SUSTAINABLE YIELDS IN FISHERIES: UNCERTAINTY, RISK-AVERSION, AND MEAN-VARIANCE ANALYSIS

Abstract

We consider a model of a fishery in which the dynamic of the unharvested fish population is given by the stochastic logistic growth equation dx(t) = [�x(t)(� − x(t))]dt + �x(t)dW(t). Similar as in the classical deterministic analogon, we assume that the fishery harvests the fish population following a constant effort strategy. In a first step we derive the effort level that leads to maximum expected sustainable yield, which is understood as the expectation of the equilibrium distribution of the stochastic dynamics. This replaces the non-zero fixed point in the classical deterministic setup. In a second step, we assume that the fishery is risk averse and that there is a trade off between expected sustainable yield and uncertainty measured in terms of the variance of the equilibrium distribution. We derive the optimal constant effort harvesting strategy for this problem. In a final step, we consider an approach which we call the mean-variance analysis to sustainable fisheries. Similar as in the now classical mean-variance analysis in Finance, going back to Markowitz (1957), we study the problem of maximizing expected sustainable yields under variance constraints, and dual to this, minimizing the variance, e.g. risk, under guaranteed minimum expected sustainable yields. We derive explicit formulas for the optimal fishing effort in all four problems considered and study the effects of uncertainty, risk aversion and mean reversion speed on fishing efforts.