On the (Ir)Relevance of Discount Factors for Future Allocations of Scarce Resources

This article is interested in future allocation of scarce resources in an environment where upper bounds and lower bounds are fixed on the stream of consumption or extractions of the scarce resource. It is shown that we can compute the optimal planning of consumptions independently from an explicit sequence of discounting factors as soon as they are decreasing at a rate smaller than a bound linked to the concavity of the utility function and the choice of the sequences of lower and upper bounds. The optimal solution is unique and exhibits two regimes with a pivotal period in the middle. Therefore, one gets plans satisfying some kind of intergenerational fairness: while the highest effort is supported by the first generations, it then decreases for the remaining ones. The argument is then extended to partially renewable resources. Finally, we consider the role of the horizon and of a potential regret after a revision for the bounds