Optimising forest management under storm risk with a Markov decision process model

Abstract

Windstorms generate windfalls that may lead to price decreases. Studies often focus on stochastic growth and price, but consider that there is no link between these two risks. In our model, we assume that storms generate windfalls and that these windfalls have an impact on timber price through volume and quality. The objective of this paper is to analyse the impact of these two effects on harvesting behaviour. We consider that the dynamic of the timber stock follows a Markov decision process and that the harvesting decision is a control variable. We solve the optimal harvesting problem under storm risk with a risk-averse forest owner and when the storm has an impact on production and price. We study the impact of a change in the storm risk distribution, the percentage of quality loss and risk aversion on the optimal harvesting decision. We show that the greater the storm risk is, the greater the harvesting will be. In addition, we observe no noticeable effect of an increase in the percentage of quality loss on harvesting. Moreover, when the forest owner’s risk aversion increases, the harvesting is reduced. Finally, we discuss our results, in particular, in relation to climate change.

Keywords:

• storm
• forest
• production risk
• price risk
• quality loss
• Markov process

Acknowledgments

We thank Joseph Buongiorno for his helpful suggestions. We are also grateful to the participants of the workshop, ‘Natural Risks, Climate Change, Natural and Renewable Resources’, held in Nancy (France) in September 2010, and to the participants of the 19th Annual Conference of the European Association of Environmental and Resource Economists organised in Prague in June 2012.

Notes

1. Many papers deal with the impact of storm risk on forest management. Similarly, some surveys on this topic exist that link the different kinds of models and hypothesis, like the paper of Pasalodos-Tato et al. (2013). The reader can also refer to the paper of Yousefpour et al. (2012) for a review of approaches of state-of-the-art methods for optimal decision-making under risk and uncertainty in forestry. In the same vein, the paper of Hanewinkel et al. (2010) presents recent approaches to model the risk of storm to forests and their integration into simulation and decision support tools.

2. The paper of Williams (2009) presents a general review of MDP applications in natural resources management.

3. The literature is very extensive on this point and we only indicate papers that analyse stochastic stumpage prices using MDPs.

4. We only consider the stock of trees of age to be harvested. The forest owner is not interested in cutting younger trees because of their low commercial value.

5. A finite state space is a standard assumption with an MDP framework (Sabaddin 2009). For simplicity, we assume that both variables have a state space of the same size but it is possible to assume different sizes for the state spaces of the two variables.

6. In the literature, only two levels of decisions are generally considered: none or all of the stock.

7. We opt for a discrete probability distribution. We do not choose a specific discrete probability distribution but for each possible value of the discrete random variable, we have associated a non-zero probability.

8. Assume that a constant growth rate appears reasonable in our model for different reasons. First, we wish to focus on the variability of the stock due to storms and not on the variability associated with the forest growth. Second, assuming a constant growth rate is an assumption that we find in various articles in the literature (Crecente-Campo et al. 2010; Cao and Strub 2008; Kohyama and Takada 1998). Finally, a constant growth rate simplifies the simulations. Indeed, such an assumption highlights the qualitative trends without increasing the resolution of the programme.

9. In France, for example, 3.5 million private forest owners hold 71% of the total forest area.

10. We also tested the impact of considering an inverse demand function that slopes downward for standing stumpage. Since we have no data for choosing and calibrating such a function, we tested the following linear decreasing inverse demand function: where and are arbitrary parameters chosen in order to guarantee positive prices. We carry out the same simulations as in Table 1 but with an inverse demand function that slopes downward and with and . We note that optimal harvesting decisions and the corresponding expected intertemporal discounted utility are modified (the number of standing stocks harvested is reduced) but the same qualitative trends are maintained. A sensitivity analysis on both parameters and confirms these results. Since no empirical data exist to calibrate a particular specification, we prefer to consider a linear price assumption in this paper.

11. The use of a CRRA specification is commonly found in economic literature (Gollier 2001; Cox and Harrison 2008).

12. This assumption simplifies the simulations and allows us to obtain results in a reasonable time.

13. As previously mentioned, such an assumption highlights the qualitative trends without increasing the resolution of the programme that could be solved for other values.

14. For forest projects, usual discount rates are comprised between 1% and 5% (Calvet, Lemoine, and Peyron 1997; Heaps and Pratt 1989).

15. There is a great deal of forest economics literature dealing with risk aversion of forest owners (Ollikainen 1993; Gong 1998; Lönnstedt and Svensson 2000; Gong and Löfgren 2003; Alvarez and Koskela 2006; Andersson 2012) but none of these papers raise the issue of the relative risk aversion coefficient of private forest owners. More generally, there is no consensus on the level of the relative risk aversion coefficient for economic agents (Cardenas and Carpenter 2008; Brick, Visser, and Burns 2012). We consider a value that is consistent with the range of reported estimates in the economic literature.

16. We also conduct a sensitivity analysis on the discount rate. We show that with or = 0.05, harvesting is always reduced or increased, compared to the best decision in the base case, and that the expected intertemporal discounted utility increases or decreases, respectively. Although the discount rate represents a time preference, it also reflects the cost of capital. Therefore, when it decreases, the owner harvests less, and when it increases, she/he harvests more.

17. According to Holt and Laury (2002), corresponds to a very risk-averse individual.

Additional information

Funding

The UMR Economie Forestière is supported by a grant from the French National Research Agency (ANR) as part of the ‘Investissements d’Avenir’ programme (ANR-11-LABX-0002-01, Laboratory of Excellence ARBRE).