The user cost of natural resources and the optimal exploitation of two non-renewable polluting resources

We study the optimal extraction of two non-renewable resources when extraction costs depend on cumulative previous extraction. We first define a complete user cost of natural resources, including environmental damages, which allows us to greatly simplify the resolution. This framework is applied to a study of oil and coal optimal extraction. The extraction cost of oil is initially lower than the one of coal but increases more rapidly. Coal leads to higher emissions. In a business as usual scenario the optimal path is to use first only oil, before using the two resources and relying a lot on cheaper coal, until the backstop becomes profitable. When the carbon price is taken into account, the optimal path relies much less on the more polluting coal, and it may be optimal to revert to an oil only trajectory.