Non-renewable input and waste stock: optimal transition towards circularity

Natural resource stocks are drastically shrinking and the contemporary debate on depletion of reserves highlights their crucial role over time. This paper deals with the management of exhaustible resources when recycling and extraction are employed, as well as the short-term effects of the economic transition towards circularity on consumption and welfare. Mathematical modeling may support effective plans for maximizing social welfare and protecting the environment. Concerning the production of a certain good, an optimal control model is studied to allocate labor between mining and recycling over a finite time horizon. A suitable scrap value function allows for reducing waste and maintaining natural stock in the forthcoming future. Well-posedness of the problem is proved and some qualitative features of the optimal solution are stated.