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

This article is interested in future allocations of scarce resources in an environment where upper bounds and lower bounds are fixed on the stream of consumptions or extractions of the scarce resource. It is shown that the optimal planning of consumptions does not depend on the 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: the upper bounds prevent the first generations from consuming too much of the resource to the detriment of future generations and the lower bounds ensure minimal consumption for these generations. We also consider the role of the horizon and of a potential regret after a revision for the bounds. The argument is then extended to partially renewable resources.