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 In this case, abatement eorts are transitional, once all the capital in a sector has been replaced by non-emiting capital, nancial eorts are not required anymore.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.

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