We study backstop adoption and carbon dioxide emission paths in a two-region model with unilateral climate policy and non-renewable resource consumption. The regions have an equal endowment of the internationally tradable resource and a backstop technology. We first study the case of a unilateral stock constraint (e.g. a 450 ppmv carbon dioxide concentration target), and show that the non-abating region makes the final switch to the backstop before the abating region does, though the latter region has two disjoint phases of backstop use if its marginal cost is sufficiently low. Furthermore, we show that the abating region has an inverse N-shaped emission path, with growing emissions in the period for which the ceiling is binding. In addition, there is a phase in which this region has a positive carbon price, but higher emissions than the non-abating region. With a global intertemporal carbon budget instead of a stock constraint, the order of definite backstop adoption is reversed and the abating region's emissions are always lower. We also show that unilateral climate policy does not lead to international carbon leakage.