Turnpikes in multi-discount rate environments and GCC policy evaluation

This note discusses the asymptotic behavior of optimal trajectories in infinite horizon control problems when the objective function is a compromise between two averaged discounted sums of the same stream of rewards. We show that the turnpike associated with the lowest discount rate is the one that will prevail. We also show that, when one discount rate is equal to 0, the problem of the best compromise between a discounted sum and the long term average reward may not admit a solution. We then discuss the implications of these findings for the definition of global climate change mitigation policies.