Fossil exploration and the energy transition

Inge van den Bijgaart∗and Mauricio Rodriguez Acosta†

December 19, 2019

PRELIMINARY

Abstract

Despite ambitious climate mitigation goals and the rapid expansion of renewable energy capac-ity, investment in new fossil exploration and development continues to be high. This raises concernsabout potential conflicts between meeting climate targets, and a need to abandon developed fossilreserves. We put forward a model of the energy transition, where we explicitly consider explorationand development of fossil reserves, fossil extraction, and investment in renewable energy capacity.We identify two distinct renewable cost thresholds that separate whether a fossil phase-out involvesleaving fossil reserves undeveloped, and developed reserves unextracted. We show that currentexploration of fossil resources need not be incompatible with an eventual phase-out, even if thisinvolves abandoning reserves. Finally, we consider a carbon budget and establish that the optimalimplementation of a binding budget requires abandoning developed reserves.

Keywords: carbon budget; energy transition; fossil exploration; nonrenewable resources; renewable energy.

JEL Classification: Q20; Q31; Q35; Q38.

∗Department of Economics, University of Gothenburg, Gothenburg, Sweden†Department of Economics, Universidad del Rosario, Bogota, Colombia

1 Introduction

Economic activity requires a continuous supply of energy. To date, the global economy is heavily reliant onfossil resources. As fossil stocks continue to be burned at a rapid pace, investments are required to satisfy theenergy needs of the future. These investments can broadly take one of two forms: exploration and developmentof fossil reserves, or investment in renewable energy technologies. When successful, the former replenishesthe stock of readily recoverable reserves of the fossil resource, while the latter creates and deploys substitutetechnologies.

Next to the need for a stable energy supply, environmental concerns have put the composition of the globalenergy mix under increased scrutiny, and decarbonization has become an important policy goal. Meeting globalclimate targets will require a rapid transition away from fossil fuels, and towards renewable sources of energy.To promote this transition, many countries have introduced incentives, ranging from electric vehicle subsidies,to feed-in tariffs for renewable energy, and carbon pricing schemes. In response to these incentives, renewableenergy capacity investment has rapidly expanded, from $40 billion in 2004, to $300 billion by 2017 (FrankfurtSchool-UNEP Centre/BNEF, 2019).

Still, concurrent with the rapid growth in renewable energy investment, investment in fossil energy contin-ues to be substantial. In 2017 alone, investment in exploration and development of new oil and gas reservesamounted to $450 billion, and for every single year to date, investment in fossil energy supply has exceededinvestment in renewable energy generation (IEA, 2018). These investments raise concerns. As documentedby Muttitt (2016), the carbon emissions embodied in currently developed reserves of oil and gas exceed thecumulative emission budget associated to the 1.5 degree target. Hence, keeping global average warming below1.5 degrees already requires abandoning developed oil and gas reserves. For the 2 degree budget, forfeiting theextraction of developed stocks of oil and gas is not yet necessary, as long as a positive share of developed coaldeposits are abandoned.1 Further investments in new fossil energy then either implies that climate goals willnot be met, or that greater amounts of developed fossil reserves will be abandoned.

In this paper, we analyze the energy transition, explicitly considering the development of new fossil reservesalongside investment in renewable energy capacity. We assess under what conditions, developed fossil reserveswill be abandoned, and rationalize exploration and development of new fossil reserves under anticipated stockabandonment. To perform this assessment, we put forward a dynamic model of the energy transition. In thismodel, the energy needs of the economy can be satisfied by fossil or renewable energy. Fossil energy generationrequires extraction of fossil resources, which need to be explored and developed before they are available forextraction. Realistic limits on the rate at which resources can be extracted provide a rationale for keepingpositive levels of developed reserves, which is consistent with empirical facts. Renewable energy capacity startsoff low, and can be expanded through investment. Rapid expansions however are costly, as they limit the extentto which firms can benefit from technological progress that reduces costs, or mitigates intermittency challenges.

We first consider the decentralized equilibrium with constant taxes on fossil resource use. We find that fossilresources are left undeveloped only if, at the margin, the cost of renewable energy is below the total costs of thelast drop of fossil energy. This total cost is equal to the sum of of exploration, extraction and use cost of one unitof fossil energy, including any taxes. Developed stocks are abandoned (i.e., left unextracted) only if a second,

1Here we consider only developed reserves. McGlade and Ekins (2015) consider both developed and undeveloped reserves, and establishthat, “globally, a third of oil reserves, half of gas reserves and over 80 per cent of current coal reserves should remain unused from 2010 to2050 in order to meet the target of 2C.” (McGlade and Ekins, 2015, p188)

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more stringent, condition is satisfied: if the cost of renewable energy is below the tax-inclusive extraction anduse cost only. These cost thresholds can be quite distinct; for conventional estimates of oil cost for instance, theextraction cost amounts to about half to two-thirds of the total per-barrel cost of oil.

An anticipated abandonment of developed reserves does not rule out positive levels of fossil explorationand development. Even if in the long run, high fossil energy taxes or low-cost renewable energy supply willlead to the abandonment of developed fossil reserves, in the short run, rapid expansions of renewable energycapacity are costly. The ensuing slow expansion of renewable energy capacity then creates demand for fossilenergy in the short run, which, if needed, will be met by increasing fossil extraction capacity through additionalexploration.

We subsequently analyze the optimal energy transition for an exogenously imposed climate target, in theform of a carbon budget. Whenever this budget is binding, the optimal fossil phase-out trajectory alwaysinvolves the abandonment of developed reserves. The implementation of this trajectory will require a tax onfossil energy that is rising at the rate of interest. This result is independent of the level of the carbon budget.Instead, the combination of a geological extraction constraint that limits the maximum rate at which fossilresources can be extracted, and an increasing fossil energy tax, give renewable energy a competitive edge. Thiscauses fossil energy to be fully displaced before developed reserves can be depleted.

Our paper builds on prior work on fossil resource exploration, development and extraction. An early con-tribution on fossil resource management that explicitly captures extraction and exploration decisions is Pindyck(1978). More recent contributions are Venables (2014) and Anderson et al. (2018).2 Venables (2014) considerstwo margins of oil extraction; field openings and depletion of open fields. He highlights the role of geologyin limiting the capacity to adjust the rate of field extraction, and puts forward and alternative rule for resourceprice growth which is independent of the interest rate. Anderson et al. (2018) present empirical evidence thatshows that oil production from existing wells is unresponsive to price shocks, while exploration and field de-velopment activities are strongly price sensitive. To explain these empirical regularities, they put forward amodel related to Venables (2014), where firms make decision regarding both resource exploration and extrac-tion. Investment in exploration increases the available stock, while extraction reduces it. Their model featuresan extraction constraint, which limits the maximal rate at which resources can be extracted. This introducesan incentive to maintain a positive stock of developed reserves, and leads to equilibria in which extraction istypically constrained.

In a related work, Bornstein et al. (2017) develop a stochastic model of the oil industry in which firmsdecide on extraction and investment in exploration capital. Their model features a lag between investment inexploration and the consequential increase in available reserves and production. Exploration investments serveto explain the differential impact of demand and productivity shocks on oil prices, production, and investmentin the short run.

Venables (2014), Anderson et al. (2018) and Bornstein et al. (2017) present a detailed models of the fossilsector, which account for both exploration and extraction decisions, and generate resource price dynamics inline with empirical regularities. They however do not explicitly incorporate renewables in their analyses andthus cannot evaluate the energy transition towards a greater share of renewable energy supply.

Vice versa, the body of literature considering resource extraction in the presence of renewable energy sub-stitutes, usually abstracts from resource exploration and development. Contributions in this literature typically

2See also Bai and Okullo (2018), Boyce and Nøstbakken (2011), and Cairns (2014). There also exists a literature that focuses on theincentives to engage in exploration, taking the extraction decision as exogenous. See for instance Nystad (1985), and Thompson (2001).

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study the energy transition towards renewables and the effects of environmental policies, such as carbon taxesand subsidies to renewables. Here one can distinguish between the frameworks that take the renewable supplycurve as constant and given (Heal and Schlenker, 2019; Tahvonen, 1997; van der Ploeg and Withagen, 2012,2014), and those accounting for the evolution of renewable energy supply over time.3 The dynamics of renew-able supply may be driven by investments in renewable energy capacity (Amigues et al., 2015; Coulomb et al.,2019), or by technological progress in the renewable energy sector. Where the latter may be due to learning-by-doing (e.g., Rezai and van der Ploeg 2017a,b) or purposeful R&D investments (see for instance, Acemogluet al., 2012; van der Meijden and Smulders, 2017, 2018). As this literature abstracts from fossil exploration anddevelopment, it does not distinguish between (the abandonment of) undeveloped and developed fossil reservesin the energy transition.

An exception is the recent contribution by van der Ploeg and Rezai (2019), who consider fossil resourceexploration, extraction, and a transition to a renewable backstop. The focus of their analysis however is differentto ours; they are primarily interested in the effect of policy on stranded assets. Additionally, by considering abackstop technology that delivers renewable energy at infinitely elastic supply, they cannot capture the importantdistinction between the present day, with insufficient renewable capacity to satisfy energy demand, and the longrun, when sufficient stocks of renewable energy capacity may have been built.

The remainder of the paper is structured as follows. The model setup and firm optimization problems arepresented in Section 2. Section 3 discusses the long and short run equilibria in the energy market. In Section 4we consider the optimal energy transition under a carbon budget. A numerical analysis of the model is presentedin Section 5. Section 6 concludes. Detailed derivations and proofs can be found in the appendix.

2 Model

We consider a stylized model of the energy market. Energy is produced from fossil and renewable resources.The extraction of fossil resources depletes developed reserves. To replenish these reserves, firms can investin fossil exploration and development. Renewable energy is generated from a renewable capital stock, whereinvestment in this capital stock will expand the stock of renewable energy capacity. Throughout this section,we assume all markets are competitive. This allows us to characterize both the fossil and renewable energysector as represented by a single firm. In the fossil energy sector, the firm chooses levels of extraction andexploration activities that maximize firm value. The firm maximization problem in the renewable energy sectoris similar, with the renewable energy firm instead choosing levels of renewable energy supply and investment innew renewable energy capacity.

2.1 Setup

Energy supply The setup is as follows. Energy can be supplied by two sources, fossil and renewable. Wedenote energy from fossil sources by EF , and energy from renewable sources by ER, such that we obtain thefollowing accounting equation for total energy supply:

E(t) = EF(t)+ER(t). (1)3van der Ploeg and Withagen (2015) present a synthesis of the implications of alternative policies when the renewable supply curve is

exogenously given.

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Fossil energy The production of each unit of fossil energy requires one unit of fossil resources, which isextracted from a developed stock of reserves, S. To convert EF units of fossil resources in to fossil energy, thefirm incurs cost CF(EF). This cost includes all cost associated with extracting the fossil resource, transformingit into energy, and any net taxes imposed on the extraction or use of the resource. Common examples of suchtaxes are environmental taxes, and royalty payments to local governments. We take CF as directly proportionalto total use of fossil energy EF , and separate the pure ’extraction and use’ cost, cF , and net taxes τ as follows:

CF(EF(t)) = (cF + τ(t))EF(t), (2)

where we assume cF is constant, but allow net taxes τ to change over time, and require EF ≥ 0.Fossil resource extraction is limited by the available extraction capacity: EF ≤ Ecap

F . In line with Andersonet al. (2018), we assume this extraction capacity is endogenously determined by the level of developed fossilreserves:

EcapF (t) = κS (t) , (3)

with κ > 0.4 The parameter κ represents the maximum rate at which the developed stock can be extracted.Conversely, κ determines the minimum amount of stock holdings required to extract a unit of fossil resourcesper unit of time, which is equal to 1/κ . Our consideration of a maximum extraction capacity that is a func-tion of remaining developed stock is motivated by the so-called decline curves for individual wells, used bypetroleum engineers. These curves relate well production rates to time since well opening, assuming continuedand maximum production from a given well.5 Equation (3) mimics this approach.

The level of developed reserves can be increased through exploration, which encompasses discoveryand development of reserves. We denote the cost of exploring and developing X units of additionalstock by CX (X (t) ,U(t)), where U(t) is the level of undeveloped, yet recoverable, resource stock at timet. The cost of stock exploration and development is convex and increasing in the level of exploration:∂CX (X(t),U(t))/∂X(t) = cX (X(t),U(t))> 0 and ∂cX (X(t),U(t))/∂X(t)> 0, where in the remainder we referto cX (·) as the marginal exploration cost. This convexity of the cost function is in line with evidence presentedby Anderson et al. (2018) and Mason and Roberts (2018), who show that there exists a positive relationship be-tween the number of oil wells drilled in a given year and the marginal cost of drilling. We take the explorationcost as equal to zero when exploration is zero, CX (0,U(t)) = 0, and require X(t)≥ 0. To capture the notion thatthe most accessible resources are exploited first, we assume that the (marginal) cost of exploration increases asmore reserves have been developed: ∂CX (·)/∂U(t)≤ 0, and ∂cX (·)/∂U(t)≤ 0.6

Given the finiteness of ultimately discoverable resources in the earth’s crust, we impose the following limiton cumulative exploration: ∫

∞

tX(ν)dν ≤U(t), (4)

4See also Cairns (2014), Cairns and Davis (2001), and Thompson (2001).5For individual oil wells, maximum per-period extraction is a function of a number of reservoir characteristics, such as stock, pressure,

porosity, and the viscosity of the resource. As extraction ensues, a reservoir’s remaining stock, pressure, and consequently maximumproduction, falls. Several techniques, such as water or steam injection, can be used to increase the amount of oil recoverable from a well.We implicitly assume that κ captures such techniques.

6It is not immediate that exploration costs are strictly decreasing in U(t): over time technological progress in resource exploration anddevelopment has dramatically reduced the cost of exploration, and made previously inaccessible deposits accessible (Boyce and Nøstbakken,2011).

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with U(0) = U0 > 0. Given exploration X and fossil resource use EF , the developed resource stock S thenevolves according to

S(t) =−EF(t)+X(t), (5)

where the dot denotes the time derivative, and we assume S(0) = S0 > 0.

Renewable energy Renewable energy, ER, is produced using renewable energy capital, K. Each unit ofrenewable energy capital generates one unit of renewable energy capacity Ecap

R , such that

ER(t)≤ EcapR (t) = K(t). (6)

Producing renewable energy has a constant marginal cost cR ≥ 0. This cost captures any cost associated tothe use of renewable energy capacity, such as the fuel cost of biomass used as an input to bioenergy, and anyoperational or maintenance cost, such as the cost of staffing a nuclear power plant. Compared to the use costof fossil energy, which includes the cost of extracting the fossil resource, cR is likely small.7 Hence, in theremainder, we will assume that cR < cF + τ(t). The implication of this assumption will be that any availablerenewable energy capacity will always be utilized prior to fossil energy capacity.

A renewable energy firm can invest to expand renewable energy capacity. The cost of investment I is CI(I(t))

and is increasing and convex in the investment level: ∂CI(I(t))/∂ I(t) = cI(I(t))≥ 0, ∂cI(I(t))/∂ I(t)> 0. Thevariable cI(·) denotes the marginal investment cost, where we assume cI(0) = 0. Due to the convexity of theinvestment cost function, it is costly to rapidly expand renewable energy capacity. This can be due to capitaladjustment cost. Also a rapid expansion of renewable energy capacity may limit the extent to which the firm cantake advantage of technical progress in renewable energy technologies.8 Renewable energy capital depreciatesat rate δ > 0, which gives

K(t) = I(t)−δK(t), (7)

with K(0) = K0 ≥ 0.

Energy demand Finally, energy is used in output production. We refrain from a detailed modeling of outputmarkets, and rather assume that there exists a continuous inverse energy demand function P(ED(t)), whichis positive, and decreasing and convex in energy demand ED(t): P′(ED(t)) < 0 and P′′(ED(t)) > 0 withlimED→0 P(ED) = ∞.

2.2 Firm optimization

Both fossil and renewable energy firms maximize the present value of profits, ΠF(t) and ΠR(t). As they bothsupply energy, they face the same energy price, denoted by pE , which they take as given. We assume firmsdiscount future profits at the, exogenously given, rate of interest r > 0.

7An exception might be biomass, which has a fuel cost comparable to fossil. The fuel cost of other renewables is (near) zero (Dowlingand Gray, 2016; EIA, 2019).

8The latter would, especially in the short run, also limit the ability to effectively deal with any imperfect substitutability between fossiland renewable energy technologies.

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Fossil firm The fossil firm chooses the path of fossil extraction and fossil exploration, [EF (ν) , X (ν)]∞ν=t ,

that maximizes ΠF(t) =∫

∞

t πF(ν)e−r(ν−t)dν , with πF(t) = (pE(t)− cF − τ(t))EF(t)−CX (X(t),U(t)), subjectto the extraction constraint (3), the constraint on cumulative exploration (4), developed stock evolution (5), andthe non-negativity constraints on exploration and extraction, X(t),EF(t) ≥ 0 ∀t. This optimization problem isrepresented by the following Hamiltonian:

HF = (pE(t)− cF − τ(t))EF(t)−CX (X(t),U(t))+µS(t) [−EF(t)+X(t)]+µU (t) [−X(t)]

+φF,cap(t) [κS(t)−EF(t)]+φF,0(t) [EF(t)]+φX ,0(t) [X(t)] , (8)

with first order conditions

[EF ] : pE(t)− cF − τ(t) = µS(t)+ [φF,cap(t)−φF,0(t)] ; φF,cap(t) [κS(t)−EF(t)] = 0; φF,0(t)EF(t) = 0 (9)

[X ] : µS(t) = cX (X(t),U(t))+µU (t)−φX ,0(t); φX ,0(t)X(t) = 0 (10)

[S] :µS(t)µS(t)

+κφF,cap(t)

µS(t)= r (11)

[U ] :µU (t)µU (t)

− ∂CX (X(t),U(t))∂U(t)

= r (12)

[TVC S] : limt→∞

µS(t)S(t)e−rt = 0 (13)

[TVC U ] : limt→∞

µU (t)U(t)e−rt = 0 (14)

These first order conditions can be interpreted as follows. (9) characterizes the firm’s tradeoff in extraction.At time t, extracting fossil reserves has an immediate net benefit of pE(t)− cF − τ , while keeping developedreserves in the ground and extracting in the future has value µS(t). Then whenever the net benefit of extractionexceeds the value of keeping developed reserves in the ground, the firm would like to choose a higher extractionlevel. From here, it immediately follows that the extraction level chosen will equal extraction capacity; theextraction constraint will be binding, as implied by a positive value of the shadow cost of the extraction capacityconstraint, φF,cap(t) > 0. Conversely, if pE(t)− cF − τ falls short of µS(t), the firm is better off choosing alower level of extraction. Extraction will then be zero, and the shadow value of the nonnegativity constraint onextraction will be positive, φF,0(t)> 0.

From (10), adding a unit of developed stock through exploration has value µS(t), and cost equal to cX plusthe shadow value of undeveloped stock µU (t). Then similar to (9), whenever the value of developed stockµS(t) falls short of the cost of the first unit of exploration, cX (0,U(t))+µU (t), the firm will not find it optimalto explore, and the non-negativity constraint on X(t) will be binding (φX ,0(t) > 0). If instead µS(t) exceedscX (0,U(t))+µU (t), the firm will choose positive exploration.

The first order condition with respect to S is a no-arbitrage condition that describes that the net return toowning fossil stock, which is the rate of change in the stock value, plus immediate profit increase associatedto relieving the extraction constraint, must be equal to the market rate of return r. Similarly, the first ordercondition with respect to U dictates that along the optimal path, the net return to leaving stock undevelopedmust equal r. This return is equal to the rate of change of the stock value, plus any exploration cost savings due

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to the higher level of U .From (11) and (13) we arrive at the following solution for the value of developed stock

µS(t) = κ

∫∞

tφF,cap(ν)e−r(ν−t)dν . (15)

This solution indicates that a unit of developed reserves is valuable only insofar the extraction constraint iseventually binding. From here it immediately follows that if the firm has an extraction plan that never hits themaximum fossil extraction capacity, then it has no incentive to engage in exploration activities. The latter canbe observed through (10): as we obtain µS(t) = 0 for all t, we must either have φX ,0 > 0 or cX (X(t),U(t)) = 0.In either case, X(t) = 0.

Renewable firm Likewise, the renewable firm chooses the path of renewable energy production andinvestment in capacity, [ER (ν) , I (ν)]∞

ν=t , that maximizes ΠR(t) =∫

∞

t πR(ν)e−r(ν−t)dν , with πR(t) =

(pE(t)− cR)ER(t)−CI(I(t)), subject to the renewable capacity constraint (6), evolution of renewable energycapacity (7) and irreversibility of renewable investment. This gives the following Hamiltonian and first orderconditions:

HR = (pE(t)− cR)ER(t)−CI(I(t))+µK(t) [I(t)−δK(t)]

+φR,cap(t) [K(t)−ER(t)]+φR,0(t) [ER(t)]+φI,0(t) [I(t)] (16)

[ER] : pE(t)− cR = [φR,cap(t)−φR,0(t)] ; φR,cap(t) [K(t)−ER(t)] = 0; φR,0(t)ER(t) = 0 (17)

[I] : µK(t) = cI(I(t))−φI,0(t); φI,0(t)I(t) = 0 (18)

[K] :µK(t)µK(t)

−δ +φR,cap(t)

µK(t)= r (19)

[TVC] : limt→∞

µK(t)K(t)e−rt = 0 (20)

The first order condition with respect to EF , (17), characterizes the decision to supply renewable energy. When-ever the net benefit of generating renewable energy from capacity, pE(t)−cR, is strictly positive, the renewableenergy firm will decide to produce at full capacity (φR,cap(t)> 0). Conversely, if the energy price falls short ofcR, the non-negativity constraint on renewable energy production will be binding, φR,0(t) > 0, and renewableenergy production will be zero. Investment can relieve the capacity constraint on renewable energy generation.As specified by (18), the firm chooses the investment level such that the marginal cost of investment cI is equalto the value of additional renewable capacity µK .9 Finally, the no-arbitrage condition (19) describes that the rateof change in the capacity’s value, net of depreciation, plus the immediate value of allowing for higher renewableenergy supply, must equal the rate of return r.

Equations (19) and (20) allow us to solve for the value of adding renewable capacity:

µK(t) =∫

∞

tφR,cap(ν)e−(r+δ )(ν−t)dν . (21)

Akin to (15) we find that an additional unit of renewable capacity has positive value only if, at some point intime, renewables are used at full capacity. Under realistic assumptions regarding initial conditions, one can

9As cI(0) = 0, the non-negativity constraint on I(t) is never binding and thus φI,0(t) = 0 for all t.

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show that renewables are always used at full capacity, ER = K.10 In the remainder of the paper we assume thatthese conditions are satisfied, which allows us to use (17) and (21) to write

µK(t) =∫

∞

t[pE(ν)− cR]e−(r+δ )(ν−t)dν . (22)

Equation (22) is straightforwardly interpreted. Each unit of renewable capacity generates one unit of renewableenergy, and depreciates at rate δ . In turn, one unit of renewable energy earns an immediate net return of pE−cR.Hence, µK captures the present value of adding a unit of renewable capacity. Equation (18) then pins down theinvestment level I.

3 Equilibrium in the energy market

This section further considers fossil firms’ equilibrium decisions regarding exploration, and extraction, andrenewable firm’s investment decisions. The discussion is structured as follows. To start, we consider the equi-librium in the long run, focusing on the type of equilibrium that will prevail. Ultimately available resources,U0 +S0, are finite. Thus a long run equilibrium with positive fossil use is not feasible; in the long run, energyuse must be fully renewable. Yet, this does not automatically imply that all fossil reserves will eventually bedeveloped, nor that all developed reserves will be extracted. Rather, we show that depending on the long-runefficiency of renewable versus fossil energy, fossil firms may cease exploration prior to U = 0. Additionally, wedetermine under what conditions, fossil firms leave developed reserves unextracted in the long run.

As the model features multiple state variables, with endogenous exploration and investment costs, andpotentially-binding constraints on exploration and extraction, a comprehensive characterization of the shortrun equilibrium requires resorting to numerical methods. Still, several analytical insights regarding exploration,extraction and renewable investment decisions in the short run can be obtained. In particular, we establish thatbeing on a trajectory towards a fossil phase-out with an anticipated abandoning of developed reserves, doesnot rule out positive fossil exploration. Despite this, supply-side interventions that restrict fossil exploration,increase the returns to renewable investment and hence incentives to invest in renewable energy.

An equilibrium consists of paths [EF (t) , X (t) , ER (t) , I (t) , S (t) ,U(t), K (t) , pE (t) , τ (t)]∞t=0 such that thefossil and renewable firms are maximizing the net present value of profits taking the path of prices and taxes asgiven, and the energy market clears, E(t) = ED(t). For now, we consider only equilibria with constant fossilfuel taxes τ . In Section 4 we consider the equilibrium where τ follows a path that optimally implements acarbon budget.

3.1 Fossil reserves in the long run

We define the long-run equilibrium as the equilibrium for which the levels of developed and undeveloped fossilstock and renewable capacity are constant, Sss = 0, U ss = 0 and Kss = 0. Here the ss superscript indicates weare in the long run equilibrium (steady state). As positive fossil resource development is incompatible with aconstant stock of undeveloped resources, it follows that in the long-run equilibrium, fossil resource exploration

10As the use cost of renewable capacity lies below the use cost of fossil, cR < cF , renewable capacity is always used first. From here itfollows that whenever a positive amount of fossil energy is used, renewables must be used at full capacity. In Appendix B.2.1 we show thatwhenever a strictly positive amount of fossil energy is used at t = 0, ER(t) = K(t) for all t. As approximately 80 percent of total energy isgenerated from fossil resources (IEA, 2018), we consider this a realistic assumption.

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must be zero: X ss = 0. From (5), the stock of developed fossil reserves is constant only if fossil resourceextraction equals exploration. Hence, with X ss = 0, also fossil resource extraction must be zero in the long run:Ess

F = 0. Finally, from (7), a constant level of the renewable capital stock requires that investment in renewablecapacity equals depreciation: Iss = δKss.

The above implies that energy supply will be fully renewable in the long run: Ess = EssR . From (17)-(19) we

then obtain the following implicit solution for the long-run equilibrium energy price:11

P(Ess) = zR(Ess), (23)

with zR(ER)≡ cR +(r+δ )cI(δER) evaluated at ER = Ess. In the long run, the energy price equals the marginalcost of producing energy from renewables, zR(ER). This marginal cost comprises the marginal use cost cR, plusa marginal cost of maintaining a renewable energy capacity of ER. The latter is equal to the rental rate r+ δ ,multiplied by the marginal cost of capital. The marginal cost of capital in turn is equal to the marginal investmentcost evaluated at the level of investment I = δER. As ∂cI/∂ I > 0, (23) establishes a positive relationshipbetween the level of steady-state renewable energy use and the long-run equilibrium price of energy; as thelong-run marginal costs of renewable energy is increasing in the level of renewable energy use, higher levels ofEss require higher prices to support them.

Even though fossil energy use is zero in the long run, undeveloped reserves may be positive, U ss > 0, andthus part of the ultimately available fossil resources may never be explored. Similarly, the long-run equilibriummay feature positive levels of developed stocks, Sss > 0: a positive amount of fossil reserves is developed, yetnever extracted.

We can then establish the following

Proposition 1. If zR(Ess)> cF +τ+(r+κ) cX (0,0)κ

, then all resources will be developed in the long run, U ss = 0.

Only if zR(Ess)≤ cF + τ +(r+κ) cX (0,0)κ

, the long run equilibrium may feature U ss > 0.

Proof. See Appendix B.1.1

Proposition 1 specifies that unless renewable energy production is sufficiently efficient relative to fossil inthe long run, all fossil resources will be eventually developed. From (23), the long run energy price is determinedby the total marginal cost of renewables, zR(Ess). Resources may then be left undeveloped only if, at this longrun energy price, it is not economically viable to continue to explore, and over time, extract fossil resources.

Whether this is the case depends on the cost associated to developing and extracting the final unit of unde-veloped reserves. This cost is equal to cF + τ +(r+κ) cX (0,0)

κ. Similar to zR(ER), this cost can be separated

into the cost associated to extracting and using a unit of fossil energy, cF + τ , and the relevant cost of fossilextraction capacity, (r+κ) cX (0,0)

κ. From (3), each unit of EF requires 1/κ units of S. This gives a per unit

cost of fossil energy extraction capacity of cX (0,0)/κ (see (10)).12 In the long run, fossil extraction will be atcapacity. With extraction equal to a share a share κ of the existent fossil capacity, this gives a net rental rate ofr+κ , from which we obtain a per unit capacity cost of (r+κ) cX (0,0)

κ. A higher level of zR(Ess) then makes it

more likely that, in the long run, all reserves will be developed. In turn, zR(Ess) is higher the higher is use costs,cR, or investment cost (r+δ )cI(δEss).

11Appendix B.2.2 establishes that the long run equilibrium is globally stable.12Note that from (12) and (14), the shadow value of the undeveloped stock, µU , will equal zero if resources remain undeveloped (U ss > 0).

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Note that the second part of Proposition 1 formulates a necessary condition for fossil reserves to remainundeveloped in the long run. As we highlight in the next subsection, exploration may be positive in the shortrun, independently of whether zR(Ess) is greater or less than cF + τ +(r+κ) cX (0,0)

κ. This implies that even

though the long run equilibrium may be consistent with positive levels of undeveloped reserves, exploration anddevelopment activities may still deplete the stock of undeveloped reserves in the short run. This is especiallylikely if the initial level of undeveloped reserves, U0 is low. Conversely, higher levels of U0 make it more likelythat reserves remain undeveloped if zR(Ess)< cF + τ +(r+κ) cX (0,0)

κ.

Proposition 1 only considers whether or not reserves remain undeveloped in the long run. Similarly, we canspecify under what conditions developed reserves may remain unextracted, i.e., abandoned. Here we establishthat if fossil use costs or taxes are relatively high, such that cF + τ > zR(Ess), then developed reserves willnot be fully depleted in the long run. If instead cF + τ < zR(Ess), the long run equilibrium is consistent onlywith Sss = 0. This result is intuitive. Fossil firms will be willing to extract any remaining stock as long as theprice of energy, pE , is sufficiently high to compensate for the cost associated with extracting and using the fuelcF + τ . In the long run, the energy price will be equal to zR(Ess). Hence, if cF + τ < zR(Ess), the fossil firmwill remain willing to extract any remaining stock in the long run. If instead cF + τ > zR(Ess), the fossil firmhas no incentive to extract any remaining developed stock in the long run. As the extraction constraint preventsthe firm to fully extract its stock in finite time, some developed stock must remain in the ground. Proposition 2summarizes this result.

Proposition 2. If zR(Ess)< cF +τ , then developed fossil stocks will remain unextracted in the long run, Sss > 0.

If zR(Ess)> cF +τ , then all developed fossil stocks will be extracted in the long run, Sss = 0. If zR(Ess) = cF +τ ,

developed fossil stocks may remain unextracted in the long run.

Proof. See Appendix B.1.2

Table 1 presents the three types of feasible long-run equilibria that can emerge. Whenever zR(Ess) exceedsthe cost associated with fossil energy, all fossil reserves will be fully developed (U ss = 0) and depleted (Sss = 0).Fossil reserves may remain undeveloped (U ss ≥ 0) once zR(Ess) falls below the cost associated with using anddeveloping new fossil reserves, equal to cF + τ + (r+κ) cX (0,0)

κ. Finally, if the long run cost of renewables

zR(Ess) lies below cF + τ , which is the cost of exploiting previously-developed reserves, then fossil firms willeventually abandon part of the developed stock of fossil reserves (Sss > 0).

Table 1: Development and depletion of reserves in the long run

zR(Ess)> cF + τ +(r+κ) cX (0,0)κ

full development: U ss = 0no abandonment of developed stock:

Sss = 0

zR(Ess) ∈(

cF + τ, cF + τ +(r+κ) cX (0,0)κ

] full or partial development: U ss ≥ 0no abandonment of developed stock:

Sss = 0

zR(Ess) = cF + τ

full or partial development: U ss ≥ 0possible abandonment of developed stock:

Sss ≥ 0

zR(Ess)< cF + τfull or partial development: U ss ≥ 0

abandonment of developed stock: Sss > 0

11

3.2 Fossil exploration in the short run

It may take many years for energy transition to be complete. Up to that point, the economy’s energy mixwill contain positive levels of fossil energy. It is however not immediately clear whether exploration will alsobe positive in the intermediate period. Suppose for instance that zR(Ess) < cF + τ , in which case the fossilfirm anticipates that it will have to abandon positive levels of developed reserves at some future point in time.Developing additional reserves would then increase the amount of stock to be abandoned, suggesting it is neveroptimal for the firm to do so. In this section, we highlight that this is only part of the story. In fact, positiveexploration can be perfectly compatible with the eventual abandonment of developed reserves. Because it takestime to build a substantial renewable energy generation capacity, current energy prices may exceed those inthe long run. Whenever this is the case, the short run return to extracting fossil stock is high, and the fossilfirm has an incentive to increase present-day extraction in response. Yet extraction constraints may limit thefirm’s ability to do so. Exploration then acts as a lever to increase extraction; it directly relieves the extractionconstraint by adding extraction capacity, hence allowing the fossil firm to take advantage of the high energyprice and increase fossil energy production nonetheless.

The above insights more formally established below, where we discuss in more detail the relationship be-tween the current returns to investing in fossil energy, µS, and the (future) slackness of the extraction constraint.

3.2.1 Extraction constraints and returns to fossil exploration and stock development

By exploring and developing fossil reserves, fossil firms invest in the option to extract stock in the future. Asexplained above, this option has positive value, µS > 0, only insofar extraction is constrained at some pointin the future; if the extraction constraint is never binding, adding developed reserves through exploration anddevelopment will not alter extraction decisions, and hence profits will remain unchanged.

Mathematically, this can be seen as follows. If the extraction constraint is never binding, the shadow value ofthe extraction constraint, φF,cap(t), is always equal to zero (see (9)). From (15) it follows that the shadow valueof the developed fossil stock also equals zero; µS(t) = 0. Setting µS(t) = φF,cap(t) = 0 in (9), and observingthat in equilibrium we must have pE = P(E), we obtain the following expression for equilibrium prices:

P(E(t)) = cF + τ−φF,0(t). (24)

If the extraction constraint is never binding, we are effectively in a scenario where developed reserves areabundant. As developed reserves are abundant, and markets are competitive, fossil energy must then sell at themarginal cost of use plus taxes, cF + τ . If equilibrium energy prices P(E(t)), lie below this marginal cost, nofossil energy will be sold, i.e., the nonnegativity constraint on extraction must be binding: φF,0(t)> 0.

From equation (24) we can immediately conclude that only under very specific circumstances the extractionconstraint will never bind. In particular, equation (24) puts a lower bound on the per-period energy use E(t) thatensures µS(t) = 0; whenever energy supply is sufficiently low, such that P(E(t)) > cF + τ , (24) cannot hold.Put differently, observing equilibrium energy prices in excess of the use cost of fossil necessarily means thatdeveloped fossil reserves command a ’scarcity rent’ in the form of a positive value of holding stock. Thoughenergy supply E(t) is endogenous, it is bounded from above. At any point in time, a renewable energy capacityof K(t) generates an equal amount of energy from renewables. Fossil energy supply in turn is at most equalto extraction capacity κS(t). Total energy supply E(t) can thus never exceed K(t) + κS(t). From here, it

12

immediately follows that for sufficiently low renewable energy capacity K(t) and fossil extraction capacityκS(t), (24) cannot be satisfied. Likewise, an outward shift of the demand function, which increases P(E(t))

for given levels of E(t), will make it more likely that extraction will be at capacity, either currently, or at somepoint in the future.

Then whenever (24) is violated, at any point in time, we must obtain that the shadow value of developedstock, µS, is strictly positive, and there exists an incentive to increase the stock of developed reserves throughexploration and stock development. Whether this indeed results in positive exploration then depends on whetherµS is higher or equal to cX .

Important to note is that equation (24) can be violated independently of which of the long run equilibria, asdescribed by Propositions 1 and 2, and Table 1 prevails.13 Put differently, the anticipation of abandonment ofdeveloped stock is not a sufficient condition for a termination of fossil resource exploration. This is summarizedby the following Proposition.

Proposition 3. Exploration may be positive even if developed reserves are eventually abandoned.

Proof. See Appendix B.1.3

The result that we observe positive exploration in the scenario with full development and no abandonment isnot surprising; the long run energy price support positive exploration, and, eventually, exploration is terminatedsimply because all reserves have already been developed. This is no longer the case when developed reserves areabandoned in the long run. Nonetheless, the fossil firm might still find it optimal to explore and develop stockduring the transition. This is the case if current energy capacity is sufficiently low. With sufficiently low currentenergy capacity, energy prices and thus the return to extraction will be very high in the short run. Explorationthen allows for greater extraction in the short run, as it relieves the extraction constraint. Then, despite the factthat the fossil firm anticipates that in the long run, energy prices will be too low to sustain exploration, andmay even force the firm to abandon part of its developed stock, the high short run energy price make currentexploration is profitable.14

4 Fossil phase-outs under a carbon budget

The previous section considers the decentralized market equilibrium for a given and constant fossil fuel tax τ .Such taxes are motivated by the substantial environmental externalities associated to the use of fossil fuels, mostnotably the climate externality. When set sufficiently high, fossil fuel taxes can prevent resources from beingdeveloped in the long run, and may even lead to an abandonment of developed reserves. Yet such a phase-outmay be slow, and thus substantial amounts of fossil fuels may be burned in the short run. Additionally, there isno ex-ante reason to presume that a constant fossil fuel tax indeed implements the socially optimal phase-outtrajectory.

13In fact, one can show that unless developed reserves may be abandoned in the long run, i.e., unless zR(Ess) ≤ cF + τ , (24) willnecessarily be violated for some t, and hence extraction must be constrained at some point in time. This follows from the fact that only ifthe cost of renewable energy use and investment are sufficiently low, we will observe an equilibrium renewable energy capacity such thatenergy prices are below cF +τ in the long run. Put differently, whenever zR(Ess)> cF +τ , µS(t)> 0 and there exists an incentive to engagein fossil exploration.

14This important role of short- to medium-run energy demand is also highlighted in IEA (2018), who point out that, unless more ambitiousclimate change policy is implemented, current levels of investment towards developing fossil reserves are insufficient, causing substantialrisk of a fossil supply shortfall, and thus high fossil energy prices, in the next decade.

13

In this section we consider the optimal energy transition, including the corresponding path of fossil fueltaxes, compatible with implementing a pre-determined climate target. Instead of explicitly modeling the carboncycle and atmospheric temperature adjustment, we adopt a carbon budget approach. Under this approach, a cu-mulative emission budget is determined based on a pre-defined warming target. Carbon budgets are commonlyemployed for evaluating temperature stabilization policies; the IPCC 5th assessment report defines cumulativeemission budgets consistent with a global temperature limit of 2 degrees relative to pre-industrial levels (Stockeret al., 2014), and similarly the IPCC special report on 1.5 degrees establishes remaining budgets of about 420and 580 GtCO2 for a two-thirds and one-half chance of limiting warming to 1.5C, respectively (Rogelj et al.,2018). The use of carbon budgets as a tool to evaluate policy exploits the insight that has emerged over the pastdecade that the maximum global mean temperature increase is approximately linear in cumulative CO2 emis-sions (Stocker et al., 2014; Allen, 2016; Rogelj et al., 2016; Dietz and Venmans, 2019), and thus independentof the exact timing of those emissions.

We consider a carbon budget that lies below the carbon embodied in ultimately available resources. Assuch, the budget could for instance resemble the budget associated with the 2 degree warming goal, yet we alsoallow for more or less ambitious goals. In any case, extracting all available fossil energy is incompatible withstaying within the carbon budget.15 We then establish that under a binding carbon budget, the optimal energytransition will involve the abandonment of a positive amount of developed reserves. This optimal transition canbe implemented by a positive carbon tax that is rising at the rate of interest.

To determine the optimal transition paths, we define the socially optimal allocation as the allocation thatmaximizes the present value sum of consumer and producer surplus∫

∞

t[CS(ν)+PS(ν)]e−r(ν−t)dν , (25)

subject to (1)-(7) and cumulative emissions remaining below a pre-determined carbon budget B:∫∞

tEF(ν)dν ≤ B(t), (26)

where we take B(0) = B0 <U0 +S0 as strictly positive and finite, and B(t) =−EF(t). Consumer surplus CS(ν)

is conventionally defined as the area under the demand curve and above prices

CS(t)≡∫ E(t)

0(P(e)− pE(t))de,

and producer surplus is the sum of fossil and renewable firm profits: PS(t) ≡ πF(t) + πR(t) +Ω(t), whereΩ(t) captures the lump sum recycling of taxes. The first order conditions corresponding to the socially optimalallocation are presented in Appendix A.1. From these first order conditions, we can then establish the followingresult regarding the implementation of the socially optimal allocation:

Proposition 4. The socially optimal allocation can be decentralized by a tax τ(t) that is positive and rising at

the rate of interest if the carbon budget is binding, and zero otherwise.

Proof. See Appendix B.1.4.

15We abstract from carbon capture and storage and other negative-emission technologies that would allow depletion of all availableresources to be compatible with the carbon budget (as is also the case, for instance, in Coulomb et al. 2019).

14

The carbon budget puts a constraint on cumulative extraction. If, in the absence of additional taxation, thisconstraint is not binding, then there is no need to introduce a fossil fuel tax to ensure cumulative extractionremains within the budget. If however the constraint is binding under τ(t) = 0 for all t, positive taxes will berequired to ensure that (26) is met. In the optimum, the tax is then equal to the shadow value of the carbonbudget. This shadow value in turn captures the increase in total surplus that can be obtained by relaxing thebinding carbon budget.

Along the optimal tax trajectory, taxes rise at the rate of interest. The intuition behind this is as follows.The carbon budget imposes an additional opportunity cost to extraction, as extracting one unit of fossil todayimplies that the budget is depleted, and at some point in the future, one fewer unit must be extracted. To ensurethat the firm is indifferent between depleting the budget today or at some point in the future, the cost of doingso, in present value, must be constant. From here, it follows that the nominal tax rate τ , must rise at the rate ofinterest.16

This rising tax rate then immediately implies that, unless energy prices rise accordingly, there must exist afinite time as of which fossil firms no longer find it profitable to extract stock. The presence of the renewablesubstitute prevents energy prices from continuously increasing in the long run; high energy prices encouragethe expansion of renewable energy capacity, which in turn dampens the energy price increase. As the extractionconstraint (3) prevents all fossil energy to be extracted in finite time, the optimal transition under a bindingcarbon budget always features abandonment of developed fossil reserves:

Proposition 5. The socially optimal allocation features abandonment of developed reserves if the carbon budget

is binding.

Proof. See Appendix B.1.5.

Proposition 5 states that the socially optimal allocation under a binding carbon budget always featuresabandonment of developed reserves. Hence observing levels of developed stock in excess of the carbon budgetis not at odds with satisfying the budget. To the contrary, an optimal transition requires that developed fossilstocks S(t), at some point in time, strictly exceed the remaining budget B(t).17

As developed reserves must strictly exceed the remaining budget at some point along the transition to arenewable-only energy supply, positive exploration may be observed despite S(t) > B(t).18 Whether positiveexploration is indeed optimal for the for relevant values of the carbon budget requires a numerical assessment. Inthe next section, we calibrate the model, and numerically evaluate time paths of fossil exploration and extraction,and renewable energy investment under several policy scenarios.

5 Numerical analysis

TO BE COMPLETED16As noted by Gollier (2018), “determining the optimal timing to consume this carbon budget is a problem equivalent to the Hotelling’s

problem of extracting a non-renewable resource” (Gollier, 2018, p2). Hence, optimally implementing a carbon budget requires a carbonprice following the Hotelling rule (Hotelling, 1931). See also Dietz and Venmans (2019) and van der Ploeg and Rezai (2019).

17As fossil fuels are phased-out and abandoned in finite time, and the budget is binding, we must observe S(T ) > B(T ) = 0, where wedefine T as the time of phase-out, i.e., the time such that for all t ≥ T , EF (t) = 0.

18In fact, whenever S(0)< B(0), there must exist some ν such that X(ν)> 0 and S(ν)≥ B(ν).

15

6 Concluding comments

Next to the rapid increase in renewable energy investment, investments in the exploration and developmentof new fossil reserves continue to be substantial. These investments in fossil resources raise concerns; thecarbon embodied in currently developed reserves already exceeds the remaining budget for the most ambitiousclimate goals. As such, the presence and role of fossil exploration and development alongside renewable energyinvestment warrants analysis. We have put forward a framework of the energy transition capturing explorationand development of new fossil reserves, and the gradual expansion of renewable energy capacity. We establishunder what conditions, the the energy transition may leave fossil reserves undeveloped, and when developedreserves will be abandoned (i.e., left unextracted). We show that an anticipated abandonment of developedreserves does not rule out positive exploration in the short run; high energy prices in the short run may generatea sufficient incentive to expand fossil extraction capacity by developing new fossil reserves.

16

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19

A Detailed derivations

A.1 Socially optimal allocation under a carbon budget

To determine the socially optimal allocation we maximize (25) subject to (1)-(7) and (26) and non-negativityconstraints on exploration, extraction, undeveloped resources and remaining budget, X(t),EF(t),U(t),B(t)≥ 0∀t. This gives the following Hamiltonian with FOC and transversality conditions:

H =∫ E(t)

0 (P(e)− pE(t))de+ pEEF(t)− cF EF(t)−CX (X(t),U(t))+ pEER(t)− cRER(t)−CI(I(t))

+λ PE (t) [EF(t)+ER(t)−E(t)]

+µPS (t) [−EF(t)+X(t)]+φ P

F,cap(t) [κS(t)−EF(t)]+φ PF,0(t) [EF(t)]

+φ PX ,0(t) [X(t)]+µP

U (t) [−X(t)]

+µPK(t) [I(t)−δK(t)]+φ P

R,cap(t) [K(t)−ER(t)]+φ PR,0(t) [ER(t)]+φ P

I,0(t) [I(t)]

+µPB (t) [−EF(t)]

(A.1)

and[E] : P(E(t))− pE(t) = λ

PE (t); λ

PE (t) [EF(t)+ER(t)−E(t)] = 0 (A.2)

[EF ] : pE(t)−cF +λPE (t)−µ

PB (t)= µ

PS (t)+

[φ

PF,cap(t)−φ

PF,0(t)

]; φ

PF,cap(t) [κS(t)−EF(t)]= 0; φ

PF,0(t)EF(t)= 0

(A.3)[X ] : µ

PS (t) = cX (X(t),U(t))+µ

PU (t)−φ

PX ,0(t); φ

PX ,0(t)X(t) = 0 (A.4)

[S] :µP

S (t)µP

S (t)+κ

φ PF,cap(t)

µPS (t)

= r (A.5)

[ER] : pE(t)− cR +λPE (t) =

[φ

PR,cap(t)−φ

PR,0(t)

]; φ

PR,cap(t) [K(t)−ER(t)] = 0; φ

PR,0(t)ER(t) = 0 (A.6)

[I] : µPK(t) = cI(I(t))−φ

PI,0(t); φ

PI,0(t)I(t) = 0 (A.7)

[K] :µP

K(t)µP

K(t)−δ +

φ PR,cap(t)

µPK(t)

= r (A.8)

[U ] :µP

U (t)µP

U (t)− ∂CX (X(t),U(t))

∂U(t)= r (A.9)

[B] :µP

B (t)µP

B (t)= r (A.10)

[TVCS] : limt→∞

µPS (t)S(t)e

−rt = 0 (A.11)

[TVCK ] : limt→∞

µPK(t)K(t)e−rt = 0 (A.12)

[TVCU ] : limt→∞

µPU (t)U(t)e−rt = 0 (A.13)

[TVCB] : limt→∞

µPB (t)B(t)e

−rt = 0 (A.14)

20

B Proofs

B.1 Proofs of Propositions

B.1.1 Proof to Proposition 1

Proposition 1 specifies a necessary condition for U ss > 0. The proof proceeds as follows. We first establish anintermediate result that if U ss > 0, µU (t) = 0 for all t, but not necessarily vice versa. Next, we establish thatµU (t) = 0 ∀t is consistent only with zF ≥ zR(Ess), where we define zF ≡ cF + τ +(r+κ) cX (0,0)

κ.

From (12) and (13) we obtain:

Lemma B.1. Either limt→∞ U(t) = 0, or µU (t) = 0 ∀t, or both.

Proof. From (12) we can write µU (t) = µ(0)ert , which with (13) gives limt→∞ µ(0)U(t) = 0. Hence, we musteither have limt→∞ U(t) = 0, or µ(0) = 0, or both. Finally, from (12), if µ(0) = 0, then µ(t) = 0 ∀t.

From (10) we can write

limt→∞

µS(t) = limt→∞

cX (0,U(t))+ limt→∞

µU (t)− limt→∞

φX ,0(t).

where we use that in steady state, X ss = 0. From (15) in turn we obtain limt→∞ µS(t) = κ

r limt→∞ φF,cap(t). With(9), this allows us to write

pssE = zF +

r+κ

κ

[limt→∞

µU (t)− limt→∞

φX ,0(t)−[cX (0,0)− lim

t→∞cX (0,U(t))

]]− lim

t→∞φF,0(t), (B.1)

where from ∂cX (·)/∂U(t)< 0 we know that cX (0,0)− limt→∞ cX (0,U(t))≥ 0. From (23), pssE = zR(Ess). From

here it directly follows that whenever zR(Ess) > zF , we must obtain limt→∞ µU (t) > 0. Then, by Lemma B.1,limt→∞ U(t) = 0. In turn, Lemma B.1 states that a solution with limt→∞ U(t) > 0 requires µU (t) = 0 for all t,which according to (B.1) requires zR(Ess)≤ zF .

B.1.2 Proof to Proposition 2

In steady state, pE is constant and given by (23). From (9) this implies that we can write

pssE = cF + τ + lim

t→∞µS(t)+

[limt→∞

φF,cap(t)− limt→∞

φF,0(t)], (B.2)

where we know pssE = zR(Ess), and by (15), µS(t)≥ 0 ∀t, with µS(t)> 0 only if φF,cap(ν)> 0 for some ν ≥ t.

1. Consider first zR(Ess) > cF + τ . Then (B.2) can only be satisfied if limt→∞ φF,cap(t) > 0. From here itfollows that the extraction constraint is binding in the long run: Ess

F = κSss. As Ess = EssR > 0 this requires

Sss = 0.

2. Consider next zR(Ess) < cF + τ . Then we require limt→∞ φF,0(t) > 0. Yet as φF,cap and φF,0 cannotsimultaneously be strictly positive, this gives limt→∞ φF,cap(t) = φ ss

F,cap = 0 and limt→∞ µS(t) = µssS = 0.

We then prove that φ ssF,cap = 0 implies Sss > 0 as follows.

First observe that by (3), X(t)≥ 0 and (5), S≥−κS. Then Sss = limt→∞ S(t) = 0 requires that for all finitet with S(t)> 0, there exists some ν > t with EF(ν)> 0. By (9), EF(ν)> 0 requires pE(ν)≥ cF + τ . In

21

turn, (17) and cR < cF +τ , imply that if pE(ν)≥ cF +τ , then ER(ν) =K(ν), and thus E(ν)≥K(ν). Nowdefine the level K′ such that P(K′) = cF + τ . Thus, for any finite t with S (t) > 0 a necessary conditionfor Sss = 0 is that there exists some ν > t such that K (ν) ≤ K′, and thus EF(ν) > 0 We next prove thatthis cannot be the case, as K(t)< Ess implies K(t)> 0.

Suppose K(t)< Ess. Then from the above, either EF(t)> 0, or E(t) = K(t)< Ess. From the above, eithercase requires pE(t) > P(Ess) = zR(Ess). Now suppose K(t) ≤ 0, so I(t) ≤ δK(t) < δEss. From (18) itthen follows that µK(t)< cI(δEss). Combining (17) with (19) we obtain

µK(t) = zR(Ess)− pE(t)+(r+δ )(µK(t)− cI(δEss))< 0.

From (18) this gives I(t) < 0 and thus K(t) < Ess and pE(t) > zR(Ess) for all t. Yet by (21), pE(t) >

zR(Ess) is inconsistent with µK(t)< cI(δEss). Hence, if K(t)< Ess, we must have that K(t)> 0.

From here it follows that K cannot be permanently below K′, and thus Sss = limt→∞ S(t) = 0 cannot hold.

3. Finally consider zR(Ess) = cF + τ . To establish that zR(Ess) = cF + τ is not a sufficient condition toestablish whether Sss = 0 or Sss > 0 we consider the following setting. Suppose also that for all ν ≥ t,φF,cap(ν) = 0. From (15) and (10), this gives µS(ν) = 0 and X(ν) = 0 for all ν ≥ t. By (9), (18) and(22), pE(ν) = pss

E , µK(ν) = µssK and thus I(ν) = Iss for all ν ≥ t. From (7) the latter implies K(ν) =

Kss− [Kss−K(t)]e−δ (ν−t), which with (1) gives

EF(ν) = [Kss−K(t)]e−δ (ν−t), (B.3)

where we use that by pE(ν) = pssE , E(ν) = Ess = Kss for all ν ≥ t. This extraction path is feasi-

ble and consistent with φF,cap(ν) = 0 for all ν ≥ t as long as EF(ν) ≤ EcapF (ν) for all ν ≥ t. From

(3), (5) and (B.3), EF(ν) ≤ EcapF (ν) for all ν ≥ t requires S(t) ≥ Y (t,ν) for all ν ≥ t with Y (t,ν) ≡

[Kss−K(t)][

1κ

e−δ (ν−t)+ 1δ

[1− e−δ (ν−t)

]]. Observe that ∂Y (·)/∂ν ≤ 0 if δ ≥ κ and ∂Y (·)/∂ν > 0 if

δ < κ . We then consider these two cases:

(a) Suppose δ ≥ κ . Then iff S(t) ≥ Y (t, t), S(t) ≥ Y (t, t) for all ν ≥ t. Hence we require S(t) ≥1κ[Kss−K(t)]. Suppose that this condition is satisfied. Then total extraction along the transition is

equal to ∫∞

tEF(ν)dν =

1δ[Kss−K(t)] , (B.4)

where we use (B.3). From here it follows that if S(t) = 1κ[Kss−K(t)] and δ = κ , then S(t) =∫

∞

t EF(ν)dν and thus Sss = 0. If instead S(t) > 1κ[Kss−K(t)], or δ > κ , or both, then S(t) >∫

∞

t EF(ν)dν and Sss > 0.

(b) Suppose δ < κ . Then iff S(t)≥ limν→∞ Y (t,ν), S(t)≥Y (t,ν) for all ν ≥ t. Hence we require S(t)≥1δ[Kss−K(t)]. Suppose that this condition is satisfied. Then total extraction along the transition is

again equal to (B.4). It then follows that if S(t) = 1δ[Kss−K(t)], then S(t) =

∫∞

t EF(ν)dν and thusSss = 0. If instead S(t)> 1

δ[Kss−K(t)], then S(t)>

∫∞

t EF(ν)dν and Sss > 0.

All in all, zR(Ess) = cF + τ is not a sufficient condition for either Sss = 0 or Sss > 0. Rather, initialconditions on S(t) and K(t) and parameter values may determine whether Sss = 0 or Sss > 0.

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B.1.3 Proof to Proposition 3

From (9) and (11) we can write

µS(t) = (r+κ)µS(t)−κ (pE(t)− (cF + τ)+φF,0(t))

and thusµS(t) = κ

∫∞

t[pE(ν)− (cF + τ)+φF,0(ν)]e−(r+κ)(ν−t)dν .

Rearranging this term gives

µS(t) = κ

∫∞

t[pE(ν)− pss

E +φF,0(ν)]e−(r+κ)(ν−t)dν +κ

r+κ[pss

E − (cF + τ)] , (B.5)

where we know that pssE = zR(Ess). From Proposition 2, developed stock will be abandoned if zR(Ess)< cF +τ .

Yet zR(Ess) < cF + τ does not preclude that pE(ν)− pssE + φF,0(ν) > 0 for some ν . Consider for instance the

case with zR(Ess) = cF + τ − εF with εF positive and arbitrarily small. Suppose also that K(t)+κS(t) < Ess,from which follows that E(t) < Ess and pE(t) > pss

E . Then we must obtain that µS(t) > 0 by the following.From (15), we can write

µS(t) = µS(t ′)e−(r+κ)(t ′−t) +κ

∫ t ′

t[pE(ν)− pss

E +φF,0(ν)]e−(r+κ)(ν−t)dν ,

with t ′ > t and we set pssE = cF +τ . Then by (15), µS(t)≥ 0 and µS(t ′)≥ 0. pE(t)> pss

E then implies µS(t)> 0.From 10, µS(t) = cX (X(t),U(t))+µU (t)−φX ,0(t). Now suppose that additionally, the marginal cost of the firstunit of exploration is zero: limX→0 cX (0,U(t)) = 0. Then µS(t) > 0 implies either X(t) > 0 or µU (t) > 0 orboth. As µU (t)> 0 requires that U ss = 0, X(ν)> 0 for some ν ≥ t as long as U(t)> 0. Hence, zR(Ess)< cF +τ

is not a sufficient condition for X(ν) = 0 ∀ν ≥ t.

B.1.4 Proof to Proposition 4

A comparison of (A.2)-(A.9) to (9)-(12) and (17)-(19) reveals that the social optimum and decentralized equi-librium coincide if τ(t) = µP

B (t). From (A.10), we then obtain

τ(t)τ(t)

=µP

B (t)µP

B (t)= r.

(A.10) and (A.14) additionally imply that µPB (t) = 0 for all t, or limt→∞ B(t) = 0.

B.1.5 Proof to Proposition 5

Suppose that the carbon budget is binding but the long run equilibrium features no abandonment of developedstock. From Proposition 4 this gives τ(t)> 0 with τ(t)/τ(t) = r > 0. Full depletion of developed stock impliesSss = limt→∞ S(t) = 0. From EF(t)≤ κS(t), and X(t)≥ 0, this gives

S(ν)≥ S(t)e−κ(ν−t) ≥ S(t)e−κ(t ′−t)−∫

ν

t ′EF(s)ds

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for any t < t ′ < ν . Hence, full depletion requires that EF(s) > 0 for some s ∈ (t ′,∞) for any t ′. From (9), anecessary condition for EF(t)> 0 is

pE(t)≥ cF + τ(t).

As τ(t)> 0 with τ(t)/τ(t) = r > 0, we thus require limt→∞ pE(t)→∞. In equilibrium, limt→∞ pE(t)→∞ onlyif limt→∞ E(t)→ 0. By (17) this in turn requires limt→∞ K(t)→ 0. Yet with limt→∞ pE(t)→ ∞, by (22), wemust obtain that µK(t) strictly positive for all t. From (18), this gives I(t) strictly positive and by (7) K(t) strictlypositive for all t. Hence, limt→∞ K(t) > 0 from which follows we cannot obtain limt→∞ pE(t)→ ∞. Hence, along run equilibrium without abandonment of developed stock is inconsistent. Instead, a binding carbon budgetimplies that exploration is terminated in finite time and hence limt→∞ S (t)> 0.

B.2 Other proofs

B.2.1 ER(t) = K(t) for all t

We first establish that φR,cap(t) = 0 requires K(t)> E with E an endogenously determined constant. In turn, weshow that as long as K(0)≤ E, K(t)≤ E, from which follows that φR,cap(t)> 0 for all t > 0, and ER(t) = K(t)

for all t.First suppose that φR,cap(t) = 0 for some t. From (17) this gives

pE(t) = cR−φR,0(t)

and thus pE(t) ≤ cR. We can then show that φR,0(t) = 0 for all t. Suppose not. Then pE(t) < cR. Yet ascR < cF +τ(t), from (9), this implies pE(t)< cF +τ(t) and thus φF,0(t)> 0 and EF(t) = 0. This gives E(t) = 0which is inconsistent with P(E(t)) = pE(t) < cR. Hence, if φR,cap(t) = 0, we must obtain that φR,0(t) = 0 andthus P(E(t)) = pE(t) = cR. This gives E(t) = E with E satisfying

P(E) = cR.

By (9), pE(t) = cR implies φF,0(t)> 0 and EF(t) = 0. From here it follows that we require ER = E ≤ K(t).Next, suppose that K(t) = E. Then pE(t) = cR. In addition, by (7), to ensure K(t) ≥ 0, we require I(t) ≥

δK(t) = δ E. Suppose this is the case. Then µK(t)≥ cI(δ E)> 0 and by (19), µK(t)> 0. From here it followsthat we must obtain I(ν) > E, K(ν) > E for all ν > t, and thus pE(ν) = cR for all ν ≥ t. Yet by (17), thelatter implies φR,cap(ν) = 0 for all ν ≥ t, which from (20) gives µK(t) = 0. Hence, K(t) = E is inconsistent withK(t)≥ 0, from which follows that as long as K(0)≤ E, K(t)< E for all t > 0, and we never obtain ER(t)<K(t).

Finally, we note that K(t) < E whenever EF(t) > 0. If EF(t) > 0, from (9), we obtain pE(t) ≥ cF + τ(t).As cF + τ(t) > cR, by P′ < 0, we must obtain E(t) < E. Additionally, as established above, EF(t) > 0 isincompatible with φR,cap(t) = 0. Hence, we must obtain ER(t) = K(t). From here it directly follows that ifEF(t)> 0, K(t)< E, and thus φR,cap(t)> 0 for all t.

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B.2.2 Global stability of the long run equilibrium

From (17)-(19), we obtain that in steady state

pssE = zR (Ess

R )− (r+δ ) limt→∞

φI,0(t)− limt→∞

φR,0(t), (B.6)

where pssE = limt→∞ pE(t) and Ess

R = limt→∞ ER(t). As pssE ≥ 0, we must obtain limt→∞ φR,0(t) = 0. In equilib-

rium, P(Ess) = pssE , where from P(0)> zR (0), we must obtain Ess > 0. Given that cR < cF + τ , (9) implies that

EssR > 0. Ess

R > 0 requires a strictly positive renewable capacity Kss > 0, and thus by (7), Iss > 0. It then followsthat limt→∞ φI,0(t) = 0. With (B.6), this gives (23).

Lemma B.2. K(t)≤ EssR for all t

Proof. As we assume K(0) < EssR , K(t) > Ess

R is feasible only if for some ν < t, K(ν) = EssR and I(ν) > Iss =

δEssR (see (7)). From (17), K(ν) ≥ Ess

R and K(t) = ER(t) for all t (see Appendix B.2.1), implies pE(ν) ≤ pssE .

Yet I(ν)> Iss, requires µK(ν)> µssK from (18), and from (19) and pE(ν)≤ pss

E , it then follows that µK(ν)> 0and thus µK(ν

′) > µssK for all ν ′ ≥ ν . This entails that I(ν ′) > Iss and K(ν ′) > Ess

R for all ν ′ > ν , and thuspE(ν) ≤ pss

E for all ν ′ ≥ ν . However from (17) and (21), pE(ν) ≤ pssE for all ν ′ ≥ ν is incompatible with

µK(ν)> µssK . Hence, K(0)< Ess

R implies that we cannot obtain K(t)> EssR for any t > 0.

Next define Z(t)≡U(t)+S(t). As we require U(t)≥ 0 and S(t)≥ 0 for all t, Z(t)≥ 0 for all t. In addition,Z(t) = U(t)+ S(t) =−EF(t)≤ 0. Then we can be in one of two situations.

1. There exists some finite t ′ such that Z(ν) = Z(t ′) for all ν ≥ t ′. From here it follows that EF(ν) = 0and E(ν) = ER(ν) for all ν ≥ t ′. From K(t) = ER(t) for all t, Lemma B.2 implies pE(ν) ≥ pss

E for allν ≥ t ′. From (17), and cR < pss

E , it follows that φR,cap(ν) > 0 for all ν > t ′ and (22) applies. This givesµK(ν) ≥ µss

K and, by (18), I(ν) ≥ Iss for all ν > t ′. By (7), for all ν > t ′, whenever K(ν) < EssR = Ess,

K(ν)> 0. Together with Lemma B.2, this implies limt→∞ K(t) = Ess.

2. There does not exist a finite t ′ such that Z(ν) = Z(t ′) for all ν ≥ t ′. As Z(t)≥ 0 for all t, we must obtainlimt→∞ Z(t) = Z ≥ 0 and thus limt→∞ EF(t) = 0, and Ess = Ess

R . From Lemma B.2, limt→∞ pE(t) ≥ pssE .

From (17), and cR < pssE , limt→∞ φR,cap(t) > 0 and from (22), limt→∞ µK(t) ≥ µss

K . By (18) and(7), limt→∞ I(t) ≥ Iss and limt→∞ K(t) ≥ Kss. By Kss = Ess

R and Lemma B.2 we must then obtainlimt→∞ K(t) = Ess.

25