Cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games: Existence theorems for a class of optimal allocation problems

This paper investigates the problem concerning the existence of a solution to a diverse class of optimal allocation problems which include models of cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games. A new formulation that encompasses all these diverse models is provided. Examples of these models for which a solution does not exist and the causes of the non-existence are studied. Two theorems are provided to tackle the existence problem under different conditions. Several analytical examples with a closed-form solution are offered to illustrate the usefulness of the existence theorems.