metadata: authors: 0: author: Hallegatte,Stephane
1: author: Meunier,Lola Guylaine Dany J C L S P P
2: author: Vogt-Schilb,Adrien Camille
count: World
docna: 0: docna: Should marginal abatement costs differ
            across sectors ? the effect of low-carbon capital accumulation
docty: Policy Research Working Paper
subtopic: Climate Change Mitigation and Green House Gases,Science of Climate Change,Climate Change and Health,Climate Change and Environment,Energy Demand,Energy and Environment,Energy and Mining,Global Environment
teratopic: Environment,Macroeconomics and Economic Growth
abstracts: cdata!: The optimal timing, sectoral
            distribution, and cost of greenhouse gas emission reductions
            is different when abatement is obtained though abatement
            expenditures chosen along an abatement cost curve, or
            through investment in low-carbon capital. In the latter
            framework, optimal investment costs differ in each sector:
            they are equal to the value of avoided carbon emissions,
            minus the value of the forgone option to invest later. It is
            therefore misleading to assess the cost-efficiency of
            investments in low-carbon capital by comparing levelized
            abatement costs, that is, efforts measured as the ratio of
            investment costs to discounted abatement. The equimarginal
            principle applies to an accounting value: the Marginal
            Implicit Rental Cost of the Capital (MIRCC) used to abate.
            Two apparently opposite views are reconciled.  On the one
            hand, higher efforts are justified in sectors that will take
            longer to decarbonize, such as urban planning; on the other
            hand, the MIRCC should be equal to the carbon price at each
            point in time and in all sectors. Equalizing the MIRCC in
            each sector to the social cost of carbon is a necessary
            condition to reach the optimal pathway, but it is not a
            sufficient condition. Decentralized optimal investment
            decisions at the sector level require not only the
            information contained in the carbon price signal, but also
            knowledge of the date when the sector reaches its full
            abatement potential.
repnme: repnme: Should marginal abatement costs differ
            across sectors ? the effect of low-carbon capital accumulation
display_title: Should marginal abatement costs
            differ across sectors the effect of low-carbon capital accumulation
listing_relative_url: /research/2013/04/17606425/marginal-abatement-costs-differ-across-sectors-effect-low-carbon-capital-accumulation
topicv3: Carbon Tax m1326372 1845,Energy and Extractives m1326595 507,Climate Change m1326829 567,Climate Change Mitigation m1327079 579,Environment, Natural Resources and Blue
            Economy m1327391 566
publishtoextweb_dt: 2013-04-01T00:00:00Z
historic_topic: Environment,Macroeconomics and Economic Growth
no_of_pages: 21
author_keys: 0: author_key: 1103816
1: author_key: 2132856
2: author_key: 1272850
model_extractions: 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 of greenhouse gas emission reductions is different when abatement is obtained though abatement expenditures chosen along an abatement cost curve, or through investment in low-carbon capital. In the latter framework, optimal investment costs differ in each sector: they are equal to the value of avoided carbon emissions, minus the value of the forgone option to invest later. It is therefore misleading to assess the cost-efficiency of investments in low-carbon capital by comparing levelized abatement costs, that is, efforts measured as the ratio of investment costs to discounted abatement. The equimarginal principle applies to an accounting value: the Marginal Implicit Rental Cost of the Capital (MIRCC) used to abate. Two apparently opposite views are reconciled. On the one hand, higher efforts are justified in sectors that will take longer to decarbonize, such as urban planning; on the other hand, the MIRCC should be equal to the carbon price at each point in time and in all sectors. Equalizing the MIRCC in each sector to the social cost of carbon is a necessary condition to reach the optimal pathway, but it is not a sufficient condition. Decentralized optimal investment decisions at the sector level require not only the information contained in the carbon price signal, but also knowledge of the date when 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<sup>1,</sup><sup>_∗_</sup> , Guy Meunier<sup>2</sup> , St ́ephane Hallegatte<sup>3</sup>

> _1CIRED, Nogent-sur-Marne, France._

> _2INRA–UR1303 ALISS, Ivry-Sur-Seine, France._

> _3The 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 classification:_ 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 system, 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, justified if multiple market failures cannot be corrected independently (Lipsey and Lancaster, 1956). For instance, if governments cannot use tariffs to discriminate imports from countries where no environmental policies 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 abatement options involve new technologies with increasing returns or learning-bydoing (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 justified in sectors with larger learning effects (Rosendahl, 2004).

> _∗_ Corresponding author _Email addresses:_ `vogt@centre-cired.fr` (Adrien Vogt-Schilb ), `guy.meunier@ivry.inra.fr` (Guy Meunier), `shallegatte@worldbank.org` (St ́ephane 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 definition 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 investments in low-carbon capital. We use an intertemporal optimization model with three characteristics. First, we do not use abatement cost functions; instead, 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 investments have convex costs : accumulating low-carbon capital faster is more expensive. For instance, retrofitting the entire building stock would be more expensive if done over a shorter period of time. Third, abatement cannot exceed 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 find 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 definitions of marginal abatement costs in a model where abatement is obtained through low-carbon capital accumulation. Here, we define 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 find 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 defined 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 suboptimal 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 define 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.<sup>1</sup> 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): For simplicity, abating capital is directly measured in terms of avoided emissions (Tab. 1).<sup>2</sup> 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 opportunity 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 retrofitted at date _t_ (the abatement _ai,t_ would then be proportional to the share of retrofitted buildings in the stock). Retrofitting 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 retrofitting buildings _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_xi_)isagrowingfunctionofthepace</sup><sup>_xi_.</sup> 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:

> 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 _xi,t_ are therefore measured as additional abating capacity (in tCO2 _/_ yr) per unit of time, i.e. in (tCO2/yr)/yr.

3
<!-- Start of picture text --> ∑ ai(t)<br>E d<br>B<br>Es<br>t time<br>/yrGtCO2<br>Emissions<br><!-- End of picture text -->

**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_<sup>_d_</sup> are measured from _Es_ .

For instance, if each vehicle is replaced by a zero-emission vehicle, all the abatement potential in the private mobility sector has been realized.<sup>3</sup> 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 simplifying 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 ceiling _B_ . Cummulative emissions have been found to be a good proxy for climate change (Allen et al., 2009; Matthews et al., 2009).<sup>4</sup> 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_<sup>_d_</sup> the emissions above _Es_ , this reads _E_<sup>_d_</sup> =<sup>�</sup> _i_<sup>_a_ ̄</sup><sup>_i_.Inotherwords,( ̄</sup><sup>_ai −ai,t_)standsforthehigh</sup> carbon capital that has not been replaced by low carbon capital yet in sector _i_ , as measured in emissions. The carbon budget reads:

> 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 find 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: The Greek letters in parentheses are the respective Lagrangian multipliers (notations 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 _μe_<sup>_rt_</sup> , which follows an Hotelling’s rule by growing at the discount rate _r_ .

The multipliers _λi,t_ are the sector-specific social costs of the sectoral potentials. 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 infinitely-lived capital**

In a first 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 defined: one single dollar invested produces an infinite amount of abatement (if a MAC was to be defined, it would be null). This issue is discussed further in the following sections.

**Definition 1.** _We call_ Marginal Investment Cost (MIC) _the cost of the last unit of investment in low-carbon capital ci_<sup>_′_</sup> ( _xi,t_ ) _._

MICs measure the economic efforts being oriented towards building and deploying 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 infinitely-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_|$/yr<br>|
| ̄_ai_|Sectoral potential in sector _i_|tCO2_/_yr<br>|
|_δi_|Depreciation rate of abating capital in sector _i_|yr<sup>_−_1</sup><br>|
|_r_|Discount rate|yr<sup>_−_1</sup><br>|
|_ai,t_|Current abatement in sector _i_|tCO2_/_yr|
|_xi,t_|Current investment in abating activities in sector _i_|(tCO2/yr)/yr|
|_νi,t_|Costate variable (present social value of green investments)|$/(tCO2/yr)|
|_λi,t_|Social cost of the sectoral potential|$/tCO2|
|_μ_|Social cost of carbon (SCC) (present value)|$/tCO2|
|_μe_<sup>_rt_</sup>|Current carbon price|$/tCO2|
|_c_<sup>_′_</sup><br>_i_|Marginal investment cost (MIC) in sector _i_|$/(tCO2/yr)|
|_li,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 infinitely-lived capital ( _δi_ = 0), the generalized Lagrangian reads: The conditions are:<sup>5</sup> The optimal MIC can be written as:<sup>6</sup> The complementary slackness condition states that the positive social cost of the sectoral potential _λi,t_ is null when the sectoral potential _a_ ̄ _i_ is not binding: 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 lim _t→∞ νi,t_ = 0 (justified 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 infinitely-lived capital ( _δi_ = 0). The dates _Ti_ denote the endogenous date when all the emitting capital in sector _i_ has been replaced by lowcarbon capital; after this date, additional investment would bring no benefit. 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 investments 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_ :<sup>7</sup> Optimal MICs equal the total social cost of the carbon — expressed in current value ( _μe_<sup>_rt_</sup> ) — that will be saved thanks to the abatement _before the sectoral potential is reached_ — the time span ( _Ti − t_ ).

The following lemma fulfills the proof. □

**Lemma 1.** _When the abating capital is infinitely-lived (δi_ = 0 _), for any cost of carbon μ, the decarbonizing date Ti is an increasing function of the sectoral_  ̄ _potential ai._

Proof. See AppendixA.

Since potentials _a_ ̄ _i_ differ across sectors, the dates _Ti_ also differ across sectors, and optimal MICs are not equal. The general shape of the optimal MICs is

> _a_  ̄ _i_ , 7thus _∀t < Txi_ ( _it,_ ) = _ai,ta_  ̇ _i,t< a_  ̄is _i_ null,= _⇒_ and, _λi,t_ = 0using(8();6)for _∀tt≥ ≥TTi,i_ the _νi,t_ abatement= 0 _,_ = _⇒ ν_  ̇ _ai,ti_ ( _t_ =) is0 constant,= _⇒ λi,t_ =equal _μ_ (5to). 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 deprecation rate of the low-carbon capital δi, and the social cost of carbon μ:_

# Proof. See AppendixB.

Equation 10 states that at each time step _t_ , each sector _i_ should invest in lowcarbon capital up to the pace at which marginal investment costs (Left-hand side term) are equal to marginal benefits (RHS term).

In the marginal benefits, the current carbon price _μe_<sup>_rt_</sup> appears multiplied by a positive time span: _δ_ <u>1</u> _i_ �1 _− e_<sup>_−δi_(</sup><sup>_Ti−t_)�</sup> . This term is the equivalent of ( _Ti − t_ ) in the case of infinitely-lived abating capital (9). We interpret _K_ as the marginal benefit 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 benefit 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 discounting may dominate<sup>8</sup> and the effort may grow exponentially; in the long term, the effect of the limited potential dominates and new capital accumulation decreases to zero (Fig. 3).

Marginal benefits also have another component, that tends to the marginal cost of maintaining abatement at its maximal level: _e_<sup>_−_(</sup><sup>_δi_+</sup><sup>_r_)(</sup><sup>_Ti−t_)</sup> _c_<sup>_′_</sup> _i_<sup>( ̄</sup><sup>_ai_).As the</sup> first term in (10) tends to 0, the optimal MIC tends to the cost of maintaining low-carbon capital at its maximal level: After _Ti_ , the abatement is constant ( _ai,t_ = ̄ _ai_ ) and the optimal MIC in sector _i_ is simply constant at _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_).</sup>

> 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 first 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 benefit 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 _δi c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_).</sup>

# _Marginal levelized abatement costs_

Our model does not feature an abatement cost function that can be differentiated 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 institutions. We find that marginal levelized abatement costs should not be equal to the carbon price, and should not be equal across sectors.

**Definition 2.** _We call_ Marginal Levelized Abatement Cost (MLAC) _and denote li,t the ratio of marginal investment costs to discounted abatements. MLACs can be expressed as:_

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.<sup>9</sup> Let us take an illustrative example: building electric vehicles (EV) to replace conventional cars. Let us say that the social cost of the last EV built at time _t_ ,

> 9 We defined _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 figure may include, in addition to the higher upfront cost of the EV, the lower discounted operation and maintenance costs — complete costs computed this way are sometimes 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 vehicle, 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.<sup>10</sup>

**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: □

**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 different abating potentials: Suppose that the two sectors take the same time to decarbonize (i.e. _T_ 1 = _T_ 2). 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 _a_ 1( _T_ 1) = _a_ 2( _T_ 2). By assumption, this last equality is not possible, as: In conclusion, different potentials _a_ ̄ _i_ have to lead to different optimal decarbonizing dates _Ti_ , and therefore to different optimal MLACs _l_<sup>_∗_.</sup> _i,t_<sup>.</sup> A similar reasoning can be done concerning two sectors with the same investment cost functions, same potentials, but different depreciation rates; or two sectors that differ only by their investment cost functions. □ This finding 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 sectors.

> 10 The MIC was computed as 7 000 $ _/_ (1 _._ 43 tCO2 _/_ yr) = 4 895 $/(tCO2 _/_ yr); and the MLAC as 0 _._ 15 yr<sup>_−_1</sup> _·_ 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 equimarginal principle: two similar goods, abatement in two different sectors, appear to have different prices. In fact, investment in low-carbon capital produce different 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 sector 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: For marginal time lapses, it tends to :

**Definition 3.** _We call_ marginal implicit rental cost of capital (MIRCC) _in sector i at a date t, denoted pi,t the following value:_

This definition extends the concept of the _implicit rental cost of capital_ to the case where investment costs are an endogenous functions of the investment pace.<sup>11</sup> It corresponds to the market rental price of low-carbon capital in a competitive equilibrium, and ensures that there are no profitable trade-offs between: (i) lending at a rate _r_ ; and (ii) investing at time _t_ in one unit of capital at cost _ci_<sup>_′_</sup> ( _xi,t_ ), renting this unit during a small time lapse _dt_ , and reselling 1 _− δdt_ units at the price _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_xi,t_+</sup><sup>_dt_)atthenexttimeperiod.</sup>

**Proposition 4.** _In each sector i, before the date Ti, the optimal implicit marginal rental cost of capital equals the current carbon price:_

Proof. In AppendixC we show that this relation is a consequence of the first order conditions.

> 11 We defined _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_<sup>_′_</sup> ( _x_<sup>_∗_</sup> _i,t_<sup>)</sup><sup>_/dt_=0.Inthiscase–andiftherearenolearning-by-doingeffectsor</sup> 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) define a differential equation that _ci_<sup>_′_</sup> ( _xi,t_ ) satisfies when the IRRC are equalized to the SCC. This differential equation has an infinity 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 firm [...] depend only on prices, the rate of interest, and the rate of change of the price of capital goods for the current period._ ”<sup>12</sup> 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 find 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_<sup>_′_</sup> ( _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 finished its decarbonizing process. In other words, if abatement costs are defined as the implicit rental cost of the low-carbon capital required to abate, MACs should be equal across sectors.

This finding required to extend the concept of implicit rental cost of capital 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 information 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 justified in the specific 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 Conference on Pricing Climate Risk held at Center for Environmental Economics and Sustainability Policy for usefull comments and suggestions. The remaining errors are the authors’ responsibility. We are grateful to Patrice Dumas for technical support. We acknowledge financial support from the Sustainable Mobility Institute (Renault and ParisTech). The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reflect the views of the World Bank, its executive directors, or the countries they represent.

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# **AppendixA. Proof of lemma 1**

Proof. As _ci_<sup>_′_</sup> is strictly growing, it is invertible. Let _χi_ be the inverse of _ci_<sup>_′_</sup> ; applying _χi_ to (9) gives: The relation between the sectoral potential ( ̄ _ai_ ), the MICs (through _χi_ ), the SCC ( _μ_ ) and the time it takes to achieve the sectoral potential _Ti_ reads: Let us define _fχi_ such that: Let us show that _fχi_ is invertible: _χi_<sup>_′_</sup> _>_ 0 as the inverse of _c_<sup>_′_</sup> _>_ 0, thus _dfdtχi >_ 0 and therefore _fχi_ is strictly growing. Finally: When the marginal cost function _ci_<sup>_′_</sup> 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
J| =| {xX

J > ~(/ | | ro / | i> J > fx= i
Before _Ti_ , (B.3) simplifies: Where _Vi_ is a constant that will be determined later. The MICs are the same quantities expressed in current value (B.4): Any _Vi_ chosen such that � _Vie_<sup>_δit_</sup> + _δ_<sup>_<u>μ</u>_</sup> _i_ � remains positive defines an investment pathway that satisfies the first order conditions. The optimal investment pathways also satisfies the following boundary conditions.

# _Boundary conditions_

At the date _Ti_ , _ai,t_ is constant and the investment _xi,t_ is used to counterbalance the depreciation of abating capital. This allows to compute _Vi_ :

# _Optimal marginal investment costs (MICs)_

Using this expression in (B.6) gives: Simplifying this expression allows to express the optimal marginal investment costs in each sector as a function of _δi_ , _a_ ̄ _i_ , _μ_ and _Ti_ : After _Ti_ , the MICs in sector _i_ are simply constant to _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_δia_ ̄</sup><sup>_i_).□</sup>

# **AppendixC. Proof of proposition 4**

The first order conditions can be rearranged. Starting from (B.4): Substituting in the definition of the implicit marginal rental cost of capital _pi,t_ (16) leads to _pi,t_ = _μe_<sup>_rt_</sup> . The solutions of (C.4), where the variable is _ci_<sup>_′_</sup> ( _xi,t_ ), are those listed in (B.6).

17
# **AppendixD. Proof of the expression of** **_li,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 infinitesimal 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): This additional investment _h_ brings current investment from _xi,t_ to ( _xi,t_ + _h_ ). The additional cost ∆ _C_ (expressed in $) that it brings reads: The MLAC _li,t_ is the division of the additional cost by the additional abatement it allows: □

# **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_ : Let _F_ ( _Kt_ ) be a classical production function (where the price of output is normalized to 1). Jorgenson (1967) defines current receipts _Rt_ as the actual cash he finds that the solution of the maximization program does not equalize the marginal productivity of capital to the investment costs _Qt_ : He defines the implicit rental cost of capital _Ct_ , as the accounting value: 18
such that the solution of the maximization program is to equalize the marginal productivity of capital and the rental cost of capital: He shows that this is consistent with maximizing discounted economic profits, where the current profit is given by the accounting rule: In this case, the (unitary) levelized cost of capital _Lt_ is given by: And the levelized cost of capital matches the optimal rental cost of capital if and only if investment costs are constant: 19