ï»¿ WPS6154
Policy Research Working Paper 6154
How Inertia and Limited Potentials Affect
the Timing of Sectoral Abatements
in Optimal Climate Policy
Adrien Vogt-Schilb
Guy Meunier
StÃ©phane Hallegatte
The World Bank
Sustainable Development Network
Office of the Chief Economist
August 2012
Policy Research Working Paper 6154
Abstract
This paper investigates the optimal timing of greenhouse authors prove that optimal marginal abatement costs
gas abatement efforts in a multi-sectoral model with should differ across sectors: they depend on the global
economic inertia, each sector having a limited abatement carbon price, but also on sector-specific shadow costs of
potential. It defines economic inertia as the conjunction the sectoral abatement potential. The paper discusses the
of technical inertiaâ€”a social planner chooses investment impact of the convexity of abatement investment costs:
on persistent abating activities, as opposed to choosing more rigid sectors are represented with more convex cost
abatement at each time period independentlyâ€”and functions and should invest more in early abatement.
increasing marginal investment costs in abating activities. The conclusion is that overlapping mitigation policies
It shows that in the presence of economic inertia, optimal should not be discarded based on the argument that they
abatement efforts (in dollars per ton) are bell-shaped and set different marginal costs (â€œdifferent carbon pricesâ€?) in
trigger a transition toward a low-carbon economy. The different sectors.
This paper is a product of the Office of the Chief Economist, the 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 vogt@centre-cired.fr and shallegatte@worldbank.org.
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Produced by the Research Support Team
How inertia and limited potentials aect the timing of
sectoral abatements in optimal climate policy
a,â€ b c,d
Adrien Vogt-Schilb , Guy Meunier , StÃ©phane Hallegatte
August 1, 2012
â€
Corresponding author (vogt@centre-cired.fr)
a
CIRED, Nogent-sur-Marne, France
b
INRAUR1303 ALISS, Ivry-Sur-Seine, France
c
The World Bank, Sustainable Development Network, Washington D.C., USA
d
Ã‰cole Nationale de la MÃ©tÃ©orologie, MÃ©tÃ©o-France, Toulouse, France
Keywords: climate change mitigation, sectoral policies, optimal policies, optimal timing, inertia,
when-exibility, how-exibility, overlapping policies.
JEL classication: L98, O21, O25, Q48, Q54, Q58
1
1 Introduction
Many countries committed to reduce their greenhouse gas (GHG) emissions to limit climate change.
The European Union, for instance, pledged to reduce its emissions to 20% below 1990 levels by 2020.
At the global level, the international community recognizes the objective of limiting the increase
â—¦
in global mean temperature to 2 C. Translating these objectives into policies is dicult. First, a
given temperature objective can be reached with dierent emissions pathways (this is known as the
when-exibility of the mitigation policies). Second, there are many sectors in which it is possible to
reduce GHG, from the use of renewable energy to better building insulation and more ecient cars
(we call this the how-exibility ). One question is the optimal distribution of emission reductions
and corresponding economic costs across these options and over time.
Economic theory establishes that the optimal policy is to introduce a price signal corresponding
to the pigovian tax and to let the market allocate the burden eciently across sectors. There may be
a controversy on which instrument is more desirable to implement the price signal, notably between
a tax or quotas (this question has been studied by a whole branch of the literature since the seminal
works of Weitzman (1974)), but it is widely accepted in the literature that the optimal approach is
to impose a unique, economy-wide carbon price leading to the same marginal abatement cost in all
sectors.
In the presence of multiple market failures or externalities, however, a welfare-maximizing climate
policy is more complicated to design (Lipsey and Lancaster, 1956). For instance, many abatement
activities use new or emerging technologies that are likely to exhibit learning spillovers. This twin-
market failure in the area of eco-innovation (Jae et al., 2005) can be addressed by combining R&D
subsidies with a carbon price (Fischer and Newell, 2008; Gerlagh et al., 2009; Grimaud et al., 2011;
Acemoglu et al., 2012).
This paper addresses another diculty in the design of climate policies: the role of economic
inertia. Many papers investigating the optimal abatement pathway (i.e. how to benet from the
exibility on when to abate) work from a xed baseline scenario, and they assume that abatement
actions are decided independently at every period, as a function of their costs and benets (Nordhaus,
1992). When path dependency is introduced, it is mainly in the form of learning by doing, which
makes abatement cost decrease with the cumulative aggregated abatement or cumulative investment
in abating capital (van der Zwaan et al., 2002; Manne and Richels, 2004).
But path dependency goes beyond learning by doing. Many authors have reframed climate
policy in the context of a transition from carbon-intensive to low-carbon economic patterns. In this
context, abatement actions at any given point in time have permanent eects. In particular, there
is a large technical inertia embedded in infrastructure patterns (e.g., transport networks, electricity
production) and spatial patterns (e.g., low-density vs. high-density cities). Ambitious abatement
actions at one point in time would thus reduce emissions over the long term. Several authors have
studied how this path dependency impacts the optimal timing of GHG reduction (Grubb et al., 1995;
Wigley et al., 1996; GrÃ¼bler and Messner, 1998; Goulder and Mathai, 2000).
1
In this paper, we compare the baseline-based and the transition-based approaches. We use a
simple multi-sector model to highlight the consequences of considering that abatement actions have
a permanent impact on emissions (transition-based approach), instead of an evanescent impact on
only one period (baseline-based approach). We call economic inertia the combination of technical
inertia in the form of permanent emission reductions and decreasing returns of abatement
investments at one point in time. When economic inertia is taken into account, the optimal marginal
abatement eort in a dollars-per-ton metric is not growing over time, like in the classical
approach with evanescent abatement actions, but is more concentrated over the shorter term. In
this case, bell-shaped abatement eorts support a transition toward low-carbon economic patterns.
We also link the question on when to abate to the question on how, i.e. in which economic
sector, to abate. We show that in our framework, the sectoral potentials the maximum amount
of abatement that can be realized in each particular sector have a strong impact on the optimal
timing of abatements at the sector level. In practice, each sector is aected by a social cost of sectoral
potentials that needs to be subtracted from the social cost of carbon to assess the optimal amount of
abatement that should be undertaken. This eect implies that, at each point in time, the marginal
costs of abatement investments should be dierent across sectors, and that the sectors that will take
longer to decarbonize should invest more per abated ton of GHG than the others. We show that this
is the case for sectors with large potentials.
1 An up-to-date literature review can be found in Cian and Massimo (2011).
2
Finally, more inert sectors also take longer to decarbonize, and we show that higher inertia
can be modeled through a higher convexity of the investment cost function. To assess precisely
optimal sectoral eorts requires information on sectoral costs and potentials, but also on inertia, an
information that is not reported separately by existing studies (IPCC, 2007; McKinsey and Company,
2007).
In policy terms, these results suggest that higher-than-average eorts are justied in high-
potential and highly inert sectors such as urban planning (Guivarch and Hallegatte, 2011).
In section 2, we briey introduce the classical approach to the denition of an optimal abatement
strategy, with a multi-sector model with evanescent abatement options and sectoral potentials, in
the spirit of Nordhaus (1992). In this model, the classical result holds: optimality requires that
marginal abatement costs are equal in all sectors. Then, we present in section 3 a slightly dierent
model, where abatement options have a permanent impact on emissions. We show that, in this
case, marginal abatement eorts should be larger in sectors that will take longer to reach their full
abatement potential. We provide an illustrative example based on IPCC abatement costs in section 4
and investigate the role of convexity on investment costs in section 5. Section 6 concludes.
2 The classical approach to optimal abatement pathways
Since the seminal article by Nordhaus (1992), optimal abatement trajectories are frequently calculated
assuming that a social planner has to choose when to abate GHG emissions. The planner does
so by spending on abatement expenditures that reduce emissions at one given point in time; we
refer to these abatement expenditures as evanescent abatements. In this framework, the amount of
abatement is decided independently at each period, and the cost at one period does not depend on
actions undertaken before. In other words, abatement costs exhibit no path dependency. Abatement
at time t, a(t), is done at a cost Î³(a(t)), where Î³ is convex, positive and twice dierentiable. These
convex costs capture increasing costs in the abatement potentials, i.e. the fact that some emissions
are cheaper to abate than others.
In this section, we expand the classical model by describing the economy as a set of sectors indexed
by i. For simplicity, we assume that these sectors do not interact with each other. The social planner
decides the non-negative abatement eort ai (t) done in each sector, using the abatement cost function
Î³i (ai (t)) which is also convex, positive and twice dierentiable:
2
Î³i > 0 (1)
Î³i > 0
The function Î³i (ai (t)) represents the marginal abatement cost (MAC), i.e. the cost of the last
unit of abatement at time t.
For instance, abatement in the transport sector can be done by changing individual vehicles.
Carbon-ecient vehicles typically have higher investment cost but lower operating costs than carbon-
intensive vehicles. For this reason, intensively-used vehicles may be replaced by zero-carbon vehicles
at a lower total cost than those driven occasionally. If vehicles are replaced in the merit order from
the cheapest to the most expensive the marginal cost of doing so Î³i (ai (t)) is growing in ai (t), and
Î³i (ai (t)) is convex. In economic terms, the convexity on Î³i comes from zero-carbon vehicles being
imperfect substitutes to thermal vehicles.
Moreover, we introduce a sectoral potential Â¯
ai in each sector i, which represents the maximum
amount of GHG emissions (in GtCO2 per year) that can be abated in this sector.
âˆ€i, âˆ€t, ai (t) â‰¤ ai
Â¯
For instance, if each vehicle is replaced by a zero-emission vehicle, all the abatement potential in the
private mobility sector has been realized. The sectoral potential may be roughly approximated by
sectoral emissions in the baseline, but they may also be smaller (if there are some fatal emissions in
the sector) or could even be higher (if negative emissions are possible).
The climate policy is modeled as a so-called carbon budget for emissions above a given level. We
assume that the environment is able to absorb a constant ow of GHG emissions E0 â‰¥ 0. Above
E0 , emissions are dangerous. The objective is to maintain cumulative dangerous emissions below a
given ceiling B (Allen et al., 2009). For simplicity, we assume that emissions would be constant in
2 All notations are reproduced in Tab. 1.
3
Emissions
GtCO2/yr
âˆ‘ ai(t)
Eabat
B
E0
t time
Figure 1: An illustration of the climate constraint. The cumulative emissions above E0 should be lower
than an intertemporal carbon budget B (this requires that the long-run emissions tend to E0 ). Dangerous
emissions Eabat are measured from E0 .
absence of abatement (Fig. 1). We assume that all dangerous emissions are abatable, and, without
loss of generality, that doing so requires to use all the abating potential in every sector. Denoting
Eabat the emissions above E0 , this reads:
Eabat = Â¯
ai (2)
i
Using another representation of climate policy objectives would not aect the qualitative results of
this analysis.
The social planner determines how and when to abate in order to minimize abatement costs
discounted at a given rate r, under the constraints set by the sectoral potentials and the carbon
budget:
âˆž
min eâˆ’rt Î³i (ai (t)) dt (3)
ai (t) 0 i
subject to ai (t) â‰¤ ai
Â¯
âˆž
Eabat âˆ’ ai (t) dt â‰¤ B
0 i
Under these assumptions, the classical result holds: the optimal strategy is to implement abatement
options such that marginal abatement costs are equal in all sectors and equal to the current carbon
price. With the objective to maintain cumulated emissions below the carbon budget, the (present
value of the) social cost of carbon is constant, we denote it Âµ. As its current value Âµert increases at
the discounting rate r, abatement eorts are increasing over time, and each sector i is progressively
decarbonized until it reaches the sectoral potential at a date denoted Ti . After this date, the sectoral
marginal abatement costs remain constant:
ert Âµ t < Ti
Î³i (ai (t)) = (4)
Î³i (Â¯i ) t â‰¥ Ti
a
These results (proved in annex A) are a straightforward application of the equimarginal principle:
unless the sectoral potentials are binding, optimal marginal eorts are equal, in all sectors, to a
unique carbon price.
3 Introducing economic inertia leads to dierent marginal costs
across sectors
The classical framework assumes that the amount of abatement is decided (and paid) at each period
independently. This is a simplifying assumption. For instance, replacing coal-red plants with gas-
red plants actually reduces annual emissions over a long period of time, while abatement costs
4
Name Description Unit
Âµ Social cost of carbon (SCC) $/tCO2
âˆ’1
r Discount rate yr
ai (t) Current abatement in sector i tCO2 /yr
Â¯
ai Sectoral potential in sector i tCO2 /yr
Î³i Abatement cost in sector i $/yr
Î³i Marginal abatement cost (MAC) in sector i $/tCO2
xi (t) Current investment in abating activities in sector i (tCO2 /yr)/yr
ci Cost of investment in sector i $/yr
ci Marginal investment cost (MIC) in sector i $/(tCO2 /yr)
rci Annualized MICs in sector i $/tCO2
Î»i (t) Social cost of the sectoral potential (SCSP) in sector i $/tCO2
Table 1: Notations and units
are mainly paid when the decision is made. In the building sector, reductions are achieved in part
by retrotting buildings, which has an almost-permanent eect on emissions. Another example is
urban development, which impacts emissions on the long term (Strand, 2011; ViguiÃ© and Hallegatte,
2012). We call this the technical inertia : abatement at time t cannot be controlled directly; instead,
it depends on the eorts that have been made before t. This introduces path dependency in the
analysis.
In our new model, abatement in each sector i starts at zero, and at each time step t, the social
planner chooses a non-negative amount of physical investment in abating capacities xi (t). Doing so
increments the current abatement ai :
ai (0) = 0 (5)
Ë™
ai (t) = xi (t) (6)
Where dotted variables represent temporal derivatives. These abatement investments are realized at
a cost ci (xi ), the functions ci are twice dierentiable, positive, increasing and convex:
ci (0) = 0 (7)
ci > 0 (8)
The values ci (xi ) are the Marginal Investment Cost (MIC) in abating activities, i.e. the cost of
the last unit of abatement investment.
The convexity bears on the investments ow, to capture the fact that additional abatements are
obtained at an increasing marginal cost (or decreasing returns) in any given sector. For instance,
xi (t) could stand for the pace measured in buildings per year at which old buildings are being
retrotted at date t. The abatement ai (t) would be proportional to the share of retrotted buildings
in the stock. In a given time lapse, retrotting buildings requires to pay scarce skilled workers. If
workers are hired in the merit order and paid at the marginal productivity, the marginal price of
retrotting buildings ci (xi ) is a growing function of xi in other words, ci is convex. In addition,
the existing literature on inertia (Ha-Duong et al., 1997; Lecocq et al., 1998; Schwoon and Tol,
2006) claims that existing studies on limited capital stock turnover justies this cost convexity: if
emitting capital is replaced by cleaner capital slower than the natural turnover rate, marginal costs
are constant; if it is replaced faster costs rise nonlinearly.
This convexity is of dierent nature than the convexity in Î³i (ai (t)) in the classical approach
presented in section 2, where convexity arises from heterogeneity in abatement options (e.g., dierent
abatement costs for frequently-driven and occasionally-driven vehicles).
We call economic inertia the combination of technical inertia (6) and convex investment costs
(8).
Proposition 1 Consider an optimal abating strategy in multiple sectors subject to economic inertia
and limited potentials. In general, at each point in time, Marginal Investment Costs (MICs) should
not be equal across sectors.
In each sector, MICs should be equal to the social cost of the abated carbon minus the value of the
forgone option to abate later.
5
Figure 2: Optimal marginal investment costs (MIC) in abating activities in a case with two sectors (i âˆˆ
{1, 2}). MICs dier across sectors, because of the social costs of the sectoral potentials. The optimal timing
of sectoral abatements comes from a trade-o between investing later in order to reduce present costs thanks
to the discounting and investing sooner to benet from the persistence of the abating eorts over time,
resulting in a bell-shaped pathway (16).
Let us sketch the proof to provide some intuitions of the mechanisms at stake. (The full proof is in
annex B). The social planner program reads:
âˆž
min eâˆ’rt ci (xi (t)) dt (9)
xi (t) 0 i
subject to ai (t) â‰¤ ai
Â¯ (10)
âˆž
Eabat âˆ’ ai (t) dt â‰¤ B (11)
0 i
Ë™
ai (t) = xi (t) (6)
The associated Lagrangian reads (see annex B):
âˆž âˆž
L(xi , ai , Î»i , Î½i , Âµ) = eâˆ’rt ci (xi (t)) dt + Î»i (t) (ai (t) âˆ’ ai ) dt
Â¯
0 i 0 i
âˆž
+Âµ Eabat âˆ’ ai (t) dt âˆ’ B
0 i
âˆž âˆž
âˆ’ Î½i (t)ai (t) dt âˆ’
Ë™ Î½i (t)xi (t) dt (12)
0 i 0 i
in which Î»i , Âµ and Î½i are the Lagrange multipliers associated with the constraints of the sectoral
potential (10), the carbon budget (11) and the abatement dynamic (6) respectively. In particular,
the present value of the carbon budget constraint is constant and equal to Âµ, we call it the social
cost of carbon (SCC); Âµert is its current value, we call it the (current) carbon price.
First-order conditions read:
âˆ‚L
= 0 â‡?â‡’ Î½i (t) = Î»i (t) âˆ’ Âµ
Ë™ (13)
âˆ‚ai
âˆ‚L
= 0 â‡?â‡’ ci (xi (t)) = ert Î½i (t) (14)
âˆ‚xi
The optimal marginal investment costs ci (xi (t)) in sector i depends on the unique SCC Âµ and on the
sector-specic social costs of the sectoral potential (SCSP) Î»i (see details in annex B). The social
planner controls xi but their integral Â¯
ai is bounded by the potentials ai , making these potentials
aect the optimal strategy before they are reached:
âˆž
ci (xi (t)) = ert (Âµ âˆ’ Î»i (Î¸)) dÎ¸ (15)
t
6
The left hand side of (15) is the cost of the last unit of investment in abating capacity, i.e. the
Marginal Investment Cost (MIC), in sector i. The integral in the right hand side is dierent across
rt
sectors. It represents the total current value (e ) of the carbon that will be saved from t onwards
thanks to the marginal investment, minus the sector-specic total SCSP from t onwards. The SCSP
Î»i (t) is null before the sectoral potential has been reached. Once the potential has been reached, the
option to abate more carbon in sector i is removed (Î»i (Î¸) = Âµ).
3
Noting Ti the dates when the sectoral potentials are reached, we can express explicitly the
optimal marginal investment costs in each sector as a function of the optimal SCC Âµ and the optimal
dates Ti (see annex B):
Âµert (Ti âˆ’ t) if t < Ti
ci (xi (t)) = (16)
0 if t â‰¥ Ti
Marginal investments should be priced at the social cost of the carbon that they will save until the
sectoral potential is reached. The longer it takes for a sector to reach its potential (i.e. the larger Ti ),
the higher should be priced investments in abating capacity.
In particular, marginal investment costs are not equal to the current carbon price Âµert , and not
equal across sectors.
The optimal investments pathways xi (t) can be found by inverting (16). They arise from a
complex trade-o: investing soon allows the planner to benet from the persistence of abatements,
and prevents to invest 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 (Fig. 2): in the short term, the eect of discounting dominates and
eorts grow exponentially; in the long term, the eect of the limited potential dominates and the
eort decrease to zero.
A corollary of (15) is that dierences in MICs come from dierent social costs of the sectoral potentials
(Fig. 2):
âˆž
ci (xi (t)) âˆ’ cj (xj (t)) = ert Î»j (Î¸) âˆ’ Î»i (Î¸) dÎ¸ (17)
t
In particular, optimal strategies may imply paying for expensive abatement while cheaper ones are
available:
Corollary 1 Assume that abatement in two sectors {1,2} exhibit the same cost structure, and that
the rst one has a greater potential than the second:
âˆ€x > 0, c1 (x) = c2 (x)
Â¯ Â¯
a1 < a2
The second sector will take longer to decarbonize:
T1 < T2
At any time, the sector with the largest potential should invest more (both in dollars per abated ton
of GHG and in physical terms) than the other.
âˆ€t, c1 (x1 (t)) < c2 (x2 (t)) and x1 (t) < x2 (t)
The proof is in annex C. Even if the cost function is the same in both sectors, dierent sectoral
potentials cause the investment along the optimal path to be dierent across sectors. This result
is explained by the combination of limited sectoral potentials and increasing marginal investments
costs without convex investment costs, the sectors would be decarbonized in the merit order: the
cheapest sector rst.
3 After T , x (t) = 0 and ci (xi (t)) = 0.
i i
7
Marginal cost Abatement potential
Î³i (1 GtCO2 /yr) [ $/tCO2 ] Â¯
ai [ GtCO2 /yr]
Waste 34 0.76
Industry 17.6 4.08
Forestry 15.9 2.75
Agriculture 11.9 4.39
Transport 11.6 2.1
Energy 10.3 3.68
Buildings 3.6 5.99
Table 2: Assumptions on marginal abatement costs and maximum potential in seven sectors of the global
economy. Numerical values derived from the abatement potential at a marginal cost below 20 $/tCO2 in
gure SPM.6, page 11 of IPCC (2007).
4 Illustrative examples using IPCC abatement costs
We use the two models (with and without economic inertia) to investigate the optimal sectoral
abatements over the 2007-2030 period. We set a policy objective over this period only,
4 and use
abatement cost information derived from the IPCC (2007). Because of data limitations, this exercise
is not supposed to suggest an optimal climate policy (see also section 5). It aims at illustrating the
impact of two extreme assumptions (abatement options are totally evanescent or totally permanent)
on the optimal abatement strategy, using the same cost estimates and the same policy objective.
We show that the optimal abatement strategy and in particular the choice of which sectors need
to abate more and earlier depends on whether abatement actions are assumed permanent or
evanescent.
4.1 Data
IPCC (2007) provides data on abatement costs and potentials until 2030, in 7 sectors of the global
economy (Tab. 2). These abatement costs have two components that are aggregated in IPCC data.
The rst component is the cost for evanescent reduction; it is captured by the model exposed in
section 2. The other component is the costs of persistent reductions, it can be captured by the
model exposed in section 3. From the same data on costs, potentials and carbon budget, we derive
two optimal reduction strategies, one from the model with no economic inertia (section 2), and the
other from the model with technical inertia and convex investment costs (section 3).
In this illustrative example, we assume abatable emissions Eabat = i Â¯
ai = 24 GtCO2 /yr.
Compared to the total baseline emissions in 2030 in the A1B scenario from Nakicenovic et al. (2000),
69 GtCO2 /yr, this gives E0 = 45 GtCO2 /yr. In this case, E0 cannot be interpreted as a denitive
safe level for emissions, but rather as an emission milestone in 2030 in a pathway toward a sustainable
longer-term target.
We call T = 23 yr the time span from 2007 the publication date of IPCC (2007) and 2030.
We set the discount rate to r = 4%/yr. We constrain the cumulative dangerous emissions over the
period as:
T
Eabat âˆ’ ai (t) â‰¤ 140 GtCO2
0 i
The two models are solved with a numerical resolution, using a time step dt = 0.1 yr.
5
4.2 Calibration
We calibrate the two models (with and without inertia) such that the discounted costs of reaching
the same target are equal in the two models (this methodology was rst used by Grubb et al. (1995)).
4 The theoretical, innite-horizon models exposed in sections 2 and 3 have to be modied; all the results exposed
still apply.
5 Data and source code are available at the corresponding author's web page. Computation and plots use Scilab
(Scilab Consortium, 2011).
8
MAC parameter MIC parameter
m $/tCO2 $/tCO2
Î³i GtCO2 /yr cm
i GtCO2 /yr3
Waste 34 3732
Industry 17.6 1932
Forestry 15.9 1745
Agriculture 11.9 1306
Transport 11.6 1273
Energy 10.3 1131
Buildings 3.6 395.2
Table 3: Calibration of the Marginal Investment Cost (MIC) functions and of the Marginal Abatement
Costs (MAC) functions. This gures are consistent with our modeling framework (18, 20) and the data from
IPCC reported in the Tab. 2.
We also ensure that relative costs (when comparing two sectors) are equal in the two models. This
way, dierences in optimal strategies arise only from the introduction of economic inertia.
We rst solve the problem exposed in section 2, assuming quadratic abatement costs. Quadratic
costs are a simple specication that grants that the Î³i are convex, and simplies the resolution as
marginal abatement costs are linear:
m
Î³i 2
âˆ€i, âˆ€x â‰¥ 0, Î³i (a) = a
2
m
âˆ€x â‰¥ 0, Î³i (a) = Î³i Â· a (18)
m
Where Î³i are parameters specic to each sector. We calibrate these costs functions using the
abatements corresponding to a 20 $/tCO2 marginal cost in gure SPM.6 of IPCC (2007) presented
in Tab. 2; the result of this calibration is presented in the rst column of Tab. 3.
The numerical resolution allows one to compute the optimal strategy in nancial (Î³i (ai (t))) and
physical (ai (t)) terms (see (34) and (36) in annex D and Fig. 3).
We call Î“ the discounted cost of the optimal mitigation strategy:
T
Î“= eâˆ’rt Î³i (ai (t))dt (19)
0 i
The numerical resolution gives Î“ = 3351 G$.
Then, we solve the model with economic inertia, still assuming quadratic costs.
cm 2
i
âˆ€i, âˆ€x â‰¥ 0, ci (x) = x (20)
2
This allows one to simplify the model (see (38) and (41) in annex E). We calibrate the cm
i parameters
by adapting the methodology from Grubb et al. (1995): rst, we assume that the relative MICs are
equal to the relative MACs from IPCC:
cm
i Î³m
âˆ€i, = i (21)
cm
1
m
Î³1
And second, we constrain the total costs of the mitigation strategy to be equal in the two models:
âˆž
eâˆ’rt ci (xi (t)) dt = Î“ (22)
i 0
A numerical resolution provides cm
i (Tab. 3), as well as the optimal abating strategy xi and ai (Fig. 3)
Finally, we introduce annualized marginal investment costs. In our modeling framework, evanescent
options are implemented paying a MAC, while permanent options are implemented paying a MIC.
Investingxi (t) at t reduces GHG emissions from t onwards, but the cost of doing so, ci (xi (t)), is paid
only at time t. To make this cost easier to compare with evanescent abatement costs, it is useful to
introduce the annualized cost of investment r Â· ci (xi (t)) and its marginal value. Indeed, paying the
9
amount ci (xi (t)) at time t is equivalent to paying the ow r Â· ci (xi (t)) from t to âˆž. Multiplying all
terms in equation (16) by r, one gets:
r(Ti âˆ’ t)Âµert if t < Ti
r Â· ci (xi (t)) = (16bis)
0 if t â‰¥ Ti
Annualized MICs are expressed in $/tCO2 so they can be compared easily to the marginal abatement
costs derived from the model without inertia (Fig. 3). Even if they can be expressed in the same
metrics, annualized MICs dier from the MACs Ã la Nordhaus (1992): as explained in sections 2 and
3, MICs depend on the decarbonizing pace in each sector, while MAC only depend on the current
abatement (also in each sector).
4.3 Results
The optimal mitigation strategy by the two models are compared sector by sector in Fig. 3, and
the aggregated pathways in terms of abatement and nancial eort are presented in Fig. 4. The
pathways suggested by these models are not optimal pathways to mitigate climate change: they only
consider a target in the 2007-2030 window and are based on crude available data.
The two models give the same result in the long run: abatement in each sector eventually reaches
its maximum potential. By construction, they also achieve aggregated abatement target at the same
discounted cost. But the optimal strategies according to the two models dier radically in terms of
aggregated abatement and nancial costs, and, more importantly, at the sector level.
The optimal abatement pathway in the model without economic inertia includes signicant
abatements as soon as the climate policy is implemented, while the model with inertia has a more
realistic abatement pathway that starts at zero and increases progressively over time.
In terms of nancial eort, the model with inertia provides an optimal pathway that starts higher
and decreases over time toward zero this is the decreasing side of the bell-shaped trajectory
discussed in the formal analysis (Fig. 2). In this case, abatement eorts are transitional, once all
the capital in a sector has been replaced by non-emiting capital, nancial eorts are not required
anymore. While the carbon price grows exponentially in both model (see the discussion after equation
(12)), the aggregated nancial eorts tend to zero when economic inertia is taken into account. This
is consistent with proposition 1: in this case, marginal investement costs in abating activities should
not be equal to the carbon price.
Taking into account economic inertia also modies the optimal sectoral burden sharing. Without
inertia, all sectors are decarbonized up to the same marginal eort until they reach their sectoral
potential. When inertia is taken into account, the optimal timing of sectoral reductions arises
from a complex trade-o. Investing soon in abating activities allows the planner to benet from
the persitency of abatements; but it brings closer the date when the sectoral potential is reached,
removing the option to invest later, when the discounted cost is lower.
To avoid paying too much at any point in time, investment should be spread over a larger period,
and longer-to-decarbonize sectors should abate at a higher cost than the others. In this example
with quadratic investment costs i.e. with the same cost convexity accross sectors , industry
is decarbonized faster (in terms of xi (t)) and at a higher cost (r Â· ci (xi (t))) than forestry, despite
forestry being a priori cheaper to decarbonize. This is because industry has a greater abatement
potential (Tab. 2) and thus takes longer to decarbonize.
These results are not policy recommendations, as we considered only a limited time period.
Moreover, the assumption of a uniform convexity accross sectors is inappropriate; we used it in
absence of data on sectoral inertia. The next section discusses the eect of dierent cost convexities
on economic inertia and optimal economic pathways.
5 The impact of cost convexity on optimal abatement strategies
The classical model (without economic inertia) suggests that buildings should be entirely decarbonized
by 2011, being the rst sector to reach its full abatement potential (bottom right panel of Fig. 3).
In the baseline-based modeling framework, this result is natural, as the model implements abating
activities in the merit order (4). As available studies nd that better insulation in the buildings
would largely pay for themselves thanks to energy savings, resulting in very low, or even negative
abatement costs (McKinsey and Company, 2007), this model will always recommend to decarbonize
buildings very fast.
10
Classical model Model with economic inertia
Figure 3: Comparison of marginal costs and optimal abatement strategies to achieve the same amount of
abatement, when the costs from table 2 are understood as marginal abatement costs (left) vs. annualized
marginal investment costs (right).
In the rst case, without economic inertia, sectoral eorts should be equal to the current carbon price, unless
they have reached the maximum potential (upper left panel). As a consequence, the abatement may jump
from 0 to a non-negative value instantaneously at the beginning of the period (as for instance in the buildings
sector, lower left panel).
In the right column, with economic inertia, the abatement starts at zero (lower panel) and increases
progressively with the accumulation of physical investments (middle). In this case, abatement in longer-
to-decarbonize sectors should be priced higher than abatement in quick-to-decarbonize ones (upper panel).
Note: in the classical model, there is no equivalent to the physical investments in abating activities xi , as
the planner controls directly the abatement level ai without having to deal with economic inertia.
11
This is not satisfying: it will arguably take decades to retrot all existing energy-consuming
buildings at the reported cost. Previous research has tackled this issue by imposing exogenous upper
bounds in the decarbonizing pace of any sector (Vogt-Schilb and Hallegatte, 2011). In our model with
economic inertia, mitigation costs depend on how fast the sector is decarbonized. We demonstrate
below that in this model, dierent convexities bearing on the investment cost function allow to take
into account dierent inertias.
We start from the model with economic inertia and linear MICs, but, this time, we choose a
particular sector b = Buildings for which we use a more general cost function (for all other sectors,
we keep the same quadratic cost functions than before), allowing to investigate the inuence of cost
convexity on the results:
Î±b
1 x
âˆ€x â‰¥ 0, cb (x) = (23)
Î±b xref
b
Where Î±b > 1 describes the convexity of the costs function. In economic terms, Î±b is the constant
cost elasticity of abating investments in sector b:
dcb cb
Â· = Î±b (24)
dxb xb
Finally,xref (in the same unit than xb ) is a scale parameter. For each value of Î±b , we calibrate
b
xb (Î±b ) such that the total mitigation cost of the building sector remains unchanged.6
ref
We investigate the impact of Î±b on the optimal timing of the mitigation strategy. First, we
compare the optimal strategies in a reference case (with Î±b = 2, replicating the case of quadratic
costs functions) and a case with higher convexity Î±b = 2.3. The corresponding optimal abatement
pathways are shown in Fig. 5. Compared to the reference case, the building sector takes much more
time to decarbonize when its costs are more convex: the date Tb moves from 2016 to 2030. To respect
the carbon budget, the social planner must abate faster in all others sectors, which is achieved with
a higher carbon cost (not shown).
We then perform a full sensitivity analysis, solving the same model for a range of posible Î±b
(Fig. 6). When Î±b tends to 1, the MIC cb does not depend on the decarbonizing pace xb anymore,
and there is no inertia. The building sector is fully decarbonized instantaneously at the beginning
of the period. On the other hand, the greater Î±b the longer the building sector takes to reach its
abating potential.
These results conrm that dierent economic inertias can be modeled through dierent convexities
of investment costs in a model with persistent abatements. In a given sector, the optimal timing
strongly depends on the convexity of its investments cost function. In this example, when Î±b varies
from 1.9 to 2.1, Tb varies from 2013 to 2021; in the rst case, decarbonizing buildings take 6 years,
in the latter 14, more than twice more.
7
These results also show that the question of how and when to abate cannot be answered based
on costs and potential information alone. In addition, information on inertia is required. Vogt-
Schilb and Hallegatte (2011) suggest to expand existing marginal abatement cost curves existing
MACCs currently report costs and potentials of a set of abatement options with information on
the implementation pace of each option. This paper conrms this need in an analytical framework.
6 Discussion and conclusion
In this paper, we modify a classical model Ã la Nordhaus (1992) and introduce economic inertia
bearing on GHG mitigation activities. We dene economic inertia as the conjunction of technical
inertia (a social planner chooses investment on persistent abating activities, as opposed to choosing
abatement at each time step independently) and increasing marginal investment costs. We investigate
the optimal timing of abatement eorts in a multisectoral model that accounts for economic inertia
in abating activities, and in which each sector has a limited abatement potential. We compare the
6 We could have decided to keep xref constant instead, without modifying the qualitative results and the message
b
from the experiment. But this would not have granted that investment costs are held constant except in the particular
case when xb = xb .
ref
7 T is less sensible to Î± if we hold xref constant instead of the total cost of decarbonizing buildings (see also
b b b
footnote 6).
12
Figure 4: Comparison of nancial and physical eorts in the models with inertia vs. without inertia. When
economic inertia is disregarded, the abatement can jump from 0 to a non-negative value instantaneously
at the beginning of the period. When economic inertia is taken into account, abatement has to grow
continuously, and short-term nancial eorts are higher than when inertia is neglected. By construction, the
total discounted cost and the cumulative abatements are equal in the two cases.
Low convexity Î±b = 2 High convexity Î±b = 2.3
Figure 5: Increasing the convexity of the building sector cost function has a direct and an indirect eect.
The direct is that it increases economic inertia of the building sectors and spreads its abatement over the
period (even if total abatement cost in the building sector is held constant). The indirect is that abatement
in the other sectors must be achieved faster to reach the carbon budget.
Figure 6: Optimal decarbonizing dates as a function of Î±Buildings . The more the cost function of the
building sector is convex, the more progressive should be the abatement in this sector: TBuildings is postponed.
Moreover, the other sectors have to decarbonize faster to cope with the global carbon budget.
13
results obtained with our new model and those coming from a classical model without economic
inertia.
These two models illustrate the dierence between two approaches to the climate change problem.
Initially, most analyses considered a given GHG emission (and development) baseline and abatement
expenditures applied to this unchanged baseline. In such a setting, abatement expenditures need to
increase over time roughly at the discount rate, and they need to do so for ever. This is what is
reproduced by our model with no inertia, a simplied version of the seminal model by Nordhaus
(1992). In another approach, it has been proposed to consider the climate change issue as a
development pathway problem (World Bank, 2010, 2012). In such a framework, what is needed
is a bifurcation toward a dierent, low-carbon development model. In this case, abatement eorts
are seen as transitional, they are required to shift from carbon-intensive to low-carbon economic
patterns.
Here, we link the question on when to abate to the question on how, i.e. in which sector, to
abate. The two models have diverging recommendation at the sector level. The model with no inertia
suggests that all sectors should implement abatement options up to a given marginal abatement cost,
equal for all sectors. This leads to unrealistic recommendations at the sector level (e.g., retrot all
old buildings by 2011). On the opposite, the model with economic inertia suggests that sectors with
higher potential (e.g., industry) should invest more than other sectors (e.g., forestry) per abated ton
at every point in time. Marginal investment costs in abating activities should be priced to a global
carbon price less a sector-specic social cost of the sectoral potential.
The dierence between the two models illustrates that the design of an optimal abatement
pathway requires reporting separately abatement costs arising from, on the one hand, imperfect
substitution between high- and low-emitting capital, and, on the other hand, transitional costs.
Costs from the second category depend on the decarbonizing pace, strongly impacting the optimal
time prole of an abating strategy.
The model with economic inertia shows that optimal marginal eorts in a given sector depend
positively on the time it will take to reach the sectoral abatement potential. In inert sectors,
investment in abating activities should be priced higher than average, and could require specic
policies such as norms and regulations (e.g., feed-in taris, eciency standards or land-use planning).
This result provides an analytical framework to support previous ndings that larger eorts are
desirable in sectors with high inertia and large potential (Jaccard and Rivers, 2007; Shalizi and
Lecocq, 2009; Vogt-Schilb and Hallegatte, 2011).
There is currently a debate on the overlapping mitigation policies, as the 3 Ã— 20 objectives of the
European Union, existing standards on automobile, buildings and household appliance, and feed-in
taris for specic low carbon technologies in the power sector. Such overlapping policies have been
criticized, partly because they set dierent marginal costs in dierent economic sectors. Previous
research has shown that dierences in marginal abatement costs can be justied, notably by learning
spillovers and other market imperfections (Braathen, 2007; Fischer and Preonas, 2010). We bring a
new argument based on the existence of limited sectoral potentials and economic inertia: investments
in abating activities should be priced at the cost of the carbon they allow to abate until the sectoral
potential is reached; in sectors that will take long to decarbonize, higher eort levels are justied.
Acknowledgments
The authors wish to thank Marianne Fay, FrÃ©dÃ©rique Ghersi, Louis-GaÃ«tan Giraudet, CÃ©line Guivarch,
Oskar Lecuyer, Baptiste Perissin Fabert, Antonin Pottier, Julie Rozenberg and Ankur Shah Delight
for useful comments. We also thank Patrice Dumas for technical support. The views expressed in
this paper are the sole responsibility of the authors. They do not necessarily reect the views of the
World Bank, its executive directors, or the countries they represent.
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16
A Proof of (4)
The associated Lagrangian reads:
âˆž âˆž
L(ai , Î»i , Âµ) = eâˆ’rt Î³i (ai (t)) dt + Î»i (t) (ai (t) âˆ’ ai ) dt
Â¯
0 i 0 i
âˆž
+Âµ Eabat âˆ’ ai (t) dt âˆ’ B (25)
0 i
The rst order condition is:
âˆ‚L
= 0 â‡?â‡’ eâˆ’rt Î³i (ai (t)) + Î»i (t) âˆ’ Âµ = 0
âˆ‚ai
â‡?â‡’ Î³i (ai (t)) = ert (Âµ âˆ’ Î»i (t)) (26)
Let Ti be the date when the abatement potential Â¯
ai is reached in sector i, such that:
âˆ€t < Ti , ai (t) < ai
Â¯
âˆ€t â‰¥ Ti , ai (t) = ai
Â¯
As the associated Lagrangian multiplier, Î»i (t) is null before the sectoral potential becomes binding:
âˆ€t Î»i (t) (ai (t) âˆ’ ai ) = 0
Â¯
=â‡’ âˆ€t < Ti , Î»i (t) = 0 (27)
Combining (27) and (26), one gets:
ert Âµ t < Ti
Î³i (ai (t)) = (28)
Î³i (Â¯i ) t â‰¥ Ti
a
This is the classical result: in every sector, current marginal abatement costs should be equal to the
rt
current carbon price (e Âµ), unless the sectoral potential has been exhausted already.
B Full proof of the proposition 1
The associated Lagrangian reads:
âˆž âˆž
Ë™
L(xi , ai , ai , Î»i , Î½i , Âµ) = eâˆ’rt ci (xi (t)) dt + Î»i (t) (ai (t) âˆ’ ai ) dt
Â¯
0 i 0 i
âˆž âˆž
+Âµ Eabat âˆ’ ai (t) dt âˆ’ B + Î½i (t) (ai (t) âˆ’ xi (t)) dt
Ë™
0 i 0 i
(29)
In the last term, Ë™
ai (t) can be removed thanks to an integration by parts:
âˆž
Î½i (t) (ai (t) âˆ’ xi (t)) dt
Ë™
t i
âˆž âˆž
= Î½i (t)ai (t) dt âˆ’
Ë™ Î½i (t)xi (t) dt
i 0 0
âˆž âˆž
= constant âˆ’ Î½i (t)ai (t) dt âˆ’
Ë™ Î½i (t)xi (t) dt
i 0 0
The transformed Lagrangian does not depend on Ë™
ai (t):
âˆž âˆž
L(xi , ai , Î»i , Î½i , Âµ) = eâˆ’rt ci (xi (t)) dt + Î»i (t) (ai (t) âˆ’ ai ) dt
Â¯ (12)
0 i 0 i
âˆž
+Âµ Eabat âˆ’ ai (t) dt âˆ’ B
0 i
âˆž âˆž
âˆ’ Î½i (t)ai (t) dt âˆ’
Ë™ Î½i (t)xi (t) dt
0 i 0 i
17
The rst order conditions read:
âˆ‚L
= 0 â‡?â‡’ Î½i (t) = Î»i (t) âˆ’ Âµ
Ë™ (13)
âˆ‚ai
âˆ‚L
= 0 â‡?â‡’ eâˆ’rt ci (xi (t)) = Î½i (t) (14)
âˆ‚xi
The abatement ai (t) are increasing over time (as ai (t) = xi (t) â‰¥ 0).
Ë™ For each sector i there is a date
Ti such that
âˆ€t < Ti , ai (t) < ai
Â¯ =â‡’ Î»i (t) = 0
âˆ€t â‰¥ Ti , ai (t) = ai
Â¯ =â‡’ Î»i (t) â‰¥ 0
The Lagrange multipliers Î»i (t) are null before Ti and strictly positive after. So, by integrating (13):
t
Î½i (t) = Î½i (Ti ) + Ë™
Î½i (t)
Ti
âˆ€t < Ti , Î½i (t) = Î½i (Ti ) + (Ti âˆ’ t)Âµ
Furthermore, for t > Ti the abatement ai (t) is constant, equal to Â¯
ai , so Ë™
xi (t) = ai (t) is null, and,
from (14) and (7):
âˆ€t â‰¥ Ti , Î½i (t) = 0
Therefore,
(Ti âˆ’ t)Âµ if t < Ti
Î½i (t) = (30)
0 if t â‰¥ Ti
Then, from (14), the marginal costs can be expressed as a function of Ti :
ert (Ti âˆ’ t)Âµ if t < Ti
ci (xi (t)) = (16)
0 if t â‰¥ Ti
Marginal investment costs should be equal to the total social cost of the carbon (Âµ) that will be
saved thanks to the abatement before the sectoral potential is reached (the time span (Ti âˆ’ t)), as
rt
expressed in current value (e ).
C Proof of corollary 1
Proof: As ci is by assumption (7) strictly growing, it is invertible. Let Ï‡i be the inverse of ci :
âˆ’1
Ï‡ i = ci (31)
Applying Ï‡i to (16) gives:
Ï‡i (ert (Ti âˆ’ t)Âµ) if t < Ti
xi (t) = (32)
0 if t â‰¥ Ti
Â¯
The relation between the sectoral potential (ai ), the MICs (through Ï‡i ), the SCC (Âµ) and the time
it takes to achieve the sectoral potential Ti reads:
Â¯
ai = ai (Ti )
Ti
= Ï‡i ert (Ti âˆ’ t)Âµ dt
0
Let us dene f Ï‡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
18
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:
ai â†’ Ti = fÏ‡i âˆ’1 (Â¯i )
Â¯ a is an increasing function
For a given marginal cost function,
8 Ti can always be found from Â¯
ai . The larger the potential, the
longer it takes for the optimal strategy to achieve it.
Noting that MICs in sector i depend increasingly on Ti (16) nishes the demonstration.
D Details on the linear MAC model
Let us dene the inverted marginal abatement cost functions Ï†i :
q
âˆ€q â‰¥ 0, Ï†i (q) = m (33)
Î³i
They allow calculating the optimal strategy by inverting (4):
Ï†i (ert Âµ) t < Ti
ai (t) =
ai
Â¯ t â‰¥ Ti
The relationship between the sectoral potential, the carbon price and the decarbonization date is:
Âµ rTi
Â¯
ai = me (34)
Î³i
And then, the emissions from sector i are:
Ti Ti
Âµ Âµ
Ei = ai âˆ’ ai (t)dt =
Â¯ erTi âˆ’ ert m dt = rÎ³ m Ti e
rTi
+ erTi âˆ’ 1 (35)
0 0 Î³i i
This allows one to express the carbon constraint in function of the decarbonization dates:
B= Ei
i
Âµ rTi
B= m Ti e + erTi âˆ’ 1 (36)
i
rÎ³i
We then calculate the total mitigation costs Î“i in the sector i before date T given SCC Âµ.
T
Î“i = eâˆ’rt Î³i (ai (t))dt
0
Ti m 2
Î³i ert Âµ m
Î³i 1
= eâˆ’rt m dt + rÂ¯2 eâˆ’rTi âˆ’ eâˆ’rT
ai
0 2 Î³i 2 r
injecting this expression of Â¯
ai (34 and 33):
Âµ2 1 rTi Âµ2 1 rTi
Î“i = m r e âˆ’1 + m e âˆ’ er(2Ti âˆ’T )
2Î³i 2Î³i r
then
Âµ2
Î“i = m 2e
rTi
âˆ’ 1 âˆ’ er(2Ti âˆ’T ) (19)
2rÎ³i
8 When c is given, Ï‡i and therefore fÏ‡i are also given.
i
19
E Details on the linear MIC model
Investment costs are quadratic:
cm 2
i
âˆ€i, âˆ€x â‰¥ 0, ci (x) =x
2
âˆ€x â‰¥ 0, ci (x) = cm Â· x
i
Let us dene the inverted marginal investment cost functions Ï‡i = ciâˆ’1 :
p
âˆ€p â‰¥ 0, Ï‡i (p) = (37)
cm
i
With this specication, it is possible to write the optimal investment strategy xi :
ert (Ti âˆ’ t)Âµ
xi (t) = (38)
cm
i
then, to link the date Ti to the sectoral potential and the cost of carbon Âµ:
Âµ
ai =
Â¯ (erTi âˆ’ 1 âˆ’ rTi ) (39)
cm r 2
i
The emissions beyond E0 from sector i are:
Ti
Ei = ai âˆ’ ai (t)dt
Â¯
0
Ti Ti t
= xi (Î¸)dÎ¸ âˆ’ xi (Î¸)dÎ¸ dt
0 0 0
Ti Ti Ti Î¸
= xi (Î¸)dÎ¸ dt = xi (Î¸)dtdÎ¸
t=0 Î¸=t Î¸=0 t=0
Ti
= Î¸xi (Î¸)dÎ¸
0
Ti
Âµ
= Î¸(Ti âˆ’ Î¸)erÎ¸ dÎ¸ ( from eq. 38 )
cm 0
i
Âµ
= m 3 rTi erTi + 1 âˆ’ 2 erTi âˆ’ 1
ci r
And the carbon budget satises:
B= Ei
i
Âµ 1
B= rTi erTi + 1 âˆ’ 2 erTi âˆ’ 1 (40)
i
cm r3
i
We calculate the total mitigation costs Ci in sector i :
âˆž
Ci = eâˆ’rt ci (xi (t)) dt
0
Ti 2
cm ert (Ti âˆ’ t)Âµ
= eâˆ’rt i
dt
0 2 cm
i
Ti
Âµ2
= ert (Ti âˆ’ t)2 dt
2cm
i 0
Âµ2 1
Ci = m 3 erÂ·Ti âˆ’ 1 âˆ’ r Â· Ti âˆ’ (r Â· Ti )2 (41)
ci r 2
20