Control theory in infinite dimension for the optimal location of economic activity: The role of the social welfare function

In this paper we deal with a family of optimal control prob- lems in infinite dimension with state constraints. We approach such problems with the dynamic programming approach identifying (in cases not yet known in the literature) a closed-form solution v of the asso- ciated Hamilton-Jacobi-Bellman (HJB) equation, which is a PDE in a suitable Hilbert space. Consequently we are able prove a verification theorem, to show that v is indeed the value function, and to provide the optimal control in closed-loop form. The abstract problem is motivated by an economic application in the context of continuous spatiotemporal growth models with capital diffusion, where a social planner chooses the optimal location of economic activity across space by maximization of an utilitarian social welfare function. From the economic point of view, we generalize previous works by considering a continuum of social welfare functions ranging from Benthamite to Millian functions. We prove that the Benthamite case is the unique case for which the optimal station- ary detrended consumption spatial distribution is uniform. Interestingly enough, we also find that as the social welfare function gets closer to the Millian case, the optimal spatiotemporal dynamics amplify the typical neoclassical dilution population size effect, even in the long-run.