metadata: authors: 0: author: Lecuyer, Oskar
1: author: Vogt-Schilb,Adrien Camille
count: World
docna: 0: docna: Optimal transition from coal to gas and
            renewable power under capacity constraints and adjustment costs
docty: Policy Research Working Paper
subtopic: Science of Climate Change,Climate Change and Health,Climate Change and Environment,Rural and Renewable Energy,Energy Policies & Economics,Energy Demand,Energy and Environment,Energy and Mining,Renewable Energy,Global Environment
teratopic: Environment,Energy
abstracts: cdata!: This paper studies the optimal
            transition from existing coal power plants to gas and
            renewable power under a carbon budget. It solves a model of
            polluting, exhaustible resources with capacity constraints
            and adjustment costs (to build coal, gas, and renewable
            power plants). It finds that optimal investment in renewable
            energy may start before coal power has been phased out and
            even before investment in gas has started, because doing so
            allows for smoothing investment over time and reduces
            adjustment costs. Gas plants may be used to reduce
            short-term investment in renewable power and associated
            costs, but must eventually be phased out to allow room for
            carbon-free power. One risk for myopic agents comparing gas
            and renewable investment is thus to overestimate the
            lifetime of gas plants -- e.g., when computing the levelized
            cost of electricity -- and be biased against renewable
            power. These analytical results are quantified with
            numerical simulations of the European Commission's 2050
            energy roadmap.
repnme: repnme: Optimal transition from coal to gas and
            renewable power under capacity constraints and adjustment costs
display_title: Optimal transition from coal to gas
            and renewable power under capacity constraints and
            adjustment costs
listing_relative_url: /research/2014/07/19896708/optimal-transition-coal-gas-renewable-power-under-capacity-constraints-adjustment-costs
topicv3: Renewable Energy m1326170 525,Energy and Extractives m1326595 507,Climate Change m1326829 567,Power and Electricity Sector m1327094 515,Environment, Natural Resources and Blue
            Economy m1327391 566
publishtoextweb_dt: 2014-07-01T00:00:00Z
historic_topic: Environment,Energy
no_of_pages: 29
author_keys: 0: author_key: 1272850
model_extractions: 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 coal power plants to gas and renewable power under a carbon budget. It solves a model of polluting, exhaustible resources with capacity constraints and adjustment costs (to build coal, gas, and renewable power plants). It finds that optimal investment in renewable energy may start before coal power has been phased out and even before investment in gas has started, because doing so allows for smoothing investment over time and reduces adjustment costs. Gas plants may be used to reduce short-term investment in renewable power and associated costs, but must eventually be phased out to allow room for carbon-free power. One risk for myopic agents comparing gas and renewable investment is thus to overestimate the lifetime of gas plants—e.g., when computing the levelized cost of electricity—and be biased against renewable power. These analytical results are quantified with numerical simulations 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<sup>1</sup> , Adrien Vogt-Schilb<sup>2, 3</sup>

> _1Department of Economics and Oeschger Centre for Climate Change Research, University of Bern, Bern, Switzerland_

> _2The World Bank, Climate Change Group, Washington D.C., USA_

> _3CIRED, 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 climate damages, which requires reaching near-zero greenhouse gas (GHG) emissions 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). Several alternatives are thus available to abate GHG emissions from electricity production. Emissions may be reduced by replacing coal plants with new gas power plants,<sup>1</sup> 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 Hallegatte, Jean-Charles Hourcade, Guy Meunier, Jean-Pierre Ponssard, Antonin Pottier, Philippe Quirion, Julie Rozenberg, Fran ̧cois Salani ́e, Ralph Winkler and one anonymous referee from 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<sup> ́</sup> 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 different fossil fuels through the lens of nonrenewable resources theory (Hotelling, 1931; Herfindahl, 1967), with little attention to the dynamics of capital accumulation (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, renewable 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.<sup>2</sup> 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) earlyscrapping existing dirty capital to reduce emissions quickly and (ii) using dirty capital at full capacity to increase short-term production. They find that earlyscraping part of the dirty capital _built before climate policies were implemented_ is optimal for achieving stringent climate targets.<sup>3</sup> 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).

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erature.<sup>4</sup> 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.

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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.<sup>5</sup> 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 surprising nor a theoretical challenge under the rational expectation hypothesis. It stresses that low-carbon capital cannot be valued from current price signals only, such valuation requires to anticipate entirely a future decarbonization strategy.<sup>6</sup> 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 ( _l_ , 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).

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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 depreciates at the constant rate _δ_ (dotted variables denote temporal derivatives):<sup>7</sup> Physical investment is made at a positive, increasing and convex cost _ci_ ( _xi,t_ ): 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: Without loss of generality, we assume low-carbon and zero-carbon capacities to be nil at the beginning ( _kl,t_ 0 = _kz,t_ 0 = 0).<sup>8</sup> 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.<sup>9</sup> The positive production _qi,t_ with technology _i_ cannot exceed the installed capacity _ki,t_ : 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.<sup>10</sup> Let _Fi_ be the carbon intensity (or emission factor) of technology _i_ . The highcarbon technology is more carbon-intensive than the low-carbon technology: The social planner is subject to an exogenous carbon budget (or emission ceiling), cumulative emissions cannot exceed a given ceiling _M_<sup> ̄</sup> :

> 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 kW _eq_ .

> 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.

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Where cumulative emissions _mt_ grow with emissions _Fi qi,t_ : 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: 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,t_ 0 = _Sh,t_ 0 = ∞). 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 social planner consists in determining the trajectories of investment _xi,t_ and production _qi,t_ that maximize discounted utility net from investment costs while complying with the carbon budget _M_<sup> ̄</sup> and the various constraints: 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).

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<!-- Start of picture text --> (a) Production and capacity (b) Output price<br><!-- End of picture text -->

**Figure 1:** Output price during the three types of phases. (1): _u_<sup>′</sup> _t_<sup>= (</sup><sup>_δ_+</sup><sup>_r_)</sup><sup>_c_′</sup> _z_<sup>(</sup><sup>_x_</sup> _z,t_<sup>) −</sup> _dt_<sup>_<u>d</u>c_</sup> _z_<sup>′(</sup><sup>_x_</sup> _z,t_<sup>)</sup>

# **2. Sequencing production from and investment in competing technologies**

# _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_<sup>+</sup><sup>_,τ_-</sup> _i_<sup>)whenproductionandinvestment</sup> with technology _i_ are strictly positive, the first-order conditions simplify to the following (see Appendix A for the complete set of equations): On the right hand side of (10), _u_<sup>′</sup> _t_<sup>isthecompetitiveelectricityprice(</sup><sup>_u_′</sup> _t_ stands for _u_<sup>′</sup> (∑ _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 implicit 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 profitable tradeoff between the two following strategies: (i) buy capital at _t_ at a cost _c_<sup>′</sup> _i_<sup>(</sup><sup>_xi,t_),rentitoutduringoneperiod</sup><sup>_dt_attherentalprice,thensellthe</sup> depreciated ( _δ_ ) capacities at _t_ + _dt_ at a price _c_<sup>′</sup> _i_<sup>(</sup><sup>_xi,t_) +</sup> _dt_<sup>_<u>d</u>c_</sup> _i_<sup>′(</sup><sup>_xi,t_)</sup><sup>_dt_or(ii)simply</sup> lend money at the interest rate _r_ .

Equation (10) can thus be seen as an application of the equimarginal principle: 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, depending on which one is the (short-term) marginal power plant:

**Lemma 1.** _The output price u_<sup>′</sup> _t_<sup>_isgivenbythevariablecostsofthemarginal_</sup> _technology, which includes a strictly positive rental cost only if capacities are fully used. Three types of situations are possible:_

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_1. When coal capacities are used at less than full capacity, electricity price is given by variable costs from coal generation:_ _2. When gas capacities are used at less than full capacity, electricity price is given by variable costs from gas generation:_ _3. If all capacities are either used at full or not used at all, rental costs adjusts such that electricity price is set by the implicit rental cost of renewable capacity:_ _In particular, it is not possible that at a given date gas and coal are both used at less than full capacity,_<sup>11</sup> _and renewable power is never used at less than full capacity._ _Proof._ The results are straightforward once it is noted that _γi,t_ , the Lagrange multiplier associated with the capacity constraint, is zero when capacity _i_ is underused, and that _γi,t_ equals the implicit rental cost of capital (Appendix A).

Lemma 1 thus defines implicitly three _types_ of phases through which electricity production may go during the transition from coal-fired to renewable power — and implies that the price may jump when the system changes phases and capacity constraint start or stop being binding (Wang and Zhao, 2013, offer a discussion of price jumps).

As discussed in the next subsection, the ordering of these three types of phases may vary. Fig. 1 illustrates a particular situation where investment in both renewable and gas power start at the beginning of the period. In a first phase, coal is progressively replaced by renewable and gas, coal capacity is underused and electricity price is given by the variable cost of coal. When coal is phased out follow two phases during which renewable power progressively replaces gas. First gas is used at full capacity, and electricity price depends on the marginal implicit rental cost of renewable capacities. At one point meeting the emission constraint requires gas production to decrease faster than its capacity depreciates: gas plants are underused and the price is set by variable costs from gas. During the final phase, all production comes from renewable power.

> 11 We disregard the case where fuel costs compensate exactly differences in carbon intensities _αl_ − _αh_ = _μ_ ( _Fh_ − _Fl_ ) as it requires a very restrictive set of assumptions.

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# _2.3. Ordering investment in low- and zero-carbon capacity_

In this subsection we study the various phases through which the system may go during the transition from coal to renewable. Three particular dates play a pivotal role.

To comply with the carbon budget constraint, all production must eventually switch to renewable power, at a first date that we denote _T_ (Fig. 2). At one point, polluting coal production is thus completely phased out, at a second date _Th_<sup>≤</sup><sup>_T_.Similarly,whilegasplantsmaybeusedduringthetransition,atone</sup> point gas production has to decline, possibly faster than the speed at which its capacity depreciates. This may result in an underutilization of gas capacity, at a third date _Tγ_ .

In Appendix B, we systematically list the various possible transition shapes, and identify three possible sequences of investment in gas and renewable power:

**Proposition 1.** _Depending on the parameters, investment phases may be ordered in three ways:_

_1. Two successive transitions. Gas first completely replaces coal, then renewable power replaces gas. In this case, investment in renewable power starts after coal is phased out, but before production from gas is phased out (Fig. 2a)._ _2. Gas and wind simultaneously replace coal. In this case, investment in renewable starts before production from any fossil resources is phased-out (Fig. 2b)._ _3. Starting with investment in renewable power. In this case, investment in renewable power starts before investment in gas power (Fig. 2c)._ _Proof._ See Appendix B.

# **3. Assessing investment in competing technologies**

In this section, we study optimal investment decisions. We first derive a general investment decision rule, showing that all future prices and the future resale value of capital are required to determine the optimal investment strategy. We then discuss the levelized cost of electricity and investigate consequences of failing to anticipate all future prices.

# _3.1. General investment decision equation_

During a time interval ( _τi_<sup>+</sup><sup>_,τ_-</sup> _i_<sup>)whenproductionandinvestmentarestrictly</sup> positive, the optimal marginal investment cost _c_<sup>′</sup> _i_<sup>(</sup><sup>_xi,t_)isthesolutionofthe</sup> differential equation (10). In general, such solution reads:<sup>12</sup>

> 12 Equation 15 is the textbook solution of equation 10 (e.g., Wikibooks contributors, 2014)

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<!-- Start of picture text --> (a) Two successive transitions<br>(b) Gas and wind simultaneously replace coal<br>(c) Starting with investment in renewable power<br><!-- End of picture text -->

**Figure 2:** Numerical simulations of three possible transition profiles. Figures on the left display capacity and production, figures on the right display optimal investment. The parameters used to produce these figures are gathered in Table B.6.

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a
# _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 consider an incremental unit of capacity built during an interval when investment is positive (at _t_ ∈( _τi_<sup>+</sup><sup>_,τ_-</sup> _i_<sup>)).Dividingitsoptimalinvestmentcosts(15)withits</sup> discounted production over the same time interval yields: 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_<sup>-,overthediscountedfutureproduction(the</sup> 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_<sup>-).</sup> 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 production 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.<sup>13</sup> In practice, using this relationship to assess investment requires to know in advance the future prices of additional capacity, resources, emissions 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).<sup>14</sup> 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.<sup>15</sup> Such an _operational LCOE_ reads: L _i,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.

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**Table 1:** Technology sets considered in the numerical model

|Set<br>|Abbreviation|Composition|
|---|---|---|
|High carbon technology|Legacy|Average<br>thermal<br>production<br>mix<br>in<br>2008: 40% gas , 50% coal , 10% oil (EN-|
|||ERDATA, 2012)|
|Low carbon technology|Gas|Efcient 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|

_Proof._ Let _x_ ̃ _z,t_ and _x_ ̃ _l,t_ be two investment schedules that equalize the operational LCOE accross technologies, such that: Comparing the allocation implied by (20) with the optimal rule stated in Corollary 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, renewable 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 described 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 maximum 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|
|---|---|---|---|---|---|---|
|_C_<sup>_m_</sup><br>_i_|Nominal<br>investment costs|$/kW|1 800|1 200|2 000|IEA (2010)|
|_Xi_|Average annual<br>new capacity in<br>Europe|GW/yr|4.2|11|10|ENERDATA (2012)|
|_Hi_|Average annual<br>operating hours|h/yr|7 500|7 500|2 000|IEA (2010)|
|_Fi_|Carbon intensity|gCO2/kWh|530|330|0|ENERDATA (2012);<br>Trotignon and Delbosc<br>(2008)|

**Table 4:** General parameter values used in the numerical application

||Description|Unit|Value|Source|
|---|---|---|---|---|
|_r_|Discount rate|%/yr|5||
| ̄<br>_M_|Carbon budget (central value)|GtCO2|22|UE (2011)|
|_D_0|Electricity demand in 2008|TWh/y|1 940|ENERDATA (2012);<br>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_ ), the marginal investment cost _Ci_<sup>_m_</sup> is equal to the OECD median value for 2010 (as found in IEA, 2010). We write the cost function as: _A_ ∈(0 _,_ 1) is a convexity parameter, equal across technologies for simplicity. When _A_ = 1, the marginal investment cost are constant, there is no adjustment cost for production capacity, and optimal investment pathways can exhibit jumps. When _A_ = 0, the marginal cost curves are linear, capacity accumu14
lated at very low speed is free (lim _xi,t_ →0; _A_ =0 _c_<sup>′</sup> _i_<sup>(</sup><sup>_xi,t_)= 0),andthecostofnew</sup> 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 _D_ 0 = 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/yr<sup>2</sup> . Demand after time _t_ is thus: Acording to Trotignon and Delbosc (2008), emission allowances allocated to the power sector in 2008 amounted to _E_ ref = 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_<sup> ̄</sup> = 22 GtCO2. A sensitivity analysis on _M_<sup> ̄</sup> is performed later. We use _r_ = 5 %/yr for the social 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 implemented (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, investment 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 before 2020 (for comparison, the Roadmap implicitly sets a budget of 22 GtCO2).

15
<!-- Start of picture text --> (a) Production and capacity (b) Operational levelized cost of elec-<br>tricity and electricity price<br>(c) Investment (d) Marginal investment cost<br><!-- End of picture text -->

**Figure 3:** Numerical application to the European electricity sector

Fig. 4c shows that the optimal maximal gas capacity increases with the convexity 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 becomes 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
<!-- Start of picture text --> (a) LCOE and price in 2008 as a func- (b) LCOE and price in 2008 as a func-<br>tion of the convexity parameter A tion of the carbon budget M ̄<br>(c) Maximal gas capacity as a function (d) Date when gas investment stops as<br>of the convexity parameter A a function of the carbon budget M ̄<br><!-- End of picture text -->

**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.<sup>16</sup> 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 energy (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 Gonzalez, 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 constraints 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 technologies such as gas, to decrease the need for more expensive renewable in the short term. The resulting plants would however need to be subsequently decommissioned 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 informs on the ordering and assessment of investment in polluting fossil-fueled and clean capital; its results could be extended to other capitalistic and energyintensive sectors, such as buildings or transportation.

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# **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 (_l_)||
|_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_<sup>′</sup><br>_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<sup>−1</sup>|
|_r_|discount rate|yr<sup>−1</sup>|
|_Fi_|emission rate of technology _i_|tCO2/GWh|
| ̄<br>_M_|carbon budget|tCO2|
|_u_(∑_i qi,t_)|consumer utility|$/yr|

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The present value Hamiltonian associated with Problem 9 reads: The conditions are: The complementary slackness conditions are: 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): 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): 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 production is eventually replaced by gas or renewable power. Let _Th_<sup>bethedatewhen</sup> high-carbon production stops: This happens necessarily before the steady state: Similarly, let _Ti_<sup>+be the date when production of technology</sup><sup>_i_∈{</sup><sup>_l,z_} starts, and</sup> _Tl_<sup>thedatewhengasproductionstops.Notethatonlylow-carbonproduction</sup> eventually stops, since zero-carbon energy is used indefinitely during the steady state.<sup>17</sup> Let _T_<sup>_β_</sup> 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_<sup>_β_</sup> coincides with the end of production from low-carbon fuel: Optimality then requires that fossil fuel prices follow the Hotelling rule: their current price _αl,t_ is increasing at the discounting rate before they are exhausted (A.6, A.9):

# _B.3. Investment and under-utilization of capacities_

For coal to be phased out, production in one of the alternative technologies must have started: Let _τi_<sup>+≤</sup><sup>_T_+</sup> _i_<sup>bethedatewheninvestmentincapacity</sup><sup>_i_starts,and</sup><sup>_τ_-</sup> _i_<sup>thedate</sup> 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.

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