metadata: authors: 0: author: Hallegatte,Stephane
1: author: Meunier,Lola Guylaine Dany J C L S P P
2: author: Vogt-Schilb,Adrien Camille
count: World
docna: 0: docna: How inertia and limited potentials affect
            the timing of sectoral abatements in optimal climate policy
docty: Policy Research Working Paper
subtopic: Climate Change Mitigation and Green House Gases,Science of Climate Change,Climate Change and Health,Climate Change and Environment,Rural and Renewable Energy,Energy Demand,Energy and Environment,Energy and Mining,Renewable Energy,Global Environment
teratopic: Environment,Transport,Macroeconomics and Economic Growth,Energy
abstracts: cdata!: This paper investigates the optimal
            timing of greenhouse gas abatement efforts in a
            multi-sectoral model with economic inertia, each sector
            having a limited abatement potential. It defines 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 period
            independently -- and increasing marginal investment costs in
            abating activities. It shows that in the presence of
            economic inertia, optimal abatement efforts (in dollars per
            ton) are bell-shaped and trigger a transition toward a
            low-carbon economy. The authors prove that optimal marginal
            abatement costs should differ across sectors: they depend on
            the global carbon price, but also on sector-specific shadow
            costs of the sectoral abatement potential. The paper
            discusses the impact of the convexity of abatement
            investment costs: more rigid sectors are represented with
            more convex cost functions and should invest more in early
            abatement. The conclusion is that overlapping mitigation
            policies should not be discarded based on the argument that
            they set different marginal costs (`"different carbon
            prices"') in different sectors.
repnme: repnme: How inertia and limited potentials affect
            the timing of sectoral abatements in optimal climate policy
display_title: How inertia and limited potentials
            affect the timing of sectoral abatements in optimal climate policy
listing_relative_url: /research/2012/08/16563648/inertia-limited-potentials-affect-timing-sectoral-abatements-optimal-climate-policy
topicv3: Renewable Energy m1326170 525,Energy and Extractives m1326595 507,Climate Change m1326829 567,Climate Change Mitigation m1327079 579,Environment, Natural Resources and Blue
            Economy m1327391 566
publishtoextweb_dt: 2012-08-01T00:00:00Z
historic_topic: Environment,Transport,Macroeconomics and Economic Growth,Energy
no_of_pages: 22
author_keys: 0: author_key: 1103816
1: author_key: 2132856
2: author_key: 1272850
model_extractions: 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 Stéphane 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 gas abatement efforts in a multi-sectoral model with economic inertia, each sector having a limited abatement potential. It defines 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 period independently—and increasing marginal investment costs in abating activities. It shows that in the presence of economic inertia, optimal abatement efforts (in dollars per ton) are bell-shaped and trigger a transition toward a low-carbon economy. The authors prove that optimal marginal abatement costs should differ across sectors: they depend on the global carbon price, but also on sector-specific shadow costs of the sectoral abatement potential. The paper discusses the impact of the convexity of abatement investment costs: more rigid sectors are represented with more convex cost functions and should invest more in early abatement. The conclusion is that overlapping mitigation policies should not be discarded based on the argument that they set different marginal costs (“different carbon prices”) in 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.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
How inertia and limited potentials aect the timing of sectoral abatements in optimal climate policy Adrien Vogt-Schilb<sup>_a,†_</sup> , Guy Meunier<sup>_b_</sup> , Stéphane Hallegatte<sup>_c,d_</sup> <mark>August 1, 2012</mark>

> _†_ Corresponding author (vogt@centre-cired.fr)

> _a_ CIRED, Nogent-sur-Marne, France

> _b_ INRAUR1303 ALISS, Ivry-Sur-Seine, France

- _c_ The World Bank, Sustainable Development Network, Washington D.C., USA _d_ École Nationale de la Météorologie, Météo-France, Toulouse, France

Keywords: climate change mitigation, sectoral policies, optimal policies, optimal timing, inertia, when-exibility, how-exibility, overlapping policies.

JEL classication: 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 in global mean temperature to 2<sup>_◦_</sup> C. Translating these objectives into policies is dicult. First, a given temperature objective can be reached with dierent 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 ecient 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 eciently 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 twinmarket failure in the area of eco-innovation (Jae 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 diculty in the design of climate policies: the role of economic inertia. Many papers investigating the optimal abatement pathway (i.e. how to benet 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 benets (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 eects. 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; Grübler and Messner, 1998; Goulder and Mathai, 2000).<sup>1</sup> 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 eort 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 eorts 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 aected 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 eect implies that, at each point in time, the marginal costs of abatement investments should be dierent 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.

> 1An 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 eorts 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 eorts are justied in highpotential and highly inert sectors such as urban planning (Guivarch and Hallegatte, 2011).

In section 2, we briey introduce the classical approach to the denition 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 dierent model, where abatement options have a permanent impact on emissions. We show that, in this case, marginal abatement eorts 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 _γ_ ( _a_ ( _t_ )) , where _γ_ is convex, positive and twice dierentiable. 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 eort _ai_ ( _t_ ) done in each sector, using the abatement cost function _γi_ ( _ai_ ( _t_ )) which is also convex, positive and twice dierentiable:<sup>2</sup> The function _γi_<sup>_′_</sup> ( _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-ecient vehicles typically have higher investment cost but lower operating costs than carbonintensive 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 _γi_<sup>_′_(</sup><sup>_ai_(</sup><sup>_t_))is growing in</sup><sup>_ai_(</sup><sup>_t_),and</sup> _γi_ ( _ai_ ( _t_ )) is convex. In economic terms, the convexity on _γi_ comes from zero-carbon vehicles being imperfect substitutes to thermal vehicles.

Moreover, we introduce a sectoral potential _a_ ̄ _i_ in each sector _i_ , which represents the maximum amount of GHG emissions (in GtCO 2 per year) that can be abated in this sector.

## _∀i, ∀t, ai_ ( _t_ ) _≤ a_  ̄ _i_

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 _E_ 0 _≥_ 0 . Above _E_ 0 , 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

> 2All notations are reproduced in Tab. 1.

3
<!-- Start of picture text --> ∑ ai(t)<br>Eabat<br>B<br>E0<br>t time<br>/yrGtCO2<br>Emissions<br><!-- End of picture text --> Figure 1: An illustration of the climate constraint. The cumulative emissions above _E_ 0 should be lower than an intertemporal carbon budget _B_ (this requires that the long-run emissions tend to _E_ 0 ). Dangerous emissions _Eabat_ are measured from _E_ 0 .

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 _E_ 0 , this reads: Using another representation of climate policy objectives would not aect 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: 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 _μ_ . As its current value _μe_<sup>_rt_</sup> increases at the discounting rate _r_ , abatement eorts 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: These results (proved in annex A) are a straightforward application of the equimarginal principle: unless the sectoral potentials are binding, optimal marginal eorts are equal, in all sectors, to a unique carbon price.

# 3 Introducing economic inertia leads to dierent 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 gasred plants actually reduces annual emissions over a long period of time, while abatement costs 4
|Name|Description<br>Unit|
|---|---|
|_μ_|Social cost of carbon (SCC)<br>$/tCO2<br>|
|_r_|Discount rate<br>yr<sup>_−_1</sup>|
|_ai_(_t_)|Current abatement in sector _i_<br>tCO2_/_yr|
| ̄_ai_|Sectoral potential in sector _i_<br>tCO2_/_yr|
|_γi_|Abatement cost in sector _i_<br>$/yr|
|_γ_<sup>_′_</sup><br>_i_|Marginal abatement cost (MAC) in sector _i_<br>$/tCO2|
|_xi_(_t_)|Current investment in abating activities in sector _i_<br>(tCO2/yr)/yr|
|_ci_|Cost of investment in sector _i_<br>$/yr|
|_c_<sup>_′_</sup><br>_i_|Marginal investment cost (MIC) in sector _i_<br>$/(tCO2/yr)|
|_rc_<sup>_′_</sup><br>_i_|Annualized MICs in sector _i_<br>$/tCO2|
|_λi_(_t_)|Social cost of the sectoral potential (SCSP) in sector _i_<br>$/tCO2|

Table 1: Notations and units are mainly paid when the decision is made. In the building sector, reductions are achieved in part by retrotting buildings, which has an almost-permanent eect on emissions. Another example is urban development, which impacts emissions on the long term (Strand, 2011; Viguié and Hallegatte, 2012). We call this the technical inertia: abatement at time _t_ cannot be controlled directly; instead, it depends on the eorts 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_ : Where dotted variables represent temporal derivatives. These abatement investments are realized at a cost _ci_ ( _xi_ ) , the functions _ci_ are twice dierentiable, positive, increasing and convex: The values _ci_<sup>_′_</sup> ( _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 retrotted at date _t_ . The abatement _ai_ ( _t_ ) would be proportional to the share of retrotted buildings in the stock. In a given time lapse, retrotting 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 retrotting buildings _c_<sup>_′_</sup> _i_<sup>(</sup><sup>_xi_)isagrowingfunctionof</sup><sup>_xi_inotherwords,</sup><sup>_ci_isconvex.Inaddition,</sup> 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 justies 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 dierent nature than the convexity in _γi_ ( _ai_ ( _t_ )) in the classical approach presented in section 2, where convexity arises from heterogeneity in abatement options (e.g., dierent 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
<!-- Start of picture text --> : a[: (Ax(0) — Ao(8))d8<br>a ra Ty<br><!-- End of picture text --> i> r(o= ) J x [x

[xX f&

|
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 dierent across sectors. It represents the total current value ( _e_<sup>_rt_</sup> ) of the carbon that will be saved from _t_ onwards thanks to the marginal investment, minus the sector-specic total SCSP from _t_ onwards. The SCSP _λi_ ( _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 ( _λi_ ( _θ_ ) = _μ_ ).

Noting _Ti_ the dates when the sectoral potentials are reached,<sup>3</sup> we can express explicitly the optimal marginal investment costs in each sector as a function of the optimal SCC _μ_ and the optimal dates _Ti_ (see annex B): 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 _μe_<sup>_rt_</sup> , 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 benet 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 eect of discounting dominates and eorts grow exponentially; in the long term, the eect of the limited potential dominates and the eort decrease to zero.

A corollary of (15) is that dierences in MICs come from dierent social costs of the sectoral potentials (Fig. 2): 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: The second sector will take longer to decarbonize: 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.

The proof is in annex C. Even if the cost function is the same in both sectors, dierent sectoral potentials cause the investment along the optimal path to be dierent 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.

> 3After _Ti_ , _xi_ ( _t_ ) = 0 and _ci′_ ( _xi_ ( _t_ )) = 0 .

7
||Marginal cost|Abat|ement potential|
|---|---|---|---|
||_γ_<sup>_′_</sup><br>_i_<sup>(1 GtCO2/yr)</sup><br>[ $/tCO2]| ̄_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 $/tCO 2 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,<sup>4</sup> 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_ =<sup>�</sup> _i_<sup>_a_ ̄</sup><sup>_i_=24GtCO2/yr.</sup> Compared to the total baseline emissions in 2030 in the A1B scenario from Nakicenovic et al. (2000), 69 GtCO 2 /yr, this gives _E_ 0 = 45 GtCO 2 /yr. In this case, _E_ 0 cannot be interpreted as a denitive 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: The two models are solved with a numerical resolution, using a time step _dt_ = 0 _._ 1 yr.<sup>5</sup>

## 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, innite-horizon models exposed in sections 2 and 3 have to be modied; all the results exposed still apply.

> 5Data 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|
|---|---|---|---|
||<sup>_m_</sup><br><br>|$/tCO2<br><br>�|_c_<sup>_m_</sup><br><br>�<br>$/tCO2<br><br>�|
||_γ_<br>_i_<br>|GtCO2_/yr_<br>|_i_<br><br>GtCO2/yr<sup>3</sup><br>|
|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, dierences 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 specication that grants that the _γi_ are convex, and simplies the resolution as marginal abatement costs are linear: Where _γi_<sup>_m_</sup> are parameters specic to each sector. We calibrate these costs functions using the abatements corresponding to a 20 $/tCO 2 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 ( _γi_<sup>_′_(</sup><sup>_ai_(</sup><sup>_t_)))and</sup> physical ( _ai_ ( _t_ ) ) terms (see (34) and (36) in annex D and Fig. 3).

We call Γ the discounted cost of the optimal mitigation strategy: The numerical resolution gives Γ = 3351 G$.

Then, we solve the model with economic inertia, still assuming quadratic costs.

This allows one to simplify the model (see (38) and (41) in annex E). We calibrate the _c_<sup>_m_</sup> _i_<sup>parameters</sup> by adapting the methodology from Grubb et al. (1995): rst, we assume that the relative MICs are equal to the relative MACs from IPCC: And second, we constrain the total costs of the mitigation strategy to be equal in the two models: A numerical resolution provides _c_<sup>_m_</sup> _i_<sup>(Tab. 3), as well as the optimal abating strategy</sup><sup>_xi_and</sup><sup>_ai_(Fig. 3)</sup> 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. Investing _xi_ ( _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 · ci_ ( _xi_ ( _t_ )) and its marginal value. Indeed, paying the 9
amount _ci_<sup>_′_</sup> ( _xi_ ( _t_ )) at time _t_ is equivalent to paying the ow _r · ci_<sup>_′_</sup> ( _xi_ ( _t_ )) from _t_ to _∞_ . Multiplying all terms in equation (16) by _r_ , one gets: Annualized MICs are expressed in $/tCO 2 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 dier from the MACs à 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 eort 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 dier 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 signicant 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 eort, 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 eorts are transitional, once all the capital in a sector has been replaced by non-emiting capital, nancial eorts are not required anymore. While the carbon price grows exponentially in both model (see the discussion after equation (12)), the aggregated nancial eorts 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 modies the optimal sectoral burden sharing. Without inertia, all sectors are decarbonized up to the same marginal eort 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 benet 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 · ci_<sup>_′_</sup> ( _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 eect of dierent 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
<!-- Start of picture text --> Classical model<br><!-- End of picture text --> <!-- Start of picture text --> Model with economic inertia<br><!-- End of picture text --> 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 eorts 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 longerto-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 retrot 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, dierent convexities bearing on the investment cost function allow to take into account dierent 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 inuence of cost convexity on the results: Where _αb >_ 1 describes the convexity of the costs function. In economic terms, _αb_ is the constant cost elasticity of abating investments in sector _b_ : Finally, _x_<sup>_ref_</sup> _b_ (in the same unit than _xb_ ) is a scale parameter. For each value of _αb_ , we calibrate _x_<sup>_ref_</sup> _b_ ( _αb_ ) such that the total mitigation cost of the building sector remains unchanged.<sup>6</sup> We investigate the impact of _αb_ on the optimal timing of the mitigation strategy. First, we compare the optimal strategies in a reference case (with _αb_ = 2 , replicating the case of quadratic costs functions) and a case with higher convexity _αb_ = 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 _αb_ (Fig. 6). When _αb_ tends to 1, the MIC _c_<sup>_′_</sup> _b_<sup>doesnotdependonthedecarbonizingpace</sup><sup>_xb_anymore,</sup> and there is no inertia. The building sector is fully decarbonized instantaneously at the beginning of the period. On the other hand, the greater _αb_ the longer the building sector takes to reach its abating potential.

These results conrm that dierent economic inertias can be modeled through dierent 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 _αb_ 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.<sup>7</sup> 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. VogtSchilb 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 conrms this need in an analytical framework.

# 6 Discussion and conclusion

In this paper, we modify a classical model à la Nordhaus (1992) and introduce economic inertia bearing on GHG mitigation activities. We dene 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 eorts 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

> 6We could have decided to keep _x_<sup>_ref_</sup> _b_ constant instead, without modifying the qualitative results and the message from the experiment. But this would not have granted that investment costs are held constant except in the particular case when _xb_ = _x_<sup>_ref_</sup> _b_ .

> 7 _Tb_ is less sensible to _αb_ if we hold _x_<sup>_ref_</sup> _b_ constant instead of the total cost of decarbonizing buildings (see also footnote 6).

12
Figure 4: Comparison of nancial and physical eorts 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 eorts are higher than when inertia is neglected. By construction, the total discounted cost and the cumulative abatements are equal in the two cases.

<!-- Start of picture text --> Low convexity αb = 2 High convexity αb = 2 . 3<br><!-- End of picture text --> Figure 5: Increasing the convexity of the building sector cost function has a direct and an indirect eect. 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 _αBuildings_ . 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.

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results obtained with our new model and those coming from a classical model without economic inertia.

These two models illustrate the dierence 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 simplied 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 dierent, low-carbon development model. In this case, abatement eorts 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., retrot 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-specic social cost of the sectoral potential.

The dierence 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 prole of an abating strategy.

The model with economic inertia shows that optimal marginal eorts 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 specic policies such as norms and regulations (e.g., feed-in taris, eciency standards or land-use planning). This result provides an analytical framework to support previous ndings that larger eorts 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 _×_ 20 objectives of the European Union, existing standards on automobile, buildings and household appliance, and feed-in taris for specic low carbon technologies in the power sector. Such overlapping policies have been criticized, partly because they set dierent marginal costs in dierent economic sectors. Previous research has shown that dierences in marginal abatement costs can be justied, 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 eort levels are justied.

# Acknowledgments

The authors wish to thank Marianne Fay, Frédérique Ghersi, Louis-Gaëtan Giraudet, Céline 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 reect the views of the World Bank, its executive directors, or the countries they represent.

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16
J J > YC =)) J > J > VC =) jse J > (/ / i/ J> J > JYC> =)J > )
The rst order conditions read: The abatement _ai_ ( _t_ ) are increasing over time (as _a_ ̇ _i_ ( _t_ ) = _xi_ ( _t_ ) _≥_ 0 ). For each sector _i_ there is a date _Ti_ such that The Lagrange multipliers _λi_ ( _t_ ) are null before _Ti_ and strictly positive after. So, by integrating (13): Furthermore, for _t > Ti_ the abatement _ai_ ( _t_ ) is constant, equal to _a_ ̄ _i_ , so _xi_ ( _t_ ) = _a_ ̇ _i_ ( _t_ ) is null, and, from (14) and (7): Therefore, Then, from (14), the marginal costs can be expressed as a function of _Ti_ : Marginal investment costs should be equal to the total social cost of the carbon ( _μ_ ) that will be saved thanks to the abatement before the sectoral potential is reached (the time span ( _Ti − t_ ) ), as expressed in current value ( _e_<sup>_rt_</sup> ).

# C Proof of corollary 1

Proof: As _ci_<sup>_′_</sup> is by assumption (7) strictly growing, it is invertible. Let _χi_ be the inverse of _ci_<sup>_′_</sup> : Applying _χi_ to (16) gives: The relation between the sectoral potential ( _a_ ̄ _i_ ), the MICs (through _χi_ ), the SCC ( _μ_ ) and the time it takes to achieve the sectoral potential _Ti_ reads: Let us dene _fχi_ such that: 18
/ » dL ~~—~~ ( flo ~~i=t~~ )

| ~~fy ot —~~ t 1 ~~—-| —|~~ |
/ J | <u>/</u> ~~—~~ / ~~t~~ C )

» ~~Ef~~ 6) / f ~~-(——) —/ —(~~ <u>-</u> <u>-</u> CY CD )