In this chapter, we study the theory of optimal pollution management in environmental economics. We consider a society consuming some good, which generates pollution as a byproduct of this consumption.

The pollution stock X(t) is only gradually degraded, and its growth rate incorporates a random shock with mean of zero and constant standard deviation r. The social welfare is defined by the utility U(c) of the consumption c net of the disutility D(x) of pollution x. The objective of the social planner is to choose time paths for consumption to maximize the social welfare with long-run average criteria.

By using the vanishing discount technique, we solve the HJB equation (10.6) associated with the long-run average problem as the limit equation when the discount rate β converges to zero. The optimal consumption policy is shown to exist in a feedback form, and the maximum value is independent of the initial condition X(0) > 0.

The Model

Consider a society consuming a homogeneous good and accumlating pollution. Define the following quantities:

1. X(t) = stock of pollution at time t.

2. r = the constant rate of pollution decay, r > 0.

3. L = the upper bound of the maximum flow of pollusion, L > 0.

4. c(t) = flow of pollution (or consumption) at time t.

5. B(t) = the standard Brownian motion.

6. σ = the nonzero diffusion constant. […]