Abstract - ecoSERVICES Group

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Journal MSP No. No. of pages: 31 PE: Matthew

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NATURAL RESOURCE M ODELLINGVolum e 21, Number x, xxx 2008

A BIOECONOMIC APPROACH TO THEFAUSTMANN–HARTMAN MODEL: ECOLOGICAL

INTERACTIONS IN MANAGED FOREST

JULIA TOUZAHelmholtz Centre of Environmental Research - UFZ, Department of Ecological

Modelling, Permoserstr. 15, 04318 Leipzig, GermanyE-mail :[email protected]

METTE TERMANSENSustainability Research Institute, School of Earth and Environment,

University of Leeds, Leeds LS2 9JT, U.K.CHARLES PERRINGS

ecoSERVICES Group, School of Life Sciences, Arizona StateUniversity, Box 874501, Tempe, AZ 85287-4501

Abstract. This paper develops a bioeconomic forestry

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model that makes it possible to take ecosystem services thatare independent of the age structure of trees into account. Wederive the Faustmann–Hartman optimal harvesting strategyas a special case. The bioeconomic model is then extended toaccount for the fact that forest harvesting decisions impacton other ecological resources, which provide benefits for thewider community. The paper focuses on impacts associatedwith disturbance caused by logging operations and habitat de-struction due to tree removal. This enables us to explore theinteractions between forest management and the dynamics ofecological resources. The optimal rotation rule is obtained asa variation on the traditional Faustmann–Hartman equation,where an additional term captures the potential benefits de-rived from the growth of the ecological resource valued at itsshadow price. The steady-state solutions to the problem andsensitivity to model parameter are identified using numericalanalysis.

Key Words: Bioeconomic model, Faustmann–Hart-man, nontimber benefits, disturbance effect, habitat effect.

1. Introduction. Forest ecosystems are the source of many directand indirect benefits. Aside from timber, they provide habitats formore terrestrial species than any other ecosystem type, watershed pro-tection, the control of soil erosion and hence siltation, microclimatic

Q3Received by the editors on xxx xxx xxxx, and in revised form on xxxx xx xxxx.

Copyright c©2008 2008 B lackwell Publish ing, Inc.

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2 J. TOUZA, M. TERMANSEN, AND C. PERRINGS

regulation, and macroclimatic regulation through their role in carbonsequestration. They also provide a wide range of nontimber products,and recreational, cultural, spiritual, and amenity benefits. It is wellrecognized that forests should be managed for the multiple benefitsthey provide (e.g., Bowes and Krutilla [1985], Hof [1993]). Moreover,much has been done to explore the implications of nontimber bene-fits for harvesting strategies within a traditional Faustman framework(Brazee, [2006]). However, the interaction between these benefits andthe implication this has for forest management are less well under-stood. Harvesting, as a perturbation to the system, and the removal ofbiomass both have significant impacts on the provision of other forestbenefits. Some of these interactive effects are best described as localshort-term effects. An example would be the impact of the temporarydisturbance due to logging in a small fraction of a forested landscape ona mobile ecological resource. Other effects are better described as locallong-term effects. An example might be a change in forest structurethat alters its value as habitat for some species. Yet other interactionswould be revealed on larger spatial scales affecting other ecosystems atsome distance from the logging activities. Again, some of these interac-tions may only be short-term effects such as temporary eutrophicationof down-steam rivers and lakes. Others may be longer-term effects likesiltation of dams and estuaries.

An awareness of the importance of these nontimber benefits has be-gun to affect the management of production forests. To this point,however, the analysis of the impact of nontimber benefits in forestmanagement has been somewhat limited by the modeling frameworkwithin which the management problem is posed. The economics offorestry has been dominated for over a century by the approach pro-posed by Faustmann (1849). The Faustmann formula identifies the ageat which an even-aged forest stand should be harvested in order tomaximize the return to forestry. This problem is formulated as themaximization of the net present value of the perpetual returns fromthe land, and it was solved by Pressler [1860] and Ohlin [1921]. Thus,the Faustmann–Pressler–Ohlin (FPO) model, or simply the Faustmannmodel, is the first valid representation of forest management practices.1

The model is intuitive and analytically simple. It constitutes the frame-work within which many subsequent forest rotation models have ad-dressed specific questions. Particular influences on the rotation age

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A BIOECONOMIC APPROACH 3

that have been modeled within the Faustmann framework are vari-

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ability in timber prices and planting costs (McConnell et al. [1983],Newman et al. [1985], Thomson [1992], Yin and Newman [1995]) andthe impact of royalties and taxation (Chang [1982], [1983], Englin andKlan [1990]). Uncertainty in the Faustmann model was explored byReed [1984], who analyzed forest fire risk, Willassen [1998], who in-cluded stochastic effects on the forest value growth, and Reed [1993]and Reed and Ye [1994], who explored the option value associated withamenity services in old-growth forest stand. Following Hartman [1976],the importance of nontimber benefits and the effect of forest-wide con-siderations have been addressed by a number of authors (Strang [1983],Paredes and Brodie [1989], Snyder and Bhattacharya [1990], Swallowet al. [1997], Koskela and Ollikainen [2001]), but in all cases the analysisis constrained by the structure of the Faustmann model. Specifically,nontimber benefits are always considered to be a direct function of theage of trees.

A second strand of the literature on the economics of forestry relieson dynamic bioeconomic models. These follow an optimal control ap-proach, and offer a more flexible mathematical framework. They havebeen used to address a range of forest management activities (Schreuder[1971], Montgomery and Adams [1995], Termansen [2007]). These mod-els analyze selective harvest management in uneven-aged forests, basedon the determination of the optimal cutting volume at each instant.This decision is based on current timber biomass and the growth rate.Thus, the models categorize forest stands by their biomass and not bythe age of the trees as contrast to the Faustmann framework, whichcharacterizes the forest stand by its age-class (Johanson and Lofgren[1985]). However, optimal control models with jumps have been usedfor age-class-based forest management within a utility maximizationframework by Tahvonen and Salo [1999] and Tahvonen [2004].

In this paper, we connect these two modeling traditions in order toanalyze the interaction between timber harvesting and nontimber bene-fits. The paper sets up an analytically simple and empirically tractablebioeconomic model which identifies optimal harvest age for a singleeven-aged stand. The Faustmann–Hartman formula is derived as theoptimal timber cutting rule from the solution of this bioeconomic prob-lem. We then extend this model to explore the effects on the optimalharvest decision of having the nontimber benefits dependent on an


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ecological resource. The model thus includes the dynamics of an eco-logical resource, which is partly determined by timber harvest activitiesbut is not directly controllable by the forest manager. In the Faustmannframework, the specification of nontimber benefits has been a functionof the stand age. We choose an alternative specification to reflect thefact that many nontimber benefits, including a number of provisioningand regulating ecosystem services, are not purely dictated by the ageof trees. Rather, they are a result of past and current managementactivities and the dynamics of the ecological resource. The paper fo-cuses two effects of harvesting. The first is a disturbance effect causinga short-term impact either on a local or on a large scale. This case iscompared to a habitat destruction effect, where the ecological resourcereduces to the carrying capacity of the postharvest forest.

By combining the traditional Faustmann framework with the bioeco-nomic framework, we show that the Faustmann–Hartman rule arises asa special case of a bioeconomic model constrained to represent harvest-ing only as clear-cutting. Furthermore, by defining nontimber benefitsas a function of the ecological resource stock we are able to evalu-ate the effect of timber management activities on the dynamics of theecological resource stock. The resulting optimal harvesting conditioncan be interpreted as a Faustmann–Hartman-like equation, which in-cludes new elements, associated with the disturbance of the ecologicalresource. To aid understanding this optimal solution is simulated foran even-aged stand of maritime pine (Pinus Pinaster) in the westernpart of Spain. The numerical simulations illustrate that the optimalrotation period depends on the impact that harvesting has on the eco-logical resource, the ability of the ecological resource to recover and theextent to which the carrying capacity of the system has been affected.

The paper is organized as follows. Section 2 sets up a bioeconomicmodel that represents clear-cutting decisions for a single even-agedstand. In Section 3, this model is extended to evaluate the interactionsbetween timber harvesting and the dynamics of ecological resources.Section 4 shows the results from the numerical simulations. In Sec-tion 5, we discuss the major findings of the paper.

2. A Basic Bioeconomic Model for an Even-Aged Stand.The model is a forest management model for a single even-aged stand,optimizing the sequence of clear cuttings. There is one state variable


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in this model, timber stock, x(t). F (x(t)) is the timber growth func-tion. This annual growth function is assumed to be concave, such thatF (0) = F (KT ) = 0 and KT > 0, where KT is the maximum volume agiven stand can accumulate. Changes in forest biomass due to harvestactivities are represented by jumps in the state variable (Seierstad andSydsaeter [1987]). These jump points, denoted by τ j , occur at discretemoments within the planning period, τ j ε [0, ∞], where j = 1, . . . , kand k may be infinite.2 Between harvests the change in the timberstock is given by the growth function. At harvest points, there is adiscontinuous change in the biomass, where x(τ−

j ) is the biomass justbefore harvest and x(τ+

j ) denotes the biomass just after harvest.

The sequence of jump points (τ j ) determines the stand rotation pe-riods. Rotation period j is denoted τ−

j − τ+j−1 . When j = 1, τ+

j−1 =τ+

0 is the initial date, and planting is assumed to occur at this date.

Let xp represent the biomass of the seedlings; and p, cp , and δ be thetimber price, the cost of planting, and the interest rate, respectively.The financial reward from harvesting the stand is the gross timber ben-efit from harvest, px(τ−

j ), minus the cost of planting, cpxp . Followingthe Faustmann–Hartman framework, re-planting activities follow im-mediately after harvest. Nontimber benefits, π(x(t)), are obtained fromthe flow of services provided by the forest ecosystem, and are specifiedto be a function of timber biomass. As the harvest is clear-cut, non-timber services are indirectly a function of the age of the trees. Themanagement objective is to maximize the value arising from both thetimber and nontimber benefits of the stand by choosing the optimumsequence of jump points (τ j ) or clear-cut harvests.

Under these conditions, the optimization problem can be expressedas,

maxτj ,k

k∑j=1

∫ τ −j

τ +j−1

π(x(t))e−δt dt +k∑

j=1

[px

(τ−j

)− cpxp

]e−δτj(1)

subject to,

x = F (x(t)) except at τj , j = 1, . . . , k(2)

x(τ+j

)− x

(τ−j

)= −x

(τ−j

)+ xp ∀ τj , j = 1, . . . , k(3)


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x(0) = xp.(4)

Equation (2) implies that between harvests the dynamics of thebiomass volume of the stand is given by the natural growth of theforest. Equation (3) relates to the magnitude of the jumps in the statevariable at the moment of harvest. Equation (4) is the initial stockconstraint.

The current-value Hamiltonian, and the first-order necessary condi-tions for the optimal solution of this problem, where µ(t) is the costatevariable for the timber stock, are as follows:

H = π(x(t)) + µ(t)F (x(t))(5)

µ = −∂H

∂x+ δµ(t) = −π′(x(t)) − µ(t)F ′(x(t)) + δµ(t)(6)

µ(τ−j

)= p(7)

H(τ−j

)= δ

[px

(τ−j

)− cpxp

]+ H

(τ+j

).(8)

Condition (6) must hold between jumps points. It states that the im-puted value of a unit of the state variable should be changing at a rateequal to its marginal contribution to the return on forestry. Because theproblem is expressed in current value terms, it will also increase withthe rate of interest. Conditions (7) and (8) need to be satisfied at thepoint of harvest. Condition (7) states that the optimal value of shadowprice of timber should equal the timber price at the optimal harvestpoint. Condition (8) characterizes the optimal choice of the sequenceof harvest, τ j . Recall that the Hamiltonian function is the sum of theinstantaneous value and the value of the growth of the capital stock.Thus in condition (8), the Hamiltonian immediately before the jump,H(τ−

j ), is the marginal increase in the objective function of postponingτ−

j ; the Hamiltonian just after the jump, H(τ+j ) is the marginal contri-

bution to the objective function of delaying τ+j , that is, of delaying by

one instant the following rotation; and the financial opportunity costof postponing harvest by one instant. This is the benefit that could beearned if the harvest were to be taken now and the net profits investedelsewhere.


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The solution of the first-order differential equations given by equa-tion (6) (see Appendix A) indicates that the shadow value of the timberstock at the beginning of each rotation, µ(τ+

j ), is determined by thetimber and nontimber benefits of the following rotation.

In order to explore the optimal cutting condition (8) further, weevaluate H(τ−

j ) and H(τ+j ) using the Hamiltonian function in equa-

tion (5).3 Substituting µ(τ+j ) (equation (A.6), Appendix A), and con-

dition (7), into the optimal harvesting rule, we derive the followingcondition,

pF(x(τ−j

))+π

(x(τ−j

))= δ

[px

(τ−j

)− cpxp

]+ pF

(x(τ−j+1

))e−δTi+1 +π

(x(τ−j+1

))e−δTi+1

+ δ

∫ τ −j+1

τ +j

π(x(t))e−δ(t−τ +j )

dt,

(9)

where T i+1 = (τ−j+1 − τ+

j ) is the harvest age of the following rotation.

Cutting condition (9) gives the harvest age for a single even-agedstand that maximizes the economic returns on forest activities. Theleft-hand side of (9) is the value of the Hamiltonian just before thejump, H(τ−), which is the marginal timber and environmental benefitsgained by delaying the harvest. The first term on the right-hand sideis the income that could be earned if revenue from harvesting wereinvested at an interest rate δ. The last three terms are the value ofthe Hamiltonian just after the jump, H(τ+). This is interpreted as theopportunity cost of postponing the future value of the timber stockand nontimber benefits. This opportunity cost is given by the marginalvalue of the timber and nontimber production at the following rotation,including the value of postponing the continuous flow of nontimberbenefits. This shows that the value of the Hamiltonian after the jumpin this dynamic framework is analogous to the concept of “site value”in the traditional Faustmann–Hartman model. Furthermore, consistentwith the Faustmann model, this cutting rule indicates that a harvestshould occur when the marginal benefit of delaying the harvest of thestand equals the marginal cost of waiting, and that this is the sum of


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the forgone interest payments and the opportunity costs of delayingthe benefits of the future rotation.

Given the model assumptions, that is, all parameters remainconstant, the forester faces an identical problem for each harvest(equation (9)). It can then be concluded immediately that all rota-tion periods must be of the same length (Clark [1990, p. 269]). Fur-thermore, Tahvonen and Salo [1999] show that stationary policies withconstant finite rotations are global optima, although a solution with afinite number of rotations and the existence of a local optimum cannotbe ruled out due to the nontimber benefit function. If we denote therotation interval as T i , where i = 1 . . .∞, so that T i = τ−

j − τ+j−1 ,

T i+1 = τ−j+1 − τ+

j , T i+2 = τ−j+2 − τ+

j+1 . . .; then T 1 = T 2 = T 3 . . .∞;and x(T 1) = x(T 2) = x(T 3) . . .∞. Excluding the nontimber benefits,the Faustmann rule can be obtained from equation (9) by dividing by

11−e−δ T i

.

Some reorganization yields

11 − e−δTi

{pF (x(Ti))

(1 − e−δTi

)− δpx(Ti)

(1 − e−δTi

)− δ

[px(Ti)e−δTi − cpxp

]}= 0.

(10)

which can be rearranged to give,

pF (x(Ti)) = δpx(Ti) +δ[px(Ti)e−δTi − cpxp

]1 − e−δTi

.(11)

This is a simple restatement of Faustmann formula. It states that it isoptimal to harvest a forest stand when the marginal benefit of delayingharvest is equal to the opportunity cost of holding on to standing trees.Similarly, if the flow of nontimber outputs are considered, the Hartmanrule can be derived from the cutting condition (9),

pF (x(Ti)) + π(x(Ti)) = δpx(Ti) +δ[px(Ti)e−δTi − cpxp

]1 − e−δTi

+δ∫ Ti

0 π(x(t))e−δt dt

1 − e−δTi.

(12)


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The left-hand side of this harvesting condition includes the additionalflow of nontimber outputs that arise if harvest is delayed. On the right-hand side, the opportunity cost derived from delaying the nontimberbenefits from future rotations is reflected in an additional “site value”term.

In the model formulation we have specified, nontimber values as afunction of biomass in contrast to the Faustmann–Hartman model(Hartman [1976], Calish et al. [1978], Swallow and Wear [1993]). Ifthe nontimber benefits were in fact a function of the age of the trees,the model would have two state variables, the timber stock and theage of the trees. The age of the trees, a(t), would be identified as thecalender time minus the time at planting, that is, a(t) = t − τ+

j , wheret ε [τ+

j , τ−j+1], where j = 1, . . . , k. This formulation is equivalent to

the specification used in this paper and can equally be written as aFaustmann–Hartman equation (Appendix B). Such a specification hasbeen used by Thavonen and Salo [1999] to study the implications offorest owner characteristics on rotation ages.

So far, we have shown that a dynamic bioeconomic framework gener-ates the traditional optimality rules in forestry economics when harvestdynamics are constrained by a clear-cut harvest rule, and all parame-ters are constant. The derivation shows that nontimber benefits at thebeginning of each rotation do not necessarily need to be equal to zero, asusually assumed (e.g., Strang [1983], Tahvonen and Salo [1999]). Thisis because the nontimber benefits at the beginning of each rotation,that is, π(x(τ+

j )), cancel out in the derivation of the optimal cuttingrule (9). We have also shown that for even-aged stands, it is equivalentto represent nontimber benefits as a function of either timber stocks ortree ages. We now turn to a model where nontimber values arise fromthe presence of a well-defined ecological resource.

3. Modeling Nontimber Benefits as a Function of an Eco-logical Resource. This model has two state variables, the timbervolume, x (t), and the stock of the ecological resource, w(t). We as-sume that the important characteristics of the forest ecosystem can becaptured in a single state variable. In practice, there may be a num-ber of state variables corresponding to the range of ecosystem services.Nontimber values are assumed to be generated by the ecological re-source stock, π(w(t)). The dynamics of the timber stock follow the

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same assumptions as stated in the model introduced in the previoussection. The stock of the ecological resource is assumed to grow accord-ing to a concave growth function G(w(t)), where G(0) = G(Kw ) = 0,where Kw is the carrying capacity for the ecological resource. Thus,the growth of the ecological resource is density-dependent limited, forexample, by resources, competitors, or predators.

We assume that the interaction between the two stocks is relatedto the harvest activities in such a way that harvest of timber resultsin a discrete change in the stock of the ecological resource. This dis-crete change may be associated with (i) a disturbance effect: loggingoperations impact on the ecological resource stock; (ii) a habitat effect:cutting activities diminishes the ecological resource’s carrying capacitydue to the reduction in resource availability (e.g., food, shelter, etc.).We specify the disturbance effect as an increasing function of the accu-mulated ecological resource stock Z(w(τ−

j )) with Z ′ > 0 and Z(0) =0. For the numerical illustrations later in the paper, we adopt a linearfunction, Z(w(τ−

j )) = ψw(τ−j ), where ψ is a constant parameter. The

habitat effect is captured by reducing the ecological resource stock toa fix number wp after the tree removal. We assume that the ecologicalresource’s carrying capacity is constant and therefore independent ofthe stage of the forest. This is undoubtedly a shortcoming of the modeland we will return to this issue in the conclusions.

We denote by w(τ−j ) and w(τ+

j ) the ecological resource stock justbefore and just after timber harvest, respectively. As in the previousmodel the forest owner’s problem is to choose the timing of clear-cutting activities, τ 1 , . . . , τ j , . . . , τ k which maximizes the timber andnontimber values. She/he has no direct control over the ecological re-source but only influences nontimber values indirectly through har-vests.

This management problem can be formulated as

maxτj

k∑j=1

∫ τ −j

τ +j−1

π(w(t))e−δt dt +k∑

j=1

[px(τ−j

)− cpxp ]e−δτj(13)

subject to,

x = F (x(t)) except at τj , j = 1, . . . , k(14)

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strong assumption


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w = G(w(t)) except at τj , j = 1, . . . , k(15)

x(τ+j

)− x

(τ−j

)= −x

(τ−j

)+ xp ∀ τj , j = 1, . . . , k(16)

w(τ+j

)− w

(τ−j

)=

{(a) −Z

(w

(τ−j

))if wp > w

(τ−j

)− Z

(w

(τ−j

))(b) −w

(τ−j

)+ wp if wp < w

(τ−j

)− Z

(w

(τ−j

))(17)

x(0) = xpw(0) = w0 .(18)

Equation (15) relates to the dynamics of the flow of the ecologicalresource between harvest instants; equation (17) indicates the magni-tude of the jumps in w(t), at the moment of harvest, and equation (18)is the initial stock constraints. Condition (17) states that if the distur-bance caused by harvesting reduces the ecological resource stock to alevel below wp then the ecological resource after harvesting is equalto the stock that “survives” the harvesting operations. Otherwise theecological resource after the harvest is equal to wp .

The procedure for solving this model is equivalent to that describedin the previous section. The corresponding current-value Hamiltonianis as follows:

H = π(w(t)) + µ1(t)F (x(t)) + µ2G(w(t)),(19)

where µ1 and µ2 are the costate variables for the timber stock and theecological resource, respectively. The first-order necessary conditionsfor the optimal solution includes the dynamics of these costate variablesbetween harvests

µ1 = −µ1(t)[δ − F ′(x(t))](20)

µ2 = −π′(w(t)) + µ2(t)[δ − G′(w(t))](21)

and the conditions to be met at optimal harvest moments, τ 1 , . . . , τ k ,

µ1(τ−j

)= p(22)

µ2(τ−j

)= µ2

(τ+j

) [1 −

∂Z(w(τ−j

))

∂w(t)

](23)


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µ2(τ−j

)= 0(24)

H(τ−j

)= δ

[px

(τ−j

)− cpxp

]e−δτj + H

(τ+j

).(25)

Equation (22) is equivalent to equation (7). Equations (23) and (24)refer to the discrete change in the imputed value of the ecological re-source and they correspond, respectively, to the two cases representedin condition (17). For the disturbance effect, condition (23) states thatthe shadow value of the ecological resource just before a clear-cut isequal to the imputed value of the remaining ecological resource alteredby the decline in the value of this resource due to harvest activities.This implies that the ecological resource costate value immediatelybefore the jump is proportional to the disturbance caused by the har-vesting activities. The higher is the disturbance, the smaller is the im-puted value assigned to the ecological resource before harvesting. Forthe habitat effect, condition (24) indicates that the ecological resourcecostate variable immediately before the harvest is equal to zero. Thisis because the ecological resource in this case is always reduced to afixed amount, wp , after the harvesting operations.

Equation (25) represents the optimal cutting decision condition. Aswith the previous model, it indicates that the optimal sequence of har-vests should balance the marginal benefits of delaying harvest (MB)and the marginal costs of waiting (MOC) (i.e., MB = MOC). Thiscutting condition can be further evaluated using the steps taken in theprevious section.

Applying the Hamiltonian function 19, together with the values ofµ1(τ−

j ), µ1(τ+j ), and µ2(τ+

j ) (equations (22), (A.7), and (A.8), respec-tively, see Appendix A) the optimal cutting condition expressed byequation (25), is now

pF(x(τ−j

))+ π

(w

(τ−j

))+ µ2

(τ−j

)G

(w

(τ−j

))= δ

[px

(τ−j

)− cpxp

]+ pF

(x(τ−j+1

))e−δTi+1

+π(w

(τ−j+1

))e−δTi+1 + δ

∫ τ −j+1

τ +j

π(w(t))e−δt dt

+µ2(τ−j+1

)G

(w

(τ−j+1

))e−δTi+1 .

(26)


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This condition generates the rotation length for a single even-agedstand that optimizes the economic returns when the nontimber benefitsdepend on the ecological resource. This condition takes into accountthe effect of the potential harvesting ages on the ecological resourcegrowth. Thus, the left-hand side of the cutting condition (26) is theincrease in the value of the forest if the clear-cutting of the stand isdelayed. This includes the value of the timber growth, the value of theproduced nontimber benefits over the time interval and the marginalincrement in the value of the ecological resource stock, evaluated at theshadow price. The first term on the right-hand side of 26 is the incomethat could be earned if revenue from cutting the stand were investedat an interest rate δ. The second, third, and fourth terms representthe opportunity cost of postponing future forest timber and nontimberreturns (including those associated with the potential growth of theecological resource), if the harvesting of the stand were delayed by oneinstant.

At the steady-state, x(τ−j ) = x(τ−

j+1) and w(τ−j ) = w(τ−

j+1), thereforeequation (26) can be written as

pF(x(τ−j

))(1 − e−δTi+1

)+ π

(w

(τ−j

))(1 − e−δTi+1

)+µ2

(τ−j

)G

(w

(τ−j

))(1 − e−δTi+1

)− δ

[px

(τ−j

)− cpxp

]− δ

∫ τ −j+1

τ +j

π(w(t))e−δt dt = 0.

(27)

The steady state implies, as before, that T i , where i = 1 . . .∞, so thatT i = τ−

j − τ+j−1 , T i+1 = τ−

j+1 − τ+j , . . . ; then T 1 = T 2 = T 3 . . .∞,

and equation (27) can be rearranged to yield

pF (x(Ti)) + π(w(Ti)) + µ2(Ti)G(w(Ti))

= δpx(Ti) +δ[px(Ti)e−δTi − cpxp ]

1 − e−δTi+

δ∫ Ti

0 π(w(t))e−δt dt

1 − e−δTi.

(28)

When the nontimber forest benefits are a function of the ecologicalresource and the disturbance effect dominates, the optimal rotationcan be defined by the Faustmann–Hartman equation in which an addi-tional term is included. Condition (28) states that the marginal benefitsderived from leaving the ecological resource to grow influence the deci-sions on when to harvest the timber. The condition also states that the


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appropriate correction is the marginal value of the ecological resourcevalued at its shadow price.

When harvesting activities generate small disturbance in the ecologi-cal resource stock, optimal rotations will be shorter than when the dis-turbance is large. For small disturbance, when the ecological resource issmall the marginal value of ecological resource will be large and it willnot be optimal to harvest. It is only when the ecological resource stockis large enough and therefore its marginal value is small enough, thatharvest might happen. However, in this situation π′(w(T i)) is smalland the optimal cutting condition will be mainly driven by the termsrelated to the timber stock. Therefore, optimal rotations will be similarto timber-only Faustmann rotations with high levels of the ecologicalresource. For large disturbances the shadow price of the ecological re-source will be small given condition (23). Therefore, the marginal valueof the ecological resource growth (the extra term) will be small and theoptimal cutting condition will be driven by the original Faustsmann-Hartman terms. However, in our model the nontimber benefits are afunction of an ecological resource.

When the habitat effect dominates, the level of the ecological resourceafter harvesting is always equal to a level, wp (see equation (17)). In thiscase, the optimal cutting condition is given by (26) where µ2(τ−

j ) =µ2(τ−

j+1) = 0. The resulting cutting condition at the steady-state istherefore similar to (28) without the additional term, that is, it yieldsa type of Faustmann–Hartman rule as shown in Section 2.

4. An Example: Pinus Pinaster in Galicia. To give a bettersense of the optimality condition in a bioeconomic framework for ongo-ing rotations of a single even-aged stand, we provide a numerical illus-tration. The model is parameterized for maritime pine (Pinus pinasterAit) in the coastal area of Galicia (Spain). Timber growth is basedon the growth model proposed by Alvarez-Gonzalez et al. [1999]. Thefunctional forms for the components of the growth function and theparameter values used are described in Table 1. The growth function,F (x ), can be derived from the specification of the volume function,x (t) (Table 1).

F (x(t)) = x(t)(−b2)(−b3)t−b3−1 + aB(t)dH(t)

dt,(29)

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TABLE 1. Biological functions and parameters.

Variable Specification Parameters

Basal area B = b1 exp(−b2 t−b3 ) b1 = 74.6515

b2 = −7.1299b3 = −0.7582

Top height H = h1Sh 2 (1 − 1/eh 3 t )h 4 h1 = 2.2878

h2 = 0.9433h3 = 0.0549h4 = 1.4061S = 154.302

Volume of timber x = aBH a = 0.4215Quadratic mean D = d1 + d2

100N d 3

+ d4H0 d1 = −1.9018diameter d2 = 1483.8

d3 = 0.75d4 = 0.8806

No of stems N = N 0 exp(−mt) N 0 = 1300m = 0.01

Growth of the ecological G(w) = rw w(t)(1 − w (t)K w

) Kw = 600resource 0.05 ≤ rw ≤ 0.25

Postharvest ecological wp wp = 250resource

Loss in w’s volume Z(w) = ψw(t) 0 < ψ < 1at timber harvest

Source: Alvarez-Gonzalez et al. [1999] and Touza [2003].Note: t represents the age of the trees, S(dm) is the site index given by the top heightof the stand at a reference age of 20 years, N (trees/ha) is the initial density, and m isthe mortality rate per year.

where B(t) is the basal area function, and H (t) is the top heightfunction. This specification produces a right skewed growth curve(Figure 1).

Maritime pine in Galicia is grown mainly for regional saw log produc-tion. The price of saw log depends on the top diameter and the qualityof the timber. For this numerical illustration, however, we exclude the


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FIGURE 1. Growth function (m3 /ha) of a maritime pine stand as a functionof stand age.

dependence on timber quality. The timber volume at different ages isallocated to three different categories of timber as a function of the topdiameter based on Rodrıguez-Soalleiro et al. [2000].4 The stumpageprices per m3 for the different timber categories were then estimatedusing the results of timber auctions carried out by the Galician for-est administration (Rodrıguez-Soalleiro et al. [2000]). The percentageyield of logs of different top diameter as a function of stand quadraticmean diameter (cm) and top height (m) was available from RoqueRodrıguez-Soalleiro (pers. com.). From this information, a stumpageprice function, P , was estimated. The price function depends on thediameter of the stand, D , and therefore on the stand’s age. Table 2 liststhe economic functions and parameter values used in the simulation.

The growth of the ecological resource, G(w(t)), is assumed to fol-low a logistic function (Table 1), where Kw is the carrying capacity ofthe ecological resource, and rw is its rate of growth. The damage oftimber harvesting on the ecological resource stock is captured throughthe discrete damage function, Z . In order to keep the analysis simple,this dependency is captured through a parameter, ψ, that takes valuesbetween zero and one (Table 1). The lower the value of this param-eter the smaller the damage, and therefore the smaller is the loss ofthe ecological resource associated with harvesting activities. The sim-ulations assume that the nontimber benefits follow a logistic function(Table 2).


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TABLE 2. Economic functions and parameters.

StandardVariable Specification Parameters errors

Stumpage net price P (t) = p1 (Dg (t))p 2 p1 = 13.1795 2.7792p2 = 1.8092 0.0614

Nontimber benefits π(w(t)) = K a1+ ce−r a ω Ka = 900

c = 10ra = 0.01

Planting and maintenance cp = 2025.48a

costs (euros/ha)Discount rate δ = 0.02

Source: Rodrıguez-Soalleiro et al. [2000] and Touza [2003].(a) Reforestation costs (euros/ha) equal planting cost (1322.27) plus discounted main-tenance cost (150.25) per year during the first five years (Rodrıguez-Soalleiro [1997]).

The optimal harvest ages are calculated using the optimal cuttingconditions given by equations (9) and (26). Recall that at the steady-state x(t), w(t), µ1(t), and µ2(t) are equal for t = τ−

j and t = τ−j+1.

The same is true for t = τ+j and t = τ+

j+1. Conditions (11) and (27)given us these cutting conditions at the steady-state. For the modelwith nontimber benefits as a function of the ecological resource (Sec-tion 3), the costate variable just before the harvest, µ2(τ−

j ) (equa-tion (23)), has a recursive nature. Therefore, we first estimate the valueof these costate variable just after the harvest µ2(τ+

j ) (equation (A.8),see Appendix A). Using µ2(τ−

j ) = µ2(τ−j+1) this variable, µ2(τ+

j ), wasexpressed as a function of the model’s parameters. The computationinvolves a search for values of the optimal rotation period and the op-timal stock of the ecological resource, w(τ+

j ) (equation (17)), for whichthe cutting condition is met.

These simulations include two possible scenarios. In the first scenario,we analyze exclusively the disturbance effect. In the second scenario,the two effects, habitat and disturbance, are combined. This meansthat the stock of the ecological resource after harvesting will dependon the strength of the disturbance versus the habitat effect (as specifiedby condition (17)).


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FIGURE 2. Optimal cutting condition. H (τ + ), opportunity cost of postponingthe following rotation benefits; H (τ−), marginal benefit of delaying the cut-ting; FOC , forgone interest payment from delaying the harvest; r(t), cuttingfunction. π(t) = K x

1 + c e −r x t , Kx = 450, c = 10, rx = 0.1, δ = 0.02.

4.1 Simulation results. We first illustrate the optimal harvestingrule in the basic bioeconomic model, equation ((9)), to illustrate theoptimal control components of this condition. Figure 2 represents thisbioeconomic condition where value of the Hamiltonian after a jumpminus the value of the Hamiltonian before a jump plus the financialopportunity cost (FOC) of harvest revenues should be equal to zero,H(τ+) − H(τ−) + FOC = 0. Recall that this condition states thatthe marginal opportunity cost of waiting, given by H(τ+) + FOC , isequal to the marginal benefit of waiting, given by H(τ−). We call thedifference between the marginal opportunity cost and marginal benefitr(t) so that r(t) = H(τ+) − H(τ−) + FOC . The Hamiltonian beforethe jump, H(τ−), which represents the marginal benefits gained byprolonging the rotation, is an increasing function of rotation length. Itincreases more rapidly in the initial stages of growth than it does whenthe trees are older. The Hamiltonian after the harvest, H(τ+), is theopportunity cost of postponing the value of timber capital and nontim-ber benefits obtained from the following rotation. It has the same shapethan H(τ−) but due to the discounting effect it has lower values. The


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financial opportunity costs (FOC) increase through the growth cycle,and at the beginning of the rotation they take negative values becausethe potential benefits from selling the timber are lower than the plant-ing costs. The decreasing slope of the cutting condition, r(t), whenthe trees are young is therefore driven by the fact that the marginalbenefits of delaying harvest increase rapidly with time, whereas theopportunity costs of the following rotation and the financial opportu-nity costs increase more slowly. When the trees get older, the cuttingcondition increases because the financial opportunity costs increase ata higher rate than the marginal benefits of waiting. When the r(t)function equals zero, it is optimal to harvest the stand.

Given the parameter values (Tables 1 and 2), the simulation showsthat it is optimal to use a 29-year rotation cycle if the stand is man-aged exclusively for timber. An increase in the stumpage price or anincrease in the interest rate produces a lower optimum rotation agesince the opportunity costs of waiting increase. For example, increas-ing the interest rate to 3% generates optimal rotations of 27 years;and increasing stumpage prices by 20% reduces the optimal rotationto 28 years. An increase in the reforestation cost extends the opti-mum rotation length because higher economic returns are necessary tocompensate for the higher initial investment. Thus, if the reforestationcosts were 20% higher, the optimal interval would be 30 years. Whennontimber benefits as a function of the timber stock are included in theanalysis, it is to be expected that the rotation period would be longer.Given the nontimber benefit function and parameter values assumed,it turns out to be optimal to harvest when the trees are 42 years of age(Figure 2).

We evaluate now the optimal rotations at steady-state for the twocases specified above, and analyze how these vary with changing val-ues of the impact of the timber harvesting ψ and the rate of growthof the ecological resource (rw ). Figure 3a shows the results for thefirst scenario, including only the disturbance effect. In this case, theoptimal rotation interval increases as ψ increases, and decrease as rw

increases. For example, high values of ψ (e.g., ψ = 0.85) give rise toa rotation that is twice as long as the Faustmann rotation age (e.g.,69 years). The results also show that if the impact of timber produc-tion on the ecological resource is very small a Faustmann harvestingage can be optimal (i.e., 29 years). In addition, rotation ages close to


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FIGURE 3. (a) Optimal harvesting age of an even aged stand, T i ; (b) opti-mal stock of the ecological resource before clear-cutting, w(τ−

j ); (c) optimalimputed value of the ecological resource after clear-cutting, µ2 (τ +

j ). Resultsfor first scenario as a function of the sensitivity of the ecological resource totimber harvesting activities (ψ) and its rate of growth (rw ).

the Faustmann rotation occur for higher values of ψ when the ecologi-cal resource has a high rate of growth. This shows that the higher theimpact of timber harvesting activities on the ecological resource—thehigher the loss in the nontimber benefits—the longer the optimal ro-tation at the steady-state. Increasing values of the rate of growth ofthe ecological resources implies faster recolonization/recovery of theecological resource after clear-cutting, and this leads to a steady-state


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with lower optimal harvest ages. From Figure 3b, we see that as ψincreases, the optimal stock of the ecological resource at the harvestpoint decreases. This means that when the timber harvesting activitieshave a small impact on the ecological resource, the optimal harvestingstrategy involves short timber rotation intervals and the ecological re-source is maintained close to its “natural” steady-state (as discussedin Section 3). From this graph, we also see that a higher growth rate ofthe ecological resource implies higher values of the ecological resourcestock before the tree removal. Figure 3c gives the value of the costatevariable of the ecological resource, just after harvest, µ2(τ+

j ). The sim-ulations show that when the growth potential of the ecological resourceis high and the negative impact from harvesting is small, the value ofadditional units of stock of the ecological resource in the beginning ofa rotation is smaller than when the growth potential is small and thenegative impact from harvesting high.

FIGURE 4. First scenario: optimal steady-state dynamics of forest manage-ment and the ecological resource stock depending on the potential disturbanceassociated with timber harvesting (ψ).


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The optimal steady-state dynamics of the timber management andthe ecological resource stock is illustrated for two combinations of pa-rameter values (Figure 4). A low disturbance of the ecological resourceon the harvesting implies shorter rotations, and a higher ecologicalresource stock, both before and after clear-cutting. When the effect ofthe logging disturbance is strong, it is optimal to manage the forest

FIGURE 5. (a) Optimal harvesting age of an even aged stand, T i ; (b) opti-mal stock of the ecological resource before clear-cutting, w(τ−

j ); (c) optimalimputed value of the ecological resource after clear-cutting, µ2 (τ +

j ). Resultsfor second scenario as a function of the sensitivity of the ecological resource totimber harvesting activities (ψ) and its rate of growth (rw ). Q5

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stand with longer rotations that allow the ecological resource stockto recover from the impact of timber production activities. The sim-ulations in Figure 4 have been performed given a growth rate of theecological resource stock equal to 0.05. Higher values of the growthparameter would lead to shorter rotations and higher stock levels ofthe ecological resource. Equally, lower values of the growth parame-ter would lead to longer rotations and lower values of the ecologicalresource.

Figure 5 shows the results for the second scenario, where the max-imum ecological resource after harvest is wp . When the disturbanceeffect is smaller than the habitat effect, the optimal rotation age (Fig-ure 5a), the ecological resource stock before removal (Figure 5b), andthe internal value of the ecological resource just after harvest (Figure5c) remain constant. This is because the level of ecological resourceafter harvesting is independent on the stock available before logging.For the parameters used in these simulations, the disturbance effectwill be greater than the habitat effect when ψ is greater than 60%. Inthese situations, the higher the impact on the ecological resource, thelonger the optimal rotation intervals (Figure 5a), and the higher is theoptimal stock of the resource at the harvest moments (Figure 5b). Thevalue of additional units of stock of the ecological resource in the begin-ning of a rotation increases when the negative impact from harvestingincreases (Figure 5c).

5. Conclusions. From a management perspective, it is importantto be able to optimize the production of timber given the ecosystemservices demanded by the wider community. The modeling approachdiscussed here would appear to be useful. While it can replicate theresults of Faustmann model, it is also able to address both the restrictedmanagement problem that motivated Faustmann and current concernsover the optimal provision of forest ecosystem services.

This paper has proposed an analytically simple bioeconomic frame-work for the analysis of clear-cutting decisions in forest management.This is achieved using an optimal control approach, which models har-vesting decisions as jump controls. The framework used has been shownto be consistent with the traditional framework in forest economics, asthe Faustmann and Hartman rules can be derived from this bioeco-nomic model with clear-cutting regimes. The resulting optimal cutting


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condition is equivalent to the solution obtained by Tahvonen and Salo’sutility maximization model (Tahvonen and Salo [1999]), where the rateof interest and the forester’s subjective time preference are assumed tobe identical, and the imputed value of nonforest assets is assumed tobe equal to one (i.e., the forest owner’s characteristics are ignored).The model proposed here is a special case of a general bioeconomicframework with jumps, which allows for different harvesting intensities(Termansen [2007]). This shows that the Faustmann–Hartman solutionarises as a special case of the modified golden rule of capital accumu-lation (Clark and Munro [1975]).

The specification of nontimber benefits used in this paper assumesthat these are indirectly derived from the state of the forest. They areaffected by harvesting activities that lead to discontinuities in the eco-logical resource stock. This specification allows a more general represen-tation of the interactions between timber and nontimber benefits thanis possible in the Faustmann–Hartman models. The optimality condi-tion derived includes an additional term that represents the marginalvalue of the ecological resource as harvest is postponed. This extraterm accounts for the disturbance effects of the logging operations.We show that when the disturbance effects are small it will be opti-mal to combine short rotations with high levels of ecological resourcestock. When disturbance effects are large, it will be optimal to chooselonger rotations. When the habitat destruction effect dominates, thepostharvest level of the ecological resource is reduced to some minimal(constant) value. This means that the shadow price of the ecologicalresource stock just before the harvest is zero and the optimal cuttingcondition is a Faustmann–Hartman rule. The optimal harvesting agewill however be influenced by the fact that the nontimber benefit de-pends on the ecological resource stock rather than the age of the trees.This result is driven by the fact that carrying capacity of the ecologicalresource is assumed independent of the forest biomass.

The numerical simulations illustrate that where the dependency (dis-turbance or habitat effects) between forest dynamics and the status ofthe ecological resource is very weak it will be optimal to maintain acontinuously high level of the ecological resource. Otherwise, the de-pendency will lead to a rotational ecological resource provision. If theeffects of forest management activities are severe enough, this mightcause local extinctions of the ecological resource when harvests take


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place. At the landscape level, many local extinction might prohibitthe recolonization process to take place with severe consequences forbiodiversity conservation. The rate of recolonization from neighboringpatches would then be one measure of the size of the disturbance ef-fect. The smaller the disturbance effect, or the the higher the rate ofrecolonization (more generally the higher recovery rate of the ecologicalresource) the shorter the optimal rotation at the steady-state.

The numerical example discussed in this paper was parameterized toreflect maritime pine forestry in western Spain, and the simulated rota-tion periods when the resource is managed for timber alone is consistentwith forestry practices applied in that location (Rodriguez-Soalleiroet al. [1997, 2000]). The simulated rotation periods when there arenontimber values that are independent of the age structure of treesshows how the wider environmental benefits of forest cover (habitat,erosion protection, hydrological regulation, water purification, carbonsequestration, etc.) might impact the optimal rotation. Given recentattention to the importance of the ecosystem services delivered by for-est systems, and given the fact that the disturbance due to harvestingmay be the most important determinant of nontimber benefits, themodeling approach developed in this paper would seem to have sometraction. The possibility of having the equation of motion of the eco-logical resource dependent on the timber stock between harvests andstochasticity issues are not addressed in this paper, and are tasks forfuture work. Modeling these aspects is not trivial, but it will providefurther insights into the impacts of harvesting on the provision of non-timber benefits.

Acknowledgments. We thank Roque Rodrıguez-Soalleiro forvaluable information to carry out the numerical simulations and JimSmart, Oliver Jakoby and two anonymous referees for useful comments.Julia Touza acknowledges financial support from the European Unionthrough the BioEcon project.

Appendix A

In Section 2, the first-order differential equation associated with theevolution of the stock costate variable between harvests, equation (5),can be solved as follows:


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µ(t) = e

∫ t

τ +j

[δ−F ′(x(t))]dt(

µ(τ+j

)−

∫ t

τ +j

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∫ t

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dt

).

(A.1)

Note that the exponential function can be solved by integrating twiceby substitution,5

e−

∫ t

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F ′(x(t))dt

= e−

∫ x (t)

x (τ +j

)

F ′(x (t))F (x (t)) dx

= e−

∫ F [x (t)]

F [x (τ +j

)]1

F (x (t)) dF (x(t))

= e−ln [F (x(t))]+ ln [F (x(τ +

j ))] =eln [F (x(τ +

j )]

eln [F (x(t))] =F

(x(τ+j

))F (x(t))

.

(A.2)

Applying this to equation (A.1) and rearranging some terms yields

µ(t) = µ(τ+j

)eδ(t−τ +

j ) F(x(τ+j

))F (x(t))

− 1F (x(t))

eδ(t−τ +

j )∫ t

τ +j

π′(x(t))F (x(t))e−δ(t−τ +j )

dt.

(A.3)

At the jump point t = τ−j+1, that is, at the end of the next rotation,

this implies

µ(τ−j+1) = µ(τ+

j )eδ(τ −j+1−τ +

j ) F(x(τ+j

))F

(x(τ−j+1

))− 1

F(x(τ−j+1

))eδ(τ −

j+1−τ +j )

×∫ τ −

j+1

τ +j

π′(x(t))F (x(t))e−δ(t−τ +j )

dt.

(A.4)

The solution to the integral is the stand nontimber benefits duringthe next rotation. τ+

j and τ−j+1 represent the points of time when the

next rotation starts and ends, respectively. Therefore, (τ−j+1 − τ+

j ) is


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the harvest age of the following rotation, that is, the next rotationinterval. This integral can be integrated by parts to yield,6

∫ τ −j+1

τ +j

π′(x(t))F (x(t))eδ(t−τ +j )

dt

= π(x(t))e−δ(t−τ +j )

∣∣∣∣τ −j+1

τ +j

+ δ

∫ τ −j+1

τ +j


dt

= π(x(t))e−δ(τ −j+1−τ +

j ) − π(x(τ+j

))+ δ

∫ τ −j+1

τ +j


dt.

(A.5)

Applying (7), that is, µ(τ−j+1) = p, and equation (A.5), to equation

(A.4), and after some rearranging, the costate value just after the har-vest can be written as a function of the timber and nontimber benefitsin the following rotation,

µ(τ+j

)= pe

−δ(τ −j+1−τ +

j ) F(x(τ−j+1

))F

(x(τ+j

))+

1F

(x(τ+j

))[π(x(t))e−δ(τ −

j+1−τ +j ) − π

(x(τ+j

))

+ δ

∫ τ −j+1

τ +j


dt

].

(A.6)

In Section 3, the first-order differential equation (20) for the timberstock costate variable can be solved following similar procedures thanabove. We found that the imputed value of the timber stock just aftereach harvest, µ1(τ+

j ), is related to the present value of the growth oftimber in the following rotation,

µ1(τ+j

)= pe−δ(Ti+1 ) F

(x(τ−j+1

))F (x

(τ+j

))

,(A.7)


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where T i+1 = τ−j+1 − τ+

j is the next rotation interval. Similarly, thesolution of the first-order differential equation (21), the shadow valueof the ecological resource just after a harvest, µ2(τ+

j ), depends onthe growth of its stock and nontimber benefits in the next rotation.The nontimber values are captured in this costate variable as in equa-tion (A.4).

µ2(τ+j

)= µ2(τ−

j+1)e−δTi+1

G(w

(τ−j+1

))G

(w

(τ+j

))+

1G

(w

(τ+j

))[∫ τ −

j+1

τ +j

π′(w(t))G(w(t))e−δ(t−τ +j )

dt

].

(A.8)

Appendix B

The age of the trees, a(t), is included as an additional state vari-able identified as the calendar time minus the time at planting,that is, a(t) = t − τ+

j , where t ε [τ+j , τ−

j+1], where j = 1, . . . , k.The associated costate variable, θ(t), is introduced and the currentvalue costate variables for timber stock and tree age at the jumppoints are µ(τ−

j ) = p, µ(τ+j ) = pe

−δτ −j+1 F (x(τ−

j+1))/F (x(τ+j )), θ(τ−

j ) =

0, θ(τ+j ) =

∫ τ −j+1

τ +j

π′(a(t))e−δ(t−τ +j )

dt, respectively.

The optimal cutting condition at the steady-state is

π(a(τ−j

))(1 − e−δ(Ti+1 )) + pF

(x(τ−j

)(1 − e−δ(Ti+1 ))

− δ[px

(τ−j

)− cpxp

]e−δτj − δ

∫ τ −j+1

τ +j

π(a(t))e−δt dt = 0,

(B.1)

where T i+1 = t − τ−j+1 is the following rotation age. This is equivalent

to equation developed in the main text.

ENDNOTES

1. Scorgie and Kennedy [1996] argued that an English agriculturalist, WilliamMarshall, correctly anticipated the Faustmann–Presser–Ohlin (FPO) condition,decades before the publication of Faustmann’s article.


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2. k �= ∞ when nontimber benefits are higher than the opportunity costs oftimber and continuously increasing with time.

3. Note that H (τ−j ) = π(x(τ−

j )) + µ(τ−j )F (x(τ−

j )); and H (τ +j ) = π(x(τ +

j )) +µ(τ +

j )F (x(τ +j )).

4. Category I: logs with a top diameter smaller than 22 cm; category II: logswith a top diameter between 22 and 35 cm; category III: logs with a top diameterhigher than 35 cm. The corresponding stumpage prices are: category I =18 euros,category II =48 euros, category III =90.15 euros.

5. First, let the integration parameter u equal x (t). Thus u = x(t) and dt =dx(t)/F (x(t)) are substituted back into the original integrand. Second, let u =F (x(t)), its derivative is du/dx = F ′(x(t)). Then, u = F (x(t)) and dx = du/F ′(x(t))are substituted into the integral.

6. Note that∫ τ

−j + 1

τ+j

π′(x(t))F (x(t))dt = π(x(t))|τ−j + 1

τ+j

.

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