Pollution control for a spatially structured economic growth system

TL;DR

The paper addresses controlling pollution in a spatially distributed economy by embedding diffusion-based pollution dynamics into a growth model and solving a regional optimal control problem. It develops a gradient-based method using forward-state and adjoint equations, combined with finite element spatial discretization and semi-implicit time stepping. Numerical experiments compare different intervention-region sizes and show how larger, region-targeted controls can sustain higher capital and reduce pollution, while the associated policy costs shape the optimal controls. The framework provides policy-relevant insights for region-specific taxation and abatement design in spatial economies and suggests directions for extensions to broader production technologies and regional layouts.

Abstract

In this paper investigations by the same authors on environmental issues concerning the control of the pollution produced by human activities have been extended to include costs related to environmental interventions. The proposed model consists of a spatially structured dynamic economic growth model which takes into account the level of pollution induced by production, a possible taxation based on the amount of produced pollution, and possible environmental interventions. It has been analyzed an optimal harvesting control problem with an objective function composed of four terms, namely the intertemporal utility of the decision maker, the space-time average of the level of pollution in the habitat, the disutility due to the imposition of taxation and the cost of environmental interventions. A specific novelty in the model proposed here is the localization of the possible interventions to a subregion of the whole habitat. Computational experiments have been carried out to exemplify the outcomes of the proposed model.

Figures (9)

• Figure 1: Initial distributions (at $t=0$) of state variables $k(x,t)$ (left) and $p(x,t)$ (right).
• Figure 2: Control regions $\omega$: small ( case 1, left) and big ( case 2, right).
• Figure 3: Case 0, no control. Space distributions of state variables $k(x,t)$ (first row) and $p(x,t)$ (second row) at three time instants.
• Figure 4: Case 1, small control region. Space distributions of the optimal state variables $k(x, t)$ (first row) and $p(x, t)$ (second row) at three time instants.
• Figure 5: Case 1, small control region. Space distributions of the optimal control variables $c(x, t)$ (first row), $\tau (x, t)$ (second row) and $\xi(x, t)$ (third row) at three time instants.