Spatial Growth Theory: Optimality and Spatial Heterogeneity

Spatiotemporal dynamics are introduced in a standard Ramsey model of optimal growth in which capital moves towards locations where the marginal productivity of capita is relatively higher. We extend Pontryagin's maximum principle to account for transition dynamics governed by a nonlinear partial differential equation emerging for spatial capital flows. The potential spatial heterogeneity of optimal growth as seen from the point of view of an optimizing social planner is examined. Our results suggest that for high utility discount rate the spatial capital flows induce the emergence of optimal spatial patterns while hor low utility discount a flat-earth steady state is socially optimal. Furthermore, when spatial heterogeneities exist due to total factor productivity differences across locations, we identify conditions under which the spatial capital flows could intensify or weaken spatial inequalities.