New insights from the canonical fisheries model – Optimal management when stocks are low

We analyse the standard optimal control fishery biomass model and derive some novel results on optimal management when fish stocks are low. We show that as long as it is not optimal to let the stock become extinct and the marginal benefit of harvesting is bounded below infinity for all harvest levels, there will always be an interval with low stock sizes where it is optimal not to harvest. This result does not depend on any assumption that marginal harvesting cost per unit increases with decreasing stock size. We then prove that under weak conditions the shadow price on the fish stock always goes to infinity as the stock approaches zero. The results are generalized to a particular class of age structured models.