Annals of Operations Research

, Volume 217, Issue 1, pp 425–445 | Cite as

The optimal harvesting problem under price uncertainty

  • Adriana Piazza
  • Bernardo K. Pagnoncelli


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.


Stochastic dynamic programming Forest management Optimal harvesting Price uncertainty 



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.


  1. Alonso-Ayuso, A., Escudero, L., Guignard, M., Quinteros, M., & Weintraub, A. (2011). Forestry management under uncertainty. Annals of Operations Research, 190, 17–39.CrossRefGoogle Scholar
  2. Alvarez, L., & Koskela, E. (2005). Wicksellian theory of forest rotation under interest rate variability. Journal of Economic Dynamics and Control, 29, 529–545.CrossRefGoogle Scholar
  3. Bastian-Pinto, C., Brandão, L. E., & Hahn, W. J. (2009). Flexibility as a source of value in the production of alternative fuels: The ethanol case. Energy Economics, 31, 411–422.CrossRefGoogle Scholar
  4. Blomvall, J., & Shapiro, A. (2006). Solving multistage asset investment problems by the sample average approximation method. Mathematical Programming, 108, 571–595.CrossRefGoogle Scholar
  5. Brazee, R., & Mendelsohn, R. (1988). Timber harvesting with fluctuating prices. Forest Science, 34, 359–372.Google Scholar
  6. Clarke, H., & Reed, W. (1989). The tree-cutting problem in a stochastic environment: The case of age-dependent growth. Journal of Economic Dynamics and Control, 13, 569–595.CrossRefGoogle Scholar
  7. Cominetti, R., & Piazza, A. (2009). Asymptotic convergence of optimal policies for resource management with application to harvesting of multiple species forest. Mathematics of Operations Research, 34, 576–593.CrossRefGoogle Scholar
  8. Cox, J., Ross, S., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of financial Economics, 7, 229–263.CrossRefGoogle Scholar
  9. Davis, L., & Johnson, K. (1987). Forest Management. New York, McGraw-Hill Book Company.Google Scholar
  10. Dixit, A. W., & Pindyck, R. (1994). Investment under Uncertainty. Princeton, NJ: Princeton University Press.Google Scholar
  11. Faustmann, M. (1995). Berechnung des Werthes, welchen Waldboden, sowie noch nicht haubare Holzbestände für die Waldwirthschaft besitzen. Allgemeine Forst- und Jagd-Zeitung; translated into english as: Calculation of the value which forest land and immature stands possess for forestry. Journal of Forest Economics, 1, 7–44.Google Scholar
  12. Gjolberg, O., & Guttormsen, A. (2002). Real options in the forest: What if prices are mean-reverting? Forest Policy and Economics, 4, 13–20.CrossRefGoogle Scholar
  13. Insley, M., & Rollins, K. (2005). On solving the multirotational timber harvesting problem with stochastic prices: A linear complementarity formulation. American Journal of Agricultural Economics, 87, 735–755.CrossRefGoogle Scholar
  14. Leuschner, W. (1990). Forest regulation, harvest scheduling, and planning techniques. New York, Wiley.Google Scholar
  15. Lohmander, P. (2000). Optimal sequential forestry decisions under risk. Annals of operations research, 95, 217–228.CrossRefGoogle Scholar
  16. McGough, B., Plantinga, A., & Provencher, B. (2004). The dynamic behavior of efficient timber prices. Land Economics, 80, 95–108.CrossRefGoogle Scholar
  17. Mitra, T., & Wan, H. (1985). Some theoretical results on the economics of forestry. The Review of Economic Studies, 52, 263–282.CrossRefGoogle Scholar
  18. Mosquera, J., Henig, M., & Weintraub, A. (2011). Design of insurance contracts using stochastic programming in forestry planning. Annals of Operations Research, 190, 117–130.CrossRefGoogle Scholar
  19. Rapaport, A., Sraidi, S., & Terreaux, J. (2003). Optimality of greedy and sustainable policies in the management of renewable resources. Optimal Control Applications and Methods, 24, 23–44.CrossRefGoogle Scholar
  20. Reeves, L., & Haight, R. (2000). Timber harvest scheduling with price uncertainty using markowitz portfolio optimization. Annals of Operations Research, 95, 229–250.CrossRefGoogle Scholar
  21. Salo, S., & Tahvonen, O. (2002). On equilibrium cycles and normal forests in optimal harvesting of tree vintages. Journal of Environmental Economics and Management, 44, 1–22.CrossRefGoogle Scholar
  22. Salo, S., & Tahvonen, O. (2003). On the economics of forest vintages. Journal of Economic Dynamics and Control, 27, 1411–1435.CrossRefGoogle Scholar
  23. Shapiro, A. (2011). Minimax and risk averse multistage stochastic programming. European Journal of Operational Research, 219, 719–726.CrossRefGoogle Scholar
  24. Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory. Philadelphia, SIAM.Google Scholar
  25. Tahvonen, O. (2004). Optimal harvesting of forest age classes: A survey of some recent results. Mathematical Population Studies, 11, 205–232.CrossRefGoogle Scholar
  26. Tahvonen, O., & Kallio, M. (2006). Optimal harvesting of forest age classes under price uncertainty and risk aversion. Natural Resource Modeling, 19, 557–585.CrossRefGoogle Scholar
  27. Thomson, T. (1992). Optimal forest rotation when stumpage prices follow a diffusion process. Land Economics, 68, 329–342.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Escuela de NegociosUniversidad Adolfo IbáñezPeñalolénChile

Personalised recommendations