WPS6415 Policy Research Working Paper 6415 Should Marginal Abatement Costs Differ Across Sectors? The Effect of Low-Carbon Capital Accumulation Adrien Vogt-Schilb Guy Meunier Stephane Hallegatte The World Bank Sustainable Development Network Office of the Chief Economist April 2013 Policy Research Working Paper 6415 Abstract The optimal timing, sectoral distribution, and cost Marginal Implicit Rental Cost of the Capital (MIRCC) of greenhouse gas emission reductions is different used to abate. Two apparently opposite views are when abatement is obtained though abatement reconciled. On the one hand, higher efforts are justified expenditures chosen along an abatement cost curve, or in sectors that will take longer to decarbonize, such as through investment in low-carbon capital. In the latter urban planning; on the other hand, the MIRCC should framework, optimal investment costs differ in each sector: be equal to the carbon price at each point in time and they are equal to the value of avoided carbon emissions, in all sectors. Equalizing the MIRCC in each sector to minus the value of the forgone option to invest later. the social cost of carbon is a necessary condition to reach It is therefore misleading to assess the cost-efficiency the optimal pathway, but it is not a sufficient condition. of investments in low-carbon capital by comparing Decentralized optimal investment decisions at the sector levelized abatement costs, that is, efforts measured as the level require not only the information contained in the ratio of investment costs to discounted abatement. The carbon price signal, but also knowledge of the date when equimarginal principle applies to an accounting value: the the sector reaches its full abatement potential. This paper is a product of the Office of the Chief Economist, Sustainable Development Network. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at shallegatte@worldbank.org and vogt@centre-cired.fr. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Should marginal abatement costs differ across sectors? The effect of low-carbon capital accumulation Adrien Vogt-Schilb 1,∗, Guy Meunier 2 , St´ ephane Hallegatte 3 1 CIRED, Nogent-sur-Marne, France. 2 INRA–UR1303 ALISS, Ivry-Sur-Seine, France. 3 The World Bank, Sustainable Development Network, Washington D.C., USA Keywords: climate change mitigation; carbon price; path dependence; sectoral policies; optimal timing; inertia; levelized costs JEL classi�cation: L98, O21, O25, Q48, Q51, Q54, Q58 1. Introduction Many countries have set ambitious targets to reduce their Greenhouse Gas (GHG) emissions. To do so, most of them rely on several policy instruments. The European Union, for instance, has implemented an emission trading sys- tem, feed-in tariffs and portfolio standards in favor of renewable power, and different energy efficiency standards on new passenger vehicles, buildings, home appliances and industrial motors. These sectoral policies are often designed to spur investments that will have long-lasting effects on emissions, but they have been criticized because they result in different Marginal Abatement Costs (MACs) in different sectors. Existing analytical assessments conclude that differentiating MACs is a second-best policy, justi�ed if multiple market failures cannot be corrected in- dependently (Lipsey and Lancaster, 1956). For instance, if governments cannot use tariffs to discriminate imports from countries where no environmental poli- cies are applied, it is optimal to differentiate the carbon tax between traded and non-traded sectors (Hoel, 1996). A government should tax emissions from households at a higher rate than emissions from the production sector, if labor supply exerts market power (Richter and Schneider, 2003). Also, many abate- ment options involve new technologies with increasing returns or learning-by- doing (LBD) and knowledge spillovers. This “twin-market failure� in the area of green innovation can be addressed optimally by combining a carbon price with a subsidy on technologies subjected to LBD and an R&D subsidy (e.g., Jaffe et al., 2005; Fischer and Newell, 2008; Grimaud and Lafforgue, 2008; Gerlagh et al., 2009; Acemoglu et al., 2012). But if the only available instrument is a carbon tax, higher carbon prices are justi�ed in sectors with larger learning effects (Rosendahl, 2004). ∗ Corresponding author Email addresses: vogt@centre-cired.fr (Adrien Vogt-Schilb ), ephane guy.meunier@ivry.inra.fr (Guy Meunier), shallegatte@worldbank.org (St´ Hallegatte) World Bank Policy Research Working Paper (2013) April 19, 2013 These studies often model a social planner who takes an abatement cost function as given, and may choose any quantity of abatement at each time step. Within this framework, MACs can be easily computed as the derivative of the cost functions; but the inertia induced by slow capital accumulation is neglected (Vogt-Schilb et al., 2012). Only a few studies explicitly model inertia or slow capital accumulation; they conclude that higher efforts are required in the particular sectors that will take longer to decarbonize, such as transportation infrastructure (Lecocq et al., 1998; Jaccard and Rivers, 2007; Vogt-Schilb and Hallegatte, 2011); but they do not provide an analytical de�nition of abatement costs. In this paper, we assess the optimal cost, timing and sectoral distribution of greenhouse gas emission reductions when abatement is obtained through in- vestments in low-carbon capital. We use an intertemporal optimization model with three characteristics. First, we do not use abatement cost functions; in- stead, abatement is obtained by accumulating low-carbon capital that has a long-lasting effect on emissions. For instance, replacing a coal power plant by renewable power reduces GHG emissions for several decades. Second, these in- vestments have convex costs : accumulating low-carbon capital faster is more expensive. For instance, retro�tting the entire building stock would be more expensive if done over a shorter period of time. Third, abatement cannot ex- ceed a given maximum potential in each sector. This potential is exhausted when all the emitting capital (e.g fossil fuel power plants) has been replaced by non-emitting capital (e.g. renewables). We �nd that it is optimal to invest more dollars per unit of low-carbon capital in sectors that will take longer to decarbonize, as for instance sectors with greater baseline emissions. Indeed, with maximum abatement potentials, investing in low-carbon capital reduces both emissions and future investment opportunities. The optimal investment costs can be expressed as the value of avoided carbon, minus the value of the forgone option to abate later in the same sector. Since the latter term is different in each sector, it leads to different investment costs and levels. There are multiple possible de�nitions of marginal abatement costs in a model where abatement is obtained through low-carbon capital accumulation. Here, we de�ne the Marginal Levelized Abatement Cost (MLAC) as the marginal cost of low-carbon capital (compared to the cost of conventional capital) divided by the discounted emissions that it abates. This metric has been simply labeled as “Marginal Abatement Cost� (MAC) by scholars and government agencies, suggesting that it should be equal to the price of carbon. We �nd that MLACs should not be equal across sectors, and should not be equal to the carbon price. Instead, the equimarginal principle applies to an accounting value: the Marginal Implicit Rental Cost of the Capital (MIRCC) used to abate. MIRCCs generalize the concept of implicit rental cost of capital proposed by Jorgenson (1967) to the case of endogenous investment costs. On the optimal pathway, MIRCCs – expressed in dollars per ton – are equal to the current carbon price and are thus equal in all sectors. If the abatement cost is de�ned as the implicit rental cost of the capital used to abate, a necessary condition for optimality is that MACs equal the carbon price. It is not a sufficient condition, as many sub- optimal investment pathways also satisfy this constraint. Optimal investment decisions to decarbonize a sector require combining the information contained in the carbon price signal with knowledge of the date when the sector reaches 2 its full abatement potential. The rest of the paper is organized as follows. We present our model in section 2. In section 3, we solve it in a particular setting where investments in low-carbon capital have a permanent impact on emissions. In section 4, we solve the model in the general case where low-carbon capital depreciates at a non-negative rate. In section 5 we de�ne the implicit marginal rental cost of capital and compute it along the optimal pathway. Section 6 concludes. 2. A model of low-carbon capital accumulation to cope with a carbon budget We model a social planner (or any equivalent decentralized procedure) that chooses when and how (i.e., in which sector) to invest in low-carbon capital in order to meet a climate target at the minimum discounted cost. 2.1. Low-carbon capital accumulation The economy is partitioned in a set of sectors indexed by i. For simplicity, we assume that abatement in each sector does not interact with the others.1 Without loss of generality, the stock of low-carbon capital in each sector i starts at zero, and at each time step t, the social planner chooses a non-negative amount of physical investment xi,t in abating capital ai,t , which depreciates at rate δi (dotted variables represent temporal derivatives): ai,0 = 0 (1) ˙ i,t = xi,t − δi ai,t a (2) For simplicity, abating capital is directly measured in terms of avoided emissions (Tab. 1).2 Investments in low-carbon capital cost ci (xi ), where the functions ci are positive, increasing, differentiable and convex. The cost convexity bears on the investments flow, to capture increasing op- portunity costs to use scarce resources (skilled workers and appropriate capital) to build and deploy low-carbon capital. For instance, xi,t could stand for the pace — measured in buildings per year — at which old buildings are being retro�tted at date t (the abatement ai,t would then be proportional to the share of retro�tted buildings in the stock). Retro�tting buildings at a given pace requires to pay a given number of scarce skilled workers. If workers are hired in the merit order and paid at the marginal productivity, the marginal price of retro�tting buildings ci (xi ) is a growing function of the pace xi . In each sector, a sectoral potential a ¯i represents the maximum amount of GHG emissions (in GtCO2 per year) that can be abated in this sector: ai,t ≤ a ¯i 1 This is not completely realistic, as abatement realized in the power sector may actually reduce the cost to implement abatement in other sectors using electric-powered capital. In order to keep things simple, we let this issue to further research. 2 Investments x i,t are therefore measured as additional abating capacity (in tCO2 /yr) per unit of time, i.e. in (tCO2 /yr)/yr. 3 Emissions GtCO2/yr ∑ ai(t) Ed B Es t time Figure 1: An illustration of the climate constraint. The cumulative emissions above Es are capped to a carbon budget B (this requires that the long-run emissions tend to Es ). Dangerous emissions E d are measured from Es . For instance, if each vehicle is replaced by a zero-emission vehicle, all the abate- ment potential in the private mobility sector has been realized.3 The sectoral potential may be roughly approximated by sectoral emissions in the baseline, but they may also be smaller (if some fatal emissions occur in the sector) or could even be higher (if negative emissions are possible). We make the simpli- fying assumption that the potentials a ¯i and the abatement cost functions ci are constant over time and let the cases of evolving potentials and induced technical change to further research. 2.2. Carbon budget The climate policy is modeled as a so-called carbon budget for emissions above a safe level (Fig. 1). We assume that the environment is able to absorb a constant flow of GHG emissions Es ≥ 0. Above Es , emissions are dangerous. The objective is to maintain cumulative dangerous emissions below a given ceil- ing B . Cummulative emissions have been found to be a good proxy for climate change (Allen et al., 2009; Matthews et al., 2009).4 For simplicity, we assume that all dangerous emissions are abatable, and, without loss of generality, that doing so requires to use all the sectoral potentials. Denoting E d the emissions above Es , this reads E d = i a ai − ai,t ) stands for the high ¯i . In other words, (¯ carbon capital that has not been replaced by low carbon capital yet in sector i, as measured in emissions. The carbon budget reads: ∞ ai − ai,t ) dt ≤ B (¯ (3) 0 i 3 This modeling approach may remind the literature on the optimal extraction rates of non renewable resources. Our maximal abatement potentials are similar to the different deposits in Kemp and Long (1980), or the stocks of different fossil fuels in the more recent literature (e.g, Chakravorty et al., 2008; Smulders and Van Der Werf, 2008; van der Ploeg and Withagen, 2012). Our results are also similar, as these authors �nd in particular that different reservoirs or different types of non-renewable resources should not necessarily be extracted at the same marginal cost. 4 Our conclusions are robust to other representations of climate policy objectives such as an exogenous carbon price, e.g. a Pigouvian tax; or a more complex climate model. 4 2.3. The social planner’s program The full social planner’s program reads: ∞ min e−rt ci (xi,t ) dt (4) xi,t 0 i ˙ i,t = xi,t − δi ai,t subject to a (νi,t ) ai,t ≤ a ¯i (λi,t ) ∞ ai − ai,t ) dt ≤ B (¯ (µ) 0 i The Greek letters in parentheses are the respective Lagrangian multipliers (no- tations are summarized in Tab. 1). The social cost of carbon (SCC) µ does not depend on i nor t, as a ton of GHG saved in any sector i at any point in time t contributes equally to meet the carbon budget. In every sector, the optimal timing of investments in low-carbon capital is partly driven by the current price of carbon µert , which follows an Hotelling’s rule by growing at the discount rate r. The multipliers λi,t are the sector-speci�c social costs of the sectoral poten- tials. They are null before the potentials a ¯i are reached (slackness condition). The costate variable νi,t may be interpreted as the present value of investments in low-carbon capital in sector i at time t. 3. Marginal investment costs (MICs) with in�nitely-lived capital In a �rst step, we solve the model in the simple case where δi = 0. This case helps to understand the mechanisms at sake. However, with this assumption, marginal abatement costs cannot be de�ned: one single dollar invested produces an in�nite amount of abatement (if a MAC was to be de�ned, it would be null). This issue is discussed further in the following sections. De�nition 1. We call Marginal Investment Cost (MIC) the cost of the last unit of investment in low-carbon capital ci (xi,t ). MICs measure the economic efforts being oriented towards building and deploy- ing low-carbon capital in a given sector at a given point in time. While one unit of investment at time t in two different sectors produces two similar goods – a unit of low-carbon capital that will save GHG from t onwards – they should not necessarily be valued equally. Proposition 1. In the case where low-carbon capital is in�nitely-lived (δi = 0), optimal MICs are not equal across sectors. Optimal MICs equal the value of the carbon they allow to save less the value of the forgone option to abate later in the same sector. Equivalently, sectors should invest up to the pace at which MICs are equal to the total social cost of emissions avoided before the sectoral potential is reached. 5 Name Description Unit ci Cost of investment in sector i $/yr ¯i a Sectoral potential in sector i tCO2 /yr δi Depreciation rate of abating capital in sector i yr−1 r Discount rate yr−1 ai,t Current abatement in sector i tCO2 /yr xi,t Current investment in abating activities in sector i (tCO2 /yr)/yr νi,t Costate variable (present social value of green investments) $/(tCO2 /yr) λi,t Social cost of the sectoral potential $/tCO2 µ Social cost of carbon (SCC) (present value) $/tCO2 µert Current carbon price $/tCO2 ci Marginal investment cost (MIC) in sector i $/(tCO2 /yr) i,t Marginal levelized abatement cost (MLAC) in sector i $/tCO2 pi,t Marginal implicit rental cost of capital (MIRCC) in sector i $/tCO2 Table 1: Notations (ordered by parameters, variables, multipliers and marginal costs). Proof. With in�nitely-lived capital (δi = 0), the generalized Lagrangian reads: ∞ ∞ L(xi,t , ai,t , λi , νi , µ) = e−rt ci (xi,t ) dt + λi,t (ai,t − a ¯i ) dt 0 i 0 i ∞ +µ ai − ai,t ) dt − B (¯ 0 i ∞ ∞ − ˙ i,t ai,t dt − ν νi,t xi,t dt 0 i 0 The �rst-order conditions are:5 ∂L ∀(i, t), ˙ i,t = λi,t − µ = 0 �⇒ ν (5) ∂ai,t ∂L ∀(i, t), = 0 �⇒ e−rt ci (x∗i,t ) = νi,t (6) ∂xi,t The optimal MIC can be written as:6 ∞ ci (x∗ i,t ) = e rt (µ − λi,θ ) dθ (7) t The complementary slackness condition states that the positive social cost of ¯i is not binding: the sectoral potential λi,t is null when the sectoral potential a ai − ai,t ) = 0 ∀(i, t), λi,t ≥ 0, and λi,t · (¯ (8) Each investment made in a sector brings closer the endogenous date, denoted 5 The same conditions can be written using a Hamiltonian. 6 We integrated ν ˙ i,t as given by (5) between t and ∞; used the relation limt→∞ νi,t = 0 (justi�ed latter); and replaced νi,t in (6). 6 Figure 2: Optimal marginal investment costs (MIC) in low-carbon capital in a case with two sectors (i ∈ {1, 2}) with in�nitely-lived capital (δi = 0). The dates Ti denote the endogenous date when all the emitting capital in sector i has been replaced by low- carbon capital; after this date, additional investment would bring no bene�t. Optimal MICs differ across sectors, because of the social costs of the sectoral potentials (λi,t ). Ti , at which all the production of this sector will come from low-carbon capital. After this date Ti , the option to abate global GHG emissions thanks to invest- ments in low-carbon capital in sector i is removed. The value of this option is the integral from t to ∞ of the social cost of the sectoral potential λi,θ ; it is subtracted from the integral from t to ∞ of µ (the value of abatement) to obtain the value of investments in low-carbon capital. Using (7) and (8) allows to express the optimal marginal investment costs as a function of Ti and µ:7 µert (Ti − t) if t < Ti ci (x∗ i,t ) = (9) 0 if t ≥ Ti Optimal MICs equal the total social cost of the carbon — expressed in current value (µert ) — that will be saved thanks to the abatement before the sectoral potential is reached — the time span (Ti − t). The following lemma ful�lls the proof. Lemma 1. When the abating capital is in�nitely-lived (δi = 0), for any cost of carbon µ, the decarbonizing date Ti is an increasing function of the sectoral ¯i . potential a Proof. See AppendixA. ¯i differ across sectors, the dates Ti also differ across sectors, Since potentials a and optimal MICs are not equal. The general shape of the optimal MICs is 7 ∀t < T , a ¯i =⇒ λi,t = 0 (8); for t ≥ Ti the abatement ai (t) is constant, equal to i i,t < a a ¯i , thus xi (t) = a ˙ i,t = 0 =⇒ λi,t = µ (5). ˙ i,t is null, and, using (6) ∀t ≥ Ti , νi,t = 0, =⇒ ν This last equality means that once the sectoral potential is binding, the associated shadow cost equals the value of the carbon that it prevents to abate. 7 displayed in Fig. 2. Vogt-Schilb et al. (2012) provide some numerical simulations calibrated with IPCC (2007) estimates of costs and abating potentials of seven sectors of the economy. 4. Marginal levelized abatement costs (MLACs) In this section, we solve for the optimal marginal investment costs in the general case when the depreciation rate is positive (δi > 0). Then, we show that the levelized abatement cost is not equal across sectors along the optimal path, and in particular is not equal to the carbon price. Optimal marginal investment costs Proposition 2. Along the optimal path, abatement in each sector i increases until it reaches the sectoral potential a¯i at a date denoted Ti ; before this date, marginal investment costs can be expressed as a function of a ¯i , Ti , the depreca- tion rate of the low-carbon capital δi , and the social cost of carbon µ: 1 − e−δi (Ti −t) ∀t ≤ Ti , ci (x∗ i,t ) = µe rt +e−(r+δi )(Ti −t) ci (δi a ¯i ) (10) δi K Proof. See AppendixB. Equation 10 states that at each time step t, each sector i should invest in low- carbon capital up to the pace at which marginal investment costs (Left-hand side term) are equal to marginal bene�ts (RHS term). In the marginal bene�ts, the current carbon price µert appears multiplied by a positive time span: δ 1 i 1 − e−δi (Ti −t) . This term is the equivalent of (Ti − t) in the case of in�nitely-lived abating capital (9). We interpret K as the marginal bene�t of building new low-carbon capital. The longer it takes for a sector to reach its potential (i.e. the further Ti ), the more expensive should be the last unit of investments directed toward low-carbon capital accumulation. K reflects a complex trade-off: investing soon allows the planner to bene�t from the persistence of abating efforts over time, and prevents investing too much in the long-term; but it brings closer the date Ti , removing the option to invest later, when the discount factor is higher. This results in a bell-shaped distribution of mitigation costs over time: in the short term, the effect of dis- counting may dominate8 and the effort may grow exponentially; in the long term, the effect of the limited potential dominates and new capital accumula- tion decreases to zero (Fig. 3). Marginal bene�ts also have another component, that tends to the marginal cost of maintaining abatement at its maximal level: e−(δi +r)(Ti −t) ci (¯ai ). As the �rst term in (10) tends to 0, the optimal MIC tends to the cost of maintaining low-carbon capital at its maximal level: ci (x∗ i,t ) −−−→ ci (δi a ¯i ) (11) t→Ti ¯i ) and the optimal MIC in sector i After Ti , the abatement is constant (ai,t = a ¯i ). is simply constant at ci (δi a 8 if either Ti or δi is sufficiently large 8 Figure 3: Ratio of marginal investments to abated GHG (MLACs) along the optimal trajectory in a case with two sectors (i ∈ {1, 2}). In a �rst phase, the optimal timing of sectoral abatement comes from a trade-off between (i) investing later in order to reduce present costs thanks to the discounting, (ii) investing sooner to bene�t from the persistent effect of the abating efforts over time, and (iii) smooth investment over time, as investment costs are convex. This results in a bell shape. After the dates Ti when the potentials a¯i have been reached, marginal abatement costs are constant to ¯i ). δi ci (δi a Marginal levelized abatement costs Our model does not feature an abatement cost function that can be differ- entiated to compute the marginal abatement costs (MACs). In this section, we compute the levelized abatement cost of the low-carbon capital. This metric is sometimes labeled “marginal abatement costs� by some scholars and institu- tions. We �nd that marginal levelized abatement costs should not be equal to the carbon price, and should not be equal across sectors. De�nition 2. We call Marginal Levelized Abatement Cost (MLAC) and de- note i,t the ratio of marginal investment costs to discounted abatements. MLACs can be expressed as: ∀xi,t , i,t = (r + δi ) ci (xi,t ) (12) Proof. See AppendixD. MLACs may be interpreted as MICs annualized using r + δi as the discount rate. This is the appropriate discount rate because, taking a carbon price as given, one unit of investment in low-carbon capital generates a flow of real revenue that decreases at the rate r + δi . Practitioners may use MLACs when comparing different technologies.9 Let us take an illustrative example: building electric vehicles (EV) to replace con- ventional cars. Let us say that the social cost of the last EV built at time t, 9 We de�ned marginal levelized costs. The gray literature simply uses levelized costs; they equal marginal levelized costs if investment costs are linear (AppendixE). 9 compared to the cost of a classic car, is 7 000 $/EV. This �gure may include, in addition to the higher upfront cost of the EV, the lower discounted oper- ation and maintenance costs — complete costs computed this way are some- times called levelized costs. If cars are driven 13 000 km per year and electric cars emit 110 gCO2 /km less than a comparable internal combustion engine ve- hicle, each EV allows to save 1.43 tCO2 /yr. The MIC in this case would be 4 900 $/(tCO2 /yr). If electric cars depreciate at a constant rate such that their average lifetime is 10 years (1/δi = 10 yr), then r + δi = 15%/yr and the MLAC is 734 $/tCO2 .10 Proposition 3. Optimal MLACs are not equal to the carbon price. Proof. Combining the expression of optimal MICs from Prop. 2 and in the expression of MLACs from Def. 2 gives the expression of optimal MLACs: ∗ ∀t ≤ Ti , i,t = (r + δi ) ci (x∗ i,t ) = µert r 1 − e−δi (Ti −t) + e−(r+δi )(Ti −t) (r + δi ) ci (δi a ¯i ) (13) Corollary 1. In general, optimal MLACs are different in different sectors. Proof. We use a proof by contradiction. Let two sectors be such that they exhibit the same investment cost function, the same depreciation rate, but dif- ferent abating potentials: ∀x > 0, c1 (x) = c2 (x), δ1 = δ2 , a ¯2 ¯1 = a Suppose that the two sectors take the same time to decarbonize (i.e. T1 = T2 ). Optimal MICs would then be equal in both sectors (10). This would lead to equal investments, hence equal abatement, in both sectors at any time (1,2), and in particular to a1 (T1 ) = a2 (T2 ). By assumption, this last equality is not possible, as: ¯1 = a a1 (T1 ) = a ¯2 = a2 (T2 ) In conclusion, different potentials a ¯i have to lead to different optimal decar- bonizing dates Ti , and therefore to different optimal MLACs ∗ i,t . A similar reasoning can be done concerning two sectors with the same in- vestment cost functions, same potentials, but different depreciation rates; or two sectors that differ only by their investment cost functions. This �nding does not necessarily imply that mitigating climate change requires other sectoral policies than those targeted at internalizing learning spillovers. Well-tried arguments plead in favor of using few instruments (Tinbergen, 1956; Laffont, 1999). In our case, a unique carbon price may induce different MLACs in different sectors. 10 The MIC was computed as 7 000 $/(1.43 tCO /yr) = 4 895 $/(tCO /yr); and the MLAC 2 2 as 0.15 yr−1 · 4 895 $/(tCO2 /yr)= 734 $/tCO2 . 10 5. Marginal implicit rental cost of capital (MIRCC) The result from the previous section may seem to contradict the equi- marginal principle: two similar goods, abatement in two different sectors, appear to have different prices. In fact, investment in low-carbon capital produce dif- ferent goods in different sectors because they have two effects: avoiding GHG emissions and removing an option to invest later in the same sector (section 3). Here we consider an investment strategy that increases abatement in a sec- tor at one date while keeping the rest of the abatement trajectory unchanged. It consequently leaves unchanged any opportunity to invest later in the same sector. From an existing investment pathway (xi,t ), the social planner may increase investment by one unit at time t and immediately reduce investment by 1 − δdt at the next period t + dt. This would allow to abate one unit of GHG between t and t + dt. The present cost (seen from t) of doing so is: 1 (1 − δi dt) P= ci (xi,t ) − c (xi,t+dt ) (14) dt (1 + rdt) i For marginal time lapses, it tends to : dci (xi,t ) P−−−→ (r + δi ) ci (xi,t ) − (15) dt→0 dt De�nition 3. We call marginal implicit rental cost of capital (MIRCC) in sector i at a date t, denoted pi,t the following value: dci (xi,t ) pi,t = (r + δi ) ci (xi,t ) − (16) dt This de�nition extends the concept of the implicit rental cost of capital to the case where investment costs are an endogenous functions of the investment pace.11 It corresponds to the market rental price of low-carbon capital in a competitive equilibrium, and ensures that there are no pro�table trade-offs be- tween: (i) lending at a rate r; and (ii) investing at time t in one unit of capital at cost ci (xi,t ), renting this unit during a small time lapse dt, and reselling 1 − δdt units at the price ci (xi,t+dt ) at the next time period. Proposition 4. In each sector i, before the date Ti , the optimal implicit marginal rental cost of capital equals the current carbon price: dci (x∗i,t ) ∀i, ∀t ≤ Ti , p∗ ∗ i,t = (r + δi ) ci (xi,t ) − = µert (17) dt Proof. In AppendixC we show that this relation is a consequence of the �rst order conditions. 11 We de�ned marginal rental costs. This differs from the proposal by Jorgenson (1963, p. 143), where investment costs are linear, and no distinction needs to be done between average and marginal costs (AppendixE). 11 Equation 17 also gives a sufficient condition for the marginal levelized abatement costs (MLACs from Def. 2) to be equal across sectors to the carbon price: this happens when marginal investment costs are constant along the optimal path: dci (x∗ i,t )/dt = 0. In this case – and if there are no learning-by-doing effects or other stock externalities – MLACs can be used labeled as MACs, and should be equal across sectors to the carbon price. But if investment costs are convex functions of the investment pace, marginal investment cost changes in time and MLACs differ from MIRCCs (9,10). Proposition 5. Equalizing MIRCC to the social cost of carbon (SCC) is not a sufficient condition to reach the optimal investment pathway. Proof. Equations (16) and (17) de�ne a differential equation that ci (xi,t ) satis�es when the IRRC are equalized to the SCC. This differential equation has an in�nity of solutions (those listed by equation B.6 in the appendix). In other words, many different investment pathways lead to equalize MIRCC and the SCC. Only one of these pathways leads to the optimal outcome; it can be selected using the fact that at Ti , abatement in each sector has to reach its maximum potential (the boundary condition used from B.7). The cost-efficiency of investments is therefore more complex to assess when investment costs are endogenous than when they are exogenous. In the latter case, as Jorgenson (1967, p. 145) emphasized: “It is very important to note that the conditions determining the values [of investment in capital] to be chosen by the �rm [...] depend only on prices, the rate of interest, and the rate of change of the price of capital goods for the current period.�12 In other words, when investment costs are exogenous, current price signals contain all the information that private agents need to take socially-optimal decisions. In contrast, in our case – with endogenous investment costs and maximum abating potentials – the information contained in prices should be complemented with the correct expectation of the date Ti when the sector is entirely decarbonized. 6. Discussion and conclusion We used three metrics to assess the social value of investments in low-carbon capital made to decarbonize the sectors of an economy: the marginal investment costs (MIC), the marginal levelized abatement cost (MLAC), and the marginal implicit rental cost of capital (MIRCC). We �nd that along the optimal path, the marginal investment costs and the marginal levelized abatement costs differ from the carbon price, and differ across sectors. This may bring strong policy implications, as agencies use levelized abatement costs labeled as “marginal abatement cost� (MAC), and existing sectoral policies are often criticized because they set different MACs (or different carbon prices) in different sectors. Our results show that levelized abatement costs should be equal across sectors only if the costs of investments in low-capital do not depend on the date or the pace at which investments are implemented. 12 In our model, these correspond respectively to the current price of carbon µert , the discount rate r, and the endogenous current change of MIC dci (xi,t )/dt. 12 In the optimal pathway, the marginal implicit rental cost of capital (MIRCC) equals the current carbon price in every sector that has not �nished its decar- bonizing process. In other words, if abatement costs are de�ned as the implicit rental cost of the low-carbon capital required to abate, MACs should be equal across sectors. This �nding required to extend the concept of implicit rental cost of capi- tal to the case of endogenous investment costs – at our best knowledge it had only been used with exogenous investment costs. A theoretical contribution is to show that when investment costs are endogenous and the capital has a maximum production, equalizing the MIRCC to the current price of the output (the carbon price in this application) is not a sufficient condition to reach the Pareto optimum. In other words, current prices do not contain all the informa- tion required to decentralize the social optimum; they must be combined with knowledge of the date when the capital reaches its maximum production. The bottom line is that two apparently opposite views are reconciled: on the one hand, higher efforts are actually justi�ed in the speci�c sectors that will take longer to decarbonize, such as urban planning and the transportation system; on the other hand, the equimarginal principle remains valid, but applies to an accounting value: the implicit rental cost of the low-carbon capital used to abate. Our analysis does not incorporate any uncertainty, imperfect foresight or incomplete or asymmetric information. We also disregarded induced technical change, known to impact the optimal timing and cost of GHG abatement; and growing abating potentials, a key factor in developing countries. A program for further research is to investigate the combined effect of these factors in the framework of low-carbon capital accumulation. Acknowledgments We thank Alain Ayong Le Kama, Marianne Fay, Jean Charles Hourcade, Christophe de Gouvello, Lionel Ragot, Julie Rozenberg, seminar participants at Cired, Universit´e Paris Ouest, Paris Sorbonne, Princeton University’s Woodrow Wilson School, and at the Joint World Bank – International Monetary Fund Seminar on Environment and Energy Topics, and the audience at the Con- ference on Pricing Climate Risk held at Center for Environmental Economics and Sustainability Policy for usefull comments and suggestions. 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Vogt-Schilb, A., Meunier, G., Hallegatte, S., 2012. How inertia and limited potentials affect the timing of sectoral abatements in optimal climate policy. Policy Research Working Paper 6154, World Bank, Washington DC, USA. AppendixA. Proof of lemma 1 Proof. As ci is strictly growing, it is invertible. Let χi be the inverse of ci ; applying χi to (9) gives: χi (ert (Ti − t)µ) if t < Ti xi,t = (A.1) 0 if t ≥ Ti ai ), the MICs (through χi ), the The relation between the sectoral potential (¯ SCC (µ) and the time it takes to achieve the sectoral potential Ti reads: ¯i = ai (Ti ) a Ti = χi ert (Ti − t)µ dt 0 Let us de�ne fχi such that: t fχi (t) = χi erθ (t − θ)µ dθ 0 t dfχi =⇒ (t) = erθ χi erθ (t − θ)µ dθ dt 0 dfχi Let us show that fχi is invertible: χi > 0 as the inverse of c > 0, thus dt >0 and therefore fχi is strictly growing. Finally: ¯i → Ti = fχi −1 (¯ a ai ) is an increasing function When the marginal cost function ci is given, χi and therefore fχi are also given. Therefore, Ti can always be found from a ¯i . The larger the potential, the longer it takes for the optimal strategy to achieve it. 15 AppendixB. Proof of proposition 2 Lagrangian The Lagrangian associated with (4) reads: ∞ ∞ ˙ i , λi , νi , µ) = L(xi , ai , a e−rt ci (xi,t ) dt + λi,t (ai,t − a ¯i ) dt 0 i 0 i ∞ +µ ai − ai,t ) dt − B (¯ 0 i ∞ + ˙ i,t − xi,t + δi ai,t ) dt νi,t (a 0 i (B.1) ˙ i,t can be removed thanks to an integration by parts: In the last term, a ∞ ˙ i,t − xi,t + δi ai,t ) dt νi,t (a t i ∞ ∞ = νi,t a ˙ i,t dt + νi,t (δi ai,t − xi,t ) dt i 0 0 ∞ ∞ = constant − ν ˙ i,t ai,t dt + νi,t (δi ai,t − xi,t ) dt i 0 0 ˙ i,t : The transformed Lagrangian does not depend on a ∞ ∞ L(xi,t , ai,t , λi , νi , µ) = e−rt ci (xi,t ) dt + λi,t (ai,t − a ¯i ) dt 0 i 0 i ∞ +µ ai − ai,t ) dt − B (¯ 0 i ∞ ∞ − ν ˙ i,t ai,t dt + νi,t (δi ai,t − xi,t ) dt 0 i 0 (B.2) First order conditions The �rst order conditions read:13 ∂L ∀(i, t), ˙ i,t − δi νi,t = λi,t − µ = 0 �⇒ ν (B.3) ∂ai,t ∂L ∀(i, t), = 0 �⇒ e−rt ci (xi,t ) = νi,t (B.4) ∂xi,t Slackness condition For each sector i there is a date Ti such that (slackness condition): ∀t < Ti , ai,t < a ¯i & λi,t = 0 (B.5) ¯i & λi,t ≥ 0 ∀t ≥ Ti , ai,t = a 13 The same conditions may be written using a Hamiltonian. 16 Before Ti , (B.3) simpli�es: ˙ i (t) − δi νi,t = −µ ∀t ≤ Ti , ν µ =⇒ νi,t = Vi eδi t + δi Where Vi is a constant that will be determined later. The MICs are the same quantities expressed in current value (B.4): µ ∀t ≤ Ti , ci (xi,t ) = ert Vi eδi t + (B.6) δi µ Any Vi chosen such that Vi eδi t + δ i remains positive de�nes an investment pathway that satis�es the �rst order conditions. The optimal investment path- ways also satis�es the following boundary conditions. Boundary conditions At the date Ti , ai,t is constant and the investment xi,t is used to counter- balance the depreciation of abating capital. This allows to compute Vi : ¯i xi (Ti ) = δi a (B.7) µ ¯i ) = Vi eδi ·Ti =⇒ e−rTi ci (δi a + (from eq. B.6) δi µ =⇒ Vi = e−δi Ti e−rTi ci (δi a ¯i ) − δi Optimal marginal investment costs (MICs) Using this expression in (B.6) gives: µ µ ∀t ≤ Ti , ci (xi,t ) = ert e−δi ·Ti e−rTi ci (δi a ¯i ) − eδi t + δi δi Simplifying this expression allows to express the optimal marginal investment ¯i , µ and Ti : costs in each sector as a function of δi , a 1 − e−δi (Ti −t) ci (x∗ i,t ) = µe rt + e−(δi +r)(Ti −t) ci (δi a ¯i ) (B.8) δi ¯i ). After Ti , the MICs in sector i are simply constant to ci (δi a AppendixC. Proof of proposition 4 The �rst order conditions can be rearranged. Starting from (B.4): ci (xi,t ) = e−rt νi,t (B.4) dci (xi,t ) =⇒ = ert (ν ˙ i,t + r · νi,t ) (C.1) dt rt = e (δi νi,t − µ + r · νi,t ) (from B.3 and B.5) (C.2) rt = (r + δ )ci (xi,t ) − µe (from B.4) (C.3) dci (xi,t ) =⇒ µert = (r + δi ) ci (xi,t ) − (C.4) dt Substituting in the de�nition of the implicit marginal rental cost of capital pi,t (16) leads to pi,t = µert . The solutions of (C.4), where the variable is ci (xi,t ), are those listed in (B.6). 17 AppendixD. Proof of the expression of i,t in Def. 2 Let h be a marginal physical investment in low-carbon capital made at time t in sector i (expressed in tCO2 /yr per year). It generates an in�nitesimal abatement flux that starts at h at time t and decreases exponentially at rate δi . For any after that, leading to the total discounted abatement ∆A (expressed in tCO2 ): ∞ ∆A = er (θ−t) h e−δi (θ−t) dθ (D.1) θ =t h = (D.2) r + δi This additional investment h brings current investment from xi,t to (xi,t + h). The additional cost ∆C (expressed in $) that it brings reads: ∆C = ci (xi,t + h) − ci (xi,t ) = h ci (xi,t ) (D.3) h→0 The MLAC i,t is the division of the additional cost by the additional abatement it allows: ∆C i,t = (D.4) ∆A i,t = (r + δi ) ci (xi,t ) (D.5) AppendixE. Levelized costs and implicit rental cost when investment costs are exogenous and linear Let It be the amount of investments made at exogenous unitary cost Qt to accumulate capital Kt that depreciates at rate δ : ˙ t = It − δ Kt K (E.1) Let F (Kt ) be a classical production function (where the price of output is normalized to 1). Jorgenson (1967) de�nes current receipts Rt as the actual cash flow: Rt = F (Kt ) − Qt It (E.2) he �nds that the solution of the maximization program ∞ max e−rt Rt dt (E.3) It 0 does not equalize the marginal productivity of capital to the investment costs Qt : ∗ FK (Kt ) = (r + δ ) Qt − Q˙t (E.4) He de�nes the implicit rental cost of capital Ct , as the accounting value: Ct = (r + δ ) Qt − Q˙t (E.5) 18 such that the solution of the maximization program is to equalize the marginal productivity of capital and the rental cost of capital: ∗ FK (Kt ) = Ct (E.6) He shows that this is consistent with maximizing discounted economic pro�ts, where the current pro�t is given by the accounting rule: Πt = F (Kt ) − Ct Kt (E.7) In this case, the (unitary) levelized cost of capital Lt is given by: Lt = (r + δ ) Qt (E.8) And the levelized cost of capital matches the optimal rental cost of capital if and only if investment costs are constant: Q˙ t = 0 �⇒ FK (Kt ∗ ) = Lt (E.9) 19