Climate change is calling for an immediate public action in order to provide incentives for carbon abatement. Prior to the design the instruments to implement, the computation of the shadow price of carbon is a measure of the total effort to make whether through taxes, pollution permits or other regulations. Abatement is mainly implemented through abatement capital accumulation. It implies that abatement takes times and this should be accounted for in the computation of the shadow price of carbon. In addition, due to uncertainties surrounding damages from climate change (see Pindyck, 2017), the cost-benefit approach to carbon pricing tend to be replaced a with cost-efficiency approach, however shifting the uncertainty problem to the concentration target. Using a theoretical model calibrated for France, we study the consequences for the shadow price of carbon to account for abatement capital accumulation and target uncertainty. One of the main lessons of this theoretical model is that the realistically calibrated optimization model leads to an emission trajectory very close to the linear trajectory. A second result concerns the optimal date of carbon neutrality between 2045 and 2053, regardless of the type of abatement technology and parameter values (including the discount rate or the size of the carbon budget) considered. We also obtain a non-monotonic path for the marginal value of the abatement capacity, while the carbon value follows a Hotelling rule. However, even if the rate of growth of the price of carbon only depends on the discount rate and the hazard rate when the target is uncertain, its initial value is affected by the characteristics of the investment technology.