Economic Growth and Pollution Diffusion

As an additional application of the methods presented in this volume, in this chapter, we introduce an environmental issue concerning the control of the pollution produced by human activities. The proposed model consists of a spatially structured dynamic economic growth model that takes into account the level of pollution induced by production and a possible taxation based on the amount of produced pollution. Then we analyze an optimal harvesting control problem with an objective function composed of three terms, namely the intertemporal utility of the decision maker, the space-time average of the level of pollution in the habitat, and the disutility due to the imposition of taxation. The system is subject to two controls, namely the level of consumption and, in addition, the taxation on physical capital rate. The optimal controls are determined by means of a sweep-type numerical algorithm that allows to solve a system of four backward–forward reaction–diffusion equations. Open problems are proposed at the end of the chapter.