A non-invariance result for the spatial AK model

This paper deals with the positivity condition of an infinite-dimensional evolutionary equation, associated with a control problem for the optimal consumption over space. We consider a spatial growth model for capital, with production generating endogenous growth and technology of the form AK. We show that for certain initial data, even in the case of heterogeneous spatial distribution of technology and population, the solution to an auxiliary control problem that is commonly used as a candidate for the original problem is not admissible. In particular, we show that initial conditions that are non-negative, under the auxiliary optimal consumption strategy, may lead to negative capital allocations over time.