Spatial dynamics and convergence: The spatial AK model

We study the optimal dynamics of an AK economy where population is uniformly distributed along the unit circle. Locations only differ in initial capital endowments. Spatio-temporal capital dynamics are described by a parabolic partial differential equation. The application of the maximum principle leads to necessary but non-sufficient first-order conditions. Thanks to the linearity of the production technology and the special spatial setting considered, the value function of the problem is found explicitly, and the (unique) optimal control is identified in feedback form. Despite constant returns to capital, we prove that the spatio-temporal dynamics, induced by the willingness of the planner to give the same (detrended) consumption over space and time, lead to convergence in the level of capital across locations in the long-run.