On the Mitra--Wan Forest Management Problem in Continuous Time

The paper provides a continuous time version of the well known discrete time Mitra-Wan model of optimal forest management, where a forest is harvested to maximize the utility of timber flow over an infinite time horizon. Besides varying with time, the state variable (describing available trees) and the other parameters of the problem vary continuously also with respect to the age of the trees. The evolution of the system is given in terms of a partial differential equation and later rephrased as an ordinary differential equation in an infinite dimensional space. The paper provides a classification of the behavior of optimal and maximal programs when the utility function is linear, convex, or strictly convex and the discount rate is positive or null. Formulas are provided for modified golden-rule configurations (uniform density functions with cutting at the ages that solve a Faustmann problem) and for Faustmann policies, and the optimality or maximality of such programs is discussed. In all different sets of data, it is shown that the optimal (or maximal) control is necessarily something more general than a function, i.e. a positive measure. In particular, in the case of strictly concave utility and null discount, when the Faustmann policy is not optimal, it is shown that optimal paths converges over time to the golden rule configuration, while in the case of strictly concave utility and positive discount the Faustmann policy is shown to be not optimal, contradicting the corresponding result in discrete time.