Optimal forest management under uncertainty

Recently, forest optimization models have accounted for uncertainty associated with forestry investments by allowing stochastic variations in, e.g. timber prices, carbon prices, timber growth and discount rates. A stochastic process can be understood as a collection of random variables, which are indexed by a time parameter (Haigh, 2005). When the distribution of these random variables satisfies certain properties, a stochastic process is referred to as a geometric Brownian motion, a diffusion process, or a mean-reverting process. For example, a geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (Ross, 2007). A mean-reverting process is a process in which the random variables tend to revert to a “normal level” with a certain speed of reversion, such as the price of oil tends to drawback towards the marginal cost of oil production (Dixit & Pindyck, 1994). These different approaches to model uncertainty have been applied to forest optimization models both at the stand and forest level.

Uncertainty has been incorporated in a single and an ongoing rotation (Faustmann formula). Depending on assumptions made, some authors found that the optimal rotation age is independent of timber price uncertainty, while others show that the optimal rotation length and the net present value are positively correlated to price uncertainty. Using continuous time, Clarke and Reed (1989) assumed timber price

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to follow a geometric Brownian motion and timber growth evolving as a diffusion process. For a single rotation, their results show that price and growth uncertainties lengthen rotation age and increase NPV value. For the Faustmann rotation, they were unable to solve their model when both stochastic price and stochastic growth were considered. When only price uncertainty was incorporated, they show that price uncertainty has a small effect on rotation age, but substantially increases the NPV.

Employing a similar methodology, though using a different specification for the timber growth process, Reed and Clarke (1990) obtained results consistent to those of Clarke and Reed (1989); except that in the single rotation, they found that price uncertainty has no effect on optimal policy. Thomson (1992) extended the Faustmann model to include timber price uncertainty. He used a binomial pricing model and specified timber price as a lognormal stochastic process. Using dynamic programming technique for both the Faustmann and his models, the author showed that price uncertainty induces forest owners to lengthen their optimal rotation age compared to the Faustmann solution. Moreover, his results suggested that NPV is positively correlated with timber price fluctuations.

Beside stochastic price and stochastic growth, other types of uncertainty have also been taking into account. Alvarez and Koskela (2006) assumed interest rate evolving as a parameterized mean-reverting process and found that interest rate uncertainty increases rotation age under risk aversion. Motoh (2004) studied the optimal rate of use of a natural resource under uncertainty with catastrophic risk. They adopted the stock of a natural resource (in quantity) following a geometric Brownian motion before catastrophic risk occurs, and the catastrophic risk is described as a Poisson process. They used stochastic dynamic programming to maximize expected discounted utility over an infinite horizon. They reported that the optimal rate of use of the natural resource increases with uncertainty or catastrophic risk.

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More recently, a real option approach has been used to deal with uncertainty in forestry economics. Insley (2002) solved a continuing rotation using the Hamilton-Jacobi-Bellman technique under a mean-reverting or geometric Brownian motion price process with deterministic volume. She argued that specifying timber price as a mean-reverting process is more rational than a geometric Brownian motion. Under a mean-reverting price, uncertainty impacts significantly on the optimal rotation period and the option value. Chladna (2007) applied a real option approach to analyse optimal single rotation for single stand forest under timber price and carbon price uncertainty. The author specified timber price as a mean-reverting process and carbon price as a geometric Brownian motion process. Employing dynamic programming technique, this paper showed that stochastic timber price and/or stochastic carbon price extend the optimal rotation length. This effect, however, is small when carbon price is low.

At the forest level, Tahvonen and Kallio (2006) extended Reed and Clarke (1989; 1990) to include risk aversion, age class structure, and planting cost. They specified timber price as a geometric Brownian motion or a mean reverting price process. Due to complexity of the problem, the authors used discrete time for their model and applied a stochastic programming technique to solve the model. In contrast to the literature, which mostly discusses single stand forests, the paper showed that price uncertainty may shorten optimal rotation age. This result follows from their assumption that only replanting cost has a significant role in determining cost factor, in contrast to earlier work that only harvesting cost is costly. The authors adopted a short planning horizon, T=13. This paper ignored

environmental values of forests.

In summary, the literature about uncertainty at both the stand and forest levels shows that uncertainty does affect the optimal rotation age of single and multiple- stand forests. The impact on the rotation age differs according to assumptions made about the type of uncertainty and the forest owners’ attitude to risk.

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