Forest Management and Carbon Sequestration in Size-Structured Forests:

The Case of Pinus Sylvestris in Spain

Renan Ulrich Goetz a

Natali Hritonenko b

Ruben Mur a

Àngels Xabadia a

Yuri Yatsenko c

a University of Girona, Campus Montilivi, s/n, 17071 Girona, Spain; renan.goetz@udg.edu

(corresponding author), Phone: +34 972-418719, Fax: +34 972-418032

b Prairie View A&M University, Box 519, Prairie View, Texas, USA; nahritonenko@pvamu.edu

c Houston Baptist University, 7502 Fondren Road, Houston, Texas, USA; yyatsenko@hbu.edu

Abstract 1

The Kyoto Protocol allows Annex I countries to deduct carbon sequestered by land use, land-use 2 change and forestry from their national carbon emissions. Thornley and Cannell (2000) demonstrated 3 that the objectives of maximizing timber and carbon sequestration are not complementary. Based on 4 this finding, this paper develops a model that takes into account the underlying biophysical and 5 economic processes, and determines empirically the optimal selective management regime. The results 6 show that sequestration costs are significantly lower than previous estimates. 7

8

Key words: Kyoto Protocol, selective logging, carbon management, dynamic optimization. 9 10 JEL classification: C61, Q23, Q54. 11

12 Acknowledgements: 13 The authors gratefully acknowledge the support of the Ministerio de Ciencia y Tecnología grant 14 AGL2007-65548, INIA grant SUM 2006-00019-C02-01, and Generalitat de Catalunya grants 15 (XREPP, and 2005 SGR213).16

1

1. Introduction 17

Pioneering work by Adams and Ek (1974), Haight et al. (1985), Haight (1987), and Getz and 18

Haight (1989) numerically determined the optimal selective-logging regime of an uneven-19

aged forest. More recently, several authors extended this line of work by considering not only 20

the net benefits of timber management but also of carbon sequestration. Many of these studies 21

were conducted to determine the costs of carbon sequestration at a national or international 22

level (Adams et al. 1999; Sohngen and Mendelsohn 2003). The decision variables used in the 23

studies included the afforested land, the length of tree rotation and the management intensity. 24

The variable rotation length has been commonly used to assess interactions between timber 25

production and carbon sequestration (van Kooten et al. 1995; Stainback and Alavalapati 26

2002). However, the variable management intensity can only be chosen from a fixed set of 27

alternatives when trees are planted, and cannot be changed while there are standing trees. 28

Hence, these studies did not provide an answer to the question of whether it is optimal to 29

harvest part of the forest during the rotation or to harvest all trees at the end of the rotation 30

length. Likewise, the optimal partial harvest and its timing require a detailed stand analysis. 31

Pohjola and Valsta (2007) analyzed these questions empirically but did not provide a 32

theoretical model and the necessary conditions for its solution. The paper by Pohjola and 33

Valsta (2007) presents an important step forward for the detailed stand analysis method but it 34

does not consider the fact that the individual trees compete among each other for scarce 35

resources such as light, space and nutrients, i.e., it does not take into account the effect of 36

density. In addition, this article contributes to the present literature on stand management by 37

considering sequestration of carbon not only in timber but also in wood products and forest 38

soil. Finally, the paper integrates further the economic and biophysical model so that 39

decisions (logging) taken during the dynamic optimization path affect the future evolution of 40

the forest and vice versa, i.e., the evolution of the forest affects future logging patterns. 41

2

The results of the study show that carbon sequestration costs for forests seem to be 42

significantly lower when the forest management regime is changed than when land use is 43

changed, i.e., a change from agriculture to forestry. Therefore, the study suggests reanalyzing 44

the role of forest carbon sequestration as a part of climate change policies within the context 45

of changing the management regime rather than changing land use. 46

47

1.1 Features of this study 48

Since the tree density (trees per hectare) and the distribution of the size within the stand 49

(Keyfitz 1968) affect the biological growth of the individual trees, the number of planted trees 50

should be determined endogenously and the effect of intraspecific competition should be 51

considered in the model. If the forest reproduces naturally, the density of the forest can be 52

regulated by thinning, i.e., by the selective logging of younger trees. Xabadia and Goetz 53

(2007) and Goetz et al. (2007) determined the optimal selective-logging regime for timber 54

production of a size-structured forest in which planting and intraspecific competition were 55

taken into account; they did not, however, consider the costs and benefits of carbon 56

sequestration. 57

The modeling of a size-structured forest can include the fact that the price of timber increases 58

with the size of the trees because the timber of larger trees can be used for higher-value 59

products, such as furniture. Consequently, the price of timber can be considered as a function 60

of the size of the tree. This aspect is not only important for correctly determining the net 61

benefits of timber production but also for accurately measuring the permanence of the 62

sequestered carbon in the final product. This study explicitly takes into account the different 63

permanence times and analyzes their economic and biological effect on the optimal logging 64

regime. 65

3

Another important part of the carbon cycle is the carbon sequestered in the soil. The potential 66

of forest soils to sequester carbon is well known (Rasse et al. 2001). Matthews (1993), for 67

instance, reports that the change from agricultural land to forest land led, after 200 years, to 68

an increase in soil carbon from 30 t C/ha initially to 70 t C/ha. 69

Hence, the density of the stand, intraspecific competition, the size of the tree, and soil carbon 70

are essential for an accurate description of the biological growth process of the tree (partial 71

integrodifferential equation), of the life cycle of carbon in the different compartments of the 72

forest, and for the correct modeling of economic factors. 73

These factors have been considered previously on their own but not together in a single study. 74

With respect to the numerical determination of the optimal logging regime, the seminal 75

contributions by Adams and Ek (1974), Haight et al. (1985), Haight (1987) and Getz and 76

Haight (1989) describe the evolution of an age (size) distributed forest by a set of difference 77

equations in time for a number of age (size) cohorts. Since the evolution of the forest over 78

time is continuous, the set of difference equations presents an approximation (discretization 79

over time and age) of the underlying biological processes. The discretization procedure 80

applied has shown to be realistic and has been employed widely; however, the set of 81

difference equations has been set up ad hoc and is not derived from the underlying partial 82

differential equation. Therefore, little can be said about the size of the error in the 83

approximation. This paper attempts to overcome this shortcoming by deriving a set of 84

difference equations from a partial integrodifferential equation, as proposed in the 85

mathematical biology literature. In a numerical analysis, de Roos (1988) shows that the 86

derived set of difference equations can be considered a good approximation of the underlying 87

partial integrodifferential equation. Moreover, the use of a single partial integrodifferential 88

equation allows the economic and biophysical model to be integrated to a high degree. The 89

previously published models incorporated data obtained from stand projection tables or stand 90

4

projection programs (biophysical model). However, partial harvesting affects the future 91

growth of the forest, so that the data must be updated whenever trees are logged. Hence, the 92

modified evolution of the forest affects the choice of the optimal partial harvesting regime. 93

The mutual interdependency of the economic and biophysical model requires the future 94

evolution of the forest within the economic decision model to be continuously updated. The 95

interlocking of the economic and biophysical model is only possible if the biophysical 96

component of the economic model not only consists of data but also of processes that govern 97

the evolution of the forest. The set of difference equations employed in the previous literature 98

can only be integrated in the economic model if the parameters of all difference equations are 99

estimated simultaneously for a sufficiently large variety of initial size distributions. However, 100

applicability may be limited since this process would require estimating a set of usually more 101

than 10 difference equations subject to constraints that restrict the movement of the trees 102

between the difference equations (cohorts). Hence, apart from the possible approximation 103

error, it already seems more reasonable from a practical point of view to estimate a single 104

partial integrodifferential equation than a large set of difference equations. 105

The paper is organized as follows: the next section describes the dynamics of a forest, and 106

presents the economic decision problem; an empirical analysis is conducted in Section 3 to 107

determine the optimal selective-logging regime of a privately owned forest with respect to 108

timber and carbon sequestration; and conclusions are presented in Section 4. 109

5

2. The bioeconomic model 110

The dynamics of the bioeconomic model reflects the change in the density of the size-111

structured trees x(t,l) over calendar time t and size l, and also the changes in soil carbon s(t) 112

over time. Before we can specify the precise mathematical presentation of these two 113

equations, we need to introduce some notation. 114

The size of the tree is measured by the diameter at breast height and is denoted by l. The set 115

of the admissible values of the variable l consists of the interval [l0, lm], where l0 corresponds 116

to the minimum vital diameter of the tree and lm is the maximum diameter that the trees can 117

reach. In the case of a completely managed forest, no natural reproduction takes place, and all 118

young trees have to be planted. Therefore, the parameter l0 corresponds to the diameter of the 119

trees when they are planted. Although the theoretical and empirical analysis is framed within 120

the context of a completely managed forest, it is quite possible to extend the framework to the 121

case in which trees reproduce naturally. In this case l0 can be interpreted as the diameter at 122

which trees are selected for upgrowth. The flux of logged trees is denoted by u(t,l), and the 123

flux of planted young trees at time t with diameter l0 by p(t). We assume that a diameter-124

distributed forest can be fully characterized by the number of trees and by the distribution of 125

the diameter of the trees. In other words, space and the local environmental conditions of the 126

trees are not taken into account.[1] 127

As the density function x(t,l) indicates the population density with respect to the structuring 128

variable l at time t, the number of trees in the forest at time t is given by 129

0

( ) ( , ) .ml

l

X t x t l dl= ∫ (1) 130

The term E(t) denotes environmental characteristics that affect the growth rate of the diameter 131

of individual trees. In the absence of pests, these environmental characteristics are given by 132

6

the local conditions where the tree is growing, and by the neighboring trees. Since our model 133

does not consider space, the term E(t) exclusively represents the competition between 134

individuals, i.e., the competition between individuals for space, light and nutrients. 135

Environmental pressure can be expressed, for example, by the total number of trees or the 136

basal area of all trees of the stand. Hence, a large (small) basal area of the stand signifies high 137

(low) intraspecific competition of the trees (Trasobares et al. 2004).[2] Thus, the change in 138

diameter of the tree over time is described by the function g(E(t),l), i.e., 139

( ( ), ),g E t ldldt

= (2) 140

where the basal area E(t) is determined based on the functional relationship between diameter 141

and basal area:

0

2( ) ( , )4

ml

lE t l x t l dlπ

= ∫ . 142

The instantaneous mortality rate, δ(E(t),l), describes the rate at which the probability of 143

survival of an l-sized tree in the presence of intraspecific competition E(t) decreases with 144

time. Based on the well-known McKendrick equation for age-structured populations 145

(McKendrick 1926), the dynamics of a diameter-distributed forest can be described by the 146

following partial integrodifferential equation as discussed by de Roos (1997) or by Metz and 147

Diekmann (1986): 148

( , ) [ ( ( ), ) ( , )] ( ( ), ) ( , ) ( , ),x t l g E t l x t l E t l x t l u t lt l

δ∂ ∂+ = − −

∂ ∂ t∈[0, T), l∈[l0, lm], (3) 149

subject to the boundary conditions x(0,l) = x0(l), l∈[l0, lm], and x(t,l0) = p(t), t∈[0, T). The first 150

two terms of Equation (3) represent the change in the tree density over time and diameter. The 151

term ( ( ) ( ))g x l∂ ⋅ ⋅ ∂ = ( ) ( ) g x l g l x∂ ∂ + ∂ ∂ takes into account the interdependence between 152

diameter and time, i.e., it represents the temporal change in diameter multiplied by the change 153

in tree density with respect to diameter plus the temporal change in diameter with respect to 154

7

diameter multiplied by the tree density.[3] Thus, the forest dynamics are described by the flux 155

of tree density with respect to diameter and time, which is equal to the terms of the right hand 156

side of Equation (3), given by tree mortality and logging. 157

To consider, in addition, the net benefits of carbon sequestration in the economic model, it is 158

necessary to analyze how the carbon content of the forest ecosystem changes over time. The 159

change in the carbon content is given by ,dz dt db dt ds dt= + and depends on the change in 160

carbon sequestered in the biomass captured by db dt , and the change in the carbon content in 161

the soil ds dt . With respect to the dynamics of soil carbon, we denote the above-ground 162

volume of the biomass of the forest by 0

0( ) ( , )ml

l

V t l x t l dlβγ= ∫ , where the strictly positive 163

parameters β and γ0 have to be chosen according to the species of the tree and the available 164

empirical data. Once function V is defined, we are in a position to specify the soil carbon 165

dynamics, which are given by 166

( )( ) ( ), ( ) ,ds t h V t s tdt

= t∈[0, T), s(0) = s0. (4) 167

Function h reflects to what extent the growth or harvest of trees and the current amount of soil 168

carbon affect the change in the soil carbon over time. 169

The amount of sequestered carbon in the biomass is given by 0

1( ) ( , )ml

l

b t l x t l dlβγ= ∫ . b(t) is 170

calculated based on the above-ground volume of the forest and the constant γ1, and the ratio γ1 171

/ γ0 reflects the carbon content per cubic meter of the biomass and is more or less constant 172

throughout the growth process of the trees. Before stating the complete bioeconomic model 173

we need to define the function ( ( , ), ( , ))B x t l u t l . This represents the net benefits of timber 174

production as a function of the standing and the harvested trees, with Bu > 0 and Bx < 0, 175

8

where a subscript of a variable with respect to a function indicates the partial derivative of the 176

function with respect to this variable. Finally, we denote the price of carbon by ρ1 and the cost 177

of planting young trees by ρ2. [4] 178

The decision problem (DP) of the manager or forest owner is to find the optimal trajectories 179

of the control variables u(t,l) and p(t), which in turn allows the optimal trajectories of the state 180

variables x(t,l) and s(t), and variables b(t), E(t) and V(t) to be obtained. We assume that the 181

owner or manager of the forest maximizes the joint net benefits of timber production and 182

carbon sequestration denoted by J. The literature currently distinguishes between two main 183

accounting methods for carbon. In the first one (the flow method) the valuation of carbon 184

sequestration is based on the flux of carbon in the ecosystem, and the price of carbon within a 185

period of time. That is, if the net storage of carbon ( ) ( )db t dt ds t dt+ is either positive or 186

above a predetermined reference value at time t [5], carbon sequestration provides a value, 187

and it is given by the net storage of carbon times the price of carbon. This approach 188

corresponds to the concept of “additionality”, which requires that only carbon sequestered 189

above the status quo should be considered. However, if the net storage of carbon is negative 190

due to extensive logging activities, or if it is below a predetermined reference value, the value 191

of the carbon balance is negative. The second accounting method (the stock method) is based 192

on the concept of a rental rate of carbon which corresponds to the value of storing one ton of 193

carbon for a period of time, and is equal to the price of carbon times the discount rate 194

(Sohngen and Mendelsohn 2003). If there is a predetermined reference value of the carbon 195

stock, then only the tons of carbon that are additionally sequestered are taken into account to 196

determine the value of carbon sequestration. In this study we opted for the flow method since 197

the sequestered carbon in the biomass and the wood product can be easily accounted for in the 198

modeling framework presented below. 199

9

The economic benefits of carbon sequestration in the presence of a strictly positive discount 200

rate reside in the fact that carbon is sequestered at time t and released at a later point in time. 201

The sequestration occurs initially in the tree and once the tree is cut the carbon remains 202

sequestered in the wood product. Therefore, we have two types of carbon parking: first in the 203

tree and second in the timber product. Unfortunately, the carbon flow db/dt only takes into 204

account the carbon sequestered in the tree, not in the wood product. According to the 205

definition of db/dt, the sequestered carbon is released immediately after the tree has been cut. 206

Thus, the term db/dt does not take into account the fact that the sequestered carbon continues 207

to be sequestered during the lifetime PT of the wood product (permanence time). When trees 208

are cut the amount of sequestered carbon is reduced. In the case of an instantaneous release of 209

carbon at time t, the loss of the discounted monetary value of the stored carbon would be 210

given by 1

0

1 1 ( , )ml

rtb

l

u e l u t l dlβρ γ−= ∫ . However, since the sequestered carbon is not released 211

immediately, but gradually over PT years and, according to the release function ( )PTω , the 212

loss of the discounted monetary value of the stored carbon of the logged trees in year t is 213

given by 2

0

( )1 1

0

( ) ( , )ml PT

r tb

l

u e PT d l u t l dlτ βω τ ρ γ− +⎛ ⎞= ⎜ ⎟

⎝ ⎠∫ ∫ .[6] The term in the inner brackets takes 214

into account the fact that the postponed release of the sequestered carbon leads to a lower loss 215

in value in comparison with the loss resulting from an immediate release of the sequestered 216

carbon after the tree is logged. Let 0

( ) [0,1]PT

re PT dτυ ω τ−= ∈∫ denote the share of the 217

sequestered carbon in the biomass that is released from the wood product if all carbon 218

releases are expressed in terms of year t. This allows the amount of the released carbon in 219

terms of year t to be calculated. Therefore, the longer the carbon is sequestered in the wood 220

product, the lower the discounted value of the released carbon. 221

10

A comparison of the payment for carbon releases shows that 222

( )1 2

0

1 1 11 ( , )ml

rt rtb b c

l

u u e l u t l dl e uβρ γ υ ρ− −− = − =∫ . The term cu is needed to account for the 223

carbon that is not released immediately after the trees are logged. Thus the term 224

( )0

1 1 ( , )ml

cl

u l u t l dlβγ υ= −∫ can be incorporated into the carbon flow equation, leading to 225

( ) ( )cdb t dt u ds t dt+ + . Collecting the previously introduced elements, the decision problem 226

takes the form of: 227

( , ), ( )max

u t l p t0

1 20

( ) ( )( ( , ), ( , )) ( ) ( )mlT

rtc

l

db t ds tJ e B x t l u t l dl u t p t dtdt dt

ρ ρ−⎧ ⎫⎪ ⎪⎡ ⎤= + + + −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

∫ ∫ (DP) 228

under the restrictions 229

( , ) [ ( ( ), ) ( , )] ( ( ), ) ( , ) ( , ),x t l g E t l x t l E t l x t l u t lt l

δ∂ ∂+ = − −

∂ ∂ t∈[0, T), l∈[l0, lm], 230

0

2( ) ( , )4

ml

l

E t l x t l dlπ= ∫ , 231

( )( ) ( ), ( ) ,ds t h V t s tdt

= t∈[0, T), s(0) = s0 , 232

0

0( ) ( , )ml

l

V t l x t l dlβγ= ∫ , 233

0

1( ) ( , )ml

l

b t l x t l dlβγ= ∫ , 234

0

1( ) (1 ( )) ( , )ml

cl

u t l l u t l dlβγ υ= −∫ , 235

x(0,l) = x0(l), l∈[l0, lm], x(t,l0) = p(t), t∈[0, T), 236

0≤ u(t,l) ≤ umax(t,l), 0 ≤ p(t) ≤ pmax(t), 237

11

As the decision problem (DP) is based on one partial integrodifferential equation, Equation 238

(3), and one ordinary integrodifferential equation, Equation (4), the necessary conditions 239

presented previously for maximizing function J cannot be applied. Hritonenko et al. (2008) 240

establish the necessary extremum conditions in the form of a maximum principle for 241

optimization problems where a partial integrodifferential equation is present. 242

The necessary conditions for a maximum yield 243

∂J/∂u(t,l)≤0 at u*(t,l)=0, ∂J/∂u(t,l)≥0 at u*(t,l)=umax(t,l), 244

∂J/∂u(t,l)=0 at 0<u*(t,l)<umax(t,l) for almost all (a.a.) t∈[0, T), l∈[l0, lm]; (5) 245

∂J/∂p(t)≤0 at p*(t)=0, ∂J/∂p(t)≥0 at p*(t)=pmax(t), ∂J/∂p(t)=0 at 0<p*(t)<pmax(t), a.a. 246

t∈[0, T), (6) 247

( , ) [ ( ( ), ) ( , )] ( ( ), ) ( , ) ( , ),x t l g E t l x t l E t l x t l u t lt l

δ∂ ∂+ = − −

∂ ∂ (7) 248

x(0,l) = x0(l), l∈[l0, lm], x(t,l0) = p(t), t∈[0, T) , (8) 249

12

1

( , ) ( , )( ( ), ) [ ( ( ), )] ( , ) ( ( , ), ( , )) ( ) ( , )

4x

t l t lg E t l r E t l t l B x t l u t l l t F t l

t lr l β

λ λ πδ λ γ γ ρ

∂ ∂+ = + − + −

∂ ∂+ , (9) 250

λ(T,l) = 0, l∈[l0, lm], λ(t,lm) = 0, t∈[0, T), (10) 251

( )( )1( ) ( ), ( ) ( )

Tr t

t

t e h V s ds

τζ ρ τ τ ζ τ τ− − ∂= +

∂∫ , (11) 252

where λ(t,l) and ( )tζ are unknown dual variables, and 253

( )0( , ) ( ) ( ), ( )F t l l t h V t s tV

βγ ζ ∂=

∂, (12) 254

0

[ ( ( ), ) ( , )]( ) { ( ( ), ) ( , )} ( , )ml

EE

l

g E t l x t lt E t l x t l t l dll

γ δ λ∂= − −

∂∫ . (13) 255

The variables λ(t,l) and ζ(t) present the in situ value of trees and sequestered carbon in the 256

soil, respectively. 257

12

Equations (5) and (6) provide, for an interior solution [7], the following necessary conditions: 258

( , )( ( , ), ( , ))rtu t le B x t l u t l λ− = , (a.a.) t∈[0, T), l∈[l0, lm]; and (14) 259

2 0( ) ( , )rte t t lρ λ−− = , t∈[0,T). (15) 260

According to Equation (14) the discounted net benefits of logging have to be equal to the in 261

situ value of the trees for almost all t and l along the optimal path. Likewise, Equation (15) 262

states that the discounted cost of planting a young tree has to be identical to the in situ value 263

of a young tree along the optimal trajectory. Equation (7) describes the dynamics of the forest 264

as discussed above, and, due to the brevity of this exposition, the discussion is not repeated 265

here. The subsequent line presents the boundary conditions of the state variable x(t,l), i.e., the 266

initial diameter distribution of the trees and the inflow of newly planted trees. Equation (9) 267

indicates that the changes in the value of a standing tree (in situ value) over time and diameter 268

have to correspond to the changes in the value if the tree were cut down. The change in the in 269

situ value over diameter in Equation (9) is a composite expression given by 270

( ( ), ) ( , )g E t l t l lλ∂ ∂ and reflects the physical growth in diameter over time evaluated by the 271

change in the in situ value over diameter. The right-hand side of Equation (9) indicates that if 272

we had cut down trees, the interest earned from the money invested from the sale of these 273

trees would have been rλ and the value of the trees that had not died naturally should have 274

been δλ . Likewise, if we had cut down trees, the maintenance cost of the remaining trees, -Bx 275

> 0, would have decreased, which would contribute positively to the net benefits. In addition, 276

the expression 2 ( )4 l tπ γ reflects the monetary value of the improved conditions for the 277

growth of the remaining trees due to the decrease in intraspecific competition resulting from 278

trees being cut down. Moreover, if we had cut down trees, we would have foregone the 279

interest paid on the money received for sequestering carbon that corresponds to the cut trees, 280

13

leading to a decrease in net benefits, which is given by the term 1 1r l βγ ρ− . Finally, if we had 281

cut down trees the volume of the forest would have decreased, leading to a non-positive 282

change in the carbon content of the soil. The monetary value of this change is given by the 283

term ( , )F t l . The transversality conditions of the decision problem are stated in Equation (10). 284

Equation (11) provides the in situ value of soil carbon, which is given by the price of 285

sequestered carbon plus the discounted value of future changes in soil carbon as its level 286

increases or decreases. 287

As the decision problem (DP) is based on one partial integrodifferential equation, Equation 288

(3), and one ordinary integrodifferential equation, Equation (4), the first order conditions 289

include a system of partial integrodifferential equations. Given the complexity of the resulting 290

equations, an analytical solution of the first order conditions cannot be obtained, and it is 291

necessary to resort to numerical techniques to obtain a solution to the forest manager’s 292

decision problem. 293

Available techniques such as the gradient projection method (Veliov 2003) or the method of 294

finite elements may be appropriate to solve a distributed optimal control problem numerically 295

(Calvo and Goetz 2001). However, all of these methods require programming complex 296

algorithms that are not widely known. Therefore, we propose adapting a different method 297

called the Escalator Boxcar Train, EBT, used by de Roos (1988) to describe the evolution of 298

physiologically-structured populations. He has shown that this technique is an efficient 299

integration technique for structured population models. In contrast to the other available 300

methods, the EBT can be implemented with standard computer software such as GAMS 301

(General Algebraic Modeling System), which is used to solve mathematical programming 302

problems. 303

14

Applying the EBT allows the partial integrodifferential equations of the problem (DP) to be 304

transformed into a set of ordinary differential equations. Besides a brief presentation of the 305

EBT method, Goetz et al. (2007) show how this approach can be extended to account for 306

optimization problems by incorporating decision variables. To transform the decision problem 307

(DP), we first divide the range of the diameter into n equal parts, and define Xi(t) as the 308

number of trees in the cohort i, with i = 0, 1, 2, 3,…, n, that is, the trees whose diameter falls 309

within the limits li and li+1 are grouped in the cohort i. Likewise, we define Li(t) as the average 310

diameter, Ui(t) as the number of cut trees within cohort i, and P(t) as the number of planted 311

trees in cohort 0. Within these provisions the decision problem (DP) can now be formulated 312

as the maximization of the objective functions subject to a set of ordinary differential 313

equations instead of one partial integrodifferential and one ordinary differential equation 314

given by 315

2( ), ( )

00 0 0 0

0

20

max ( ( ), ( )) ( ( ))

subject to

( ) ( )( ( ), ) ( ) ( ), ( ( ), )

( ( ), ) ( ) ( ( ), ) ( ) ( )

( )

)

( )( )

( ( ,

Trt

U ct P t

i ii i i i

B X t U t P t e dt

dX t dL tE L X t U t g E Ldt dt

dX dE t l X t E t l L t P

db t ds tu

tdt dl

dL g E tdt

tdt dt

t t

ρ

δ

δ δ

ρ −⎧ ⎫−⎨ ⎬

⎩ ⎭

= − − =

= −

⎡ ⎤+ + +⎢ ⎥⎣ ⎦

− +

=

∫

0 0 0 0 0 0

00 0

) ( ) ( ( ), ) ( ) ( ( ), ) ( )

(0) , ( ( ), ) ( , ) ( ), ( ), ( ) 0 ( ) ( )i i i i

dl X t g E t l L t E t l L tdl

X X g E t l x t l P t U t P t U t X t

δ+ −

= = ≥ ≤ 316

( ) ( )20

0

( )( ) , ( ), ( ) , (0)4

n

i ii

ds tE t L X h V t s t s sdt

π=

= =∑ , 317

( ) ( )0 10 0

( ) , ( )n n

i i i ii i

V t L X b t L Xβ βγ γ= =

= =∑ ∑ , 318

15

where X and U denote the vectors 1, , nX X X= … and 1, , nU U U= … , respectively, and 319

0X the initial number of tress in each cohort. For numerical reasons the cohort with the 320

smallest trees has to be treated differently than the other cohorts. Hence, we obtain a separate 321

ordinary differential equation for 0X and 0L . Once the system of differential equations is 322

written as a system of difference equations, standard mathematical programming software 323

such as GAMS (General Algebraic Modeling System, (Brooke et al. 1992)) can be employed 324

to solve the problem. 325

However, before the decision problem can be solved, all functions have to be specified. While 326

the specification of the benefit and cost functions is straightforward, the biological functions 327

have to be estimated based on data generated by a biophysical forest growth simulator. For 328

this purpose we employed the bio-physical simulation model GOTILWA (Growth Of Trees Is 329

Limited by Water).[8] This not only simulates the development of the trees, but also the 330

amount of sequestered carbon in the biomass and in the soil. The next section describes the 331

simulation of forest growth by GOTILWA in detail, and presents the solution to the forest 332

manager’s decision problem, i.e., the optimal short- and long-term trajectories of the logging 333

and planting decisions and the evolution of the standing trees in the forest. 334

335

3. Empirical analysis 336

The purpose of the empirical analysis is to demonstrate the applicability of the presented 337

theoretical model for management and policy issues but not a detailed analysis of the land use 338

and land-use change options within existing international agreements with respect to climate 339

change. Initially we determine the selective-logging regime that maximizes the discounted 340

private net benefits from timber production and carbon sequestration of a stand of Pinus 341

16

sylvestris over a time horizon of 200 years. Thereafter, the analysis concentrates on the main 342

aspect of the empirical study by establishing to what extent a change in the price of carbon 343

affects the optimal logging regime. We specify the parameters and the functions in Section 344

3.1, and analyze the optimal management regime in Section 3.2. 345

3.1. Data and specification of functions 346

The net benefit function of the economic model, ( ( , ), ( , ))B x t l u t l , consists of the net revenue 347

from the sale of timber at time t, minus the costs of maintenance, which comprise clearing, 348

pruning, and grinding the residues. The net revenue is given by the sum of the revenue of the 349

timber sale minus logging costs defined as: ( )( ) ( )( )( ) ( )( ) ( ) ( )( )0

n

i i i ii

L t lc L t tv L t U t mc X tρ=

⎡ ⎤⎡ ⎤− −⎢ ⎥ ⎣ ⎦

⎣ ⎦∑ 350

where ( ) ( )0

n

ii

X t X t=

= ∑ . The terms in the first square brackets denote the sum of the revenue of 351

the timber sale minus the cutting costs of each cohort i, and the term in the second square 352

bracket, ( )( )mc X t , accounts for the maintenance costs. The function ( )iLρ denotes how the 353

timber price per cubic meter of wood varies with diameter; ( )itv L is the total volume of a tree 354

as a function of its diameter; ( )ilc L is logging cost. Timber price per cubic meter was taken 355

from a study by Palahí and Pukkala (2003), who analyzed the optimal management of a Pinus 356

sylvestris forest in a clear-cutting regime. They estimated a polynomial function given by 357

( ) 23.24 13.63 , 86.65L Min Lρ ⎡ ⎤= − +⎣ ⎦ , which is an increasing and strictly convex function, for a 358

diameter lower than 65cm. At L = 65 the price reaches its maximum value, and for 65L > it is 359

considered constant. The logging cost comprises logging, pruning, cleaning the underbrush, 360

and collecting and removing residues. Based on the work by Palahí and Pukkala (2003), the 361

logging cost is given by the function ( )( ) 6 exp 4.292 0.506lnlc L L= + −⎡ ⎤⎣ ⎦ , measured in euros 362

per cubic meter of logged timber. Data about maintenance costs were provided by the 363

17

consulting firm Tecnosylva, which elaborates forest management plans throughout Spain. 364

According to the data supplied by Tecnosylva, the maintenance cost function is approximated 365

by ( ( ))mc X t , and is given by 2( ( )) 10 0.0159 0.0000186mc X t X X= + + . The planting cost 366

is linear in the amount of planted trees and is given by C(P) = 0.73P. The thinning and 367

planting period, Δt, is set at 10 years, which is a common practice for a Pinus sylvestris forest 368

(Cañellas et al. 2000). 369

To proceed with the empirical study, various initial diameter distributions of a forest were 370

chosen. These distributions were specified as a transformed beta density function ( )lθ since it 371

is defined over a closed interval and allows a large variety of different shapes of the initial 372

tree diameter distributions to be defined (Mendenhall et al. 1990). The initial forest consists of 373

a population of trees with diameters within the interval 0 cm ≤ l ≤ 50 cm. The density function 374

of the diameter of trees, ( ); ;lθ γ ϕ , is defined over a closed interval, and thus the integral 375

( )1 1

1

1

; ,l

l dlθ γ ϕ+

∫ (16) 376

gives the proportion of trees lying within the range [ , 1)i il l + . We defined l0 = 0 and lm = 80 as 377

the minimum and maximum tree diameters, respectively. For the simulation of the forest 378

dynamics, we concentrate on the diameter interval [0, 50], because thereafter the tree growth 379

rate is very small. This interval was divided into 10 subintervals of identical length. 380

Therefore, the diameters of the trees of each cohort differ at most by 5 cm, and the size of the 381

trees of each cohort can be considered homogeneous. 382

To determine the forest and soil dynamics, the growth of a diameter-distributed stand of Pinus 383

sylvestris without thinning was simulated with the bio-physical simulation model GOTILWA. 384

This model simulates growth and mortality and explores how the life cycle of an individual 385

tree is influenced by the climate, the characteristics of the tree itself, and environmental 386

18

conditions. The model is defined by 11 input files specifying more than 90 parameters related 387

to the site, soil composition, tree species, photosynthesis, stomatal conductance, forest 388

composition, canopy hydrology, and climate. We simulated the growth of the forest based on 389

the previously specified initial diameter distributions. 390

The data generated from the series of simulations allow function ( ), ig E L , which describes the 391

change in diameter over time, to be estimated. This type of function was specified as a von 392

Bertalanffy growth curve (von Bertalanffy 1957), generalized by Millar and Myers (1990), 393

which allows the growth rate of the diameter to vary with environmental conditions specified 394

as the total basal area of all trees. The precise specification of the function is given by 395

( ) ( )( )0 1, i m ig E L l L Eβ β= − − . The exogenous variables of this function are diameter at breast 396

height (Li) and basal area (E) provided by GOTILWA. The parameters β0 and β1 are 397

proportionality constants and were estimated by OLS based on the pooled data. The 398

estimation yielded the growth function ( , ) (80 )(0.0070177 -0.000043079 )i ig E L L E= − . Other 399

functional forms of ( ), ig E L were evaluated as well, but they explained the observed variables 400

to a lesser degree. As GOTILWA only simulates the survival or death of an entire cohort, not 401

the survival or death of an individual tree, it was not possible to obtain an adequate estimation 402

of function ( ), iE Lδ , which describes the mortality of the forest. Nevertheless, information 403

provided by the company Tecnosylva suggests that the mortality rate in a managed forest can 404

be considered almost constant over time and independent of the diameter. Thus, according to 405

the data supplied by Tecnosylva, ( ), iE Lδ was made constant over time and equal to 0.01 for 406

each cohort. 407

The value of the tree volume parameters, ( )itv L , has also been estimated using the data 408

generated with GOTILWA. The tree volume is based on the allometric relation 409

19

( ) 1.7450870.00157387tv L L= . In addition, the data obtained from the GOTILWA simulations 410

allowed us to estimate the change in soil carbon over time as a function of the forest tree 411

volume and the amount of soil carbon. The estimation yielded the 412

function (212.12 ( ))( 0.0322 0.0003385 ( ))h s t V t= − − + . Row and Phelps (1990) suggest that 413

the release of sequestered carbon in the wood product can be described by an extended 414

logistic decay function, which Eggers (2002) specified by ( )

1.2( ) 1.21 5 c PTPT

eω = −

+. In 415

accordance with van Kooten and Bulte (1999) and Eggers (2002) we assumed that a long 416

permanence time corresponds to a half-life period of 50 years. Hence we employed the report 417

values for the function ( )c PT by Eggers (2002) for a long (50), medium (16) and short (1) 418

half-life period. Hence, we obtain for a long, medium and short half-life period that 1 υ− is 419

given by 0.58, 0.28, and 0.08, respectively. This expresses in terms of year t the share of 420

carbon that is retained in the wood product from time t to the end of the lifetime of the 421

product. 422

For a given initial diameter distribution of the trees, and given specifications of the economic 423

and biophysical functions of the model, a numerical solution of the decision problem (DP) 424

can be obtained. To proceed with the empirical study, three different initial diameter 425

distributions of a forest were chosen. They were obtained by varying the parameters γ and ϕ 426

of the beta density function, and are depicted in Figure 1. They stand for a young forest, a 427

mature forest, and a forest in which trees are distributed uniformly over the diameter range. 428

The initial basal area of all three distributions is identical and equal to 25m2/ha so that the 429

resulting optimal management regimes can be compared. 430

Figure 1 here 431

3.2. Analysis of the optimal selective-logging regime 432

20

In this part of the empirical study we determine the optimal management regime of a size-433

distributed forest and examine how the optimal management regime is affected by variations 434

in the carbon price. The optimizations were carried out with the CONOPT3 solver available 435

in the GAMS optimization package. For a given initial distribution, the numerical solution of 436

the problem determines the optimal trajectories of the decision variables: logging, iU , and 437

planting, P; and of the state variables: number of trees, iX , and diameter, iL . Consequently, it 438

also determines the value of the economic variables, such as the net revenue from timber 439

sales, logging costs, planting costs, maintenance costs and revenues from carbon storage. All 440

optimizations were carried out on a per-hectare basis. 441

The empirical analysis begins by calculating the optimal selective-logging regime for the 442

young forest distribution, given a long half-life period and a discount rate of 2%.[9] Figure 2 443

depicts the evolution of the number of standing trees and carbon in the forest over time for 444

different levels of the carbon price. It shows that the price of sequestered carbon, expressed in 445

terms of CO2, has a significant influence on the optimal selective-logging regime. When the 446

price increases from zero (base case) to 20 €/CO2 ton, it is optimal to increase the investment 447

in the forest. Therefore, the number of planted trees increases from 0 to 880 in the first period, 448

and the number of logged trees decreases from 115 to 107.[10] As a result, the number of 449

standing trees in the forest increases, especially in the initial periods of the time horizon, and 450

the amount of carbon sequestered increases in the long run (see Figure 2a). For a price of 0 451

€/CO2 ton, the amount of carbon in the forest ecosystem at the end of the planning horizon, 452

including soil carbon and the carbon stored in the above-ground and below ground biomass, is 453

108.3 tons/ha. However, it increases to 210.6 tons/ha when the carbon price is 20 €/CO2 ton, 454

that is, the carbon stored in the ecosystem in the long run is almost doubled (see Figure 2b). 455

Figure 2 here 456

21

Tables 1 and 2 present a more detailed analysis of the results for the base case, in which net 457

benefits only originate from timber (price of 0 €/CO2 ton) and for the case in which the price 458

is 10 €/CO2 ton. Table 1 contains the evolution of the physical variables over the first 100 459

years, while Table 2 presents the monetary values resulting from optimal forest management. 460

In particular, it is important to analyze the discounted sum of the net benefits obtained from 461

forest management, which we will refer to as simply the net benefits in the remaining part of 462

the paper. It is possible to see in Table 2 that when carbon sequestration does not form part of 463

the management objective, the net benefits over 200 years are about 6763 €/ha. When the 464

benefits and costs of carbon sequestration with a carbon price of 10 €/CO2 ton are 465

incorporated in the formulation of the decision problem, the net benefits increase to 7703 466

€/ha. Although the discounted sum of the net timber benefits decreases from 6762 €/ha to 467

6108 €/ha, the net benefits of carbon sequestration compensates the lower net benefits 468

obtained from the timber sale and the higher maintenance costs due to an increase in the tree 469

density. For an exact calculation of the net benefit of the carbon sequestration it is important 470

to consider only the carbon that is sequestered in addition to the carbon sequestered by the 471

forest that maximizes the net benefits of the timber product. In this way the empirical study 472

only takes into account the “additional” sequestered carbon. 473

Table 1 and 2 here 474

To analyze to what extent this result can be generalized, we conducted a sensitivity analysis 475

for the three initial forest distributions considered, and for different carbon prices, ranging 476

from 0 to 25 €/CO2 ton. Figure 3 depicts the NPV of the different optimization scenarios. The 477

figure corroborates the previously obtained result by showing that the NPV of the benefits of 478

forest management increases with an increase in the price from 0 to 25 €/CO2 ton. However, 479

the increase in the net benefits depends on the shape of the initial forest distribution. For low 480

carbon prices (0 - 10 €/CO2 ton), both a mature and a uniformly distributed forest produces 481

22

higher net benefits compared to a young forest. In contrast, for high carbon prices (over 15 482

€/CO2 ton), the result is reversed. This is due to the fact that for low carbon prices, the timber 483

price is more decisive, so that mature or uniform forests generate higher net benefits than 484

young forests. However, if the carbon price is relatively high its price becomes decisive, and 485

the carbon net benefits of a young and highly productive forest outweigh the reduction in net 486

benefits of timber production. 487

Figure 3 here 488

One might suppose that the results obtained depend strongly on the permanence time, PT, of 489

the sequestered carbon in the wood products since this determines the amount that the forest 490

owner has to pay for the release of carbon once the trees have been logged. To analyze this 491

point in more detail, we conducted a sensitivity analysis with respect to PT for the young 492

distribution, and we calculated the optimal logging regime, given a short lifespan (half-life 493

period equal to 1 year), a medium lifespan (half-life period equal to 16 years) and a long 494

lifespan (half-life period of 50 years). The results show that the PT does not affect the net 495

benefits obtained from forest management. In order to save space we chose not to present 496

these results in graphical form. 497

In addition, we conducted a sensitivity analysis of the discount rate. Figure 4 depicts the 498

change in the net benefits over a time horizon of 200 years for different carbon prices, given 499

discount rates of 2% and 4%. Figure 4 shows that an increase in the carbon price may lead to 500

a substantial increase in the net benefits if the chosen discount rate is low. For instance, for a 501

discount rate of 2%, a price of 5 €/ton of CO2 leads to an increase in the net benefits of 5% 502

compared to the base case, and a price of 25 €/ton of CO2 leads to an increase in the net 503

benefits of 20%. However, the higher the discount rate, the smaller the gains from carbon 504

sequestration. This tendency is amplified even more if the carbon price is relatively small. For 505

23

instance, for a discount rate of 4%, the carbon price needs to be 15 €/ton to obtain a 6% 506

increase in the net benefits compared to the base case. Hence, if the discount rate of the 507

private owner of a forest is above the discount rate used by a social planner, the amount of 508

sequestered carbon is below the socially optimal amount. 509

Figure 4 here 510

Our calculations show that an increase in the carbon price induces management regimes that 511

augment the sequestered carbon in the biomass and soil. However, the increase in the 512

sequestered carbon goes along with a decrease in the net benefits from timber production. The 513

relation between the decrease in the net benefits of timber production and the increase in 514

sequestered carbon (price of carbon) over time horizons of 100 and 200 years is presented in 515

Figure 5a. This figure also shows that the decrease in the net benefits of timber is more 516

pronounced over 100 years than over 200 years. This result can be explained by the fact that a 517

large share of the timber net benefits can only be obtained after 100 years because only then 518

the trees have achieved the optimal logging diameter. The decrease in net benefits of timber 519

production results from an increase in the carbon price, which in turn leads to a change in the 520

management regime that favors biomass production at the cost of timber production. 521

Moreover, Figure 5a allows the costs of sequestered carbon to be calculated per ton, and those 522

costs presented in Figure 5b, which shows that the costs of sequestered carbon per ton are 523

between 8.5 and 48.1 € (2.3 – 13.1 €/ton of CO2). Based on an extensive literature review and 524

a meta-analysis, van Kooten and Sohngen (2007) state that the carbon sequestration costs in 525

Europe range between 120 and 710 €/tC.[11] In this respect the sequestration costs presented 526

in this study are clearly at the lower end of previous studies. It may seem that the results of 527

our study cannot be compared directly with the results of the previous literature since 528

previous studies predominantly determined the carbon sequestration costs for afforested land 529

and, consequently, the opportunity costs of the land have to form part of the carbon 530

24

sequestration costs. However, since we are considering a change in the management regime of 531

a standing forest, the opportunity costs of the land affect the aggregate discounted net benefit 532

of timber but not a change in these benefits. In other words, the opportunity costs of the land 533

are independent of the forest management regime and therefore do not need to be taken into 534

account when the costs for additionally sequestered carbon are calculated. 535

Figure 5 here 536

4. Conclusions 537

In this paper we present a theoretical model that allows us to determine the optimal 538

management regime of a diameter-distributed forest. The theoretical model can be formulated 539

as a distributed optimal control problem in which the control variables and the state variable 540

depend on two factors: time and the diameter of the tree. As the resulting necessary conditions 541

of this problem include a system of partial integrodifferential equations, the problem cannot 542

be solved analytically. For this reason, a numerical method is employed. The Escalator 543

Boxcar Train technique can be used to solve a distributed optimal control problem by 544

transforming the independent factor, diameter, into a state variable of the system that evolves 545

over time. In this way, the partial integrodifferential equation is decoupled into a system of 546

ordinary differential equations, converting the distributed optimal control problem into a 547

classic optimization problem that can be solved by utilizing standard mathematical 548

programming techniques. 549

An empirical analysis is conducted to determine the optimal selective-logging regime, that is, 550

the selective-logging scheme that maximizes the present value of net benefits from timber 551

production and carbon sequestration of a privately owned Pinus sylvestris forest, and to 552

evaluate how the optimal management of the forest is affected by variation in the market 553

prices of carbon sequestration. The study is characterized by the fact that the complex growth 554

25

process of the trees and the storage of carbon in the soil and in the biomass are taken into 555

account. 556

The results of our calculations show that an increase in the carbon price leads to an increase in 557

the number of trees in the forest and to an increase in the carbon sequestered in the forest 558

ecosystem. Moreover, it underlines that the accurate assessment of the carbon in the biomass 559

and the soil, and the considered time horizon are crucial for the correct determination of the 560

optimal forest and carbon management regime. In contrast, the permanence time of the 561

sequestered carbon in wood products does not affect the optimal management regime. If the 562

carbon price is relatively low, mature and uniform forests lead to a higher increase in net 563

benefits than young forests. This result is reversed for the case of a young forest. The overall 564

costs of carbon sequestration as a result of a change in the forest management regime are 565

relatively small in comparison with the results presented in the existing literature. Part of this 566

difference can be explained by the fact that the additionally sequestered carbon does not 567

require a change in the land use. Hence, in contrast with afforestation, a change in the 568

management regime does not require the opportunity cost of the land to be imputed. 569

Consequently, carbon sequestration costs are expected to be lower when the forest 570

management regime is changed than when land use is changed. This study presents the 571

magnitude of the decrease in the costs. 572

573

Endnotes574 [1] More complex models which take into account space, for instance, in the form of local

conditions can be solved by heuristic approaches, but not by efficient optimization

techniques (Weintraub et al. 2007).

26

[2] We do not consider space and assume that trees with different sizes are distributed evenly

in the stand. However, if this is not the case, distance dependent competition indices would

be more appropriate (García-Abril et al. 2007).

[3] If we had chosen age as the structuring variable the function g would be constant and

equal to 1 since (age) 1d dt g= = and therefore the aging of the tree by one year would

correspond to one year of calendar time.

[4] In the case of natural reproduction (Karlsson and Örlander 2000) it can be assumed that

the number of seeds that turn into seedlings is sufficiently large, which allows the forest

manager to choose the number of trees with a diameter of l0 by removing any additional

trees. In this case, p(t) denotes the number of upgrowing trees and ρ2 the “nursing” costs.

[5] For example, the reference value may be given by the net storage of carbon resulting from

maximizing the net benefits of timber production alone for an already existing or a newly

planted stand. The management regime that only maximizes the net benefits of timber

production follows from setting the price of carbon equal to zero.

[6] An example of the specification of the function ( )PTω is presented in Section 3.1: Data

and Specification of Functions.

[7] The numerical solution of the problem showed that interior and boundary solutions occur

along the optimal trajectory. In the latter case the Lagrange multipliers take on positive

values.

[8] This program has been developed by C. Gracia and S. Sabaté, University of Barcelona,

Department of Ecology and CREAF (the Centre for Ecological Research and Forestry

Applications), Autonomous University of Barcelona, respectively.

[9] Palahí and Pukkala (2003) and Díaz-Balteiro and Romero (2003) choose a discount rate

of 2% while Creedy and Wurzbacher (2001) choose a discount rate of 4%. Thus, we start

the empirical analysis with a discount rate of 2% and subsequently conduct a sensitivity

27

analysis to evaluate the effect of different discount rates on the optimal management

regime.

[10] These results demonstrate how important it is to include planting as a separate decision

variable.

[11] Van Kooten and Sohngen present their results in US$ which we converted into € based

on an exchange rate of 1.45$/€.

28

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30

0 10 20 30 40diameter at breast height (cm)

50

100

150

200

250

300

num

ber

oftr

ees mature

forest

youngforest

uniform

distribution

Figure 1. Initial distributions used in the study

31

2a)

0 50 100 150 200time

500

1000

1500

2000N

umbe

rof

stan

ding

tree

s

CO2 price of 20 Euro

CO2 price of 10 Euro

CO2 price of 0 Euro

2b)

0 50 100 150 200time

50

100

150

200

250

300

Car

bon

inth

efo

rest

ecos

yste

m(t

ons/

ha)

CO2 price of 20 Euro

CO2 price of 10 Euro

CO2 price of 0 Euro

Figure 2: Variation in the evolution of the main variables over time for different levels of the

carbon price

32

0 5 10 15 20 25Price of CO2 (euros/Tm)

4000

6000

8000

10000

12000

14000

Netbenefits(euros)

matureforest

uniform

distribution

youngforest

Figure 3: Net benefits over 200 years as a function of the CO2 price for different initial

distributions of the forest

0 5 10 15 20 25Price of CO2 (euros/Tm)

5

10

15

20

25

Incrementofthenetbenefits(%)

r=0.04

r=0.02

Figure 4: Change in the net benefits over 200 years as a function of the CO2 price for different

levels of the discount rate

33

a) Total costs

20 30 40 50 60 70 80Increase in sequestered carbon averaged over time (tons)

500

1000

1500

2000

2500

3000

3500D

ecre

ase

inne

tben

efits

(Eur

o)

200 years

100 years

b) Unitary costs

20 30 40 50 60 70 80Increase in sequestered carbon averaged over time (tons)

500

1000

1500

2000

2500

3000

3500

Dec

reas

ein

netb

enef

its(E

uro)

200 years

100 years

Figure 5: Carbon sequestration costs as a function of the increment in the sequestered carbon.

34

Table 1: Optimal selective-logging regime of a young stand (physical variables)

Considering only timber (price =0 €/CO2 ton) Logged

Year Number of trees

Planted trees

Logged trees

volume (m3/ha)

0 820 0 115 98.78 10 797 0 15 9.14 20 763 26 52 38.56 30 773 102 84 58.18 40 780 119 104 67.64 50 796 151 127 79.77 60 815 170 143 87.33 70 806 93 94 58.56 80 768 127 157 112.63 90 801 55 14 8.25 100 789 101 105 64.57

Considering timber and carbon sequestration (price =10 €/CO2 ton) 0 1250 402 87 79.45 10 1341 149 46 33.34 20 1267 0 61 43.15 30 1276 115 93 60.29 40 1303 123 84 53.73 50 1319 156 127 76.86 60 1322 175 158 90.72 70 1286 222 245 134.88 80 1250 243 266 172.15 90 1267 139 110 81.07 100 1282 151 124 91.66

35

Table 2: Optimal selective-logging regime of a young stand (monetary values)

Considering only timber (price =0 €/CO2 ton)

Year

Net revenue from timber

sale (€/ha)(b)

Maintenance cost

(€/ha)

Planting cost

(€/ha)

Carbon in the forest (tons/ha)

Carbon revenue (€/ha)

Net benefit (€/ha)

Discounted net benefit

(€/ha) 0 3019.96 355.74 0 134.35 - 2664.21 2664.21 10 218.92 345.26 0 146.93 - -126.33 -103.64 20 1042.79 329.71 18.7 153.19 - 694.37 467.29 30 1496.9 334.1 74.4 154.15 - 1088.4 600.88 40 1670.21 337.53 86.94 152.13 - 1245.75 564.19 50 1909.38 344.85 110.54 146.26 - 1453.99 540.2 60 2050.62 353.6 124.1 137.79 - 1572.92 479.4 70 1401.55 349.46 67.9 137.13 - 984.19 246.07 80 2979.38 332 92.52 121.42 - 2554.86 524.03 90 191.78 347.14 40.07 134.69 - -195.42 -32.88 100 1525.91 341.63 73.73 133.775 - 1110.55 153.29

Discounted sum over 200 years: 8900.34 1873.29 264.29 - 6762.75 Considering timber and carbon sequestration (price =10 €/CO2 ton)

0 2525.16 590.18 293.49 140.81 0 1641.5 1641.5 10 901.06 648.64 108.87 152.54 262.64 406.19 333.22 20 1131.89 600.47 0 166.5 541.06 1072.47 721.74 30 1493.96 606.6 83.8 177.3 699.73 1503.3 829.93 40 1307.8 623.64 90.09 190.29 479.12 1073.2 486.04 50 1787.5 633.74 113.92 196.67 434.68 1474.51 547.82 60 2027.26 636.05 127.68 198.47 393.82 1657.35 505.13 70 2917.77 612.71 161.87 186.79 -15.09 2128.09 532.08 80 4227.54 590.25 177.58 162.89 2.95 3462.66 710.22 90 2194.64 600.88 101.6 164.22 -66.75 1425.41 239.84 100 2484.47 610.36 110.45 164.08 166.39 1930.05 266.41

Discounted sum over 200 years: 10206.29 3373.43 724.22 1484.82 7703.56