## 1 Introduction

In Niger, fuelwood is still the most important source of energy for cooking food. Because of drought, of a growing urban population, and of a partially uncontrolled access to the resource, pressure on forests around cities such as Niamey is high. Since 1989, a new governmental strategy has led to the protection of some forest areas, where harvesting is managed to ensure sustainability. The yield is then sold exclusively in neighboring state-approved markets. A differential tax system encourages traders to buy fuelwood from these structures instead of organizing the harvesting of uncontrolled forests [1].

Two management systems have been selected. In the ‘oriented rural markets’, only dead trees are harvested. In the ‘controlled rural markets’, apart from the dead trees, some live trees are also cut and the cutting is organized as a cycle to allow spatial rotation. A lower bound for the diameter of the live trees to be cut is sometimes fixed. It may take several cycles for a live tree to reach this diameter, so that dead trees form in general the main part of each harvest. Typically, cycle length is fixed between 5 and 10 years.

The harvesting policy in these ‘controlled rural markets’ is therefore quite similar to the selective logging practised in uneven-aged forests of other parts of the world. But the importance of the dead trees is rather unusual because of the destination (dead wood is more appropriate than recently cut green wood for cooking food), the absence of transformation, and its availability (the Sahelian climate induces a high death rate for the trees).

Despite the large literature dealing with various extensions of Faustmann's classical formula [2,3] for the optimal rotation problem ([4], and over 300 papers listed in [5]), no simple closed formula has been proposed for the optimal cutting cycle and diameter of uneven-aged stands subject to a high natural death rate, where dead trees form the main part of the harvest to be used as fuelwood. In [6], a closed formula was derived but for even-aged stands subject to a low catastrophic death rate such as fire. The model used continuous time, and the formula was found using the theory of renewal reward processes. It should be emphasized that for high natural death rate, isolated trees die from time to time and there may be a delay before they are cut during the next planned harvest. On the contrary, for a catastrophic death rate such as fire, the whole forest is affected almost instantly and the trees are removed shortly after. In the first case, simple spatial rotation can be maintained while in the second case, as stressed in [6], it cannot.

In [7], a closed formula generalizing Faustmann's formula to uneven-aged stands was derived, but the model focused only on the economics and did not consider any particular biological growth model. In [8], this formula was used in combination with a discrete-time matrix model similar to [9,10] for the growth, but no simple formula for the optimal cutting cycle could be found so that the results had to rely on simulations. Many references such as [11,12] stress the fact that Faustmann's model and many of its generalizations are particular examples of Markov decision processes, for which dynamic programming techniques can be used. The focus is then more on numerical algorithms than on simple closed formulas. In all these references, little attention was generally paid to natural death rate, because in the European or North-American context, it is rather negligible compared to several other factors.

The goal of this paper is therefore to derive – in the framework of a simple model for uneven-aged stands subject to a high natural death rate – a closed formula for the optimal cycle length and cutting diameter. The optimization criterion will be the same as in Faustmann's model, namely the discounted income over an infinite horizon. The interest is mainly theoretical since obviously, despite [13], much field work is still needed to obtain sufficient data to calibrate the model for the forests of Niger. However, as more and more ‘controlled rural markets’ are being organized in Niger and in the whole Sahelian region, more data may become available and the interest may shift from ‘how to manage the forest’ to ‘how to optimize the management’. The model is also intended to serve as the basis for the development of more complex and more realistic models, to take into account, for example, the price fluctuations reported in [14].

The plan of the paper is the following. In Section 2, the notations of the model are introduced. In Section 3, a formula is derived for the optimal cutting cycle and diameter, and it is discussed using a simple mathematical expression for the trees' growth curve. In Section 4, it is shown that Faustmann's classical formula corresponds to the limit of a negligible death rate ($\alpha \to 0$) and of a cutting diameter $D=0$ (clear-cut). It is also shown that the formula for the average yield of even-aged stands subject to a low but catastrophic death rate such as fire given in [6] corresponds to the limit of a cutting cycle length $T\to 0$ with a discounting factor $\beta \to 0$. Further extensions of the formula – to an age-dependent death rate and to a possible difference in price between dead trees and recently cut green trees – are also presented. They may be useful in other contexts.

## 2 A simple model

Suppose that a stand of trees has a fixed number N of spots for trees to grow (N is large). Suppose that $V(x)$ is the volume of a tree with age x, and that $V(0)=0$. Suppose that the price P per unit volume of the wood is constant. Let c be the cost of preparation for the growth of one new tree, and ${c}^{\prime}$ be a fixed cost per harvest. Let α be the death rate of the trees: during a short interval of time dt, $\alpha \phantom{\rule{0.2em}{0ex}}\mathrm{d}t$ is the percentage of the trees which die (if we think about the model as being deterministic), or the probability for one tree of dying (if we think about the model as being stochastic). For the moment, α is supposed to be independent of age, but this assumption will be relaxed in Section 4.

Suppose that the length of the cutting cycle is T and that every T year, the harvest consists of all dead trees and of those live trees which have a diameter greater than D. Let X be the age at which trees reach the diameter D. Regeneration is supposed to be purely artificial: trees that have been cut are replaced without delay by new trees with age 0, so that the total number of trees stays constant. Natural regeneration is supposed to be negligible during one rotation. Finally, let β be a discounting factor such that income at time $t=nT$ is weighted by the factor ${\mathrm{e}}^{-\beta nT}$ in the optimization criterion. The problem is to choose T and X in order to optimize the expected discounted income over all future harvests.

Formally, if $p(x,t)$ is the expected age-specific population density of live trees at time t, if $q(x,t)$ is the expected age-specific population density of dead trees at time t (when they die, trees stop aging), and if ${K}_{n}$ is the expected income of the harvest at time $t=nT$, then the problem can be formulated as a system of partial differential equations with impulse control. Recall that these partial differential equations are just the continuous-time equivalent of the discrete-time age-structured matrix models. Such an approach was adopted, for example, in [15,16], but for steady instead of periodic harvesting of forests. Between the harvests ($nT<t<(n+1)T$), the functions satisfy

$$\frac{\partial p}{\partial t}+\frac{\partial p}{\partial x}+\alpha p(x,t)=0$$ |

$$\frac{\partial q}{\partial t}=\alpha p(x,t)$$ |

$$p(x,n{T}^{+})=[\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\infty}{\int}}q(\xi ,n{T}^{-})\phantom{\rule{0.2em}{0ex}}\mathrm{d}\xi +\underset{X}{\overset{\infty}{\int}}p(\xi ,n{T}^{-})\phantom{\rule{0.2em}{0ex}}\mathrm{d}\xi ]{\delta}_{x=0}+p(x,n{T}^{-}){1}_{x\in (0,X)}$$ |

$${K}_{n}=\underset{0}{\overset{\infty}{\int}}q(x,n{T}^{-})[PV(x)-c]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\underset{X}{\overset{\infty}{\int}}p(x,n{T}^{-})[PV(x)-c]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-{c}^{\prime}$$ |

$$\sum _{n=1}^{\infty}{\mathrm{e}}^{-\beta nT}{K}_{n}$$ |

## 3 Optimal cutting cycle and diameter

Whatever the age structure of the initial conditions, it can be easily realized that the age specific densities $p(x,t)$ and $q(x,t)$ converge after a certain number of years to T-periodic solutions $\stackrel{\u02c6}{p}(x,t)$ and $\stackrel{\u02c6}{q}(x,t)$ which satisfy

$$\stackrel{\u02c6}{p}(x,n{T}^{-})=N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}\sum _{k=1}^{i}{\mathrm{e}}^{-k\alpha T}{\delta}_{x=kT}$$ |

$$\stackrel{\u02c6}{q}(x,n{T}^{-})=N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}\alpha {\mathrm{e}}^{-\alpha x}{1}_{x\in (0,iT)}$$ |

$$\stackrel{\u02c6}{p}(x,n{T}^{+})=N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}\sum _{k=0}^{i-1}{\mathrm{e}}^{-k\alpha T}{\delta}_{x=kT}$$ |

$$N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}\underset{0}{\overset{iT}{\int}}\alpha {\mathrm{e}}^{-\alpha x}[PV(x)-c]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x$$ |

$$N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}{\mathrm{e}}^{-i\alpha T}[PV(iT)-c]$$ |

$${K}_{n}=N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}[P\underset{0}{\overset{iT}{\int}}{\mathrm{e}}^{-\alpha x}{V}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-c]-{c}^{\prime}$$ |

$$\frac{1}{{\mathrm{e}}^{\beta T}-1}\{N\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}[P\underset{0}{\overset{iT}{\int}}{\mathrm{e}}^{-\alpha x}{V}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-c]-{c}^{\prime}\}$$ | (1) |

$$NP{V}_{\infty}\frac{1}{{\mathrm{e}}^{\beta T}-1}\{\frac{1-{\mathrm{e}}^{-\alpha T}}{1-{\mathrm{e}}^{-i\alpha T}}[\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{iT}{\int}}{\mathrm{e}}^{-\alpha x}\frac{{V}^{\prime}(x)}{{V}_{\infty}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{c}{P{V}_{\infty}}]-\frac{{c}^{\prime}}{NP{V}_{\infty}}\}$$ |

To discuss the formula, consider a simple mathematical expression for the growth curve of the trees $V(x)$, or equivalently for its derivative ${V}^{\prime}(x)$, for example,

$${V}^{\prime}(x)={V}_{\infty}\frac{{(\gamma /\tau )}^{\gamma +1}}{\mathrm{\Gamma}(\gamma +1)}{x}^{\gamma}{\mathrm{e}}^{-\gamma x/\tau}$$ |

Fig. 1 illustrates how the expected discounted income can vary as a function of the length T of the cutting cycle for different values of the cutting diameter (corresponding to $i=1\dots 4$). The curve most to the right is the one for $i=1$ (clear-cutting and even-aged management). The optimal strategy is obtained with ${T}^{\ast}\simeq 9.5$ and $i=2$, i.e., by cutting every ${T}^{\ast}$ years the dead trees and the live trees aged $2{T}^{\ast}$ or more. Notice that all curves become negative when T approaches 0 because of the nonzero fixed cost per harvest. The parameter values used for this figure are $\alpha =0.05$ per year (life expectation of $1/\alpha =20$ years for the trees), $\tau =10$ years (age at which the growth is the fastest) and $\gamma =2$ for the growth curve, a discounting factor $\beta =0.03$ per year, costs $c/(P{V}_{\infty})=0$ and ${c}^{\prime}/(NP{V}_{\infty})=0.1$, and a normalized maximum forest value $NP{V}_{\infty}=1$.

Fig. 2 illustrates how the optimal cutting cycle and diameter can change when the death rate α, the discounting factor β, and the fixed cost per harvest ${c}^{\prime}$ vary. The values for the fixed parameters are the same as for Fig. 1. As can be expected, uneven-aged management with a short cutting cycle (represented by areas labelled $i=2$ and $i=3$) appears more profitable than even-aged management ($i=1$) when the death rate is high (top-left and top-right diagram), when the discounting factor is high (top-left and bottom diagram), or when the fixed cost per harvest is low (top-right and bottom diagram). Notice from the top-left (respectively, top-right) diagram that even in the limit $\alpha \to 0$ (the case of Faustmann's model), uneven-aged management becomes more profitable than even-aged management if the discounting factor (respectively, fixed cost per harvest) is above (respectively, below) a certain threshold value.

Fig. 2 also shows the level curves of the optimum length of the cutting cycle. Notice that there is no simple relationship between the optimal length of the cutting cycle and the death rate: the former may be an increasing or a decreasing function of the latter, depending on parameter values, as in the top-right diagram.

## 4 Limit cases, extensions

If one sets the critical diameter D to cut the live trees to 0 (clear-cut), then the corresponding age is $X=0$. So $i=1$, and formula (1) reduces to

$$\frac{1}{{\mathrm{e}}^{\beta T}-1}\{N[P\underset{0}{\overset{T}{\int}}{\mathrm{e}}^{-\alpha x}{V}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-c]-{c}^{\prime}\}$$ |

$$N\frac{PV(T)-c}{{\mathrm{e}}^{\beta T}-1}$$ |

Another limiting case is obtained by letting the discounting factor β and the cutting cycle length T tend to 0, which corresponds to looking at the average income with a permanent screening of the forest to harvest dead trees and live trees aged over X. Recalling the relationship between average income per year and discounted income over all future harvests

$$\underset{\nu \to \infty}{\mathrm{lim}}\frac{1}{\nu T}\sum _{n=1}^{\nu}{K}_{n}=\underset{\beta \to 0}{\mathrm{lim}}\beta \sum _{n=1}^{\infty}{\mathrm{e}}^{-\beta nT}{K}_{n}$$ |

$$N\alpha \frac{P{\int}_{0}^{X}{\mathrm{e}}^{-\alpha x}{V}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-c}{1-{\mathrm{e}}^{-\alpha X}}$$ | (2) |

$$\stackrel{\u02c6}{p}(x)=N\frac{\alpha {\mathrm{e}}^{-\alpha x}}{1-{\mathrm{e}}^{-\alpha X}}{1}_{x\in (0,X)}$$ |

Extensions. Suppose now that the death rate, instead of being constant, depends on age. Set $\mathrm{\Delta}(x)=\mathrm{exp}(-{\int}_{0}^{x}\alpha (\xi )\phantom{\rule{0.2em}{0ex}}\mathrm{d}\xi )$. Suppose that the price of dead wood is Q, which can differ from the price of recently cut green wood P. The expected discounted income over all future harvests is then

$$\frac{1}{{\mathrm{e}}^{\beta T}-1}\{N(\frac{\mathrm{\Delta}(iT)[PV(iT)-c]}{{\sum}_{n=0}^{i-1}\mathrm{\Delta}(nT)}-\frac{{\int}_{0}^{iT}{\mathrm{\Delta}}^{\prime}(x)[QV(x)-c]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{{\sum}_{n=0}^{i-1}\mathrm{\Delta}(nT)})-{c}^{\prime}\}$$ |

$$N\frac{\mathrm{\Delta}(X)[PV(X)-c]-{\int}_{0}^{X}{\mathrm{\Delta}}^{\prime}(x)[QV(x)-c]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{{\int}_{0}^{X}\mathrm{\Delta}(x)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}$$ |

## 5 Conclusion

In the framework of a simple model, a formula for the optimal cutting cycle and diameter was derived when the harvesting policy consists in cutting every T year the dead trees and the live trees with a diameter greater than a fixed value D. The model was designed for forests subject to a high natural death rate, where dead trees form the main part of the harvest and are used as fuelwood. It was seen that uneven-aged management was more profitable under such circumstances. For a low natural death rate, a low discounting factor, or a high fixed cost per harvest, even-aged management can be more profitable.

Forest management in Niger and in other Sahelian countries is still in its early phase. Rotation has been introduced only a decade ago. Field data are still very limited. Though a considerable amount of research has been accumulated during more than 150 years concerning the optimal rotation problem in developed countries, adaptation to local conditions is necessary. In this study, we focused on only one specific aspect, namely the high natural death rate. Hopefully, this microeconomic model will be combined with the macroeconomic models [17–19] to provide a more complete modelling tool to decision makers.

## Acknowledgments

The second author would like to thank Édith Perrier (IRD), Alain Bertrand, Martine Antona, Christophe Le Page (CIRAD) and Aboubacar Ichaou (Environment Department of Niger). This paper is dedicated to the memory of Ovide Arino.