Analytic solution of a dynamic pool model incorporating constant marginal cost and discount rates

Abstract

A simple dynamic pool model is extended to describe the economics of a single-species fishery by incorporating constant marginal cost and discount rates. Assuming that the population has already come to equilibrium under an initial fishing mortality rate and that any change in that rate is to be sustained indefinitely, the model can be solved analytically to yield the optimal fishing mortality rate. When this rate is expressed as a proportion of the natural mortality rate, the solution takes the form of a third-degree polynomial whose coefficients are simple functions of four other parameters. The solution exhibits positive conservation effects as long as all four parameters are sufficiently high. These conservation effects may be great enough to warrant closing the fishery when the marginal cost rate exceeds a well-defined limit.