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