. We assume for simplicity that all technologies have the same depreciation rate δ , calibrated as δ = 1 lifetime assuming a lifetime of 30 years (IEA, 2010). We consider that Europe is price-taker on the fossil resources costs αi,t , which follows the trajectory assumed in the European 2050 Roadmap (Table 2). Note that αh corresponds to the average price of the resources of the legacy technology mix (Table 1). To calibrate the cost functions, we assume that when investment equals the actual average annual investment flow in Europe between 2009 and 2011 (Xi ), m the marginal investment cost Ci is equal to the OECD median value for 2010 (as found in IEA, 2010). We write the cost function as: xi,t 1 − A xi,t 2 ci (xi,t ) = Ci m ⋅ Xi ⋅ A + (21) Xi 2 Xi i (Xi ) = Ci ⇒ c′ m (22) A ∈ (0, 1) is a convexity parameter, equal across technologies for simplicity. When A = 1, the marginal investment cost are constant, there is no adjust- ment cost for production capacity, and optimal investment pathways can ex- hibit jumps. When A = 0, the marginal cost curves are linear, capacity accumu- 14 i (xi,t ) = 0), and the cost of new lated at very low speed is free (limxi,t →0;A=0 c′ capacity doubles when the investment pace doubles. For intermediate value A ∈ (0, 1), new capacity is always costly, and the marginal cost of new capacity increases with the investment pace. We first assume quadratic costs (that is linear marginal costs and A = 0), and perform later a sensitivity anaysis on A. The demand is assumed inelastic in the short-term, and follows an exogenous long-term growth. Its value in 2008 is calibrated as the reference fossil energy production (from coal, oil and gas), that is D0 = 1 940 TWh/yr (ENERDATA, 2012). The central scenario of the Roadmap forecasts a 700 TWh/yr increase of electricity consumption between 2008 and 2050, that we model as a linear growth of G = 16.5 TWh/yr2 . Demand after time t is thus: Dt = D0 + t ⋅ G (23) Acording to Trotignon and Delbosc (2008), emission allowances allocated to the power sector in 2008 amounted to Eref = 1.03 GtCO2 /yr , leading to a reference emission rate of 530 tCO2 /GWh. A linear decrease of these emissions until 2050, as planned in the Roadmap, yields a carbon budget of M ¯ = 22 GtCO2 . A ¯ is performed later. We use r = 5 %/yr for the social sensitivity analysis on M discount rate. 4.2. Results Fig. 3 shows production, investment, the investment cost, the electricity price and the operational LCOE obtained in the numerical application. The following lists several interesting results. The social planner does not invest in the legacy capacity; moreover, existing legacy plants start being decommissioned as soon as the climate policy is imple- mented (Fig. 3a). In the central scenario, legacy fossil-fueled plants are entirely phased out in 2028. With our technology assumptions, the carbon budget is too loose to justify a complete transition by 2050, but instead phases out all fossil production by 2064. Investment in renewable starts at the beginning of the transition, at the same time as gas and before coal or gas are phased out (Fig. 3c). It grows over time, and only decreases for a short period from 2055 to 2064 when gas is phased out, as most of the power plants have already been replaced (Fig. 3a). Investment in renewable starts at 22 GW/yr in 2008, more than twice the actual average investment rate observed between 2009 and 2011, and reaches 70 GW/yr in 2060. Investment grows again after 2064 to maintain the installed renewable capacity and meet the growing demand. Gas plants operate as a temporary bridge between coal and renewable. As seen in Fig. 3c, investment in gas starts since the beginning of the transition in 2008, lowering the need for investment in renewable. In 2035, however, in- vestment in gas stops and gas capacities start being underused, allowing gas to be replaced faster by renewable power. The sensitivity analysis (Fig. 4d) shows that it is optimal to build gas power plants transiently for any positive carbon budget ranging up to 45 GtCO2 : investment always starts in 2008 and, with quadratic investment costs, plants are underused immediately after investment stops. Only carbon budgets lower than 17 GtCO2 make gas investment stop be- fore 2020 (for comparison, the Roadmap implicitly sets a budget of 22 GtCO2 ). 15 (a) Production and capacity (b) Operational levelized cost of elec- tricity and electricity price (c) Investment (d) Marginal investment cost Figure 3: Numerical application to the European electricity sector Fig. 4c shows that the optimal maximal gas capacity increases with the con- vexity of the investment functions (when A tends to 0). The more convex the investment costs, the more renewable investment must be smoothed and the more gas is needed. In contrast, for constant investment costs (when A tends to 1), the social planner would replace all the stock of capital with renewable power plants at once. The capacity cost is always higher for renewable than for gas — as predicted by Corollary 1. Fig. 3d displays the marginal costs for new capacities (gas and renewable) along the period. They essentially follow the investment trajectories. Gas investment costs decrease from 2015 on, as the average power plants be- comes less and less carbon-intensive, making investment in low carbon capacity less and less profitable. Electricity prices are displayed in Fig. 3b. In a first phase, before 2028, the marginal capacity is the legacy dirty technology. The endogenous carbon price is 137 $/tCO2 , representing 73 $/MWh or 63 % of the electricity price during the phase while the legacy power plant is marginal. After the dirty technology has been phased out, gas becomes the marginal technology and is under-used from 16 (a) LCOE and price in 2008 as a func- (b) LCOE and price in 2008 as a func- tion of the convexity parameter A ¯ tion of the carbon budget M (c) Maximal gas capacity as a function (d) Date when gas investment stops as of the convexity parameter A ¯ a function of the carbon budget M Figure 4: Sensitivity analysis on the carbon budget and the convexity parameter 2036 to 2064. In between and after fossil technologies have been phased out, the electricity price equals the rental cost of renewable power plants (Lemma 1). Finally, along the simulated optimal path, the levelized cost of electricity computed against current prices and current returns (the operational LCOEs ) is several times higher for renewable power than for gas. Fig. 3b shows the LCOEs and the electricity price. The LCOE is higher than the electricity price for renewable plants, and lower for gas plants. Fig. 4a and 4b show the operational LCOEs of renewable and gas in 2008, along with the electricity price for a range of parameters.16 For a wide range of carbon budget and convexity parameter values, the operational LCOE is 2 to 5 times higher for renewable power than for gas. This difference comes from the wrong anticipations one makes when considering only current signals. These are biased toward gas for two reasons: the future increase in fuel and carbon price is not accounted for, and gas is assumed to produce at full capacity for its whole lifetime. These simulations thus suggest that if an agent was to assess investment in new capacity for the European electricity market by computing LCOEs from current prices, she would recommend to invest too much in gas capacity, and not enough in renewable capacity. 16 Note that computing the LCOE only make sense if investment is strictly positive. 17 5. Conclusion Our analysis makes several simplifications. For instance, we do not account for knowledge accumulation and directed technical change, which have been found to play a key role in the optimal transition from fossil to renewable en- ergy (Tahvonen and Salo, 2001; Acemoglu et al., 2012; Andr´ e and Smulders, 2014). Knowledge spillovers would tend to increase the short-term gap between optimal investment in renewable and gas power (Rosendahl, 2004; del Rio Gon- zalez, 2008), adding to the effect of adjustment costs studied here. We also disregarded uncertainty, known to play a key role in optimal accumulation of capital (Pindyck, 1991), optimal emission reduction pathways (Ha Duong et al., 1997), and optimal extraction of several energy sources (Gaudet and Lasserre, 2011). Finally, we did not account for the possibility to retrofit existing plants (notably with carbon capture and storage), and the possibility to use cleaner fuels (in particular derived from biomass) in existing power plants. Notwithstanding these limitations, our analysis suggests that capacity con- straints and adjustment costs play an essential role in the transition from coal to gas and renewable energy. In particular, investment in renewable power should start before coal and gas resources are phased out, and should always be higher than investment in gas power. The transition to carbon-free power may nonetheless benefit from temporary investment in intermediate technolo- gies such as gas, to decrease the need for more expensive renewable in the short term. The resulting plants would however need to be subsequently decommis- sioned before the end of their natural lifetime, to give room to more carbon-free power in the medium term. Our results shed light on technical choices (e.g. investors can consider gas plants with shorter scheduled lifetimes or an option to retrofit) as well as policy decisions (in setting milestones for carbon-free power and preparing for the social consequences of decommissioning fossil-fueled plants). Moreover, we warn that the levelized cost of electricity (LCOE) computed against current price signals is biased toward gas power and against renewable power. An accurate valuation of different types of power plants requires to anticipate a comprehensive long-term investment and production pathway, and in particular that gas built to replace coal in the short term may itself be replaced with renewable in the medium term. Finally, while this paper focuses on power generation, it more broadly in- forms on the ordering and assessment of investment in polluting fossil-fueled and clean capital; its results could be extended to other capitalistic and energy- intensive sectors, such as buildings or transportation. References Acemoglu, D., Aghion, P., Bursztyn, L., Hemous, D., 2012. The environment and directed technical change. American Economic Review 102 (1), 131–166. Allen, M., Frame, D., Huntingford, C., Jones, C., Lowe, J., Meinshausen, M., Meinshausen, N., 2009. Warming caused by cumulative carbon emissions to- wards the trillionth tonne. Nature 458 (7242), 1163–1166. Alvarez, R. A., Pacala, S. W., Winebrake, J. J., Chameides, W. L., Ham- burg, S. P., 2012. Greater focus needed on methane leakage from natural gas 18 infrastructure. Proceedings of the National Academy of Sciences 109 (17), 6435–6440. Ambec, S., Crampes, C., 2012. Electricity provision with intermittent sources of energy. Resource and Energy Economics 34 (3), 319–336. Amigues, J.-P., Ayong Le Kama, A., Chakravorty, U., Moreaux, M., 2013. Equilibrium transitions from non renewable energy to renewable energy under capacity constraints. LERNA Working Papers 11.07.341. Amigues, J.-P., Favard, P., Gaudet, G., Moreaux, M., 1998. On the optimal order of natural resource use when the capacity of the inexhaustible substitute is limited. Journal of Economic Theory 80 (1), 153 – 170. e, F. J., Smulders, S., 2014. Fueling growth when oil peaks: Directed Andr´ technological change and the limits to efficiency. European Economic Review. Arrow, K. J., 1964. Optimal capital policy, the cost of capital, and myopic decision rules. Annals of the Institute of Statistical Mathematics 16 (1), 21– 30. Bazilian, M., Onyeji, I., Liebreich, M., MacGill, I., Chase, J., Shah, J., Gie- len, D., Arent, D., Landfear, D., Zhengrong, S., 2013. Re-considering the economics of photovoltaic power. Renewable Energy 53, 329–338. Cairns, R. D., 1998. The microeconomics of mineral extraction under capacity constraints. Nonrenewable Resources 7 (3), 233–244. Campbell, H. F., 1980. The effect of capital intensity on the optimal rate of extraction of a mineral deposit. The Canadian Journal of Economics / Revue canadienne d’Economique 13 (2), 349–356. Chakravorty, U., Moreaux, M., Tidball, M., 2008. Ordering the extraction of polluting nonrenewable resources. The American Economic Review 98 (3), 1128–1144. Collins, M., Knutti, R., Arblaster, J. M., Dufresne, J.-L., Fichefet, T., Friedling- stein, P., Gao, X., Gutowski, W. J., Johns, T., Krinner, G., 2013. Long-term climate change: projections, commitments and irreversibility. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Dasgupta, P., Heal, G., 1974. The optimal depletion of exhaustible resources. The Review of Economic Studies 41, pp. 3–28. del Rio Gonzalez, P., 2008. Policy implications of potential conflicts between short-term and long-term efficiency in CO2 emissions abatement. Ecological Economics 65 (2), 292–303. EIA, 2013. Levelized cost of new generation resources in the annual energy outlook 2013. Tech. rep., Energy Information Administration. ENERDATA, 2012. Global energy & CO2 database. Consulted May 2012. 19 EU, 2011. Impact assessment accompanying the communication from the commission: Energy roadmap 2050. Commission Staff Working Document Sec(2011) 1565 final, European Commission. Fischer, C., Withagen, C., Toman, M., 2004. Optimal investment in clean pro- duction capacity. Environmental and Resource Economics 28 (3), 325–345. Gaudet, G., 1983. Optimal investment and adjustment costs in the economic theory of the mine. The Canadian Journal of Economics / Revue canadienne d’Economique 16 (1), 39–51. Gaudet, G., 2014. Modelling the use of nonrenewable resources with multiple demands and multiple sources. Gaudet, G., Lasserre, P., 2011. The efficient use of multiple sources of a non- renewable resource under supply cost uncertainty. International Economic Review 52 (1), 245–258. Golombek, R., Greaker, M., Hoel, M., 2010. Carbon taxes and innovation with- out commitment. The B.E. Journal of Economic Analysis & Policy 10 (1). Gould, J. P., 1968. Adjustment costs in the theory of investment of the firm. The Review of Economic Studies 35 (1), pp. 47–55. Ha Duong, M., Grubb, M., Hourcade, J.-C., 1997. Influence of socioeconomic in- ertia and uncertainty on optimal CO2-emission abatement. Nature 390 (6657), 270–273. Helm, D., Hepburn, C., Mash, R., 2003. Credible carbon policy. Oxford Review of Economic Policy 19 (3), 438–450. Herfindahl, O. C., 1967. Depletion and economic theory. Extractive resources and taxation, 63–90. Holland, S. P., 2003. Extraction capacity and the optimal order of extraction. Journal of Environmental Economics and Management 45 (3), 569–588. Hotelling, H., 1931. The economics of exhaustible resources. The Journal of Political Economy 39 (2), 175, 137. IEA, 2010. Projected cost of generating electricity, 2010 edition. Tech. rep., International Energy Agency. IEA, 2014. Harnessing Electricity’s Potential. Energy Technology Perspectives. International Energy Agency. IPCC, 2014. Technical summary. In: Climate Change 2014: Mitigation of Cli- mate Change. Intergovernmental Panel on Climate Change. Iyer, G., Hultman, N., Eom, J., McJeon, H., Patel, P., Clarke, L., 2014. Diffusion of low-carbon technologies and the feasibility of long-term climate targets. Technological Forecasting and Social Change (forthcoming). Jorgenson, D., 1967. The theory of investment behavior. In: Determinants of investment behavior. NBER, pp. 129–188. 20 Joskow, P. L., 2011. Comparing the costs of intermittent and dispatchable elec- tricity generating technologies. American Economic Review: Paper and Pro- ceedings 101 (3), 238–41. Kemp, M. C., Long, N. V., 1980. On two folk theorems concerning the extraction of exhaustible resources. Econometrica 48 (3), 663–673. Kost, C., Jessica Thomsen, Sebastian Nold, Johannes Mayer, 2012. Levelized cost of electricity renewable energies. Tech. rep., Fraunhofer ISE. Kriegler, E., Weyant, J. P., Blanford, G. J., Krey, V., Clarke, L., Edmonds, J., Fawcett, A., Luderer, G., Riahi, K., Richels, R., Rose, S. K., Tavoni, M., Vuuren, D. P. v., 2014. The role of technology for achieving climate policy objectives: overview of the EMF 27 study on global technology and climate policy strategies. Climatic Change 123 (3-4), 353–367. Lasserre, P., 1985. Exhaustible-resource extraction with capital. In: Progress in natural resource economic, oxford: clarendon press, Edition. Anthony Scott, pp. 178–202. Loulou, R., 2008. ETSAP-TIAM: the TIMES integrated assessment model. Computational Management Science 5 (1-2), 41–66. Matthews, H. D., Gillett, N., Stott, P., Zickfeld, K., 2009. The proportionality of global warming to cumulative carbon emissions. Nature 459 (7248), 829–832. Mussa, M., 1977. External and internal adjustment costs and the theory of aggregate and firm investment. Economica 44 (174), pp. 163–178. Pindyck, R. S., 1991. Irreversibility, uncertainty, and investment. Journal of Economic Literature 29 (3), 1110–1148. Rogelj, J., Hare, W., Lowe, J., van Vuuren, D. P., Riahi, K., Matthews, B., Hanaoka, T., Jiang, K., Meinshausen, M., 2011. Emission pathways consistent with a 2‰ global temperature limit. Nature Climate Change 1 (8), 413–418. Rosendahl, K. E., 2004. Cost-effective environmental policy: implications of in- duced technological change. Journal of Environmental Economics and Man- agement 48 (3), 1099–1121. Rozenberg, J., Vogt-Schilb, A., Hallegatte, S., 2014. Transition to clean capi- tal, irreversible investment and stranded assets. World Bank Policy Research Working Paper (6859). Seebregts, A., Goldstein, G., Smekens, K., 2002. Energy/environmental mod- eling with the MARKAL family of models. In: Chamoni, P., Leisten, R., Martin, A., Minnemann, J., Stadtler, H. (Eds.), Operations Research Pro- ceedings 2001. Vol. 2001 of Operations Research Proceedings 2001. Springer Berlin Heidelberg, pp. 75–82. Slechten, A., 2013. Intertemporal links in cap-and-trade schemes. Journal of Environmental Economics and Management 66 (2), 319–336. 21 Smulders, S., van der Werf, 2008. Climate policy and the optimal extraction of high- and low-carbon fossil fuels. The Canadian Journal of Economics / Revue canadienne d’Economique 41 (4), 1421–1444. Solow, R. M., 1974. Intergenerational equity and exhaustible resources. The Review of Economic Studies 41, 29–45. Spiro, D., 2012. Resource prices and planning horizons. Mimeo. Steinacher, M., Joos, F., Stocker, T. F., 2013. Allowable carbon emissions low- ered by multiple climate targets. Nature 499 (7457), 197–201. Stiglitz, J., 1974. Growth with exhaustible natural resources: Efficient and op- timal growth paths. The Review of Economic Studies 41, 123–137. Tahvonen, O., Salo, S., 2001. Economic growth and transitions between renew- able and nonrenewable energy resources. European Economic Review 45 (8), 1379 – 1398. ´ Trotignon, R., Delbosc, A., 2008. Echanges de quotas en p´eriode d’essai du marche europ´ ev` een du CO2 : ce que r´ ele le CITL. Etude climat 13, Caisse des epots, Mission climat. d´ UE, 2011. A roadmap for moving to a competitive low carbon economy in 2050. Communication from the Commission COM(2011) 112 final, European Com- mission. Ueckerdt, F., Hirth, L., Luderer, G., Edenhofer, O., 2013. System LCOE: What are the costs of variable renewables? Energy 63, 61 – 75. Ulph, A., Ulph, D., Oct. 2013. Optimal climate change policies when govern- ments cannot commit. Environmental and Resource Economics 56 (2), 161– 176. van der Ploeg, F., Withagen, C., 2012. Too much coal, too little oil. Journal of Public Economics 96 (12), 62–77. van der Ploeg, F., Withagen, C., 2014. Growth, renewables, and the optimal carbon tax. International Economic Review 55 (1), 283–311. Vogt-Schilb, A., Hallegatte, S., de Gouvello, C., 2014a. Long-term mitigation strategies and marginal abatement cost curves: a case study on Brazil. World Bank Policy Research (6808). Vogt-Schilb, A., Meunier, G., Hallegatte, S., 2014b. Optimal timing, cost and sectoral dispatch of emission reductions: abatement cost curves vs. abatement capital accumulation. Waisman, H., Rozenberg, J., Sassi, O., Hourcade, J.-C., 2012. Peak oil profiles through the lens of a general equilibrium assessment. Energy Policy 48, 744– 753. Wang, M., Zhao, J., 2013. Monopoly extraction of a nonrenewable resource fac- ing capacity constrained renewable competition. Economics Letters 120 (3), 503–508. 22 Weyant, J. P., 2004. Introduction and overview to the EMF 19 study on tech- nology and climate change policy. Energy Economics 26 (4), 501–515. Wikibooks contributors, 2014. First order linear 1. In: Ordinary Differential Equations. Wikibooks. Williams, J. H., DeBenedictis, A., Ghanadan, R., Mahone, A., Moore, J., Mor- row, W. R., Price, S., Torn, M. S., 2012. The technology path to deep green- house gas emissions cuts by 2050: The pivotal role of electricity. Science 335 (6064), 53–59. Williams, R., 2010. Setting the initial time-profile of climate policy: The eco- nomics of environmental policy phase-ins. Working Paper 16120, National Bureau of Economic Research. Wilson, C., Grubler, A., Bauer, N., Krey, V., Riahi, K., 2013. Future capacity growth of energy technologies: are scenarios consistent with historical evi- dence? Climatic Change 118 (2), 381–395. A. Efficiency conditions Table A.5: Variables and parameters notations used in the model. Description Possible units i technology index h high-carbon technology (h) l low-carbon technology ( ) z zero-carbon technology (z ) ki,t capacity of technology i at time t GW qi,t production of technology i at time t GW xi,t physical investment in technology i at time t GW/yr νi,t shadow price of new capacity i $/GW µt cost of emissions $/tCO2 αi,t cost of resource used by technology i $/MWh γi,t shadow rental cost of existing capacity i $/(GW⋅ yr) u′ t output price $/GWh ci (xi,t ) monetary investment in technology i at time t $/yr mt stock of atmospheric carbon tCO2 δ depreciation rate yr−1 r discount rate yr−1 Fi emission rate of technology i tCO2 /GWh M¯ carbon budget tCO2 u (∑i qi,t ) consumer utility $/yr 23 The present value Hamiltonian associated with Problem 9 reads: H = e−rt u qi,t − ci (xi,t ) − νi,t (δ ki,t − xi,t ) i i i − µt ¯ − Ri qi,t − ηt mt − M (αi,t qi,t − βi,t Si,t ) (A.1) i i − γi,t (qi,t − ki,t ) + λi,t qi,t i i The first-order conditions are: ∂H =0 ⇐⇒ i (xi,t ) = νi,t c′ (A.2) ∂xi ∂H =0 ⇐⇒ t = γi,t λi,t − µt Ri − αi,t + u′ (A.3) ∂qi ∂H −d e−rt νi,t = ⇐⇒ ˙ i,t = γi,t (δ + r) νi,t − ν (A.4) ∂ki dt ∂H −d e−rt µt = ⇐⇒ ˙ t − r µt = −ηt µ (A.5) ∂mt dt ∂ H −d e αi,t −rt = ⇐⇒ ˙ i − r αi,t = −βi,t α (A.6) ∂Si dt The complementary slackness conditions are: ∀i, t, λi,t ≥ 0, qi,t ≥ 0 and λi,t qi,t = 0 (A.7) ∀i, t, ηt ≥ 0, M − mt ≥ 0 ¯ and ηt M ¯ − mt = 0 (A.8) ∀i, t, βi,t ≥ 0, Si,t ≥ 0 and βi,t Si,t = 0 (A.9) ∀i, t, γi,t ≥ 0, ki,t − qi,t ≥ 0 and γi,t (ki,t − qi,t ) = 0 (A.10) The transversality condition is replaced by the terminal condition that at some point the atmospheric carbon reaches its ceiling (6). B. Investment and production phases B.1. Carbon budget constraint and steady state Let Tη be the date when the ceiling on atmospheric carbon is reached. The carbon-free atmosphere can be seen as a non renewable resource depleted by GHG emissions. In this context, the optimal current carbon price follows Hotelling’s rule, i.e. grows at the discount rate r (A.5, A.8): ∀t < Tη , µt = µert > 0 (B.1) where µ is the carbon price at t = 0. Eq. B.1 reflects the fact that abatement realized at any time contributes equally to meet the carbon budget. We focus on the case where the carbon budget is binding, so that the carbon price µt is strictly positive. Let T be the date when the system reaches a steady state. During the steady state, the clean backstop produces all the output. Indeed, atmospheric carbon 24 is stable, hence emissions and production from high- and low-carbon fossil fuels must be nil (4, 6, 7): ˙t=0 ∀t ≥ T, m ⇒ q ,t = qh,t = 0 (B.2) When the carbon budget is binding, the date when the ceiling on atmospheric carbon is reached and the steady state necessarily coincide: T = Tη . B.2. Output production and fossil fuel deposits Initially, all the output is produced from coal by assumption. Coal produc- - tion is eventually replaced by gas or renewable power. Let Th be the date when high-carbon production stops: ∀t ≥ Th - , qh,t = 0 (B.3) This happens necessarily before the steady state: - Th ≤T (B.4) Similarly, let Ti+ be the date when production of technology i ∈ { , z } starts, and T - the date when gas production stops. Note that only low-carbon production eventually stops, since zero-carbon energy is used indefinitely during the steady state.17 Let T β be the date when the low-carbon fossil fuel deposit is depleted. If it exists (i.e. if the gas deposit is eventually depleted), T β coincides with the end of production from low-carbon fuel: ∃T β ⇒ Tβ = T- (B.5) Optimality then requires that fossil fuel prices follow the Hotelling rule: their current price α ,t is increasing at the discounting rate before they are exhausted (A.6, A.9): ∀t < T β , α ,t = α ert > 0 (B.6) B.3. Investment and under-utilization of capacities For coal to be phased out, production in one of the alternative technologies must have started: - Th ≥ min(T + , Tz + ) (B.7) Let τi+ ≤ Ti+ be the date when investment in capacity i starts, and τi- the date when it stops. Note that only the investment in gas stops eventually, since renewable capacity is used indefinitely during the steady state and must be maintained as δ > 0. If gas is used during the transition, production from gas must decline before the steady state, either because gas reserves are depleted, or to reduce GHG emissions to zero. If it declines faster than the depreciation rate of capacities, 17 As the marginal utility tends to infinity when energy consumption tends to zero energy production is always strictly positive. 25 installed gas capacities may be underused. Let Tγ ∈ [T + , T - ] be the date when gas production is lower than its capacity: ∀t ≥ Tγ , k ,t >0 ⇒ q ,t