On the Mitra-Wan forest management problem in continuous time

The theory of optimal forest management has been developed in terms of discrete time while a consistent continuous time version of the MitraWan model has never been developed in the literature. No specific economic reason dictates that the discrete time framework should be preferred to the continuous time one, so that it may be the technical complexity that continuous time introduces into the analysis of the model that explains such lack of contributions. In this paper the discrete time Mitra-Wan model is reformulated in continuous time, so that also age of capital goods (i.e. age of trees) vary continuously, their evolution in time is described by means of a partial differential equation. Other features of the problem (e.g. the boundary condition containing the control) make it mathematically challenging, and one option to perform its analysis is to reformulate it as an infinite dimensional control problem, and to develop ad hoc techniques. Besides the new continuous time modeling, the paper provides a golden-rule configuration (i.e., an uniform density function with cutting at some age M > 0) which turns out to be optimal for the problem, and which is something more general than a function, i.e. a positive measure. Moreover, if the golden-rule configuration is unique, then undiscounted maximal (or optimal) paths exist from any given initial configuration and, provided the utility function is strictly concave, converge over time to the golden rule configuration.