_{1}

^{*}

A new Faustmann optimal rotation harvesting stands’ problem under Brown geometric price and Logistic and Gompertz wood stock, diffusions is presented. The optimal cut policies for the stochastic Faustmann model and the single harvest rotation or Vicksell model are evaluated in the case of a Chilean Radiata pine forest company. The company cut policy validates the Vicksell model, its optimal cut policies overestimate the company policy cut in 1.2%, in the Gompertz case, and underestimate it in 2.3%, in the Logistic case. The Faustmann optimal cut policies present a larger underestimation of the company cut policy in 10.1%, in the Gompertz case, and in 21.5%, in the Logistic case. The preference for shorter evaluation period that the company shows is due to the organizational risk that the forest economic sectors has in Chile.

The optimal forest harvest models development is marked by the emergence of two controversial issues. The first issue was formulated in its early beginning by Faustmann in 1849 . He formulated a multiple rotation version considering the dynamic effect that the future renovations had in the rotation period of the cut and planting of the trees. The issue was not considered relevant by researchers and forester management, who preferred a simple rotation or Vicksell model. The Faustmann model was finally rediscovered, first by Gaffney in 1957 , and finally by Samuelson in 1976 , who validated Faustmann’s deterministic formula as the correct one, since it was the only one to consider the cost of land rent. An increasing number of researchers continued extending this model (Brazee, 2001; Chang, 2001; Amacher et al., 2011) . The second issue was strongly formulated by Samuelson (1976) , calling to replace the simple notion of stationary equilibrium by the notion of perpetual Brownian motion of wood price. Many researchers followed his recommendation and considered the single rotation problem as an American call option, taking into account wood price as a geometric Brown diffusion (see, Newman, 2002; Clarke & Reed, 1989 ; Thomson, 1992; Platinga et al., 1998). Other authors also considered the wood stock as a second geometric Brown diffusion (Morck & Schwartz, 1989; Insley, 2002) , or as a Logistic diffusion (Alvarez & Koskela, 2007; Navarrete, 2011) . Willassen (1998) introduced uncertainty to the Faustmann model, incorporating the forest growth as a stochastic Markov diffusion process and characterizing the properties of the optimal solutions. Sodal (2002) simplified Willassen’s approach in a closed-form rotation formula for the same state stochastic variable and Insley & Rollins (2005) proposed a numeric algorithm to solve its Hamilton-Jacobi- Bellman solution. Navarrete & Bustos (2013) extended the Faustmann model considering, price as a geometric Brown and wood stock as a Logistical or Gompertz, diffusions processes, transforming the model into an equivalent optimal stopping problem, in spite of its contribution. This paper requires some improvement which will be done in the present paper. Therefore this paper has two basic objectives. In section 2.3, the proof of the equivalence of the Faustmann rotation model under Brown Price and ITO wood stock independent diffusion processes and a one dimensional modified wood stock Optimal Stopping problem is formalized as a reformulation lemma. In section 4, a new algorithm is developed to obtain the Faustmann equivalent one dimensional optimal stopping solution, intersecting the optimal condition curve with the expected wood stock growth feasibility restriction. Finally new stands growth data for the middle age 11 and 12 years is incorporated in Section 3, in order to validate Vicksell and Faustmann stands cut models, under price and wood stock uncertainty.

The basic model considers two independent stochastic processes: an ITO diffusion for the wood stock, Equation (1) and a geometric Brownian diffusion, Equation (2). The functional objective for the single rotation is given by Equation (3) and for the multiple rotations by Equation (4) see Johnson (2006) :

Model notation:

Deterministic state variable Economic Parameter

t = Wood stock age C = Stand regeneration cost

Stochastic sate variables P_{0} = Price at time 0

V_{t} = Wood stock c = C/P_{0}

P_{t} = Wood stumpage price at time t R, Q = Probabilistic metrics

Diffusion Parameters F, Z = Functional objective

W_{v} = Wiener wood stock i = Risky rate of return

μ(V) = Wood stock diffusion drift rate parameter r_{T} =

σ(V) = Wood stock volatility parameter Optimal Parameters

W_{p} = Wiener price T = Optimal cut time

α = Wood price diffusion drift rate

β = Wood price volatility

The stochastic model (1) (2) and (3) is difficult to solve due to both diffusions. But it is equivalent to the following one dimensional diffusion Optimal Stopping problem, (5) and (6) see Navarrete (2011) .

The solution of this problem is given by the (HJB) Equation (7), see Navarrete (2011) .

Assuming the existence of a frontier V^{*} that divides the zone into a continuation zone (no-cutting) and a stopping zone (immediate-cutting), the solution to the equation HJB is given by (8) for the continuation zone (8) and stopping zone (9). A solution of (8) is given by (10) (see Johnson, 2006 ).

where ^{*}, equations (11) and eliminating the constant A gives Equation (12).

The Faustmann optimal rotation model (1) (2) and (4) its equivalent to the following Optimal Stopping one dimensional diffusion problem (13) and (14), see Lemma 1.

Lemma 1. Reformulation Lemma:

The Faustmann optimal rotation problem is equivalent to the following Optimal Stopping one dimensional diffusion problem (13) and (14).

Proof:

Given the independence of both variables the expectation can be calculated as the product of two independent expectation “

_{0}c in the functional objective (4) results in the objective (15) and reduce the Faustmann problem to the following one dimensional diffusion problem (15) and (16).

Dividing the objective functional (15) by the constant P_{0} and applying the Thijssen version of the Girsanov theorem to Equations (15) and (16), for the Martingale M_{t} and the Radom-Nykodym derivative

The formulation of the HJB equation for this problem is given by Equation (17) for the capitalized risky rate of return

In this case the differential equation for the continuation region

The solution to the ordinary differential equation, (18) under the initial condition for a given risky rate of return r_{T} is given in Equation (20), with

In this case the smooth pasting condition for each parameter r_{T} are given by Equation (21):

So _{T}.

The basic requirement of a pine stand growing diffusion is its sigmoid pattern (Garcia, 2005) . The logistic diffusion, Equation (23) is a special case of the sigmoid model given by

The integration of the value of V is given by Equation (24) (Kloeden & Platen, 1992: p. 125) and its expected value is given by Equation (25)

Other important sigmoid diffusion is the Gompertz geometrical diffusion, which is given by Equation (26), which is integrated to the expression (27), and its expected values given by expression (28) (see Gutierrez, 2009).

The Geometric Brown Price diffusion is given by Equation (29), and integrates in Equation (30).

The experimental data belonged to 122 harvest pine stands of a Chilean Forest Company in the Araucania Chile region, between years 1999 and 2005 (see Appendix A), and came from different sample plots, with site indexes between 30 and 35 meters, representing sites with high forest aptitude and a tree average initial volume of 32 m^{3}/ha at the first 4 years after initial seed cultivation period. The additional 20 point for years 11 and 12 were taken from Alvarez et al. (2012). The business harvest cut data for a Logistic diffusion for the 95% confidence range is given by data point plotted in

The logistic diffusion parameter cannot be adjusted by maximum verisimilitude, see (Beskos et al., 2006) , so it was fitted using a logistical nonlinear regression and a Monte Carlo/Bootstrap simulation sampling method, implemented by Meyer et al. (Loglet Lab.1 software, 1999) . Choosing V_{0} = 1/2γ = half saturation volume, and T_{0} = T_{m}, time to achieve that volume, Equation (25) is transformed into the more conventional expression (31).

With; 1/g = saturation volume, m = growth rate parameter, V_{s} = Saturation volume and t_{m} = time to achieve the midpoint of the saturation volume. The standard deviation Sd(∞), at the saturation zone is constant, and given by Equation (32).

Since the saturation volume V_{s} is also constant, sigma can easily be estimated by Equation (33) and the summary of the parameter fitting is shown in

The Gompertz model can be fitted by common statistic features, such as maximum verisimilitude (see Gutierrez et al., 2008 ), but the lack of equal time distribution of data make it difficult. So a quadratic fitting method was development using the SSPS software. By taking natural logarithm and arranging it we get Equation (34).

with

Given a value for k, a quadratic fitting for _{0} = 32 (m^{3}/ha) and the results are summarized in

The stumpage stand price Brownian diffusion parameters are estimated from saw logs and pulp logs exportation prices (see Appendix B). The summary of Brown diffusion parameters for the stumpage price and the regeneration cost of radiata Pine Stands are given in

The deterministic optimal solution given by

Models | Drift Parameter µ | Saturation Volume | Saturation Parameter γ | Volatility Parameter σ |
---|---|---|---|---|

Stochastic | 0.191 | 576.0 | 0.00174 | 0.12 |

Deterministic | 0.191 | 576.0 | 0.00174 |

Parameters | Saturation | Drift | Drift | Volatility | |
---|---|---|---|---|---|

Models | V_{s} | k | Θ | σ | R^{2} |

Gompertz | 653.3 | 0.102 | 6.538 | 0.151 | 0.992 |

Deterministic | 653.3 | 0.102 | 6.538 |

Price stumpage drift | α | 2.9% |
---|---|---|

Price stumpage volatility | β | 15.9% |

Actual stumpage log price | P_{T} | 39.74 US$/ha |

Initial stumpage price | P_{0} | 21.43 US$/ha |

Risky rate of Capital | WACC | 12% |

Stands regeneration cost | C | 882 US$/ha |

Stands cost per unit initial price | c = C/P_{0} | 41.16 |

Source: Appendix B.

The solution of the Vicksell and Faustmann model under Brown price and Logistic wood stock diffusion requires the solution of the differential Equations (8) and (19) for _{T} = i in the Vicksell case.

The solution of Equation (37) is given by the Kummer’s confluent hyper geometric function, expression (38) with the positive root θ by Equation (39), ψ(V) also must fulfill the smooth pasting condition (12), which was programmed in Mapple 15, see Navarrete (2013).

The Faustmann stochastic optimum requires the solution of the differential Equation (16), so the positive function ψ(V) is the solution of its homogenous Equation (37) for the parameter r_{T} and also fulfill the smooth pasting condition, (22). For a given value of the capitalized interest _{T}. The optimum solution is obtained intersecting the optimal parametric solution for different T with the corresponding expected value logistical modify diffusion, (14) for

The simple optimal cut is a better explanation of the company cut policy, since it only underestimates it in a 2.5%. As expected the Faustmann optimal cut is lower and underestimate the company policy in 11.5%. The deterministic optimal cuts are even worse and underestimate the company policy in 23.4% in the simple model and in 25.6% in the multiple cases. Finally the Logistic wood stock underestimates the saturation Volume in 4.2%.

The deterministic optimal solution, is given by

Optimum Policy | Simple m^{3}/ha | Harvest % | Multiple m^{3}/ha | Harvest % | Saturation m^{3}/ha | Volume % |
---|---|---|---|---|---|---|

Company | 392.9 | 100.0 | 392.9 | 100.0 | 601.6 | 100 |

Deterministic | 300.9 | −23.4 | 292.3 | −25.6 | ||

Stochastic | 383.1 | −2.5 | 347.6 | −11.5 | 576 | −4.3 |

The solution of the Vicksell and Faustmann model under Brown price and Gompertz wood stock diffusion requires the solution of the differential Equations, (8) and (16) for _{T} = i for the simple rotation case and the capitalized rate of interest r_{T} for the Faustmann case.

Replacing the parameters given in Equations (42) in (41), gives the differential Equation (43) of the Exponential, Ornstein-Uhlembeck diffusion whose positive increasing solution ψ(V) is given by Equation (44) (see Johnson, 2005), with:

The optimal solutions also requires that ψ(V) fulfills the smooth pasting condition (38), which is programmed in Mapple 15 generating one optimal cut solution for the Vicksell model and a family of optimal cut condition parametrized by T for the Faustmann case. In this late case the optimal solution is obtain intercepting the family with the expected volume of the Gompertz modify diffusion (20), see

The deterministic and stochastic optimum were programed in Maple 15, using in this case the KummerU function of the program, the results are summarized in

1) The optimal cut company policy validates the use of the simple stochastic rotations model under Bown price and Logistic or Gompertz wood stock diffusion.

2) The discrepancy in the theoretical and practical cut policy can be explained by the preference that the business policy gives to shorter rotations periods 25 or less years due to the high organizational risk of the industrial sector in Chile.

3) The Gompertz and Logistic diffusion models present small estimation differences in the growing phase of wood stock, but significant differences in the saturation volume of the wood stock, which should be crucial in the model diffusion selection.

The author acknowledge the collaboration of P. Santibañez of MININCO (2006) forest company for providing with the Radiata pine stands harvest near Temuco and to “Modelo Nacional de Simulación de Pino Radiata” for the complementary stands data of the region.

Optimum Policy | Simple mts^{3}/ha | Rotation % | Multiple mts^{3}/ha | Rotation % | Saturation mts^{3}/ha | Volume% |
---|---|---|---|---|---|---|

Company | 392.9 | 100.0 | 392.9 | 100.0 | 601.6 | 100 |

Deterministic | 283.2 | −27.9 | 262.8 | −33.1 | ||

Stochastic | 397.6 | 1.2 | 353.2 | −10.1 | 653.3 | 8.6 |

EduardoNavarrete, (2015) Optimal Stochastic Pine Stands Harvest Rotation Policies. Open Journal of Forestry,05,593-606. doi: 10.4236/ojf.2015.56053

Notes: TVOL = Total Volume (mts3/ha); TVOL = POD + DEBO + INDUST + COMERC + PULPA POD + DEBO = Wood thinning cuts; Logs composition: Saw logs = INDUST + COMERC = 83.9%; Pulp logs = PULP + POD + DEBO = 16.1%; Saturation volume Vs = 601.6 m^{3}/ha, (3% stands biggest TVOL average).

YEARS | Saw logs | Pulp logs | YEARS | Saw logs | Pulp logs |
---|---|---|---|---|---|

US$/mts^{3} | US$/mts^{3} | US$/mts^{3} | US$/mts^{3} | ||

1985 | 32.0 | 27.0 | 1997 | 62.0 | 55.0 |

1986 | 34.0 | 28.0 | 1998 | 52.0 | 54.0 |

1987 | 39.0 | 27.0 | 1999 | 49.0 | 53.0 |

1988 | 45.0 | 27.0 | 2000 | 46.0 | 42.0 |

1989 | 43.0 | 27.0 | 2001 | 48.0 | 34.0 |

1990 | 49.0 | 32.0 | 2002 | 46.0 | 41.6 |

1991 | 51.0 | 40.0 | 2003 | 45.9 | 37.4 |

1992 | 47.0 | 40.0 | 2004 | 48.6 | 33.0 |

1993 | 85.0 | 38.0 | 2005 | 57.0 | 33.5 |

1994 | 63.0 | 46.0 | 2006 | 60.0 | 36.0 |

1995 | 67.0 | 43.0 | 2007 | 63.0 | 40.0 |

1996 | 65.0 | 52.0 |

Source CONAF-INFOR Chile.

The Price diffusion parameters can very easily be calculated by making the following logarithmic transformation

Summary | Stumpage logs | Saw logs | Pulp logs |
---|---|---|---|

Percentage | 100% | 83.9 | 16.1 |

Price drift α | 2.9% | 3.08 | 1.79 |

Price volatility β | 15.9% | 16.52 | 12.74 |

Average Price | 39.74 |