Optimal Use of a Polluting Non-Renewable Resource Generating both Manageable and Catastrophic Damages

We consider a model with two energy sources, a non-renewable one, cheap but polluting, and a renewable one, expensive but clean, let's say coal and solar. The aim of environmental policy is to maintain atmospheric carbon concentration under a given ceiling, chosen to prevent an excessive rise of the temperature and catastrophic damages. Before the ceiling damages exist but remain small, hence manageable. We show first that costs matter a lot. Whatever abundant or rare, if solar is more expensive than coal at the ceiling, it will never be used before the end of the phase at the ceiling, when coal becomes so scarce that the ceiling will never be reached again. On the contrary, if solar is less expensive than coal at the ceiling, it may even be sufficiently cheap to be exploited before the ceiling, in which case first coal is exploited alone, next both resources are used together before, at and after the ceiling, and finally solar is exploited alone, after the exhaustion of coal. Second, the carbon shadow value is first increasing until the ceiling is reached, next decreasing during the phase at the ceiling, and finally is stabilized at a constant level after the ceiling. The initial carbon shadow cost is an increasing function of the value of the marginal damage, and a decreasing function of the ceiling. Lastly, contrary to intuition, higher marginal damages and/or a lower ceiling induce a delay in the penetration of solar and also a delay in the transition towards a totally clean energy.