a1 Catholic University of Louvain and University of Glasgow
a2 Catholic University of Louvain
a3 University of Luxembourg
We study a Ramsey problem in infinite and continuous time and space. The problem is discounted both temporally and spatially. Capital flows to locations with higher marginal return. We show that the problem amounts to optimal control of parabolic partial differential equations (PDEs). We rely on the existing related mathematical literature to derive the Pontryagin conditions. Using explicit representations of the solutions to the PDEs, we first show that the resulting dynamic system gives rise to an ill-posed problem in the sense of Hadamard. We then turn to the spatial Ramsey problem with linear utility. The obtained properties are significantly different from those of the nonspatial linear Ramsey model due to the spatial dynamics induced by capital mobility.
We are grateful to Cuong Le Van, Omar Licandro, Dominique Peeters, Jacques Thisse and Vladimir Veliov for stimulating comments. We acknowledge the financial support of the Belgian French-speaking community (Grant ARC 03/08-302) and of the Belgian Federal Government (Grant PAI P5/10)). The third author also acknowledges financial support from the German Science Foundation (DFG, grant GRK1134/1).