The Optimum Rotation Problem in Forestry

Abstract

One of the most interesting problems in forestry economics is the selection of the optimal time to cut a tree or an even-aged forest. An even-aged forest refers to timber stands composed of trees of only one age group. This problem has been known since the work of Faustmann in the mid-nineteenth century (Faustmann, 1849). As a tree ages, the volume of wood it produces increases up to a certain point, beyond which the tree begins to decay from old age. Figure 6.1 shows the growth cycle of a representative tree. After the tree has become firmly established the volume of wood grows at an increasing rate up to a point a. In terms of mathematics, between points 0 and a, the first and second derivatives of the volume function are positive, i.e. d V/dt, d²V/dt² > 0. Beyond a the volume increase continues but at a slower rate until β when the maximum timber output is achieved. That is, between a and β the first derivative of the function is positive, dV/dt > 0, but the second derivative is negative, d² V/dt² < 0. At point β, which is the turning point of the volume curve, the derivatives are zero.