A Linear Bang-Bang Model of Firm Behavior and Water Quality

This paper considers the problem of a firm located on a lake in a situation wherein its production process pollutes the lake, while the pollution, in turn, increases the firm's production costs. It is assumed that the pollution caused is proportional to the production rate, that the lake has an exponential self-cleaning ability, and that the effectiveness of the direct pollution control dollars is proportional to the existing pollution level. The optimal control problem is to determine the optimal levels of production and pollution abatement expenditures in a way that maximizes the present value of net profit streams over an infinite horizon. The optimal control is characterized by a continuum of stationary optimal equilibria. It is also shown that a policy of periodic production and shutdowns can never be optimal