Optimal location of economic activity and population density: The role of the social welfare function

In this paper, we consider a spatiotemporal growth model where a social planner chooses the optimal location of economic activity across space by maximization of a spatiotemporal utilitarian social welfare function. Space and time are continuous, and capital law of motion is a parabolic partial differential diffusion equation. The production function is AK. We generalize previous work by considering a continuum of social welfare functions ranging from Benthamite to Millian functions. Using a dynamic programming method in infinite dimension, we can identify a closed-form solution to the induced HJB equation in infinite dimension and recover the optimal control for the original spatiotemporal optimal control problem. Optimal stationary spatial distributions are also obtained analytically. We prove that the Benthamite case is the unique case for which the optimal stationary 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.