The spatial AK model and the Pontryagin maximum principle

We are concerned with the endogenous growth model, namely the spatial AK model, that has recently been proposed and analyzed by Boucekkine et al. (2013a,b). From the mathematical standpoint, this model consists of an infinite-horizon parabolic optimal control problem, which is excellently solved in Boucekkine et al. (2013b) by means of dynamic programming. Nevertheless, one of the main aims of Boucekkine et al. (2013a,b) is also to show that the spatial AK model cannot be dealt with using the maximum principle of Pontryagin. More precisely, according to the analysis carried out by Boucekkine, Camacho and Fabbri, the Pontryagin conditions, albeit necessary, would not allow one to determine the unique solution of the optimal control problem. In the present paper, we show that such a conclusion needs to be reconsidered. In particular, if a Michel-type transversality condition is imposed and the fact that the adjoint variable must be non-negative is taken into account, the maximum principle is capable of yielding the unique solution of the spatial AK model.