The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

Notes

  1. 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. 2.

    The expressions in bold print represent vectors.

  3. 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. 4.

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

  5. 5.

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

  6. 6.

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

  7. 7.

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

  8. 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. 9.

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

  10. 10.

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

References

  1. Alvarez LHR, Koskela E (2006) Does risk aversion accelerate optimal forest rotation under uncertainty? J For Econ 12:171–184

    Google Scholar 

  2. Alvarez LHR, Koskela E (2007) Optimal harvesting under resource stock and price uncertainty. J Econ Dynam Control 31:2461–2485

    Article  Google Scholar 

  3. Alvarez LHR, Koskela E (2007) Taxation and rotation age under stochastic forest stand value. J Environ Econ Manag 54:113–127

    Article  Google Scholar 

  4. Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228

    Article  Google Scholar 

  5. Birge J, Louveaux F (1997) Introduction to stochastic programming. Springer, USA

    Google Scholar 

  6. Blomvall J, Shapiro A (2006) Solving multistage asset investment problems by the sample average approximation method. Math Program 108:571–595

    Article  Google Scholar 

  7. Chen S, Insley M (2012) Regime switching in stochastic models of commodity prices: an application to an optimal tree harvesting problem. J Econ Dynam Control 36:201–219

    Article  Google Scholar 

  8. Clarke W, Reed H (1989) The tree-cutting problem in a stochastic environment: the case of age-dependent growth. J Econ Dynam Control 13:569–595

    Article  Google Scholar 

  9. Di Corato L, Moretto M, Vergalli S (2013) Land conversion pace under uncertainty and irreversibility: too fast or too slow? J Econ 110:45–82

    Article  Google Scholar 

  10. Delbaen F, Drapeau S, Kupper M (2011) A von Neumann–Morgenstern representation result without weak continuity assumption. J Math Econ 47:401–408

    Article  Google Scholar 

  11. Dixit A, Pindyck R(1994) Investment under uncertainty, Princeton UP

  12. Faustmann M (1995) Berechnung des Wertes welchen Waldboden sowie noch nicht haubare Holzbestände für die Waldwirtschaft besitzen, Allg. Forst-und Jagdzeitung 15 (1849) 441–455. Translated by Linnard W Calculation of the value which forest land and immature stands possess for forestry. J For Econ 1:7–44

  13. Föllmer H, Schied A (2011) Stochastic finance: an introduction in discrete time. De Gruyter, Germany

    Google Scholar 

  14. Gjolberg O, Guttormsen A (2002) Real options in the forest: what if prices are mean-reverting? For Policy Econ 4:13–20

    Article  Google Scholar 

  15. Gong P, Löfgren K (2003) Risk-aversion and the short-run supply of timber. For Sci 49:647–656

    Google Scholar 

  16. Gong P, Löfgren K (2008) Impact of risk aversion on the optimal rotation with stochastic price. Nat Res Model 21:385–415

    Article  Google Scholar 

  17. Hildebrandt P, Kirchlechner P, Hahn A, Knoke T, Mujica R (2010) Mixed species plantations in Southern Chile and the risk of timber price fluctuation. Eur J For Res 129:935–946

    Article  Google Scholar 

  18. Insley M (2002) A real option approach to the valuation of a forestry investment. J Environ Econ Manag 44:471–492

    Article  Google Scholar 

  19. Khajuria RP, Kant S, Laaksonen-Craig S (2009) Valuation of timber harvesting options using a contingent claims approach. Land Econ 85:655–674

    Google Scholar 

  20. Khan MA, Piazza A (2012) On the Mitra–Wan forestry model: a unified analysis. J Econ Theor 147:230–260

    Article  Google Scholar 

  21. Knoke T, Stimm B, Ammer C, Moog M (2005) Mixed forests reconsidered: a forest economics contribution on an ecological concept. For Ecol Manag 213:102–116

    Article  Google Scholar 

  22. Laeven R, Denuit M, Dhaene J, Goovaerts M, Kaas R (2006) Risk measurement with equivalent utility principles. Stat Decis 24:1–25

    Article  Google Scholar 

  23. Latta G, Sjølie H, Solberg B (2013) A review of recent developments and applications of partial equilibrium models of the forest sector. J For Econ 19:250–360

    Google Scholar 

  24. Markowitz H (1952) Portfolio selection. J Financ 7:77–91

    Google Scholar 

  25. Mei B, Clutter M, Harris T (2010) Modeling and forecasting pine sawtimber stumpage prices in the US South by various time series models. Can J For Res 40:1506–1516

    Article  Google Scholar 

  26. Mitra T, Roy S (2012) Sustained positive consumption in a model of stochastic growth: the role of risk aversion. J Econ Theor 147:850–880

    Article  Google Scholar 

  27. Mitra T, Wan H (1985) Some theoretical results on the economics of forestry. Rev Econ Stud 52:263–282

    Article  Google Scholar 

  28. Mitra T, Wan H (1986) On the Faustmann solution to the forest management problem. J Econ Theor 40:229–249

    Article  Google Scholar 

  29. Murota K (2003) Discrete convex analysis. SIAM

  30. Pagnoncelli BK, Piazza A (2012) The optimal harvesting problem under risk aversion. Available at http://www.optimization-online.org. Accessed 8 Oct 2013

  31. Piazza A, Pagnoncelli BK (2014) The optimal harvesting problem under price uncertainty. Ann Oper Res 217:425–445

    Article  Google Scholar 

  32. Penttinen M (2006) Impact of stochastic price and growth processes on optimal rotation age. Eur J For Res 125:335–343

    Article  Google Scholar 

  33. Peters H (2012) A preference foundation for constant loss aversion. J Math Econ 48:21–25

    Article  Google Scholar 

  34. Plantinga A (1998) The optimal timber rotation: an option value approach. For Sci 44:192–202

    Google Scholar 

  35. Riedel F (2004) Dynamic coherent risk measures. Stoch Process Appl 112:185–200

    Article  Google Scholar 

  36. Rockafellar R, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–42

    Google Scholar 

  37. Rockafellar R, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26:1443–1471

    Article  Google Scholar 

  38. Roessiger J, Griess V, Knoke T (2011) May risk aversion lead to near-natural forestry? a simulation study. Forestry 84:527–537

    Article  Google Scholar 

  39. Ruszczyński A, Shapiro A (2006) Conditional risk mappings. Math Oper Res 31:544–561

    Article  Google Scholar 

  40. Salo S, Tahvonen O (2002) On equilibrium cycles and normal forests in optimal harvesting of tree vintages. J Environ Econ Manag 44:1–22

    Article  Google Scholar 

  41. Salo S, Tahvonen O (2003) On the economics of forest vintages. J Econ Dynam Control 27:1411–1435

    Article  Google Scholar 

  42. Schechter L (2007) Risk aversion and expected-utility theory: a calibration exercise. J Risk Uncertain 35:67–76

    Article  Google Scholar 

  43. Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory, vol 9. SIAM, Philadelphia

  44. Tahvonen O, Kallio M (2006) Optimal harvesting of forest age classes under price uncertainty and risk aversion. Nat Res Model 19:557–585

    Article  Google Scholar 

  45. Thomson T (1992) Optimal forest rotation when stumpage prices follow a diffusion process. Land Econ 68:329–342

    Article  Google Scholar 

  46. Von Neumann J, Morgenstern O (2007) Theory of games and economic behavior. Commemorative edn, Princeton University Press, USA

  47. Yoshimoto A, Shoji I (1998) Searching for an optimal rotation age for forest stand management under stochastic log prices. Eur J Oper Res 105:100–112

    Article  Google Scholar 

  48. Zhou M, Buongiorno J (2006) Space-time modeling of timber prices. J Agric Res Econ 31:40–56

    Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Adriana Piazza.

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):

$$\begin{aligned} Q_T(\{{\varvec{x}}_t\})\!=\!- p_{0}c_{0} \!+\! \delta \rho _{|p_{0}}[ -p_{1}c_{1} + \ldots +\delta \rho _{|p_{T-2}}[-p_{T-1}c_{T-1} + \delta \rho _{|p_{T-1}}[-p_Tc_T ]]] \end{aligned}$$

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,

$$\begin{aligned} Q_T(\{{\varvec{x}}_t\})&= - p_{0}c_{0} + \delta \rho _{|p_{0}}[ -p_{1}c_{1} + \ldots +\delta \rho _{|p_{T-2}}[-p_{T-1}(c_{T-1} - \delta e^{\mu } c_T) ]]\\&\vdots \\&= -p_0 \sum _{t=0}^T (\delta e^\mu )^t c_t \ge -p_0 \sum _{t=0}^T (\delta e^\mu )^t = -p_0 \frac{1-(\delta e^\mu )^{T+1}}{1-\delta e^\mu } \end{aligned}$$

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

$$\begin{aligned} \tau _a=\frac{r^a}{1-r^a} ~~\text {with } a=1\dots , n \end{aligned}$$
(13)

and where

$$\begin{aligned} {\varvec{q}}=\Bigg (0,\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _1},\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _2},\dots \dots , \underbrace{\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _{{\theta _{\mathbb {E}}}-1}}}_{{\theta _{\mathbb {E}}}\text{- }th~coord.},f_{\theta _{\mathbb {E}}},\dots \dots ,f_{n} \Bigg ). \end{aligned}$$

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

$$\begin{aligned} c_t +\sum _{a=1}^n q_ax_{a,t+1} - \frac{1}{r}\sum _{a=1}^n q_ax_{a,t}\le \sum _{a=1}^{n}q_{a+1}\left( x_{a,t}-\frac{x_{a+1,t}}{r}\right) \qquad \forall t, \end{aligned}$$
(14)

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

$$\begin{aligned}&c_t + \sum _{a=1}^n q_ax_{a,t+1} - \frac{1}{r}\sum _{a=1}^n q_ax_{a,t}~=~c_t + \sum _{a=1}^n q_a\left( x_{a,t+1} - \frac{1}{r}x_{a,t}\right) \\&= \sum _{a=1}^{n} f_a(x_{a,t}-x_{a+1,t+1}) + \sum _{a=1}^{{\theta _{\mathbb {E}}}-1} \frac{f_{\theta _{\mathbb {E}}} \tau _{\theta _{\mathbb {E}}}}{\tau _a} \left( x_{a+1,t+1}-\frac{x_{a+1,t}}{r}\right) \\&+\sum _{a={\theta _{\mathbb {E}}}}^{n-1}f_a\left( x_{a+1,t+1}-\frac{x_{a+1,t}}{r}\right) \\&= \sum _{a=1}^{{\theta _{\mathbb {E}}}-1} \left[ f_a(x_{a,t}-x_{a+1,t+1}) +\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _a}\left( x_{a+1,t+1} \!-\!x_{a,t} \!+\! x_{a,t}\!-\!\frac{x_{a+1,t}}{r} \right) \right] \\&+\sum _{a={\theta _{\mathbb {E}}}}^{n}f_a\left( x_{a,t}\!-\!\frac{x_{a+1,t}}{r}\right) \\&= \sum _{a=1}^{{\theta _{\mathbb {E}}}-1}\left[ \frac{1}{\tau _a}(f_a\tau _a-f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}})(x_{a,t}-x_{a+1,t+1}) +\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _a}\left( x_{a,t}\!-\! \frac{x_{a+1,t}}{r}\right) \right] \\&+\sum _{a={\theta _{\mathbb {E}}}}^{n}f_a\left( x_{a,t}\!-\!\frac{x_{a+1,t}}{r}\right) \\&= \sum _{a=1}^{{\theta _{\mathbb {E}}}-1} \frac{1}{\tau _a}(f_a\tau _a-f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}})(x_{a,t}-x_{a+1,t+1}) +\sum _{a=1}^{n}q_{a+1}\left( x_{a,t}-\frac{x_{a+1,t}}{r}\right) \end{aligned}$$

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

$$\begin{aligned} \sum _{a=1}^{n}q_{a+1}\left( x_{a}-\frac{x_{a+1}}{r}\right) \le \frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _1}, \end{aligned}$$
(15)

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

$$\begin{aligned}&\sum _{a=1}^{n}q_{a+1}\left( x_a-\frac{x_{a+1}}{r}\right) ~=~ \sum _{a=1}^{n}q_{a+1}x_a-\sum _{a=2}^{n+1}q_{a}\frac{x_{a}}{r}\\&= q_2x_1+\sum _{a=2}^{n}\left( q_{a+1}-\frac{q_{a}}{r}\right) x_a\\&= \frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _1}x_1 + \sum _{a=2}^{{\theta _{\mathbb {E}}}}f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}} \left( \frac{1}{\tau _{a}}-\frac{1}{r\tau _{a-1}}\right) x_a + \sum _{a={\theta _{\mathbb {E}}}+1}^{n}\left( f_{a}-\frac{f_{a-1}}{r}\right) x_a\\&= \sum _{a=1}^{{\theta _{\mathbb {E}}}}\frac{f_{\theta _{\mathbb {E}}}\tau _{\theta _{\mathbb {E}}}}{\tau _1}x_a + \sum _{a={\theta _{\mathbb {E}}}+1}^{n}\left( f_{a}-\frac{f_{a-1}}{r}\right) x_a, \end{aligned}$$

where in the last step we use the following equality:

$$\begin{aligned} \frac{1}{\tau _a}-\frac{1}{r\tau _{a-1}}=\frac{1}{\tau _1}, \end{aligned}$$
(16)

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

$$\begin{aligned} f_a-\frac{f_{a-1}}{r}\le \frac{f_{{\theta _{\mathbb {E}}}}\tau _{{\theta _{\mathbb {E}}}}}{\tau _1} \text{ for } \text{ all } a>\theta _{\mathbb {E}}. \end{aligned}$$
(17)

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

$$\begin{aligned}&f_{n}-{f_{n-1}}<\dots <f_{a_{M}+1}-f_{a_{M}}<0<f_{a_{M}}- f_{a_{M}-1}<\dots < f_2-f_1\\&\Rightarrow 0<f_a-{f_{a-1}}< f_{a-1}-{f_{a-2}}< \frac{f_{a-1}-f_{a-2}}{r}~~~~~\forall a\le a_{M}\\&\Rightarrow f_a-\frac{f_{a-1}}{r}<f_{a-1}-\frac{f_{a-2}}{r} ~~~~~\forall a\le a_{M}. \end{aligned}$$

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

$$\begin{aligned} f_{\theta _{\mathbb {E}}+1}-\frac{f_{\theta _{\mathbb {E}}}}{r}< \frac{f_{{\theta _{\mathbb {E}}}}\tau _{{\theta _{\mathbb {E}}}}}{\tau _1}, \end{aligned}$$

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

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

$$\begin{aligned} c_t +\sum _{a=1}^n q_ax_{a,t+1} - \frac{1}{r}\sum _{a=1}^n q_ax_{a,t}\le \frac{f_{{\theta _{\mathbb {E}}}}\tau _{{\theta _{\mathbb {E}}}}}{\tau _1} \qquad \forall t \end{aligned}$$
(18)

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

$$\begin{aligned} \sum _{t=1}^Tr^tp_tc_t-\sum _{t=1}^Tr^tp_tc^*_t \le r^{T-1}\sum _{a=1}^nq_a(x^*_{a,T}-x_{a,T})p_0 \end{aligned}$$

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,

$$\begin{aligned} F'_r(a)=0\iff \frac{f'(a)}{f(a)}=\frac{\ln (1/r)}{1-r^{a}}. \end{aligned}$$
(19)

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

$$\begin{aligned} F''_r(a^*)=\frac{r^{a^*}}{1-r^{a^*}}f''(a^*)-\frac{r^{a^*}\ln ^2(r)}{(1-r^{a^*})^2}~f(a^{*}) <0. \end{aligned}$$

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 \(r\).

Fig. 3
figure3

Monotonicity of \(a^*(r)\)  \((\text{ with } r<r')\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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

JEL Classification

  • Q23
  • C61
  • D81