This paper investigates the optimal timing of greenhouse gas abatement efforts in a multi-sectoral model with economic inertia, each sector having a limited abatement potential. It defines economic inertia as the conjunction of technical inertia --- a social planner chooses investment on persistent abating activities, as opposed to choosing abatement at each time period independently --- and increasing marginal investment costs in abating activities. It shows that in the presence of economic inertia, optimal abatement efforts (in dollars per ton) are bell-shaped and trigger a transition toward a low-carbon economy. The authors prove that optimal marginal abatement costs should differ across sectors: they depend on the global carbon price, but also on sector-specific shadow costs of the sectoral abatement potential. The paper discusses the impact of the convexity of abatement investment costs: more rigid sectors are represented with more convex cost functions and should invest more in early abatement. The conclusion is that overlapping mitigation policies should not be discarded based on the argument that they set different marginal costs (''different carbon prices'') in different sectors.