lated at very low speed is free (lim _xi,t_ →0; _A_ =0 _c_<sup>′</sup> _i_<sup>(</sup><sup>_xi,t_)= 0),andthecostofnew</sup> capacity doubles when the investment pace doubles. For intermediate value _A_ ∈(0 _,_ 1), new capacity is always costly, and the marginal cost of new capacity increases with the investment pace. We first assume quadratic costs (that is linear marginal costs and _A_ = 0), and perform later a sensitivity anaysis on _A_ .

The demand is assumed inelastic in the short-term, and follows an exogenous long-term growth. Its value in 2008 is calibrated as the reference fossil energy production (from coal, oil and gas), that is _D_ 0 = 1 940 TWh/yr (ENERDATA, 2012). The central scenario of the Roadmap forecasts a 700 TWh/yr increase of electricity consumption between 2008 and 2050, that we model as a linear growth of _G_ = 16 _._ 5 TWh/yr<sup>2</sup> . Demand after time _t_ is thus: Acording to Trotignon and Delbosc (2008), emission allowances allocated to the power sector in 2008 amounted to _E_ ref = 1 _._ 03 GtCO2/yr , leading to a reference emission rate of 530 tCO2/GWh. A linear decrease of these emissions until 2050, as planned in the Roadmap, yields a carbon budget of _M_<sup> ̄</sup> = 22 GtCO2. A sensitivity analysis on _M_<sup> ̄</sup> is performed later. We use _r_ = 5 %/yr for the social discount rate.

# _4.2. Results_

Fig. 3 shows production, investment, the investment cost, the electricity price and the operational LCOE obtained in the numerical application. The following lists several interesting results.

The social planner does not invest in the legacy capacity; moreover, existing legacy plants start being decommissioned as soon as the climate policy is implemented (Fig. 3a). In the central scenario, legacy fossil-fueled plants are entirely phased out in 2028. With our technology assumptions, the carbon budget is too loose to justify a complete transition by 2050, but instead phases out all fossil production by 2064.

Investment in renewable starts at the beginning of the transition, at the same time as gas and before coal or gas are phased out (Fig. 3c). It grows over time, and only decreases for a short period from 2055 to 2064 when gas is phased out, as most of the power plants have already been replaced (Fig. 3a). Investment in renewable starts at 22 GW/yr in 2008, more than twice the actual average investment rate observed between 2009 and 2011, and reaches 70 GW/yr in 2060. Investment grows again after 2064 to maintain the installed renewable capacity and meet the growing demand.

Gas plants operate as a temporary bridge between coal and renewable. As seen in Fig. 3c, investment in gas starts since the beginning of the transition in 2008, lowering the need for investment in renewable. In 2035, however, investment in gas stops and gas capacities start being underused, allowing gas to be replaced faster by renewable power. The sensitivity analysis (Fig. 4d) shows that it is optimal to build gas power plants transiently for any positive carbon budget ranging up to 45 GtCO2: investment always starts in 2008 and, with quadratic investment costs, plants are underused immediately after investment stops. Only carbon budgets lower than 17 GtCO2 make gas investment stop before 2020 (for comparison, the Roadmap implicitly sets a budget of 22 GtCO2).

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