|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.

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