The optimal harvesting problem with price uncertainty ∗

In this paper we study the exploitation of a one species forest plantation when timber price is governed by a stochastic process. The work focuses on providing closed expressions for the optimal harvesting policy in terms of the parameters of the price process and the discount factor. We assume that harvest is restricted to mature trees older than a certain age and that the growth after maturity is neglected as well as the natural mortality. We use stochastic dynamic programming techniques to characterize the optimal policy for two important cases: when prices follow a geometric Brownian motion we completely characterize the optimal policy for all possible choices of drift and discount factor. If prices are governed by a mean-reverting (Ornstein-Uhlenbeck) process we provide sufficient conditions, based on explicit expressions for reservation prices at every time period above which harvesting everything available is optimal. In both cases we solve the problem for every initial condition and the best policy is obtained endogenously, that is, without imposing any ad hoc restrictions such as maximum sustained yield or convergence to a predefined final state. ∗This research was partially supported by Project Anillo ACT-88 and Programa Basal PFB 03, CMM. Adriana Piazza also acknowledges the financial support of Fondecyt under Project 11090254. The authors thank Marcos Goycoolea, Alexander Shapiro and Andrés Weintraub for fruitful conversation and encouragement.