3. Renewable Resources

Turning to renewable resources, the new feature is some process for regeneration, accomplished by introducing a biological growth function, the widely applicable logistic law, into the optimal control model for exhaustible resources. Although a quite different result might be expected, and this is in fact the case, it turns out that the optimal management regime can be characterized in a way that also centrally involves the rate of interest. Depending on assumptions about the behavior of the growth function and the interest rate, a variety of outcomes are derived. In particular, under certain conditions the bio-economic optimum can be shown to be equivalent to what might be called a purely biological optimum, the maximum sustainable yield. This analysis has been for a generic renewable resource. The chapter also develops a model for the optimal harvesting of a particular resource, timber, or more generally forest management. This involves no mathematics beyond the elementary calculus used in the modeling of optimal extraction of an exhaustible resource, though now in a setting of continuous time and, following the conventional treatment, the decision variable is the optimal harvest date or rotation period rather than the amount of timber to be harvested at each point in time. Four cases are considered: the optimal harvest date for a one-shot harvest (which might be appropriate for a very slow-growing species); the optimal rotation period where repeated harvests are indicated, the conventional approach; a purely biological model, in which the objective is to maximize not value but the sustainable physical yield, which is contrasted with the bioeconomic model as in the case of a generic renewable resource; and the same analyses (though it is sufficient to look at how results are affected in the first case, the one-shot harvest) taking into account that a standing forest can also provide value. This leads quite naturally to the next chapter, on environmental dynamics.