## Abstract

We extend the classic Mitra and Wan forestry model by assuming that prices follow a geometric Brownian motion. We move one step further in the model with stochastic prices and include risk aversion in the objective function. We prove that, as in the deterministic case, the optimal program is periodic both in the risk neutral and risk averse frameworks, when the benefit function is linear. We find the optimal rotation ages in both stochastic cases and show that they may differ significantly from the deterministic rotation age. In addition, we show how the drift of the price process affects the optimal rotation age and how the degree of risk aversion shortens it. We illustrate our findings for an example of a biomass function and for different values of the model’s parameters.

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## Notes

- 1.
However, one difference should be noted. In their treatment, Mitra-Wan take \(n\) to be the age at which the timber volume coefficient per unit of land is maximized, claiming that “for any reasonable objective function for the economy, trees will never be allowed to grow beyond age \(n\)” (Mitra and Wan 1986), p. 332. It was pointed out by Khan and Piazza (2012), that the concavity of the benefit function favors a homogeneously configured forest and that it may be optimal to postpone harvesting beyond age \(n\) in order to reshape the forest into a more homogeneous state. Following Khan and Piazza (2012), we circumvent this by assuming \(n\) to be the age after which a tree dies.

- 2.
The expressions in bold print represent vectors.

- 3.
Under the hypotheses that there are no costs of harvesting or plantation, it is optimal to plant all the available land at every time period.

- 4.
In particular, this property is satisfied by any discrete mid-point concave function, see Murota (2003).

- 5.
In Lemma 1 we prove that the nested objective function is well-defined in the cases we are interested in.

- 6.
The computations can be found in Pagnoncelli and Piazza (2012).

- 7.
We have \(\rho (a) = a\) for any constant \(a\).

- 8.
In Mitra and Wan (1985), the authors assume that the solution of (5) is unique. For some values of \(r\), (8) may have multiple solutions. We will see later that Assumption 2 implies that (8) has at most two solutions that happen to be consecutive. In such a case, both Faustmann policies yield exactly the same value for Problem (3) for every initial condition. We take the convention of choosing the largest value as the solution.

- 9.
Where \(\lfloor a \rfloor \) stands for the integer part of \(a\).

- 10.
For formal definitions and a brief discussion of multiple existing criteria we refer the reader to Khan and Piazza (2012).

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## Acknowledgments

This research was partially funded by Basal Project CMM, Universidad de Chile. A. Piazza acknowledges the financial support of FONDECYT under Project 1140720 and of CONICYT Anillo ACT1106. B. K. Pagnoncelli acknowledges the financial support of FONDECYT under Projects 1120244 and 11130056.

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## Appendix

### Appendix

*Proof of Lemma 1.* Given any program \(\{{\varvec{x}}_t\}\) (not necessarily optimal) and \(\{c_t\}\) the corresponding sequence of harvesting, we consider the finite horizon value of problem (3):

Due to the fact that \(p_tc_t \ge 0\) for all \(t\) (when prices follow a GBM) and the monotonicity of any coherent risk measure we know that \(Q_T(\{{\varvec{x}}_t\})\ge Q_{T+1}(\{{\varvec{x}}_t\})\), hence the sequence \(Q_T(\{{\varvec{x}}_t\})\) either converges to the limit or diverges to \(-\infty \) when \(T\rightarrow \infty \). We prove now that \(Q_T(\cdot )\) is bounded below for all \(T\) and therefore it has a limit. If \(\rho =\mathbb E\), we have \(\rho _{p_{t-1}}[-p_tc_t]=-e^\mu p_{t-1}c_t\) and hence,

where the inequality follows because \(c_t\le 1\) for all \(t\). If \(\delta e^\mu <1\) we get \(Q_T> -p_0 \frac{1}{1-\delta e^\mu } >-\infty \) for all \(T\). This implies that the sequence \(Q_T\) converges when \(T\) goes to infinity. This limit is the value associated with program \(\{{\varvec{x}}_t\}\) in the infinite time horizon formulation. As the bound does not depend on the particular program we get that the minimal value is also well defined and finite.

When \(\rho =\text{ CVaR }_{\alpha }\) the proof is the same, substituting \(\delta e^\mu \) by \(\delta e^\mu \kappa \).

*Proof of Theorem 1*. We denote by \(\{{\varvec{x}}^*_t\}\) and \(\{c^*_t\}\) (resp. \(\{{\varvec{x}}_t\}\) and \(\{c_t\}\) the program and the harvesting sequence generated by the proposed optimal policy (resp. any alternative optimal policy) from the initial state \({\varvec{x}}^o\).

We treat first the risk neutral case. We let \(r\) represent \(\delta e^\mu \) and define the auxiliary constants

and where

We recall that \(n\) is the age after which die, hence, it is suboptimal to let trees grow beyond \(n\) and without loss of generality, we assume that \(c^n_t=x^n_t\) for all \(\{x_t\}\). To simplify the calculation below we add the auxiliary \((n+1)-\) coordinate to the state vector \({\varvec{x}}_t\), assuming that \(x_{n+1,t}=0\) for all \(t\).

Let us first show that

with equality iff \(\{{\varvec{x}}_t\}=\{{\varvec{x}}^*_t\}\) and \(\{c_t\}=\{c^*_t\}\). We have

The first sum is always less than or equal to zero, with equality if and only if \(x_{a,t}=x_{a+1,t+1}\) for \(a<{\theta _{\mathbb {E}}}\) which is equivalent to \(c_{a,t}=0\) for all \(a<{\theta _{\mathbb {E}}}\). Hence, we have obtained (14).

We focus now in getting a bound independent of \({\varvec{x}}_t\). We claim that

for every state \({\varvec{x}}\), with equality if and only if \(x_a=0\) for all \(a>{\theta _{\mathbb {E}}}\). Indeed,

where in the last step we use the following equality:

which follows from (13). To finish the proof of (15) we only need to show that

This is evidently true if the left hand side is negative, hence we will only deal with the cases where \(f_a-\frac{f_{a-1}}{r}>0\). This means, in particular, that we only need to consider values of \(a\) such that \(f_{a}-f_{a-1}>0\). Let \(a_{M}\) denote \(\arg \max {f_a}\). Thanks to Assumption 1 we know that

Using (16) and the fact that \(f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}>f_{\theta _{\mathbb {E}}+1}\tau _{\theta _{\mathbb {E}}+1}\) we obtain

which finishes the proof of (17) as well as (15).

Putting (14) and (15) together we readily get

In summary, we have that (14) and (18) are fulfilled along any feasible program. Besides, along the proposed optimal program (14) is satisfied with equality for all \(t\) and (18) is satisfied with equality for all \(t\ge 2\).

To finish the proof we compare the benefits obtained by *any* program and the proposed optimal program up until some \(T\). As prices follow a GBM we know that \(\mathbb E_{|p_t}[p_{t+1}]=e^\mu p_t\) and we can write the objective function of (3) as with \(r=\delta e^\mu \).

Using (14) to bound the difference corresponding to \(t=1\) and (15) for the other terms we get

Given that the product \(q'({\varvec{x}}^*_T-{\varvec{x}}_T)p_0\) is bounded for all \(T\) and that obviously \(r^{T-1}\rightarrow 0\) when \(T\rightarrow \infty \) we get that

and hence the proposed harvesting policy is optimal.

The proof in the risk averse case follows analogously. Indeed, using that the conditional CVaR is positive homogeneous and that \(\text {CVaR}_{p_t}[p_{t+1}]=e^\mu \kappa p_t\) when prices follow a GBM, taking \(r=\delta e^\mu \kappa \), we can write the objective function as . Hence, it suffices to take \(r=\delta e^{\mu }\kappa \) and substitute \(\theta _{\mathbb {E}}\) by \(\theta _{\rho }\) throughout the proof.

*Proof of Theorem 2*. The function \(F_r(a)\) is non-negative and twice differentiable in \([{a_0}, n]\). Its derivative is \(F'_r(a)=\frac{r^a}{1-r^a}\left( f'(a)+\frac{f(a)\ln (r)}{1-r^a} \right) \) and, hence,

A priori, there might be several points satisfying (19). We claim that it is unique. Indeed, evaluating \(F''_r\) in \(a^*\) such that \(F'_r(a^*)=0\) we get

It implies that every zero of the first derivative is a local maximum and, as there cannot be two local maxima without a minimum, we conclude that the first derivative has a unique zero. We denote this point as \(a^*_r\) to point out its dependence with the parameter \(r\).

To see the variation of \(a^*_r\) with respect to \(r\) let us consider (19). We know that \(F'_r(a)\) is zero only at \(a^*_r\) and that \(F''_r(a^*_r)<0\), which implies that \(F'_r(a)>0\) whenever \(a<a^*(r)\). Hence, we know that the graph of \(\frac{f'(a)}{f(a)}\) is above the one of \(\frac{\ln (1/r)}{1-r^{a}}\) whenever \(a<a^*(r)\). While the left hand side of (19) is independent of \(r\), its right hand side is decreasing with respect to \(r\). From Fig. 3 it is easy to see that \(a^*_r\) is increasing with \(r\).

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Piazza, A., Pagnoncelli, B.K. The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases.
*J Econ* **115, **175–194 (2015). https://doi.org/10.1007/s00712-014-0414-4

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### Keywords

- Forestry
- Dynamic programming
- Risk analysis
- Coherent risk measures

### JEL Classification

- Q23
- C61
- D81