ï»¿ 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
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Produced by the Research Support Team
Should marginal abatement costs diï¬€er across sectors?
The eï¬€ect of low-carbon capital accumulation
Adrien Vogt-Schilb 1,âˆ—, Guy Meunier 2 , StÂ´
ephane Hallegatte 3
1 CIRED, Nogent-sur-Marne, France.
2 INRAâ€“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 classiï¬?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 tariï¬€s and portfolio standards in favor of renewable power, and
diï¬€erent energy eï¬ƒciency 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 eï¬€ects on emissions, but they
have been criticized because they result in diï¬€erent Marginal Abatement Costs
(MACs) in diï¬€erent sectors.
Existing analytical assessments conclude that diï¬€erentiating MACs is a
second-best policy, justiï¬?ed if multiple market failures cannot be corrected in-
dependently (Lipsey and Lancaster, 1956). For instance, if governments cannot
use tariï¬€s to discriminate imports from countries where no environmental poli-
cies are applied, it is optimal to diï¬€erentiate 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 â€œtwin-market failureâ€? 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., Jaï¬€e
et al., 2005; Fischer and Newell, 2008; Grimaud and Laï¬€orgue, 2008; Gerlagh
et al., 2009; Acemoglu et al., 2012). But if the only available instrument is
a carbon tax, higher carbon prices are justiï¬?ed in sectors with larger learning
eï¬€ects (Rosendahl, 2004).
âˆ— Corresponding author
Email addresses: vogt@centre-cired.fr (Adrien Vogt-Schilb ),
ephane
guy.meunier@ivry.inra.fr (Guy Meunier), shallegatte@worldbank.org (StÂ´
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 eï¬€orts 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 deï¬?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 eï¬€ect 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, retroï¬?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 ï¬?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 diï¬€erent in each sector, it leads to diï¬€erent
investment costs and levels.
There are multiple possible deï¬?nitions of marginal abatement costs in a
model where abatement is obtained through low-carbon capital accumulation.
Here, we deï¬?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 â€œMarginal Abatement Costâ€? (MAC) by scholars and government agencies,
suggesting that it should be equal to the price of carbon. We ï¬?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 â€“ expressed in dollars per ton â€“ are equal to the current carbon price
and are thus equal in all sectors. If the abatement cost is deï¬?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 suï¬ƒcient 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 deï¬?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 Î´i (dotted variables represent temporal derivatives):
ai,0 = 0 (1)
Ë™ i,t = xi,t âˆ’ Î´i 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, diï¬€erentiable and convex.
The cost convexity bears on the investments ï¬‚ow, 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 â€” measured in buildings per year â€” at which old buildings are being
retroï¬?tted at date t (the abatement ai,t would then be proportional to the
share of retroï¬?tted buildings in the stock). Retroï¬?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 retroï¬?tting buildings ci (xi ) is a growing function of the pace xi .
In each sector, a sectoral potential a Â¯i represents the maximum amount of
GHG emissions (in GtCO2 per year) that can be abated in this sector:
ai,t â‰¤ a
Â¯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
âˆ‘ 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 Â¯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 ï¬‚ow of GHG emissions Es â‰¥ 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 âˆ’ ai,t ) stands for the high
Â¯i . In other words, (Â¯
carbon capital that has not been replaced by low carbon capital yet in sector i,
as measured in emissions. The carbon budget reads:
âˆž
ai âˆ’ ai,t ) dt â‰¤ B
(Â¯ (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 diï¬€erent deposits
in Kemp and Long (1980), or the stocks of diï¬€erent 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 ï¬?nd in particular that diï¬€erent reservoirs
or diï¬€erent 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 plannerâ€™s program
The full social plannerâ€™s program reads:
âˆž
min eâˆ’rt ci (xi,t ) dt (4)
xi,t 0 i
Ë™ i,t = xi,t âˆ’ Î´i ai,t
subject to a (Î½i,t )
ai,t â‰¤ a
Â¯i (Î»i,t )
âˆž
ai âˆ’ ai,t ) dt â‰¤ B
(Â¯ (Âµ)
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) Âµ 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 Âµert , which follows an
Hotellingâ€™s rule by growing at the discount rate r.
The multipliers Î»i,t are the sector-speciï¬?c social costs of the sectoral poten-
tials. They are null before the potentials a Â¯i are reached (slackness condition).
The costate variable Î½i,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 inï¬?nitely-lived capital
In a ï¬?rst step, we solve the model in the simple case where Î´i = 0. This case
helps to understand the mechanisms at sake. However, with this assumption,
marginal abatement costs cannot be deï¬?ned: one single dollar invested produces
an inï¬?nite amount of abatement (if a MAC was to be deï¬?ned, it would be null).
This issue is discussed further in the following sections.
Deï¬?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 eï¬€orts 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 diï¬€erent sectors produces two similar goods â€“ a
unit of low-carbon capital that will save GHG from t onwards â€“ they should not
necessarily be valued equally.
Proposition 1. In the case where low-carbon capital is inï¬?nitely-lived (Î´i = 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
Â¯i
a Sectoral potential in sector i tCO2 /yr
Î´i Depreciation rate of abating capital in sector i yrâˆ’1
r Discount rate yrâˆ’1
ai,t Current abatement in sector i tCO2 /yr
xi,t Current investment in abating activities in sector i (tCO2 /yr)/yr
Î½i,t Costate variable (present social value of green investments) $/(tCO2 /yr)
Î»i,t Social cost of the sectoral potential $/tCO2
Âµ Social cost of carbon (SCC) (present value) $/tCO2
Âµert 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 inï¬?nitely-lived capital (Î´i = 0), the generalized Lagrangian reads:
âˆž âˆž
L(xi,t , ai,t , Î»i , Î½i , Âµ) = eâˆ’rt ci (xi,t ) dt + Î»i,t (ai,t âˆ’ a
Â¯i ) dt
0 i 0 i
âˆž
+Âµ ai âˆ’ ai,t ) dt âˆ’ B
(Â¯
0 i
âˆž âˆž
âˆ’ Ë™ i,t ai,t dt âˆ’
Î½ Î½i,t xi,t dt
0 i 0
The ï¬?rst-order conditions are:5
âˆ‚L
âˆ€(i, t), Ë™ i,t = Î»i,t âˆ’ Âµ
= 0 â‡?â‡’ Î½ (5)
âˆ‚ai,t
âˆ‚L
âˆ€(i, t), = 0 â‡?â‡’ eâˆ’rt ci (xâˆ—i,t ) = Î½i,t (6)
âˆ‚xi,t
The optimal MIC can be written as:6
âˆž
ci (xâˆ—
i,t ) = e
rt
(Âµ âˆ’ Î»i,Î¸ ) dÎ¸ (7)
t
The complementary slackness condition states that the positive social cost of
Â¯i is not binding:
the sectoral potential Î»i,t is null when the sectoral potential a
ai âˆ’ ai,t ) = 0
âˆ€(i, t), Î»i,t â‰¥ 0, and Î»i,t Â· (Â¯ (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 Î½ Ë™ i,t as given by (5) between t and âˆž; used the relation limtâ†’âˆž Î½i,t = 0
(justiï¬?ed latter); and replaced Î½i,t in (6).
6
Figure 2: Optimal marginal investment costs (MIC) in low-carbon capital in a case
with two sectors (i âˆˆ {1, 2}) with inï¬?nitely-lived capital (Î´i = 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 beneï¬?t. Optimal
MICs diï¬€er across sectors, because of the social costs of the sectoral potentials (Î»i,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 âˆž of the social cost of the sectoral potential Î»i,Î¸ ; it
is subtracted from the integral from t to âˆž of Âµ (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 Âµ:7
Âµert (Ti âˆ’ t) if t < Ti
ci (xâˆ—
i,t ) = (9)
0 if t â‰¥ Ti
Optimal MICs equal the total social cost of the carbon â€” expressed in current
value (Âµert ) â€” that will be saved thanks to the abatement before the sectoral
potential is reached â€” the time span (Ti âˆ’ t).
The following lemma fulï¬?lls the proof.
Lemma 1. When the abating capital is inï¬?nitely-lived (Î´i = 0), for any cost
of carbon Âµ, the decarbonizing date Ti is an increasing function of the sectoral
Â¯i .
potential a
Proof. See AppendixA.
Â¯i diï¬€er across sectors, the dates Ti also diï¬€er across sectors,
Since potentials a
and optimal MICs are not equal. The general shape of the optimal MICs is
7 âˆ€t < T , a Â¯i =â‡’ Î»i,t = 0 (8); for t â‰¥ Ti the abatement ai (t) is constant, equal to
i i,t < a
a
Â¯i , thus xi (t) = a Ë™ i,t = 0 =â‡’ Î»i,t = Âµ (5).
Ë™ i,t is null, and, using (6) âˆ€t â‰¥ Ti , Î½i,t = 0, =â‡’ Î½
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 (Î´i > 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 aÂ¯i at a date denoted Ti ; before this date,
marginal investment costs can be expressed as a function of a Â¯i , Ti , the depreca-
tion rate of the low-carbon capital Î´i , and the social cost of carbon Âµ:
1 âˆ’ eâˆ’Î´i (Ti âˆ’t)
âˆ€t â‰¤ Ti , ci (xâˆ—
i,t ) = Âµe
rt
+eâˆ’(r+Î´i )(Ti âˆ’t) ci (Î´i a
Â¯i ) (10)
Î´i
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 beneï¬?ts (RHS term).
In the marginal beneï¬?ts, the current carbon price Âµert appears multiplied by
a positive time span: Î´ 1
i
1 âˆ’ eâˆ’Î´i (Ti âˆ’t) . This term is the equivalent of (Ti âˆ’ t)
in the case of inï¬?nitely-lived abating capital (9). We interpret K as the marginal
beneï¬?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 reï¬‚ects a complex trade-oï¬€: investing soon allows the planner to beneï¬?t
from the persistence of abating eï¬€orts 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 eï¬€ect of dis-
counting may dominate8 and the eï¬€ort may grow exponentially; in the long
term, the eï¬€ect of the limited potential dominates and new capital accumula-
tion decreases to zero (Fig. 3).
Marginal beneï¬?ts also have another component, that tends to the marginal
cost of maintaining abatement at its maximal level: eâˆ’(Î´i +r)(Ti âˆ’t) ci (Â¯ai ). As the
ï¬?rst term in (10) tends to 0, the optimal MIC tends to the cost of maintaining
low-carbon capital at its maximal level:
ci (xâˆ—
i,t ) âˆ’âˆ’âˆ’â†’ ci (Î´i a
Â¯i ) (11)
tâ†’Ti
Â¯i ) and the optimal MIC in sector i
After Ti , the abatement is constant (ai,t = a
Â¯i ).
is simply constant at ci (Î´i a
8 if either Ti or Î´i is suï¬ƒciently large
8
Figure 3: Ratio of marginal investments to abated GHG (MLACs) along the optimal
trajectory in a case with two sectors (i âˆˆ {1, 2}). In a ï¬?rst phase, the optimal timing
of sectoral abatement comes from a trade-oï¬€ between (i) investing later in order to
reduce present costs thanks to the discounting, (ii) investing sooner to beneï¬?t from
the persistent eï¬€ect of the abating eï¬€orts 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 aÂ¯i have been reached, marginal abatement costs are constant to
Â¯i ).
Î´i ci (Î´i a
Marginal levelized abatement costs
Our model does not feature an abatement cost function that can be diï¬€er-
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 â€œmarginal abatement costsâ€? by some scholars and institu-
tions. We ï¬?nd that marginal levelized abatement costs should not be equal to
the carbon price, and should not be equal across sectors.
Deï¬?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:
âˆ€xi,t , i,t = (r + Î´i ) ci (xi,t ) (12)
Proof. See AppendixD.
MLACs may be interpreted as MICs annualized using r + Î´i 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 ï¬‚ow of real revenue
that decreases at the rate r + Î´i .
Practitioners may use MLACs when comparing diï¬€erent 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 deï¬?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 ï¬?gure may include,
in addition to the higher upfront cost of the EV, the lower discounted oper-
ation and maintenance costs â€” 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/Î´i = 10 yr), then r + Î´i = 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:
âˆ—
âˆ€t â‰¤ Ti , i,t = (r + Î´i ) ci (xâˆ—
i,t )
= Âµert r 1 âˆ’ eâˆ’Î´i (Ti âˆ’t) + eâˆ’(r+Î´i )(Ti âˆ’t) (r + Î´i ) ci (Î´i a
Â¯i ) (13)
Corollary 1. In general, optimal MLACs are diï¬€erent in diï¬€erent 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:
âˆ€x > 0, c1 (x) = c2 (x), Î´1 = Î´2 , a Â¯2
Â¯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:
Â¯1 = a
a1 (T1 ) = a Â¯2 = a2 (T2 )
In conclusion, diï¬€erent potentials a Â¯i have to lead to diï¬€erent optimal decar-
bonizing dates Ti , and therefore to diï¬€erent optimal MLACs âˆ— i,t .
A similar reasoning can be done concerning two sectors with the same in-
vestment cost functions, same potentials, but diï¬€erent depreciation rates; or
two sectors that diï¬€er only by their investment cost functions.
This ï¬?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;
Laï¬€ont, 1999). In our case, a unique carbon price may induce diï¬€erent MLACs
in diï¬€erent 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 yrâˆ’1 Â· 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 diï¬€erent sectors, appear
to have diï¬€erent prices. In fact, investment in low-carbon capital produce dif-
ferent goods in diï¬€erent sectors because they have two eï¬€ects: 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 âˆ’ Î´dt
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 âˆ’ Î´i dt)
P= ci (xi,t ) âˆ’ c (xi,t+dt ) (14)
dt (1 + rdt) i
For marginal time lapses, it tends to :
dci (xi,t )
Pâˆ’âˆ’âˆ’â†’ (r + Î´i ) ci (xi,t ) âˆ’ (15)
dtâ†’0 dt
Deï¬?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 + Î´i ) ci (xi,t ) âˆ’ (16)
dt
This deï¬?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 proï¬?table trade-oï¬€s 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 âˆ’ Î´dt 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 (xâˆ—i,t )
âˆ€i, âˆ€t â‰¤ Ti , pâˆ— âˆ—
i,t = (r + Î´i ) ci (xi,t ) âˆ’ = Âµert (17)
dt
Proof. In AppendixC we show that this relation is a consequence of the ï¬?rst
order conditions.
11 We deï¬?ned marginal rental costs. This diï¬€ers 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 suï¬ƒcient 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 (xâˆ—
i,t )/dt = 0. In this case â€“ and if there are no learning-by-doing eï¬€ects or
other stock externalities â€“ 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 diï¬€er from MIRCCs (9,10).
Proposition 5. Equalizing MIRCC to the social cost of carbon (SCC) is not a
suï¬ƒcient condition to reach the optimal investment pathway.
Proof. Equations (16) and (17) deï¬?ne a diï¬€erential equation that ci (xi,t )
satisï¬?es when the IRRC are equalized to the SCC. This diï¬€erential equation
has an inï¬?nity of solutions (those listed by equation B.6 in the appendix). In
other words, many diï¬€erent 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-eï¬ƒciency 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: â€œIt is very important to note that
the conditions determining the values [of investment in capital] to be chosen by
the ï¬?rm [...] depend only on prices, the rate of interest, and the rate of change
of the price of capital goods for the current period.â€?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 â€“ with endogenous investment costs and maximum abating potentials â€“
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 ï¬?nd that along the optimal path, the marginal investment costs and the
marginal levelized abatement costs diï¬€er from the carbon price, and diï¬€er across
sectors. This may bring strong policy implications, as agencies use levelized
abatement costs labeled as â€œmarginal abatement costâ€? (MAC), and existing
sectoral policies are often criticized because they set diï¬€erent MACs (or diï¬€erent
carbon prices) in diï¬€erent 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 Âµert , 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 ï¬?nished its decar-
bonizing process. In other words, if abatement costs are deï¬?ned as the implicit
rental cost of the low-carbon capital required to abate, MACs should be equal
across sectors.
This ï¬?nding required to extend the concept of implicit rental cost of capi-
tal to the case of endogenous investment costs â€“ 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 suï¬ƒcient 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 eï¬€orts are actually justiï¬?ed in the speciï¬?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 eï¬€ect 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, UniversitÂ´e Paris Ouest, Paris Sorbonne, Princeton Universityâ€™s Woodrow
Wilson School, and at the Joint World Bank â€“ 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. The remaining
errors are the authorsâ€™ responsibility. We are grateful to Patrice Dumas for tech-
nical support. We acknowledge ï¬?nancial 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 reï¬‚ect 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 is strictly growing, it is invertible. Let Ï‡i be the inverse of ci ;
applying Ï‡i to (9) gives:
Ï‡i (ert (Ti âˆ’ t)Âµ) if t < Ti
xi,t = (A.1)
0 if t â‰¥ Ti
ai ), the MICs (through Ï‡i ), the
The relation between the sectoral potential (Â¯
SCC (Âµ) and the time it takes to achieve the sectoral potential Ti reads:
Â¯i = ai (Ti )
a
Ti
= Ï‡i ert (Ti âˆ’ t)Âµ dt
0
Let us deï¬?ne fÏ‡i such that:
t
fÏ‡i (t) = Ï‡i erÎ¸ (t âˆ’ Î¸)Âµ dÎ¸
0
t
dfÏ‡i
=â‡’ (t) = erÎ¸ Ï‡i erÎ¸ (t âˆ’ Î¸)Âµ dÎ¸
dt 0
dfÏ‡i
Let us show that fÏ‡i is invertible: Ï‡i > 0 as the inverse of c > 0, thus dt >0
and therefore fÏ‡i is strictly growing. Finally:
Â¯i â†’ Ti = fÏ‡i âˆ’1 (Â¯
a ai ) is an increasing function
When the marginal cost function ci is given, Ï‡i and therefore fÏ‡i are also given.
Therefore, Ti can always be found from a Â¯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:
âˆž âˆž
Ë™ i , Î»i , Î½i , Âµ) =
L(xi , ai , a eâˆ’rt ci (xi,t ) dt + Î»i,t (ai,t âˆ’ a
Â¯i ) dt
0 i 0 i
âˆž
+Âµ ai âˆ’ ai,t ) dt âˆ’ B
(Â¯
0 i
âˆž
+ Ë™ i,t âˆ’ xi,t + Î´i ai,t ) dt
Î½i,t (a
0 i
(B.1)
Ë™ i,t can be removed thanks to an integration by parts:
In the last term, a
âˆž
Ë™ i,t âˆ’ xi,t + Î´i ai,t ) dt
Î½i,t (a
t i
âˆž âˆž
= Î½i,t a
Ë™ i,t dt + Î½i,t (Î´i ai,t âˆ’ xi,t ) dt
i 0 0
âˆž âˆž
= constant âˆ’ Î½
Ë™ i,t ai,t dt + Î½i,t (Î´i ai,t âˆ’ xi,t ) dt
i 0 0
Ë™ i,t :
The transformed Lagrangian does not depend on a
âˆž âˆž
L(xi,t , ai,t , Î»i , Î½i , Âµ) = eâˆ’rt ci (xi,t ) dt + Î»i,t (ai,t âˆ’ a
Â¯i ) dt
0 i 0 i
âˆž
+Âµ ai âˆ’ ai,t ) dt âˆ’ B
(Â¯
0 i
âˆž âˆž
âˆ’ Î½
Ë™ i,t ai,t dt + Î½i,t (Î´i ai,t âˆ’ xi,t ) dt
0 i 0
(B.2)
First order conditions
The ï¬?rst order conditions read:13
âˆ‚L
âˆ€(i, t), Ë™ i,t âˆ’ Î´i Î½i,t = Î»i,t âˆ’ Âµ
= 0 â‡?â‡’ Î½ (B.3)
âˆ‚ai,t
âˆ‚L
âˆ€(i, t), = 0 â‡?â‡’ eâˆ’rt ci (xi,t ) = Î½i,t (B.4)
âˆ‚xi,t
Slackness condition
For each sector i there is a date Ti such that (slackness condition):
âˆ€t < Ti , ai,t < a
Â¯i & Î»i,t = 0 (B.5)
Â¯i & Î»i,t â‰¥ 0
âˆ€t â‰¥ Ti , ai,t = a
13 The same conditions may be written using a Hamiltonian.
16
Before Ti , (B.3) simpliï¬?es:
Ë™ i (t) âˆ’ Î´i Î½i,t = âˆ’Âµ
âˆ€t â‰¤ Ti , Î½
Âµ
=â‡’ Î½i,t = Vi eÎ´i t +
Î´i
Where Vi is a constant that will be determined later. The MICs are the same
quantities expressed in current value (B.4):
Âµ
âˆ€t â‰¤ Ti , ci (xi,t ) = ert Vi eÎ´i t + (B.6)
Î´i
Âµ
Any Vi chosen such that Vi eÎ´i t + Î´ i
remains positive deï¬?nes an investment
pathway that satisï¬?es the ï¬?rst order conditions. The optimal investment path-
ways also satisï¬?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 :
Â¯i
xi (Ti ) = Î´i a (B.7)
Âµ
Â¯i ) = Vi eÎ´i Â·Ti
=â‡’ eâˆ’rTi ci (Î´i a + (from eq. B.6)
Î´i
Âµ
=â‡’ Vi = eâˆ’Î´i Ti eâˆ’rTi ci (Î´i a
Â¯i ) âˆ’
Î´i
Optimal marginal investment costs (MICs)
Using this expression in (B.6) gives:
Âµ Âµ
âˆ€t â‰¤ Ti , ci (xi,t ) = ert eâˆ’Î´i Â·Ti eâˆ’rTi ci (Î´i a
Â¯i ) âˆ’ eÎ´i t +
Î´i Î´i
Simplifying this expression allows to express the optimal marginal investment
Â¯i , Âµ and Ti :
costs in each sector as a function of Î´i , a
1 âˆ’ eâˆ’Î´i (Ti âˆ’t)
ci (xâˆ—
i,t ) = Âµe
rt
+ eâˆ’(Î´i +r)(Ti âˆ’t) ci (Î´i a
Â¯i ) (B.8)
Î´i
Â¯i ).
After Ti , the MICs in sector i are simply constant to ci (Î´i a
AppendixC. Proof of proposition 4
The ï¬?rst order conditions can be rearranged. Starting from (B.4):
ci (xi,t ) = eâˆ’rt Î½i,t (B.4)
dci (xi,t )
=â‡’ = ert (Î½
Ë™ i,t + r Â· Î½i,t ) (C.1)
dt
rt
= e (Î´i Î½i,t âˆ’ Âµ + r Â· Î½i,t ) (from B.3 and B.5) (C.2)
rt
= (r + Î´ )ci (xi,t ) âˆ’ Âµe (from B.4) (C.3)
dci (xi,t )
=â‡’ Âµert = (r + Î´i ) ci (xi,t ) âˆ’ (C.4)
dt
Substituting in the deï¬?nition of the implicit marginal rental cost of capital pi,t
(16) leads to pi,t = Âµert . 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 inï¬?nitesimal
abatement ï¬‚ux that starts at h at time t and decreases exponentially at rate Î´i .
For any after that, leading to the total discounted abatement âˆ†A (expressed
in tCO2 ):
âˆž
âˆ†A = er (Î¸âˆ’t) h eâˆ’Î´i (Î¸âˆ’t) dÎ¸ (D.1)
Î¸ =t
h
= (D.2)
r + Î´i
This additional investment h brings current investment from xi,t to (xi,t + h).
The additional cost âˆ†C (expressed in $) that it brings reads:
âˆ†C = ci (xi,t + h) âˆ’ ci (xi,t ) = h ci (xi,t ) (D.3)
hâ†’0
The MLAC i,t is the division of the additional cost by the additional abatement
it allows:
âˆ†C
i,t = (D.4)
âˆ†A
i,t = (r + Î´i ) 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 Î´ :
Ë™ t = It âˆ’ Î´ Kt
K (E.1)
Let F (Kt ) be a classical production function (where the price of output is
normalized to 1). Jorgenson (1967) deï¬?nes current receipts Rt as the actual
cash ï¬‚ow:
Rt = F (Kt ) âˆ’ Qt It (E.2)
he ï¬?nds that the solution of the maximization program
âˆž
max eâˆ’rt Rt dt (E.3)
It 0
does not equalize the marginal productivity of capital to the investment costs
Qt :
âˆ—
FK (Kt ) = (r + Î´ ) Qt âˆ’ QË™t (E.4)
He deï¬?nes the implicit rental cost of capital Ct , as the accounting value:
Ct = (r + Î´ ) Qt âˆ’ QË™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:
âˆ—
FK (Kt ) = Ct (E.6)
He shows that this is consistent with maximizing discounted economic proï¬?ts,
where the current proï¬?t is given by the accounting rule:
Î t = F (Kt ) âˆ’ Ct Kt (E.7)
In this case, the (unitary) levelized cost of capital Lt is given by:
Lt = (r + Î´ ) Qt (E.8)
And the levelized cost of capital matches the optimal rental cost of capital if
and only if investment costs are constant:
QË™ t = 0 â‡?â‡’ FK (Kt
âˆ—
) = Lt (E.9)
19