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Energy substitutions, climate change and carbon sinks
EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 58 9 –5 97
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n
ANALYSIS
Energy substitutions, climate change and carbon sinks ☆ Gilles Lafforguea,⁎, Bertrand Magnéb , Michel Moreauxc a
Toulouse School of Economics (INRA and LERNA), France Paul Scherrer Institute (LEA), Switzerland c Toulouse School of Economics (IDEI and LERNA), France b
AR TIC LE I N FO
ABS TR ACT
Article history:
We determine the optimal exploitation time-paths of two energy resources, one being
Received 4 September 2007
depletable and polluting, namely a fossil fuel, the other being renewable and clean. These
Received in revised form
optimal paths are considered along with the two following features. First, the cumulative
14 January 2008
atmospheric pollution stock is set not to exceed some critical threshold and second, the
Accepted 14 January 2008
polluting emissions produced by the use of fossil fuel can be reduced at the source and
Available online 4 March 2008
stockpiled in several carbon sinks of limited capacity. We show that, if the renewable resource flow is abundant, the optimal path requires that sequestration is implemented
Keywords:
only once the ceiling is reached. Moreover, the reservoirs should be completely filled by
Non-renewable resource
increasing order of their respective sequestration costs.
Pollution target
© 2008 Elsevier B.V. All rights reserved.
Climate change Carbon sequestration and storage Hotelling
1.
Introduction
Greenhouse gas emissions essentially result from the use of fossil, carbon-based, energy resources.1 Numerous technical and regulatory devices allowing for emission abatement are readily available. Their implementation may arise sooner or later depending on their respective costs. Atmospheric pollution abatement consists of capturing the dirty emissions at their source and storing them underground, either in natural
☆ We thank two anonymous referees for their helpful comments on a first version of the paper and for numerous suggestions to improve it. ⁎ Corresponding author. 21 Allée de Brienne, 31000 Toulouse, France. Tel.: +33 5 61 12 88 76; fax: +33 5 61 12 85 20. E-mail addresses: [email protected] (G. Lafforgue), [email protected] (B. Magné), [email protected] (M. Moreaux). 1 Among others, animal and human wastes, the reduction of forest cover and the extensive agricultural practices also constitute non-negligible emissions sources and favor carbon release from shallow soils.
reservoirs or in depleted oil fields and mine sites that will be called carbon sinks in the remainder of the text.2 According to the recent IPCC special report (IPCC, 2005) three industrialscale storage projects are already in operation: the Sleipner project operated by the Norwegian oil company Statoil in an offshore saline formation, the Weyburn Enhanced Oil Recovery project in Canada,3 and the In Salah project in a gas field in Algeria. Other projects are planned. In the present study, we first intend to determine the optimal starting date and pace of geological carbon sequestration. Second, we want to characterize how the recourse to such an abatement option alters the optimal time path of fossil resource exploitation, when the cumulative atmospheric 2 The term carbon sinks also refer to biomass carbon storage, e.g. in soils, plants and especially trees (see for example IPCC, 2001). Biomass carbon storage is not the focus of the present analysis. 3 In this case, captured gases are injected in oil deposits so as to enhance oil recovery, and thus increasing the oil resource base. The economic analysis of such a storage process poses some specific problems that are eluded here.
0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2008.01.008
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concentration of carbon has to be maintained below some given critical threshold, in accordance to the United Nations Framework Convention on Climate Change.4,5 The possibility of sequestering some fraction of the carbon dioxide emitted by the combustion of fossil resources has motivated numerous empirical studies (see for example the results of integrated assessment models from McFarland et al., 2003, Edenhofer et al., 2005, Edmonds et al., 2004, Gerlagh, 2006, Gerlagh and van der Zwaan, 2006). The level of complexity of such operational models, aimed at defining some climate policy, may be required so as to take into account the numerous interactions at hand. Some models of this kind explicitly address the important question of the choice of sinks according to their specific characteristics, such as storage capacities or access costs (e.g. Kurosawa, 2004, Riahi et al., 2004, Gitz et al., in press). We here take a different approach and use a stylized model so as to exhibit the main driving forces that should determine the location choice of suitable carbon storage deposits. At a regional scale, such choices appear critical. For example, the Alberta Basin and the Western Texas regions were estimated in 2005 to contain 43 and 70 suitable CO2 storage sites, respectively (IPCC, 2005). To go straight to the point in our modeling analysis, we assume that there are final users that consume two kinds of primary energy: a polluting and scarce fossil resource, and a clean and renewable backstop resource. The users derive their utility from the use of energy in its final form, and thus face an energy cost that comprises all the costs incurred through the transformation process of the primary energy source. Hence, the two types of energy are perfect substitutes for the final users. Moreover, in accordance with current technologies, we assume that the useful energy obtained from the non-renewable resource is cheaper than the energy obtained from the renewable one. To assume the existence of an atmospheric concentration ceiling, this pollution stock being in addition partially eliminated owing to natural decay or absorption,6 implicitly con4 Before the Kyoto Protocol, the UNFCCC had already formulated the objective: “to achieve stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system” (UNFCCC, 1992, Article 2). 5 Even though the ultimate goal of the IPCC consists in stabilizing the atmospheric concentration of greenhouse gases, the Kyoto Protocol does not prescribe the level of such a target for the 2008-12 commitment period. It rather prescribes some constraint on the maximum flow of carbon emissions for the industrialized countries. More precisely, their emissions shall be reduced down to their levels of the reference year 1990, over the commitment period, i.e. 2008–12. Were an atmospheric stabilization target set, this kind of agreement would remain ineffective until all countries participate. This classical question of the cost of a public good and its breakdown among agents is made more complex in the present setting because of the dynamic nature of the problem. Since the key variable is the accumulation of greenhouse gases (we will restrain our analysis to the case of carbon dioxide), the most straightforward way of tackling the problem is to set an upper bound on the state variable. 6 This process can be interpreted as some natural carbon sequestration by a sink of very large size, e.g. typically the oceans (for more details, see IPCC, 2001).
strains the instantaneous rate of consumption of the polluting resource once the threshold is reached, in the absence of pollution capture and storage option. Two observations then deserve to be mentioned. First, along the optimal path, the time interval during which the flow of fossil consumption is constrained is endogenous. Because the carbon ceiling is reached at a date that is a function of the whole fossil consumption path from the initial time period on, then the cap imposed on the carbon accumulation affects the entire time path of the fossil resource exploitation. Second, two options are available to society if it wants to relax the constraint on fossil fuel use once at the ceiling. It may either substitute the clean resource for the dirty one, or sequester some fraction of the polluting flow generated by the fossil resource use. Each of those options entails some monetary cost. The renewable resource is more expensive than the non-renewable one. To capture the carbon dioxide is also costly. But each of these options also exhibits some specific opportunity cost. A scarcity rent is to be associated to the consumption of the fossil resource, as is the case for every non-renewable resource. To this rent augmented by the extraction and processing cost of the fossil fuel, we must add now the shadow cost of the atmospheric carbon stock and, when pollution is abated at the source of emissions, the cost of capturing the carbon augmented by the shadow cost of the sink.7 To set a cap on the accumulation of pollution, possibly removable by sequestration, results in having de facto two available carbon storage deposits. The first one is the atmospheric reservoir of temporary bounded capacity, but of infinite long run capacity thanks to the natural progressive regeneration. But to take advantage of this infinite long run capacity, the carbon emission flow has to be restrained when the temporary capacity is saturated. Hence some rent has to be charged for the use of this capacity, even before it is reached as we shall show. The second reservoir is the sink, which itself can be of limited capacity so that another rent must be charged for its use. Clearly at each point of time, some part of the emission flow can be sent into one reservoir, thus having to bear the corresponding rent, while the other part is sent into the other reservoir, having to bear the other rent. The dynamics of these two rents obey two different rules and their trajectories diverge over time. Note that the renewable resource would also entail a scarcity rent if the exploitable carbon-free flow is not sufficiently abundant. For the sake of simplicity, we assume here that the renewable substitute is abundant. The article is organized as follows. Section 2 presents the model. In Section 3, we examine the optimal paths when carbon can be only stockpiled in a single sink. In Section 4, we generalize our analysis to the case of multiple storage deposits that are differentiated by their access costs and their storage capacities. We show that those deposits have to be exploited
7 In this respect, the problem is similar to the case of residential waste management treated in Gaudet et al. (2001), with the exception that the waste discharge incurs a cost that includes a Hotelling rent (Hotelling, 1931) before it is effectively disposed. Note also that, contrary to Gaudet et al. (2001), we do not assume any sunk fixed cost for making the discharge available.
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in increasing order of access cost, no matter their capacity, and we identify the most costly deposit having to be used. We also show how this multiple deposits model can be interpreted as the approximation of a more realistic model where the cost of sequestration in each deposit is an increasing function of the cumulative stored carbon. We briefly conclude in Section 5.
2.
The model
2.1.
Assumptions and notations
However, let us assume that some carbon sequestration device is available. The potential pollution flow can be reduced at the source of emission8 and stockpiled in n carbon sinks, or reservoirs, indexed by i, i = 1,…, n. Each sink i is characterized ― by its unitary sequestration cost csi and its capacity S i. By convention, the n reservoirs are ranked by strictly increasing order of costs: cs1 b … b csi b … b csn. Capacities and sequestration costs are independent a priori and the theory developed here remains compatible with any form of relationship. Let S0i be the initial stock of pollutant in reservoir i. Without any loss of generality, we postulate that S0i = 0, i = 1,…, n. We thus have for any t ≥ 0: n X
We consider an economy in which the instantaneous gross surplus, or utility, generated by an instantaneous energy consumption qt is given by u(qt), where function u(.) has the following standard properties:
zt ¼ fxt
Assumption (A.1). u: IR++ → IR+ is a function of class C2 strictly increasing and strictly concave, satisfying the Inada conditions.
: Sit ¼ sit and Si0 ¼ 0; i ¼ 1; N ; n and t z 0
sit z 0;
: Xt ¼ xt ;
X0 ¼ X0 N 0 given:
ð1Þ
Using coal potentially generates a pollutant flow. Let ζ be the unitary carbon content of the fossil resource so that, without any abatement activity, the instantaneous carbon flow released into the atmosphere would be equal to ζxt. Let Zt be the stock of pollutant in the atmosphere at time t, zt the flow of emissions and α, α N 0, the instantaneous proportional rate of natural regeneration, assumed to be constant for the sake of simplicity (see for instance Kolstad and Krautkraemer, 1993) so that: : ð2Þ Zt ¼ zt aZt : We assume that this stock of carbon cannot be larger than ― some threshold Z . Let Z0 the stock of carbon in the atmosphere at the beginning of the planning period, assumed to be smaller ― than Z . We thus have for any t ≥ 0: P
Z Zt z 0; Z0 ¼ Z0 given:
ð3Þ
ð4Þ
i¼1
where sit is the part of the potential carbon emission flow that is sequestered into reservoir i at time t, so that:
P
We also use p to denote the marginal surplus u′ as well as, by a slight abuse of notation, the marginal surplus function: p(q) = u′(q), where u′(q) ≡ du / dq. The direct demand function d (p) is the inverse of p(q), as usually defined. We denote by u″ the second derivative of u. Under Assumption (A.1), u′(q) N 0 and u″(q) b 0, ∀q N 0. Energy needs may be supplied by two resources: a dirty nonrenewable resource (fossil), e.g. coal, and a clean renewable resource, e.g. solar. We assume that these two energy sources are perfect substitutes, so that if xt denotes the instantaneous consumption of the fossil resource and yt, the instantaneous consumption of the renewable resource, then the surplus generated by the total consumption is u(xt + yt), i.e. qt = xt + yt. The average cost of transforming coal into directly usable energy, also equal to the marginal cost, is constant and de~ denote the flow of fossil resource to be noted by cx. Let x consumed in order to equalize the marginal surplus to the marginal monetary cost of the resource. Thus, x~ is the solu~ = d(c ). Let X0 be the initial fossil tion of u′(x) = cx, that is, x x resource stock and Xt, the stock available at time t, so that:
591
ð5Þ
Si Sit z 0; i ¼ 1; N ;n and t z 0
ð6Þ
sit z 0; i ¼ 1; N ;n and t z 0:
ð7Þ
― ˙ ≤ 0. Hence, the Once the ceiling Z is reached, we must have Z t upper bound on the fossil fuel consumption without se_ ― ― questration is ― x =αZ /ζ. Let p denote the marginal utility of x : ― ― p =u′( x ). In the case of sequestration in reservoir i, we assume that the total marginal cost of a “clean” consumption of the ― fossil resource, cx +csiζ, is lower than p . If not, we would always _ be better off remaining at x rather than relaxing the constraint by sequestering any part of the emission flow in reservoir i.
_ Assumption (A.2). ∀i = 1,…, n: cx + csiζ b p. The other resource is a non-polluting renewable resource that can be made available to the end users at a constant average cost cy. The cost of the renewable resource is the total cost of supplying the good to the final users, so that the nonrenewable and the renewable resources are perfect substitutes _ for the users. We assume that cy N p since, if cy was lower _ ― than p, condition Z − Zt would never constrain the energy consumption. _ Let y be the constant instantaneous flow of this renewable ~ the energy available at each point of time. We denote by y flow of renewable resource that the society would choose to _ consume once the fossil resource is exhausted, provided that y ~ is the solution of u′(y) = c : ~ is sufficiently large. Then, y y y = d(cy). We assume that the available flow of the renewable resource ~ , so that no rent has to be is abundant, i.e. at least equal to y charged for the use of this resource.9 _ _ Assumption (A.3). p b cy and ~y b y. Let us assume that the instantaneous social rate of discount, ρ N 0, is constant. The objective of the social planner is to choose 8
Despite the fact that the carbon capture primarily applies to emissions from power regeneration processes, we here assume that the entire carbon emission flow can be abated. 9 The case of a constrained flow of renewable resource is analyzed in Lafforgue et al. (in press).
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the resource and sequestration trajectories that maximize the sum of the discounted instantaneous net surplus.
2.2.
Problem formulation and optimality conditions
The social planner problem (P) can be expressed as follows: # Z l" n X uðxt þ yt Þ csi sit cx xt cy yt eqt dt ðPÞ max fðsit ;i¼1; N ;n;xt ;yt Þ;tz0g
0
i¼1
subject to Eqs. (1)–(7) and subject to the following inequality constraints: Xt ≥ 0, xt ≥ 0 and yt ≥ 0, t ≥ 0. Let Lt be the current valued Lagrangian for the problem (P): n n X X csi sit cx xt cy yt kt xt þ git sit Lt ¼ uðxt þ yt Þ i¼1 i¼1 " # n n X X P P þ At fxt sit aZt þ mit Si Sit þ mZt Zt Zt i¼1 i¼1 " # n n X X P sit þ git sit þ gxt xt þ gyt yt : þ g st fxt i¼1
i¼1
The static and the dynamic first-order conditions of (P) are: g st þ git ; ALt =Asit ¼ 0 f csi ¼ git At P
i ¼ 1; N ;n
ALt =Axt ¼ 0 f uVðxt þ yt Þ ¼ cx þ kt At f P g st f gxt
ð8Þ ð9Þ
ALt =Ayt ¼ 0 f uVðxt þ yt Þ ¼ cy gyt ;
ð10Þ
: : k t ¼ qkt ALt =AXt f k t ¼ qkt f kt ¼ k0 eqt
ð11Þ
: : g it ¼ qgit ALt =ASit f g it ¼ qgit þ mit ; i ¼ 1; N ;n
ð12Þ
: : A t ¼ qAt ALt =AZit f A it ¼ ða þ qÞAt þ mZt ;
ð13Þ
with the usual complementary slackness conditions and the associated transversality conditions. First, we note that ηit ≤ 0, ηit being the instantaneous marginal cost of the carbon stock which is already sequestered into sink i at time t or, alternatively, − ηit being the marginal value of this part of sink i which is still available for carbon storage. If the storage capacity of this sink is limited and if at date t, Sit was increased by an exogenous amount dSit N 0, then the optimized value of the objective function of (P) would diminish. Second, as long as reservoir i is not completely filled, i.e. for any ― t such that S i − Sit N 0, νit = 0, so that from Eq. (12) it comes: P
Si Sit N 0 Z git ¼ gi0 eqt :
ð14Þ
By the same type of argument, μt ≤ 0, μt being the instantaneous marginal cost of the carbon stock in the atmosphere at date t ― or, alternatively, − μt being the marginal value of Z − Zt, this part of the atmospheric carbon reservoir which has not been used at time t. If at date t, Zt was increased by an exogenous amount dZt N 0, the optimized value of the objective function of (P) would, at best, remain constant, and would decrease in the ― worst case. Furthermore, since Z0 b Z , there must exist an initial time interval during which the stock of carbon in the atmosphere is below the ceiling, hence νZt = 0 so that, by integrating Eq. (13), we get: At ¼ A0 eðaþqÞt :
ð15Þ
In fact, from the planner's point of view, the possibility of sequestering carbon emissions is double: either emissions are
stockpiled in non-regenerating and costly reservoirs whose capacity is limited, i.e. the geological carbon sinks, or they are released in a natural reservoir, i.e. the atmosphere, whose capacity is limited by the ceiling constraint, but which is autoregenerating and free of charge. Then, the difference in the dynamics of ηit, i=1,…, n and of μt can be explained as follows. On the one hand, the decision to resort to the storage capacity of reservoir i is irreversible since Sit is a monotonous and nondecreasing function. On the other hand, owing to the natural regeneration, the stock of pollutant in the atmosphere Zt can either increase or decrease. On that account, whatever the sequestration policy may be, Zt will be ultimately induced to decrease since the fossil resource stock is finite. As it will be seen in the next sections, such an asymmetry strongly governs the optimal solution. ― ― Finally, if there exists t such that for any t ≥ t the ceiling ― constraint is slack for all future time periods, then μt = 0, t ≥ t . Absent the ceiling constraint as well as the sequestration device, the optimal solution would be the well-known Herfindahl-Hotelling path.10,11 For any λ0 ∈ (0; cy − cx), define pH t (λ0) as the price path along which the coal rent is increasing ρt H at the discount rate ρ, pH t (λ0) = cx + λ0e ; T (λ0) as the time at H which pt (λ0) is equal to the average cost of the abundant renewable substitute, TH(λ0) = [ln(cy − cx) − ln λ0] / ρ; dH t (λ0) as the H corresponding instantaneous energy demand, dH t (λ0) ≡ d(pt (λ0)) H H H and zt (λ0) as the generated polluting flow, zt (λ0) = ζdt (λ0); last Rt H H DH t (λ0) as the cumulative demand, Dt ðk0 Þ ¼ 0 ds ðk0 Þds. ―H Trivial calculations would show that D (λ0)≡DTHH(λ0) (λ0) is a strictly decreasing function with limλ0↓0DH(λ0)=+∞ and limλλ0↑(cy− ― cx)DH(λ0)=0, so that the equilibrium equation D H(λ0) =X0 has a H unique solution λ0 which is the optimal value of λ0. Thus the optimal path is a two-phase path. During the first phase [0, TH(λH 0 )], the energy demand is supplied only by the fossil fuel, and the energy consumption is decreasing from dH 0 (λ0) =d H H (cx +λ0) down to dH TH(λH0) = d(cy). From T (λ0 ) onwards, the energy consumption is stationary and equal to d(cy), supplied by the renewable source. The pollution emission flow determines the pollution stock trajectory. Let ZH t (λ0) be the pollution stock induced by the emission flow: 8 H H > < fdt ðk0 Þ ; ta½0;T ðk0 ÞÞ H ð16Þ zt ðk0 Þ ¼ > :0 ; ta TH ðk0 Þ; þ lÞ; that is12: ( ZH t ðk0 Þ ¼
0 at
Z e
þ
R
t H aðtsÞ ds 0 ds ðk0 Þe R T H ðk Þ aðtTH ðk0 ÞÞ aðtsÞ e ds f 0 0 dH s ðk0 Þe
Z0 eat þ f
;
ta½0;TH ðk0 ÞÞ ; ta TH ðk0 Þ; þ lÞ:
ð17Þ 10
Herfindahl (1967), Hotelling (1931). 11 In such a case, the objective function of program (P) writes: Rl uðxt þ yt Þ cx xt cy yt eqt dt and the constraints (2)–(7) vanish. 0 The conditions (8), (12) and (13) also vanish and, from Eq. (11), the condition (9) becomes: u′(xt + yt) = cx + λ0eρt − γxt. The condition (10) remains the same. 12 0 –αt At time t, ZH t (λ0) is the sum of Z e , this part of the initial pollution stock not yet naturally regenerated at this time, and the sequence of −α(t − τ) ζdH , τ∈[0,t], these parts of the successive emission flows of τ (λ0)e time τ not yet regenerated. After TH(λ0), there is no new emission and H the stock smoothly declines according to eaðtT ðk0 ÞÞ ZH TH ðk0 Þ ðk0 Þ, down to 0.
EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 58 9 –5 97
―
―
H Let ZH(λ0)≡max{ZH t (λ0),t≥0}. Clearly, dZ (λ0)/dλ0 b 0 and there ―0 ― ― ― ― 0 exists some critical value X of X such that ZH(X0 )=Z and DH(λ0)= ―0 0 ―0 H H ― X . For X bX , the ceiling constraint is never active, Zt (λ0 )bZ, ― and for X 0 NX 0 there must exist some time interval within which ― H ZH t (λ0 )NZ thus violating the constraint. We assume from now that the initial amount of fossil fuel is sufficiently large in such a way that, without any ceiling constraint, the critical level of the pollution stock would be overshot over some time interval.
―
Assumption (A.4). X0 N X 0.
3.
The case of a single reservoir
In this section, we assume that the sequestration device consists in a single carbon sink, so that we can neglect the index i in conditions (8) and (12). The capacity of the sink can be said “large” or “small” according to the role played by the reservoir capacity constraint (5), together with Eq. (6). If this constraint is never active (i.e. ηt = 0, t ≥ 0), meaning that no rent has to be charged for the use of the sink, then the reservoir capacity is said to be large. In that case, we would get quite similar insights in terms of abatement option as in Chakravorty et al. (2006). Clearly, the most interesting case is the case of a “small” reservoir. If the sink capacity is sufficiently small, its use implies an opportunity cost in addition to the sequestration cost cs. But contrary to cs, this opportunity cost is not constant over time. We know (cf. Section 2) that, as long as the reservoir is not filled, the absolute value of ηt is increasing at ― the social rate of discount: St b S ⇒ ηt = η0eρt. Substituting for ηt into condition (8) and given that γst = 0 provided st N 0, we get: g st Z At ¼ cs g0 eqt þ P g st : cs ¼ g0 eqt At P
ð18Þ
Next, we can substitute for − μt into Eq. (9). Given that γxt = 0 and provided that xt N 0, and after simplifications, we must have: uVðxt Þ ¼ cx þ k0 eqt þ cs g0 eqt f;
ð19Þ
where cs − η0eρt, the full marginal cost of sequestration, is increasing over time since η0 b 0. If for any phase during which only the fossil resource is ― used, the ceiling constraint Z − Zt ≥ 0 is never active and a part of the carbon emissions is sequestered, then the resource consumption xt must be equal to d[cx + λ0eρt + (cs − η0ept)ζ] so as to satisfy the first-order conditions. However, it does not mean that the totality of the emission flow is necessarily captured. Minimization of costs implies that only part of emissions that exceeds the ceiling has to be stockpiled, so that: st ¼ f d cx þ k0 eqt þ cs g0 eqt f P x :
593
Fig. 1 illustrates why, at the ceiling, it is optimal first to abate and to sequester, and next to stop the abatement. Since the full marginal cost, cx + λ0eρt + (cs − η0eρt)ζ, is increasing over time, the sequestration phase should precede a phase during which the fossil resource exploitation is constrained by the natural regeneration capacity of the atmosphere at the ceiling, ― ― ― that is a phase during which qt = xt = x and pt = p for Zt = Z . In Fig. 1 the marginal cost curves are drawn for three dates t, t′, t″, t b t′ b t″, sufficiently spaced but not too much, when the economy is at the ceiling. At date t, the inverse demand function is crossing the upper branch of the marginal cost ― curve at xt, implying that the part ζ[xt − x ] of the emission flow has to be sequestered. At t′, the inverse demand function ― intercepts the two steps of the marginal cost curve at x , ― implying that the optimal consumption must amount to x and that abatement is too costly. At time t″, the inverse demand curve is crossing the lowest part of the marginal cost curve, so ― that the optimal consumption is lower than x . Then, Zt is decreasing and the path is the Hotelling path forever. The optimal path consists in five phases, as illustrated in Fig. 2. During the first phase [0,t1), the pollution stock is increasing, the resource price is equal to the full marginal cost ^ pt, ^ pt ≡cx + ^t). At the λ0eρt −μ0e(α + ρ)tζ, and only the fossil fuel is used: qt =xt =d(p ~ , where ~ pt ≡cx +λ0eρt +(cs −η0eρt)ζ is the end of the phase, pt =p t marginal cost of consuming the “emissions-free” coal, and the ceiling is attained. The second phase [t1,t2) is a phase at the ceiling during which only the fossil resource is used and some part of the ~ ) and pt, qt = xt = d(p potential emission flow is sequestered: pt = ~ t ― ― ~ st = ζ[d(p t) − x ]. At the end of the phase, pt = p . The third phase [t2,t3) is still a phase at the ceiling during ― which pt = p , and only the fossil resource is consumed, qt = xt = ― x , but the emission flow is no longer stockpiled, being just balanced by the natural regeneration. During this phase, |μt| is decreasing and becomes nil at the end of the phase. The fourth phase [t3,t4) is a pure Hotelling phase, pt = pH t (λ0), only the fossil resource has to be used, qt = xt = d(pH t ) and since ― xt b x the pollution stock starts to decrease. At the end of this phase, the energy price is just equal to the marginal cost of the renewable resource cy and the fossil resource is exhausted. The last phase [t4,∞) is a phase during which only the re~ newable resource is used: qt = yt = y and pt = cy.
ð20Þ
―
When Zt = Z , the full marginal cost of the energy satisfying the ceiling constraint is a two-step function, equal to the ― monetary extraction cost augmented by the rent if xt ≤ x , at which the sequestration cost (cs − η0eρt) ζ must be added if ― xt N x , given that the sequestration capacity of the sink is not saturated: cm ¼
; if xt V P x cx þ k0 eqt cx þ k0 eqt þ ðcs g0 eqt Þf ; if xt N P x:
ð21Þ
Fig. 1 – Full marginal cost of the fossil resource and optimal consumption at the ceiling.
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EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 5 8 9 –5 97
Consequently, we would follow a continuous and increasing ― marginal cost path, from cx + λ0 at t = 0 up to p at t2, and then: • either cx + λ0eρt + cs2ζ N cx + λ0eρt + (cs1 − η10eρt)ζ over the interval [t1,t2) and in this case, reservoir 2 will never be used. The full marginal cost of the now “clean” fossil resource, owing to sequestration, into reservoir 1 would always be smaller than the marginal cost when sequestering into reservoir 2; ― • or there exists a date t b t2, such that cx + λ0eρt1 + cs2ζ b cx + λ0eρ + ― ρt (cs1 − η10e )ζ, t b t , and in this case, reservoir 2 must also be used though the sequestration cost in this sink, cs2, is larger than the sequestration cost in reservoir 1, cs1. The point to be noticed here is that, at the optimum, the two sinks should never be used simultaneously. If it was not the case, i.e. if sit N 0, i = 1, 2 over an interval (t′,t″), t′ b t″, we would have γit = 0, and νit = 0,13 i = 1, 2 over the same interval and then, from Eq. (8): g st ; At ¼ csi gi0 eqt þ P
taðtV;t WÞ;
i ¼ 1; 2:
ð24Þ
Substituting for μt in Eq. (9), we obtain: uVðxt þ yt Þ ¼ cx þ k0 eqt þ csi gi0 eqt f;
taðtV; t WÞ;
i ¼ 1; 2
ð25Þ
so that: cs1 g10 eqt ¼ cs2 g20 eqt ; Fig. 2 – Optimal price, consumption and sequestration paths — the single reservoir case. The values of the seven endogenous variables λ0, μ0, η0, t1, t2, t3 and t4 characterizing this type of optimal path are determined by solving a seven-equation system that ensure the continuity of the various time-paths at each transition date. Such a system of equations is given in Appendix A for the more complex case of multiple reservoirs.
4.
The multiple reservoirs case
We first examine the case of the optimal use of two reservoirs and generalize the analysis to any number of reservoirs.
4.1.
The case of two reservoirs
Consider the case where two different sequestration devices can be used, namely reservoirs 1 and 2. As long as reservoir 2 is not used, results of Section 3 apply. In particular, as in the single reservoir case, the two first phases of the optimal price paths would be pt = cx + λ0eρt − μ0e(α + ρ)tζ over [0,t1) and pt = cx + λ0eρt − μ0e(α + ρ)tζ + (cs1 − η10eρt)ζ over [t1,t2), with the following associated continuity conditions: cx þ k0 eqt1 A0 eðaþqÞt1 f ¼ cx þ k0 eqt1 A0 eðaþqÞt1 f þ cs1 g10 eqt1 f
taðtV;t WÞ
ð26Þ
which is clearly impossible if cs1 b cs2 for any − ηi0 ≥ 0, i = 1, 2. The only case in which we can have in one phase the full marginal cost of the clean fossil resource with sequestration in reservoir 1 smaller than the cost involved with sequestering in reservoir 2, and another phase during which the inverted inequality holds, is when −η20 b −η10. Thus the stockpiling phase into reservoir 1 must always precede the stockpiling phase into reservoir 2. Taking this remark into account, the optimal path consists now in six phases: • Phase 1, [0,t1), rise of the pollution stock to the ceiling without any sequestration; • Phase 2, [t1,t2), at the ceiling with active sequestration in reservoir 1; • Phase 3, [t2,t3), at the ceiling with active sequestration in reservoir 2; • Phase 4, [t3,t4), at the ceiling without any sequestration; • Phase 5, [t4,t5), pure Hotelling path, with a pollution stock under the ceiling and forever; • Phase 6, [t5,∞), renewable resource exploitation. The values of the nine variables λ0, μ0, η10, η20, t1, t2, t3, t4 and t5 characterizing this type of path are determined by solving the nine-equation system characterized in Appendix A. The corresponding optimal price path is illustrated in Fig. 3.
4.2.
The case of n reservoirs
ð22Þ The previous analysis can be easily extended to the case of n reservoirs. Let the sinks be indexed by strictly increasing order
and p: cx þ k0 eqt2 A0 eðaþqÞt2 f þ cs1 g10 eqt2 f ¼ P
ð23Þ
13
―
If sit N 0 over (t′,t″), then Sit b S i from which vit = 0 and ηit = ηi0eρt.
EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 58 9 –5 97
Fig. 3 – Optimal price path — the case of two reservoirs.
question leads to the first exploitation of the sub-sink whose cost is c1sii and whose capacity is ΔS1sii, then if necessary, the second sub-sink whose cost is c2sii and whose capacity is ΔS2sii, and so and so forth. In other words, it is necessary to exploit the sub-sinks in the natural order of filling the sink i. Let us consider now the case of n different sinks, their respective cost functions being themselves approximated by a step function. Let us build the sequence of theoretical sinks j = 1,…, m by sorting again the whole steps of the various sinks indexed by increasing order of their costs and by grouping the capacities of the steps whose costs are identical. We denote by cjs the average cost of the theoretical j-ranked ― sink whose sequestration capacity is S j: c1s b … b cjs b … b cm s. Those theoretical sinks are built by recurrence as follows. For j = 1: c1s ¼ min c1s i ;
of sequestration costs: cs1 b … b csi b … b csn. Reservoir m, for any m b n, should be used over [tm, tm + 1) and at the end of this sequestration phase, we must have cx + λ0eptm + 1 + (csm − ηm0eρtm + 1) ― ζ = p . Hence, there are two alternatives: • either c x + λ0 eρt + c sm + 1 ζ ≥ c x + λ0 eρt + (csm − η m0 eρt)ζ over the interval [tm, tm + 1) and in this case, the m + 1th reservoir should not be used; ― • or there exists a date t b tm + 1, such that cx + λ0eρt + csm + 1ζ b cx + ― λ0eρt + (csm − ηm0eρt)ζ, t N t , so that the m + 1th sink should also be used. We thus have to implement again the sequestration opportunity test as described above. For a finite number of reservoirs, we obtain an algorithm that converges to a finite number of steps.
4.3. The model as a model of sequestration cost increasing with the cumulated sequestration Until now, we have assumed that the average cost of sequestration in each sink i, csi, is constant and thus equal to the marginal cost. Since the higher the amount of stored carbon the higher is the pressure in the reservoir, and since this high pressure makes the incremental carbon unit more costly to be stored, one has to assume that the sequestration cost is an increasing function of the carbon mass already injected. In other words, one may state that for each deposit i, csi = csi(Si) and dcsi / dSi N 0, where Si is the carbon mass already stored into the sink i. The model analyzed so far could be seen as a model where the functions csi(.) are approximated by step functions, the more numerous the steps, the more accurate the approximation. Let mi be the number of steps chosen to approximate the function csi(.) and let 1i,…, hi,…, mi be the indexes of those various steps, by increasing order of average sequestration cost hi i for each step, c1sii b … b chsii b … b cm si , csi being the average (marginal) cost of sequestration at step hi. To each step is associated Phi P P i ―h some absorption capacity ΔS i i , so that m hi ¼1 DSi ¼ Si . If there was a single sequestration sink, the sink i, the identification to the preceding model is straightforward. The optimal exploitation rule for differentiated average sequestration costs prescribes that it is necessary to exploit them by increasing order of their costs. Let us apply this rule to the only sink i, decomposed in mi differentiated sub-sinks. The rule in
595
i ¼ 1; N ;m :
ð27Þ
Let I(1) be the set of indexes of sinks for which c1si = c1s. Then: P1
S ¼
X
P1i
D Si :
ð28Þ
iaIð1Þ
Let us take out the steps whose costs are c1s and start the procedure again. Since the cardinal of the set of steps is finite, the defined procedure includes a finite number of stages. The P number of theoretical sinks, m, is at most equal to i¼1 mi . It would precisely be equal to this sum if the costs at the steps of the effective sinks were all differentiated. The order by which one shall optimally use the theoretical sinks is the order of their costs. By proceeding in such a manner, the order of exploitation of the various steps of a single sink is the natural order of filling this reservoir. Moreover, all the effective steps that constitute a theoretical j-rank sink can be used simultaneously. At the optimum, the carbon shall thus be stored in several sinks at a time. Note that some part of a given reservoir may be used at some stage and next, other reservoirs used before going back the first one.
5.
Conclusion
We have studied the economic rationale of capturing and sequestering the carbon, so as to maintain its atmospheric stock below some threshold level. Consuming a fossil resource, and capturing and sequestering the pollution which is generated, is like consuming a resource coming simultaneously out of two mining sites: the proper underground site of extraction and the sequestering site, the rent to be charged for each one having to grow at the interest rate since both must satisfy the Hotelling arbitrage condition. In this paper, we insisted upon the multiplicity of sequestering sites. Clearly, a generalization along the Herfindahl (1967) line of analysis of the case of different mining sites is immediate. We have shown that such a policy ought to be implemented only once this critical level has been reached, whatever the cost of sequestration is, i.e. independently of the number of deposits, their access costs and their retention capacities. Moreover, it stems from our analysis in Section 4 that the optimal resources exploitation and the sequestration implementation, obtained with constant average costs, are robust
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with respect to other costs specifications that would depend on the flow and/or the cumulative sequestered carbon and/or the resource extraction as far as the carbon deposit and the exhaustible resource are respectively concerned (see Heal, 1976). The absence of sequestration in the short run does not mean the absence of environmental policy in the short run. On the contrary, even before its implementation, the sequestration option affects the optimal pace at which exhaustible resource exploitation is reduced even before the ceiling is reached. The consumption reduction is attributable both to the opportunity cost of emitted pollution before the ceiling and to the opportunity cost of pollution sequestration once the ceiling is reached, those costs adding up to the total delivery cost of the resource. Finally, our definition of the storage process does not include the possibility of some leakage that would result in sending the carbon back to the atmosphere (see Herzog et al., 2003; and Paccala, 2003). This leakage phenomenon, would it be continuous over time, would not have any incidence on the optimal solution in the short run. In this case, only the length of the capture and storage phase would be extended to the entire phase at the ceiling, the sequestration activity exactly compensating the leakage at each date.
Appendix A. Continuity equation system of the optimal path in the multiple reservoir case • The cumulative fossil resource consumption-initial stock balance equation, here: Z
t1
0
Z t2
d cx þ k0 eqt A0 eðaþqÞt f dt þ d cx þ k0 eqt þ cs1 g10 eqt f dt ð29Þ t1 Z t3 P þ d cx þ k0 eqt þ cs2 g20 eqt f dt þ ½t4 t3 x t2 Z t5 þ d cx þ k0 eqt dt ¼ X0 : t4
• The price continuity equation at t1: cx þ k0 eqt1 A0 eðaþqÞt1 f ¼ cx þ k0 eqt1 þ cs1 g10 eqt1 f:
ð30Þ
• The pollution stock continuity equation at t1: Z0 eat1 þ f
Z
t1 0
P d cx þ k0 eqt fA0 eðaþqÞt eaðtt1 Þ dt ¼ Z:
ð31Þ
• The saturation condition for reservoir 1 at t2: Z f
t2
t1
P P d cx þ k0 eqt cs1 g1t2 eqt2 f x dt ¼ S1 :
ð32Þ
• The price continuity equation at t2: cx þ k0 eqt2 þ cs1 g10 eqt2 f ¼ cx þ k0 eqt2 þ cs2 g20 eqt2 f:
ð33Þ
• The saturation condition for reservoir 2 at t3: Z f
t3
t4
P P d cx þ k0 eqt þ cs2 g20 eqt f x dt ¼ S2 :
ð34Þ
• The price continuity equations at t3, t4 and t5: P cx þ k0 eqt3 þ cs2 g20 eqt2 f ¼ p; P
ð35Þ
cx þ k0 eqt4 ¼ p;
ð36Þ
cx þ k0 eqt5 ¼ cy :
ð37Þ
Note that the single reservoir case developed in Section 3 is ― ― ― ― ― the special case for which cs1 = cs2 ≡ cs, S 1 = S 2 and S 1 + S 2 ≡ S , which directly implies η10 = η20 ≡ η0. The date t3 works as t2, t4 as t3 and t5 as t4. Condition (33) vanishes, and Eqs. (32) and (34) are replaced by a single saturation condition: Z f
t2 t1
P P d cx þ k0 eqt þ cs g0 eqt f x dt ¼ S:
ð38Þ
REFERENCES Chakravorty, U., Magné, B., Moreaux, M., 2006. A Hotelling model with a ceiling on the stock of pollution. J. Econ. Dyn. Control 30, 2875–2904. Edenhofer, O., Bauer, N., Kriegler, E., 2005. The impact of technological change on climate protection and welfare: insights from the model MIND. Ecol. Econ. 54, 277–292. Edmonds, J., Clarke, J., Dooley, J., Kim, S.H., Smith, S.J., 2004. Stabilization of CO2 in a B2 world: insights on the roles of carbon capture and disposal, hydrogen, and transportation technologies. Energy Econ. 26, 517–537. Gaudet, G., Moreaux, M., Salant, S., 2001. Intertemporal depletion of resource sites by spatially-distributed users. Am. Econ. Rev. 91, 1149–1159. Gerlagh, R., 2006. ITC in a global growth-climate model with CCS. The value of induced technical change for climate stabilization. Energy J. 1, 55–72 Special issue. Gerlagh, R., van der Zwaan, B.C., 2006. Options and instruments for a deep Cut in CO2 emissions: carbon capture or renewable, taxes or subsidies? Energy J. 27, 25–48. Gitz, V., Ambrosi, P., Magné, B., Ciais, P., in press. Is there an optimal timing for sequestration to stabilize future climate? Forthcoming in: American Geophysical Union, Geophysical Monograph, Washington, DC. Heal, G., 1976. The relationship between price and extraction cost for a resource with a backstop technology. Bell J. Econ. 7, 371–378. Herfindahl, O.L., 1967. Depletion and economic theory. In: Gaffney, M. (Ed.), Extractive Resources and Taxation. University of Wisconsin Press, Madison, pp. 63–90. Herzog, H., Caldeira, K., Reilly, J.M., 2003. An issue of permanence: assessing the effectiveness of temporary carbon storage. Clim. Change 59, 293–310. Hotelling, H., 1931. The economics of exhaustible resources. J. Polit. Econ. 39, 137–175. IPCC, 2001. Climate change 2001. The Scientific Basis, Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge Univ. Press, Cambridge. IPCC, 2005. Special Report on Carbon Dioxide Capture and Storage, Contribution of Working Group III, Report of the Intergovernmental Panel on Climate Change. Cambridge Univ. Press, Cambridge. Kolstad, C.D., Krautkraemer, J.A., 1993. Natural resource use and the environment. In: Kneese, A.V., Sweeney, J.L. (Eds.), Handbook of Natural Resource and Energy Economics, Vol. 3. Elsevier Science Publishers. Chap. 26.
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Kurosawa, A., 2004. Carbon concentration target and technological choice. Energy Econ. 26, 675–684. Lafforgue, G., Magné, B., Moreaux, M., in press. The optimal sequestration policy with a ceiling on the stock of carbon in the atmosphere. Forthcoming in: R. Guesnerie and H. Tulkens (Eds.), Design of Climate Policy, CESifo Seminar Series, MIT Press, Boston. McFarland, J.R., Herzog, H.J., Reilly, J.M., 2003. Economic modeling of the global adoption of carbon capture and sequestration technologies. In: Gale, J., Kaya, Y. (Eds.), Proceedings of the
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Sixth International Conference on Greenhouse Gas Control Technologies. Elsevier Science, Oxford. Paccala, S.W., 2003. Global constraints on reservoir leakage. In: Gale, J., Kaya, Y. (Eds.), Proceedings of the Sixth International Conference on Greenhouse Gas Control Technologies. Elsevier Science, Oxford. Riahi, K., Rubin, E.S., Taylor, M.R., Schrattenholzer, L., Hounshell, D., 2004. Technological learning for carbon capture and sequestration technologies. Energy Econ. 26, 539–564. UNFCCC, 1992. UN Framework Convention on Climate Change, Earth Summit in Rio.
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n
ANALYSIS
Energy substitutions, climate change and carbon sinks ☆ Gilles Lafforguea,⁎, Bertrand Magnéb , Michel Moreauxc a
Toulouse School of Economics (INRA and LERNA), France Paul Scherrer Institute (LEA), Switzerland c Toulouse School of Economics (IDEI and LERNA), France b
AR TIC LE I N FO
ABS TR ACT
Article history:
We determine the optimal exploitation time-paths of two energy resources, one being
Received 4 September 2007
depletable and polluting, namely a fossil fuel, the other being renewable and clean. These
Received in revised form
optimal paths are considered along with the two following features. First, the cumulative
14 January 2008
atmospheric pollution stock is set not to exceed some critical threshold and second, the
Accepted 14 January 2008
polluting emissions produced by the use of fossil fuel can be reduced at the source and
Available online 4 March 2008
stockpiled in several carbon sinks of limited capacity. We show that, if the renewable resource flow is abundant, the optimal path requires that sequestration is implemented
Keywords:
only once the ceiling is reached. Moreover, the reservoirs should be completely filled by
Non-renewable resource
increasing order of their respective sequestration costs.
Pollution target
© 2008 Elsevier B.V. All rights reserved.
Climate change Carbon sequestration and storage Hotelling
1.
Introduction
Greenhouse gas emissions essentially result from the use of fossil, carbon-based, energy resources.1 Numerous technical and regulatory devices allowing for emission abatement are readily available. Their implementation may arise sooner or later depending on their respective costs. Atmospheric pollution abatement consists of capturing the dirty emissions at their source and storing them underground, either in natural
☆ We thank two anonymous referees for their helpful comments on a first version of the paper and for numerous suggestions to improve it. ⁎ Corresponding author. 21 Allée de Brienne, 31000 Toulouse, France. Tel.: +33 5 61 12 88 76; fax: +33 5 61 12 85 20. E-mail addresses: [email protected] (G. Lafforgue), [email protected] (B. Magné), [email protected] (M. Moreaux). 1 Among others, animal and human wastes, the reduction of forest cover and the extensive agricultural practices also constitute non-negligible emissions sources and favor carbon release from shallow soils.
reservoirs or in depleted oil fields and mine sites that will be called carbon sinks in the remainder of the text.2 According to the recent IPCC special report (IPCC, 2005) three industrialscale storage projects are already in operation: the Sleipner project operated by the Norwegian oil company Statoil in an offshore saline formation, the Weyburn Enhanced Oil Recovery project in Canada,3 and the In Salah project in a gas field in Algeria. Other projects are planned. In the present study, we first intend to determine the optimal starting date and pace of geological carbon sequestration. Second, we want to characterize how the recourse to such an abatement option alters the optimal time path of fossil resource exploitation, when the cumulative atmospheric 2 The term carbon sinks also refer to biomass carbon storage, e.g. in soils, plants and especially trees (see for example IPCC, 2001). Biomass carbon storage is not the focus of the present analysis. 3 In this case, captured gases are injected in oil deposits so as to enhance oil recovery, and thus increasing the oil resource base. The economic analysis of such a storage process poses some specific problems that are eluded here.
0921-8009/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2008.01.008
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concentration of carbon has to be maintained below some given critical threshold, in accordance to the United Nations Framework Convention on Climate Change.4,5 The possibility of sequestering some fraction of the carbon dioxide emitted by the combustion of fossil resources has motivated numerous empirical studies (see for example the results of integrated assessment models from McFarland et al., 2003, Edenhofer et al., 2005, Edmonds et al., 2004, Gerlagh, 2006, Gerlagh and van der Zwaan, 2006). The level of complexity of such operational models, aimed at defining some climate policy, may be required so as to take into account the numerous interactions at hand. Some models of this kind explicitly address the important question of the choice of sinks according to their specific characteristics, such as storage capacities or access costs (e.g. Kurosawa, 2004, Riahi et al., 2004, Gitz et al., in press). We here take a different approach and use a stylized model so as to exhibit the main driving forces that should determine the location choice of suitable carbon storage deposits. At a regional scale, such choices appear critical. For example, the Alberta Basin and the Western Texas regions were estimated in 2005 to contain 43 and 70 suitable CO2 storage sites, respectively (IPCC, 2005). To go straight to the point in our modeling analysis, we assume that there are final users that consume two kinds of primary energy: a polluting and scarce fossil resource, and a clean and renewable backstop resource. The users derive their utility from the use of energy in its final form, and thus face an energy cost that comprises all the costs incurred through the transformation process of the primary energy source. Hence, the two types of energy are perfect substitutes for the final users. Moreover, in accordance with current technologies, we assume that the useful energy obtained from the non-renewable resource is cheaper than the energy obtained from the renewable one. To assume the existence of an atmospheric concentration ceiling, this pollution stock being in addition partially eliminated owing to natural decay or absorption,6 implicitly con4 Before the Kyoto Protocol, the UNFCCC had already formulated the objective: “to achieve stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system” (UNFCCC, 1992, Article 2). 5 Even though the ultimate goal of the IPCC consists in stabilizing the atmospheric concentration of greenhouse gases, the Kyoto Protocol does not prescribe the level of such a target for the 2008-12 commitment period. It rather prescribes some constraint on the maximum flow of carbon emissions for the industrialized countries. More precisely, their emissions shall be reduced down to their levels of the reference year 1990, over the commitment period, i.e. 2008–12. Were an atmospheric stabilization target set, this kind of agreement would remain ineffective until all countries participate. This classical question of the cost of a public good and its breakdown among agents is made more complex in the present setting because of the dynamic nature of the problem. Since the key variable is the accumulation of greenhouse gases (we will restrain our analysis to the case of carbon dioxide), the most straightforward way of tackling the problem is to set an upper bound on the state variable. 6 This process can be interpreted as some natural carbon sequestration by a sink of very large size, e.g. typically the oceans (for more details, see IPCC, 2001).
strains the instantaneous rate of consumption of the polluting resource once the threshold is reached, in the absence of pollution capture and storage option. Two observations then deserve to be mentioned. First, along the optimal path, the time interval during which the flow of fossil consumption is constrained is endogenous. Because the carbon ceiling is reached at a date that is a function of the whole fossil consumption path from the initial time period on, then the cap imposed on the carbon accumulation affects the entire time path of the fossil resource exploitation. Second, two options are available to society if it wants to relax the constraint on fossil fuel use once at the ceiling. It may either substitute the clean resource for the dirty one, or sequester some fraction of the polluting flow generated by the fossil resource use. Each of those options entails some monetary cost. The renewable resource is more expensive than the non-renewable one. To capture the carbon dioxide is also costly. But each of these options also exhibits some specific opportunity cost. A scarcity rent is to be associated to the consumption of the fossil resource, as is the case for every non-renewable resource. To this rent augmented by the extraction and processing cost of the fossil fuel, we must add now the shadow cost of the atmospheric carbon stock and, when pollution is abated at the source of emissions, the cost of capturing the carbon augmented by the shadow cost of the sink.7 To set a cap on the accumulation of pollution, possibly removable by sequestration, results in having de facto two available carbon storage deposits. The first one is the atmospheric reservoir of temporary bounded capacity, but of infinite long run capacity thanks to the natural progressive regeneration. But to take advantage of this infinite long run capacity, the carbon emission flow has to be restrained when the temporary capacity is saturated. Hence some rent has to be charged for the use of this capacity, even before it is reached as we shall show. The second reservoir is the sink, which itself can be of limited capacity so that another rent must be charged for its use. Clearly at each point of time, some part of the emission flow can be sent into one reservoir, thus having to bear the corresponding rent, while the other part is sent into the other reservoir, having to bear the other rent. The dynamics of these two rents obey two different rules and their trajectories diverge over time. Note that the renewable resource would also entail a scarcity rent if the exploitable carbon-free flow is not sufficiently abundant. For the sake of simplicity, we assume here that the renewable substitute is abundant. The article is organized as follows. Section 2 presents the model. In Section 3, we examine the optimal paths when carbon can be only stockpiled in a single sink. In Section 4, we generalize our analysis to the case of multiple storage deposits that are differentiated by their access costs and their storage capacities. We show that those deposits have to be exploited
7 In this respect, the problem is similar to the case of residential waste management treated in Gaudet et al. (2001), with the exception that the waste discharge incurs a cost that includes a Hotelling rent (Hotelling, 1931) before it is effectively disposed. Note also that, contrary to Gaudet et al. (2001), we do not assume any sunk fixed cost for making the discharge available.
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in increasing order of access cost, no matter their capacity, and we identify the most costly deposit having to be used. We also show how this multiple deposits model can be interpreted as the approximation of a more realistic model where the cost of sequestration in each deposit is an increasing function of the cumulative stored carbon. We briefly conclude in Section 5.
2.
The model
2.1.
Assumptions and notations
However, let us assume that some carbon sequestration device is available. The potential pollution flow can be reduced at the source of emission8 and stockpiled in n carbon sinks, or reservoirs, indexed by i, i = 1,…, n. Each sink i is characterized ― by its unitary sequestration cost csi and its capacity S i. By convention, the n reservoirs are ranked by strictly increasing order of costs: cs1 b … b csi b … b csn. Capacities and sequestration costs are independent a priori and the theory developed here remains compatible with any form of relationship. Let S0i be the initial stock of pollutant in reservoir i. Without any loss of generality, we postulate that S0i = 0, i = 1,…, n. We thus have for any t ≥ 0: n X
We consider an economy in which the instantaneous gross surplus, or utility, generated by an instantaneous energy consumption qt is given by u(qt), where function u(.) has the following standard properties:
zt ¼ fxt
Assumption (A.1). u: IR++ → IR+ is a function of class C2 strictly increasing and strictly concave, satisfying the Inada conditions.
: Sit ¼ sit and Si0 ¼ 0; i ¼ 1; N ; n and t z 0
sit z 0;
: Xt ¼ xt ;
X0 ¼ X0 N 0 given:
ð1Þ
Using coal potentially generates a pollutant flow. Let ζ be the unitary carbon content of the fossil resource so that, without any abatement activity, the instantaneous carbon flow released into the atmosphere would be equal to ζxt. Let Zt be the stock of pollutant in the atmosphere at time t, zt the flow of emissions and α, α N 0, the instantaneous proportional rate of natural regeneration, assumed to be constant for the sake of simplicity (see for instance Kolstad and Krautkraemer, 1993) so that: : ð2Þ Zt ¼ zt aZt : We assume that this stock of carbon cannot be larger than ― some threshold Z . Let Z0 the stock of carbon in the atmosphere at the beginning of the planning period, assumed to be smaller ― than Z . We thus have for any t ≥ 0: P
Z Zt z 0; Z0 ¼ Z0 given:
ð3Þ
ð4Þ
i¼1
where sit is the part of the potential carbon emission flow that is sequestered into reservoir i at time t, so that:
P
We also use p to denote the marginal surplus u′ as well as, by a slight abuse of notation, the marginal surplus function: p(q) = u′(q), where u′(q) ≡ du / dq. The direct demand function d (p) is the inverse of p(q), as usually defined. We denote by u″ the second derivative of u. Under Assumption (A.1), u′(q) N 0 and u″(q) b 0, ∀q N 0. Energy needs may be supplied by two resources: a dirty nonrenewable resource (fossil), e.g. coal, and a clean renewable resource, e.g. solar. We assume that these two energy sources are perfect substitutes, so that if xt denotes the instantaneous consumption of the fossil resource and yt, the instantaneous consumption of the renewable resource, then the surplus generated by the total consumption is u(xt + yt), i.e. qt = xt + yt. The average cost of transforming coal into directly usable energy, also equal to the marginal cost, is constant and de~ denote the flow of fossil resource to be noted by cx. Let x consumed in order to equalize the marginal surplus to the marginal monetary cost of the resource. Thus, x~ is the solu~ = d(c ). Let X0 be the initial fossil tion of u′(x) = cx, that is, x x resource stock and Xt, the stock available at time t, so that:
591
ð5Þ
Si Sit z 0; i ¼ 1; N ;n and t z 0
ð6Þ
sit z 0; i ¼ 1; N ;n and t z 0:
ð7Þ
― ˙ ≤ 0. Hence, the Once the ceiling Z is reached, we must have Z t upper bound on the fossil fuel consumption without se_ ― ― questration is ― x =αZ /ζ. Let p denote the marginal utility of x : ― ― p =u′( x ). In the case of sequestration in reservoir i, we assume that the total marginal cost of a “clean” consumption of the ― fossil resource, cx +csiζ, is lower than p . If not, we would always _ be better off remaining at x rather than relaxing the constraint by sequestering any part of the emission flow in reservoir i.
_ Assumption (A.2). ∀i = 1,…, n: cx + csiζ b p. The other resource is a non-polluting renewable resource that can be made available to the end users at a constant average cost cy. The cost of the renewable resource is the total cost of supplying the good to the final users, so that the nonrenewable and the renewable resources are perfect substitutes _ for the users. We assume that cy N p since, if cy was lower _ ― than p, condition Z − Zt would never constrain the energy consumption. _ Let y be the constant instantaneous flow of this renewable ~ the energy available at each point of time. We denote by y flow of renewable resource that the society would choose to _ consume once the fossil resource is exhausted, provided that y ~ is the solution of u′(y) = c : ~ is sufficiently large. Then, y y y = d(cy). We assume that the available flow of the renewable resource ~ , so that no rent has to be is abundant, i.e. at least equal to y charged for the use of this resource.9 _ _ Assumption (A.3). p b cy and ~y b y. Let us assume that the instantaneous social rate of discount, ρ N 0, is constant. The objective of the social planner is to choose 8
Despite the fact that the carbon capture primarily applies to emissions from power regeneration processes, we here assume that the entire carbon emission flow can be abated. 9 The case of a constrained flow of renewable resource is analyzed in Lafforgue et al. (in press).
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the resource and sequestration trajectories that maximize the sum of the discounted instantaneous net surplus.
2.2.
Problem formulation and optimality conditions
The social planner problem (P) can be expressed as follows: # Z l" n X uðxt þ yt Þ csi sit cx xt cy yt eqt dt ðPÞ max fðsit ;i¼1; N ;n;xt ;yt Þ;tz0g
0
i¼1
subject to Eqs. (1)–(7) and subject to the following inequality constraints: Xt ≥ 0, xt ≥ 0 and yt ≥ 0, t ≥ 0. Let Lt be the current valued Lagrangian for the problem (P): n n X X csi sit cx xt cy yt kt xt þ git sit Lt ¼ uðxt þ yt Þ i¼1 i¼1 " # n n X X P P þ At fxt sit aZt þ mit Si Sit þ mZt Zt Zt i¼1 i¼1 " # n n X X P sit þ git sit þ gxt xt þ gyt yt : þ g st fxt i¼1
i¼1
The static and the dynamic first-order conditions of (P) are: g st þ git ; ALt =Asit ¼ 0 f csi ¼ git At P
i ¼ 1; N ;n
ALt =Axt ¼ 0 f uVðxt þ yt Þ ¼ cx þ kt At f P g st f gxt
ð8Þ ð9Þ
ALt =Ayt ¼ 0 f uVðxt þ yt Þ ¼ cy gyt ;
ð10Þ
: : k t ¼ qkt ALt =AXt f k t ¼ qkt f kt ¼ k0 eqt
ð11Þ
: : g it ¼ qgit ALt =ASit f g it ¼ qgit þ mit ; i ¼ 1; N ;n
ð12Þ
: : A t ¼ qAt ALt =AZit f A it ¼ ða þ qÞAt þ mZt ;
ð13Þ
with the usual complementary slackness conditions and the associated transversality conditions. First, we note that ηit ≤ 0, ηit being the instantaneous marginal cost of the carbon stock which is already sequestered into sink i at time t or, alternatively, − ηit being the marginal value of this part of sink i which is still available for carbon storage. If the storage capacity of this sink is limited and if at date t, Sit was increased by an exogenous amount dSit N 0, then the optimized value of the objective function of (P) would diminish. Second, as long as reservoir i is not completely filled, i.e. for any ― t such that S i − Sit N 0, νit = 0, so that from Eq. (12) it comes: P
Si Sit N 0 Z git ¼ gi0 eqt :
ð14Þ
By the same type of argument, μt ≤ 0, μt being the instantaneous marginal cost of the carbon stock in the atmosphere at date t ― or, alternatively, − μt being the marginal value of Z − Zt, this part of the atmospheric carbon reservoir which has not been used at time t. If at date t, Zt was increased by an exogenous amount dZt N 0, the optimized value of the objective function of (P) would, at best, remain constant, and would decrease in the ― worst case. Furthermore, since Z0 b Z , there must exist an initial time interval during which the stock of carbon in the atmosphere is below the ceiling, hence νZt = 0 so that, by integrating Eq. (13), we get: At ¼ A0 eðaþqÞt :
ð15Þ
In fact, from the planner's point of view, the possibility of sequestering carbon emissions is double: either emissions are
stockpiled in non-regenerating and costly reservoirs whose capacity is limited, i.e. the geological carbon sinks, or they are released in a natural reservoir, i.e. the atmosphere, whose capacity is limited by the ceiling constraint, but which is autoregenerating and free of charge. Then, the difference in the dynamics of ηit, i=1,…, n and of μt can be explained as follows. On the one hand, the decision to resort to the storage capacity of reservoir i is irreversible since Sit is a monotonous and nondecreasing function. On the other hand, owing to the natural regeneration, the stock of pollutant in the atmosphere Zt can either increase or decrease. On that account, whatever the sequestration policy may be, Zt will be ultimately induced to decrease since the fossil resource stock is finite. As it will be seen in the next sections, such an asymmetry strongly governs the optimal solution. ― ― Finally, if there exists t such that for any t ≥ t the ceiling ― constraint is slack for all future time periods, then μt = 0, t ≥ t . Absent the ceiling constraint as well as the sequestration device, the optimal solution would be the well-known Herfindahl-Hotelling path.10,11 For any λ0 ∈ (0; cy − cx), define pH t (λ0) as the price path along which the coal rent is increasing ρt H at the discount rate ρ, pH t (λ0) = cx + λ0e ; T (λ0) as the time at H which pt (λ0) is equal to the average cost of the abundant renewable substitute, TH(λ0) = [ln(cy − cx) − ln λ0] / ρ; dH t (λ0) as the H corresponding instantaneous energy demand, dH t (λ0) ≡ d(pt (λ0)) H H H and zt (λ0) as the generated polluting flow, zt (λ0) = ζdt (λ0); last Rt H H DH t (λ0) as the cumulative demand, Dt ðk0 Þ ¼ 0 ds ðk0 Þds. ―H Trivial calculations would show that D (λ0)≡DTHH(λ0) (λ0) is a strictly decreasing function with limλ0↓0DH(λ0)=+∞ and limλλ0↑(cy− ― cx)DH(λ0)=0, so that the equilibrium equation D H(λ0) =X0 has a H unique solution λ0 which is the optimal value of λ0. Thus the optimal path is a two-phase path. During the first phase [0, TH(λH 0 )], the energy demand is supplied only by the fossil fuel, and the energy consumption is decreasing from dH 0 (λ0) =d H H (cx +λ0) down to dH TH(λH0) = d(cy). From T (λ0 ) onwards, the energy consumption is stationary and equal to d(cy), supplied by the renewable source. The pollution emission flow determines the pollution stock trajectory. Let ZH t (λ0) be the pollution stock induced by the emission flow: 8 H H > < fdt ðk0 Þ ; ta½0;T ðk0 ÞÞ H ð16Þ zt ðk0 Þ ¼ > :0 ; ta TH ðk0 Þ; þ lÞ; that is12: ( ZH t ðk0 Þ ¼
0 at
Z e
þ
R
t H aðtsÞ ds 0 ds ðk0 Þe R T H ðk Þ aðtTH ðk0 ÞÞ aðtsÞ e ds f 0 0 dH s ðk0 Þe
Z0 eat þ f
;
ta½0;TH ðk0 ÞÞ ; ta TH ðk0 Þ; þ lÞ:
ð17Þ 10
Herfindahl (1967), Hotelling (1931). 11 In such a case, the objective function of program (P) writes: Rl uðxt þ yt Þ cx xt cy yt eqt dt and the constraints (2)–(7) vanish. 0 The conditions (8), (12) and (13) also vanish and, from Eq. (11), the condition (9) becomes: u′(xt + yt) = cx + λ0eρt − γxt. The condition (10) remains the same. 12 0 –αt At time t, ZH t (λ0) is the sum of Z e , this part of the initial pollution stock not yet naturally regenerated at this time, and the sequence of −α(t − τ) ζdH , τ∈[0,t], these parts of the successive emission flows of τ (λ0)e time τ not yet regenerated. After TH(λ0), there is no new emission and H the stock smoothly declines according to eaðtT ðk0 ÞÞ ZH TH ðk0 Þ ðk0 Þ, down to 0.
EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 58 9 –5 97
―
―
H Let ZH(λ0)≡max{ZH t (λ0),t≥0}. Clearly, dZ (λ0)/dλ0 b 0 and there ―0 ― ― ― ― 0 exists some critical value X of X such that ZH(X0 )=Z and DH(λ0)= ―0 0 ―0 H H ― X . For X bX , the ceiling constraint is never active, Zt (λ0 )bZ, ― and for X 0 NX 0 there must exist some time interval within which ― H ZH t (λ0 )NZ thus violating the constraint. We assume from now that the initial amount of fossil fuel is sufficiently large in such a way that, without any ceiling constraint, the critical level of the pollution stock would be overshot over some time interval.
―
Assumption (A.4). X0 N X 0.
3.
The case of a single reservoir
In this section, we assume that the sequestration device consists in a single carbon sink, so that we can neglect the index i in conditions (8) and (12). The capacity of the sink can be said “large” or “small” according to the role played by the reservoir capacity constraint (5), together with Eq. (6). If this constraint is never active (i.e. ηt = 0, t ≥ 0), meaning that no rent has to be charged for the use of the sink, then the reservoir capacity is said to be large. In that case, we would get quite similar insights in terms of abatement option as in Chakravorty et al. (2006). Clearly, the most interesting case is the case of a “small” reservoir. If the sink capacity is sufficiently small, its use implies an opportunity cost in addition to the sequestration cost cs. But contrary to cs, this opportunity cost is not constant over time. We know (cf. Section 2) that, as long as the reservoir is not filled, the absolute value of ηt is increasing at ― the social rate of discount: St b S ⇒ ηt = η0eρt. Substituting for ηt into condition (8) and given that γst = 0 provided st N 0, we get: g st Z At ¼ cs g0 eqt þ P g st : cs ¼ g0 eqt At P
ð18Þ
Next, we can substitute for − μt into Eq. (9). Given that γxt = 0 and provided that xt N 0, and after simplifications, we must have: uVðxt Þ ¼ cx þ k0 eqt þ cs g0 eqt f;
ð19Þ
where cs − η0eρt, the full marginal cost of sequestration, is increasing over time since η0 b 0. If for any phase during which only the fossil resource is ― used, the ceiling constraint Z − Zt ≥ 0 is never active and a part of the carbon emissions is sequestered, then the resource consumption xt must be equal to d[cx + λ0eρt + (cs − η0ept)ζ] so as to satisfy the first-order conditions. However, it does not mean that the totality of the emission flow is necessarily captured. Minimization of costs implies that only part of emissions that exceeds the ceiling has to be stockpiled, so that: st ¼ f d cx þ k0 eqt þ cs g0 eqt f P x :
593
Fig. 1 illustrates why, at the ceiling, it is optimal first to abate and to sequester, and next to stop the abatement. Since the full marginal cost, cx + λ0eρt + (cs − η0eρt)ζ, is increasing over time, the sequestration phase should precede a phase during which the fossil resource exploitation is constrained by the natural regeneration capacity of the atmosphere at the ceiling, ― ― ― that is a phase during which qt = xt = x and pt = p for Zt = Z . In Fig. 1 the marginal cost curves are drawn for three dates t, t′, t″, t b t′ b t″, sufficiently spaced but not too much, when the economy is at the ceiling. At date t, the inverse demand function is crossing the upper branch of the marginal cost ― curve at xt, implying that the part ζ[xt − x ] of the emission flow has to be sequestered. At t′, the inverse demand function ― intercepts the two steps of the marginal cost curve at x , ― implying that the optimal consumption must amount to x and that abatement is too costly. At time t″, the inverse demand curve is crossing the lowest part of the marginal cost curve, so ― that the optimal consumption is lower than x . Then, Zt is decreasing and the path is the Hotelling path forever. The optimal path consists in five phases, as illustrated in Fig. 2. During the first phase [0,t1), the pollution stock is increasing, the resource price is equal to the full marginal cost ^ pt, ^ pt ≡cx + ^t). At the λ0eρt −μ0e(α + ρ)tζ, and only the fossil fuel is used: qt =xt =d(p ~ , where ~ pt ≡cx +λ0eρt +(cs −η0eρt)ζ is the end of the phase, pt =p t marginal cost of consuming the “emissions-free” coal, and the ceiling is attained. The second phase [t1,t2) is a phase at the ceiling during which only the fossil resource is used and some part of the ~ ) and pt, qt = xt = d(p potential emission flow is sequestered: pt = ~ t ― ― ~ st = ζ[d(p t) − x ]. At the end of the phase, pt = p . The third phase [t2,t3) is still a phase at the ceiling during ― which pt = p , and only the fossil resource is consumed, qt = xt = ― x , but the emission flow is no longer stockpiled, being just balanced by the natural regeneration. During this phase, |μt| is decreasing and becomes nil at the end of the phase. The fourth phase [t3,t4) is a pure Hotelling phase, pt = pH t (λ0), only the fossil resource has to be used, qt = xt = d(pH t ) and since ― xt b x the pollution stock starts to decrease. At the end of this phase, the energy price is just equal to the marginal cost of the renewable resource cy and the fossil resource is exhausted. The last phase [t4,∞) is a phase during which only the re~ newable resource is used: qt = yt = y and pt = cy.
ð20Þ
―
When Zt = Z , the full marginal cost of the energy satisfying the ceiling constraint is a two-step function, equal to the ― monetary extraction cost augmented by the rent if xt ≤ x , at which the sequestration cost (cs − η0eρt) ζ must be added if ― xt N x , given that the sequestration capacity of the sink is not saturated: cm ¼
; if xt V P x cx þ k0 eqt cx þ k0 eqt þ ðcs g0 eqt Þf ; if xt N P x:
ð21Þ
Fig. 1 – Full marginal cost of the fossil resource and optimal consumption at the ceiling.
594
EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 5 8 9 –5 97
Consequently, we would follow a continuous and increasing ― marginal cost path, from cx + λ0 at t = 0 up to p at t2, and then: • either cx + λ0eρt + cs2ζ N cx + λ0eρt + (cs1 − η10eρt)ζ over the interval [t1,t2) and in this case, reservoir 2 will never be used. The full marginal cost of the now “clean” fossil resource, owing to sequestration, into reservoir 1 would always be smaller than the marginal cost when sequestering into reservoir 2; ― • or there exists a date t b t2, such that cx + λ0eρt1 + cs2ζ b cx + λ0eρ + ― ρt (cs1 − η10e )ζ, t b t , and in this case, reservoir 2 must also be used though the sequestration cost in this sink, cs2, is larger than the sequestration cost in reservoir 1, cs1. The point to be noticed here is that, at the optimum, the two sinks should never be used simultaneously. If it was not the case, i.e. if sit N 0, i = 1, 2 over an interval (t′,t″), t′ b t″, we would have γit = 0, and νit = 0,13 i = 1, 2 over the same interval and then, from Eq. (8): g st ; At ¼ csi gi0 eqt þ P
taðtV;t WÞ;
i ¼ 1; 2:
ð24Þ
Substituting for μt in Eq. (9), we obtain: uVðxt þ yt Þ ¼ cx þ k0 eqt þ csi gi0 eqt f;
taðtV; t WÞ;
i ¼ 1; 2
ð25Þ
so that: cs1 g10 eqt ¼ cs2 g20 eqt ; Fig. 2 – Optimal price, consumption and sequestration paths — the single reservoir case. The values of the seven endogenous variables λ0, μ0, η0, t1, t2, t3 and t4 characterizing this type of optimal path are determined by solving a seven-equation system that ensure the continuity of the various time-paths at each transition date. Such a system of equations is given in Appendix A for the more complex case of multiple reservoirs.
4.
The multiple reservoirs case
We first examine the case of the optimal use of two reservoirs and generalize the analysis to any number of reservoirs.
4.1.
The case of two reservoirs
Consider the case where two different sequestration devices can be used, namely reservoirs 1 and 2. As long as reservoir 2 is not used, results of Section 3 apply. In particular, as in the single reservoir case, the two first phases of the optimal price paths would be pt = cx + λ0eρt − μ0e(α + ρ)tζ over [0,t1) and pt = cx + λ0eρt − μ0e(α + ρ)tζ + (cs1 − η10eρt)ζ over [t1,t2), with the following associated continuity conditions: cx þ k0 eqt1 A0 eðaþqÞt1 f ¼ cx þ k0 eqt1 A0 eðaþqÞt1 f þ cs1 g10 eqt1 f
taðtV;t WÞ
ð26Þ
which is clearly impossible if cs1 b cs2 for any − ηi0 ≥ 0, i = 1, 2. The only case in which we can have in one phase the full marginal cost of the clean fossil resource with sequestration in reservoir 1 smaller than the cost involved with sequestering in reservoir 2, and another phase during which the inverted inequality holds, is when −η20 b −η10. Thus the stockpiling phase into reservoir 1 must always precede the stockpiling phase into reservoir 2. Taking this remark into account, the optimal path consists now in six phases: • Phase 1, [0,t1), rise of the pollution stock to the ceiling without any sequestration; • Phase 2, [t1,t2), at the ceiling with active sequestration in reservoir 1; • Phase 3, [t2,t3), at the ceiling with active sequestration in reservoir 2; • Phase 4, [t3,t4), at the ceiling without any sequestration; • Phase 5, [t4,t5), pure Hotelling path, with a pollution stock under the ceiling and forever; • Phase 6, [t5,∞), renewable resource exploitation. The values of the nine variables λ0, μ0, η10, η20, t1, t2, t3, t4 and t5 characterizing this type of path are determined by solving the nine-equation system characterized in Appendix A. The corresponding optimal price path is illustrated in Fig. 3.
4.2.
The case of n reservoirs
ð22Þ The previous analysis can be easily extended to the case of n reservoirs. Let the sinks be indexed by strictly increasing order
and p: cx þ k0 eqt2 A0 eðaþqÞt2 f þ cs1 g10 eqt2 f ¼ P
ð23Þ
13
―
If sit N 0 over (t′,t″), then Sit b S i from which vit = 0 and ηit = ηi0eρt.
EC O L O G IC A L E C O N O M IC S 6 7 ( 2 0 08 ) 58 9 –5 97
Fig. 3 – Optimal price path — the case of two reservoirs.
question leads to the first exploitation of the sub-sink whose cost is c1sii and whose capacity is ΔS1sii, then if necessary, the second sub-sink whose cost is c2sii and whose capacity is ΔS2sii, and so and so forth. In other words, it is necessary to exploit the sub-sinks in the natural order of filling the sink i. Let us consider now the case of n different sinks, their respective cost functions being themselves approximated by a step function. Let us build the sequence of theoretical sinks j = 1,…, m by sorting again the whole steps of the various sinks indexed by increasing order of their costs and by grouping the capacities of the steps whose costs are identical. We denote by cjs the average cost of the theoretical j-ranked ― sink whose sequestration capacity is S j: c1s b … b cjs b … b cm s. Those theoretical sinks are built by recurrence as follows. For j = 1: c1s ¼ min c1s i ;
of sequestration costs: cs1 b … b csi b … b csn. Reservoir m, for any m b n, should be used over [tm, tm + 1) and at the end of this sequestration phase, we must have cx + λ0eptm + 1 + (csm − ηm0eρtm + 1) ― ζ = p . Hence, there are two alternatives: • either c x + λ0 eρt + c sm + 1 ζ ≥ c x + λ0 eρt + (csm − η m0 eρt)ζ over the interval [tm, tm + 1) and in this case, the m + 1th reservoir should not be used; ― • or there exists a date t b tm + 1, such that cx + λ0eρt + csm + 1ζ b cx + ― λ0eρt + (csm − ηm0eρt)ζ, t N t , so that the m + 1th sink should also be used. We thus have to implement again the sequestration opportunity test as described above. For a finite number of reservoirs, we obtain an algorithm that converges to a finite number of steps.
4.3. The model as a model of sequestration cost increasing with the cumulated sequestration Until now, we have assumed that the average cost of sequestration in each sink i, csi, is constant and thus equal to the marginal cost. Since the higher the amount of stored carbon the higher is the pressure in the reservoir, and since this high pressure makes the incremental carbon unit more costly to be stored, one has to assume that the sequestration cost is an increasing function of the carbon mass already injected. In other words, one may state that for each deposit i, csi = csi(Si) and dcsi / dSi N 0, where Si is the carbon mass already stored into the sink i. The model analyzed so far could be seen as a model where the functions csi(.) are approximated by step functions, the more numerous the steps, the more accurate the approximation. Let mi be the number of steps chosen to approximate the function csi(.) and let 1i,…, hi,…, mi be the indexes of those various steps, by increasing order of average sequestration cost hi i for each step, c1sii b … b chsii b … b cm si , csi being the average (marginal) cost of sequestration at step hi. To each step is associated Phi P P i ―h some absorption capacity ΔS i i , so that m hi ¼1 DSi ¼ Si . If there was a single sequestration sink, the sink i, the identification to the preceding model is straightforward. The optimal exploitation rule for differentiated average sequestration costs prescribes that it is necessary to exploit them by increasing order of their costs. Let us apply this rule to the only sink i, decomposed in mi differentiated sub-sinks. The rule in
595
i ¼ 1; N ;m :
ð27Þ
Let I(1) be the set of indexes of sinks for which c1si = c1s. Then: P1
S ¼
X
P1i
D Si :
ð28Þ
iaIð1Þ
Let us take out the steps whose costs are c1s and start the procedure again. Since the cardinal of the set of steps is finite, the defined procedure includes a finite number of stages. The P number of theoretical sinks, m, is at most equal to i¼1 mi . It would precisely be equal to this sum if the costs at the steps of the effective sinks were all differentiated. The order by which one shall optimally use the theoretical sinks is the order of their costs. By proceeding in such a manner, the order of exploitation of the various steps of a single sink is the natural order of filling this reservoir. Moreover, all the effective steps that constitute a theoretical j-rank sink can be used simultaneously. At the optimum, the carbon shall thus be stored in several sinks at a time. Note that some part of a given reservoir may be used at some stage and next, other reservoirs used before going back the first one.
5.
Conclusion
We have studied the economic rationale of capturing and sequestering the carbon, so as to maintain its atmospheric stock below some threshold level. Consuming a fossil resource, and capturing and sequestering the pollution which is generated, is like consuming a resource coming simultaneously out of two mining sites: the proper underground site of extraction and the sequestering site, the rent to be charged for each one having to grow at the interest rate since both must satisfy the Hotelling arbitrage condition. In this paper, we insisted upon the multiplicity of sequestering sites. Clearly, a generalization along the Herfindahl (1967) line of analysis of the case of different mining sites is immediate. We have shown that such a policy ought to be implemented only once this critical level has been reached, whatever the cost of sequestration is, i.e. independently of the number of deposits, their access costs and their retention capacities. Moreover, it stems from our analysis in Section 4 that the optimal resources exploitation and the sequestration implementation, obtained with constant average costs, are robust
596
EC O LO GIC A L E CO N O M ICS 6 7 ( 2 00 8 ) 5 8 9 –5 97
with respect to other costs specifications that would depend on the flow and/or the cumulative sequestered carbon and/or the resource extraction as far as the carbon deposit and the exhaustible resource are respectively concerned (see Heal, 1976). The absence of sequestration in the short run does not mean the absence of environmental policy in the short run. On the contrary, even before its implementation, the sequestration option affects the optimal pace at which exhaustible resource exploitation is reduced even before the ceiling is reached. The consumption reduction is attributable both to the opportunity cost of emitted pollution before the ceiling and to the opportunity cost of pollution sequestration once the ceiling is reached, those costs adding up to the total delivery cost of the resource. Finally, our definition of the storage process does not include the possibility of some leakage that would result in sending the carbon back to the atmosphere (see Herzog et al., 2003; and Paccala, 2003). This leakage phenomenon, would it be continuous over time, would not have any incidence on the optimal solution in the short run. In this case, only the length of the capture and storage phase would be extended to the entire phase at the ceiling, the sequestration activity exactly compensating the leakage at each date.
Appendix A. Continuity equation system of the optimal path in the multiple reservoir case • The cumulative fossil resource consumption-initial stock balance equation, here: Z
t1
0
Z t2
d cx þ k0 eqt A0 eðaþqÞt f dt þ d cx þ k0 eqt þ cs1 g10 eqt f dt ð29Þ t1 Z t3 P þ d cx þ k0 eqt þ cs2 g20 eqt f dt þ ½t4 t3 x t2 Z t5 þ d cx þ k0 eqt dt ¼ X0 : t4
• The price continuity equation at t1: cx þ k0 eqt1 A0 eðaþqÞt1 f ¼ cx þ k0 eqt1 þ cs1 g10 eqt1 f:
ð30Þ
• The pollution stock continuity equation at t1: Z0 eat1 þ f
Z
t1 0
P d cx þ k0 eqt fA0 eðaþqÞt eaðtt1 Þ dt ¼ Z:
ð31Þ
• The saturation condition for reservoir 1 at t2: Z f
t2
t1
P P d cx þ k0 eqt cs1 g1t2 eqt2 f x dt ¼ S1 :
ð32Þ
• The price continuity equation at t2: cx þ k0 eqt2 þ cs1 g10 eqt2 f ¼ cx þ k0 eqt2 þ cs2 g20 eqt2 f:
ð33Þ
• The saturation condition for reservoir 2 at t3: Z f
t3
t4
P P d cx þ k0 eqt þ cs2 g20 eqt f x dt ¼ S2 :
ð34Þ
• The price continuity equations at t3, t4 and t5: P cx þ k0 eqt3 þ cs2 g20 eqt2 f ¼ p; P
ð35Þ
cx þ k0 eqt4 ¼ p;
ð36Þ
cx þ k0 eqt5 ¼ cy :
ð37Þ
Note that the single reservoir case developed in Section 3 is ― ― ― ― ― the special case for which cs1 = cs2 ≡ cs, S 1 = S 2 and S 1 + S 2 ≡ S , which directly implies η10 = η20 ≡ η0. The date t3 works as t2, t4 as t3 and t5 as t4. Condition (33) vanishes, and Eqs. (32) and (34) are replaced by a single saturation condition: Z f
t2 t1
P P d cx þ k0 eqt þ cs g0 eqt f x dt ¼ S:
ð38Þ
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