# The optimal harvesting problem under price uncertainty

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## Abstract

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, with finite and infinite time horizon. We assume that harvest is restricted to mature trees older than a certain age and that growth and natural mortality after maturity are neglected. We use stochastic dynamic programming techniques to characterize the optimal policy and we model price using a geometric Brownian motion and an Ornstein–Uhlenbeck process. In the first case we completely characterize the optimal policy for all possible choices of the parameters. In the second case we provide sufficient conditions, based on explicit expressions for reservation prices, assuring that harvesting everything available is optimal. In addition, for the Ornstein–Uhlenbeck case we propose a policy based on a reservation price that performs well in numerical simulations. 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.

### Keywords

Stochastic dynamic programming Forest management Optimal harvesting Price uncertainty## Notes

### Acknowledgements

This research was partially supported by Programa Basal PFB 03, CMM. Adriana Piazza acknowledges the financial support of Fondecyt under Projects 11090254 and 1140720 and Project Anillo ACT-1106. Bernardo Pagnoncelli acknowledges the financial support of Fondecyt under Project 1120244. The authors thank Roberto Cominetti, Marcos Goycoolea, Alexander Shapiro and Andrés Weintraub for fruitful conversation and encouragement.

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