Optimal structure and development of uneven-aged Norway spruce forests
Olli Tahvonen
Department of Forest Sciences
University of Helsinki
Latokartanonkaari 7
000140 University of Helsinki
Finland
[email protected]
tel. +358504156565
fax: +358919158100
Optimal structure and development of uneven-aged Norway spruce forests
Abstract
Optimal harvesting of Norway spruce forests is studied applying a single-tree model for
uneven-aged management. Optimization is carried out by gradient-based, large-scale
interior point methods. Assuming volume maximization and natural regeneration, it is
optimal to apply uneven-aged management. Allowing artificial regeneration the result is
the reverse. Economically optimal solutions with a 20-year harvesting interval produce an
annual sawn timber output of 4.4-2.4m3 depending on thermal zone and interest rate.
Before harvest basal area varies between 18 and 12m2 and the diameter of harvested trees
between 15 and 33 cm. In contrast with the classic inverted J-structure, optimal steadystate size structure resembles a serrate form. Profitability of even- and uneven-aged
management is compared assuming the initial stand state represents an optimal unevenaged steady state. A switch to even-aged management is optimal given most favorable
growth conditions and interest rate below 1-2%. In other cases, it is economically optimal
to continue uneven-aged management although volume output remains lower than under
even-aged management.
Keywords: uneven-aged forestry, even-aged forestry, individual tree model,
optimal harvesting, age-structured models, size structured models
1. Introduction
Forest policy in Nordic countries is strongly oriented toward managing boreal conifer
forests as even-aged stands (Siiskonen 2007). The main argument for the prevailing evenaged management system has been its economic superiority compared to selective
cutting. However, scientific evidence is incomplete. To date there are no convincing
studies showing that for shade tolerant species like Norway spruce, even-aged
management is superior to alternatives like selective cuttings. In contrast with the present
practices, a recent survey on forest policy legitimacy (Valkeapää 2009) found that a
majority of Finnish citizens and large share of forest owners prefer to have alternatives in
forest management and do not favor clearcuts. This study sheds some now light to this
puzzle by extending the existing understanding applying a single tree model together with
economic optimization.
Most earlier studies such as Andreassen (1995) and Andreassen and Oyen (2002)
have performed silvicultural experiments where a mature coastal Norway spruce stand
has been managed applying clearcuts, single-tree selection, or group selection. According
to the results, single-tree selection yields about 85% of the income obtained by clearcuts.
In contrast, studies based on optimization have shown that uneven-aged Norway
spruce stands may yield higher net present values of revenue compared to even-aged
management (Pukkala et al. 2010; Tahvonen et al. 2010). In Wikström (2000) the result is
the reverse: uneven-aged management yields about 90% of the even-aged revenues.
However, none of these studies have yet applied unrestricted single-tree models together
with general dynamic optimization approach. The aim of this study is to solve
economically optimal structure and development for uneven-aged Norway spruce stands
applying a single tree growth model, detailed biological and economic data and most
general economic optimization principles.
3
Existing economic studies on uneven-aged management based on single-tree
models are rare. Exceptions include Haight (1987) and Haight and Monserud (1990,
1991) who apply multiple-species, single-tree models in the western United States. Their
optimization method is the Hooke and Jeeves (1961) algorithm. The present study
includes differentiable functions and the optimization can be based on the most efficient
gradient-based algorithms for large-scale nonlinear problems. This allows for the use of
many state variables (up to 60) and long time horizons (e.g., 400 periods) without any
need to simplify the single-tree model. In this study, the single-tree model determines
ingrowth endogenously, in contrast to fixed ingrowth in Wikström (2000). The
optimization problem is solved as a general dynamic problem, in contrast to theoretically
problematic investment-efficient steady states in Pukkala et al. (2010). Dynamic
optimization yields both theoretically correct steady states and paths converging toward
long-run equilibria. Tahvonen et al. (2010) used transition matrix model with fixed size
classes and concentrate to one thermal zone only. This study applies a more detailed
single-tree model and extends the results over five thermal zones.
Because of the above differences with existing literature this study presents
several new results on economically optimal uneven-aged management and its
profitability with respect to traditional even-aged management. It is shown that in spite of
the large number of state variables and nonlinearities the volume maximization unevenaged solutions converge to steady states with constant harvest and stand structure.
However, independently of terminal zones volume output is maximized applying artificial
regeneration and even-aged management. In contrast, economic outcome is maximized
applying uneven-aged management the exceptions being the most favorable thermal
zones and interest rates below 1-2%. Economically optimal solutions converge to steady
states where the tree size distributions resemble a serrate form instead to the traditional
4
inverted J-structure. In addition, results yield information on the advantages of using a
more complex single tree model compared to transition matrix model with less
demanding computational requirements.
The paper is organized as follows. The next section introduces the optimization
problem, the empirical specification used and the numerical solution methods. The results
section shows first the outcomes based on volume maximization and then proceeds to
economically optimal solutions. Next the economic solutions are compared to outcomes
in even-aged management. Discussion section compares the results to those in the
previous studies. Finally, the conclusions section presents some forest policy remarks and
open questions to future studies.
2. The model and empirical specifications
The optimization problem
Let xst ,s = 1,...,n + 1, t = 0,1, 2 ,... denote the number of trees in age class s and variables
d st , s = 1,..., n, t = 0,1, 2,... the diameter (cm) of age class s trees. Both are given in the
beginning of period t. Harvested trees are hst , s = 1,..., n, t = 0,1, 2,.... Natural regeneration
is given by the ingrowth function φ ( xt ,dt ) . The fraction α s (xt , dt ), s = 1,..., n of trees
survives to the next period. Thus the numbers of trees evolve according to
x 0 given,
(1)
x1,t +1 = φ ( xt ,dt ) − h1t , t = 0 ,1,...
(2)
xs +1,t +1 = xstα s ( xt ,dt ) − hst , s = 1,...,n, t = 0 ,1,....
(3)
Note that harvest occurs at the end of each period. In (3) trees die naturally if they reach
age class n and are not harvested.
5
The tree diameter in the youngest age class is δ 0 ( xt ,dt ) and the diameter
increment in age class s in period t is δ s (xt , dt ) . Thus, the tree diameter distribution
evolves according to
d 0 given
(4)
d1,t +1 = δ 0 ( xt ,dt ), t = 0,1,...
(5)
d s +1,t +1 = d st + δ s ( xt ,dt ), s = 1,...,n − 1, t = 0,1,...
(6)
The volumes of saw logs and pulpwood are denoted by functions νi (d st ), i = 1, 2 (m3 ) .
Let pi , 1, 2 refer to the associate prices. The harvesting cost is C (ht , dt ) and the discount
factor b(=1/(1+r), where r is the interest rate. The problem is to
max
{ hst ,t =1,...,s =1,...,n }
 n
 t
∑ [ p1v1( d s +1,t +1 )hst + p2 v2 ( d s +1,t +1 )hst ] − C( ht ,dt +1 ) b
∑
t = 0  s =1

∞
(7)
subject to (1)-(6) and the constraints ht ≥ 0, xt ≥ 0, t = 0,1,.... Note that harvest occurs at
the end of each period and the diameter of a tree harvested from age class s equals its
level in age class s+1 in the beginning of period t+1. When the aim is to maximize timber
volume production, C = 0, pi = 1, i = 1, 2 , and b=1. The cost function includes fixed cost
with the implication that it may not be optimal to harvest the stand every period. This will
be taken into account by computing the model by different harvesting intervals, i.e.,
assuming that
hst = 0, s = 1,...,n for t ≠ ik , where i = 0,1,... and k ≥ 1 is an integer.
(8)
6
Empirical specifications
In their study on Finnish uneven-aged forests, Pukkala et al. (2009) estimated the
ingrowth, survival, diameter growth, and height functions for Scots pine, silver birch, and
Norway spruce. In this study these functions are used only for Norway spruce and
Myrtillus (MT) site types. The estimated ingrowth function is
φ ( xt ,dt ) = e4.121−0.712
Bt + 0.083 Nt
− 1,
(9)
where Bt is the stand total basal area and Nt is the total number of trees. The fractions of
trees that survive over the next five year period are given as
{
α s ( dt , xt ) = 1 + e − [ 4.418+1.423
d st −1.046 ln Bt − 0.0954 Bst ]
}
−5 / 6
,
(10)
where Bst is the basal area of trees with a larger diameter than age class s. The function
that gives the growth of the tree diameter takes the form
δ s ( dt , xt ) = 1.124e−5.317 −0.043 B
st − 0.486 ln Bt
+ 0.455 d st −
0.000927 d st2 − 0.18 + 0.823 lnTsum ,
(11)
where Tsum is the temperature sum in degree days (d.d.). It varies between 900 and 1,300
d.d. The diameter of the youngest trees is given by
d 0 ( xt ,dt ) = e 2.004−0.101ln Bt −0.0176 .
(12)
7
The height of trees is a function of diameter:
he ( d st ) =
39.691
.
1 + 25.683 / d st + 37.785 / d s2t
(13)
The functions for saw timber and pulpwood volumes are obtained using Heinonen (1994)
and equation (13). The maximum and minimum lengths for saw logs are 5.5 and 4.3
meters, respectively. The minimum diameters for saw logs is 15cm and for pulp logs 6
cm. In order to maintain smooth model properties the volumes (m3) are given as:
ν 1 = 10−3 (116.0906 − 31.1854 d st + 1.9407 d st 2 − 0.0121 d st 3 ) ,
ν 2 = 10−3 ( 0.0068176 d st 3 − 0.660699 d st 2 + 18.2853 d st − 72.8905 ) ,
(14a,b)
where 12 ≤ d st ≤ 40 . The roadside price for saw logs is 52€ and for pulpwood 28€.
Harvesting costs are from Kuitto et al. (1994). These detailed models (Appendix
A) were originally estimated for even-aged management. They contain separate models
for cutting and hauling costs and for thinnings and clearcuts. Following Surakka and
Siren (2004), the cost model for uneven-aged management is formed by taking the
hauling cost from the thinning model and the logging costs from the clearcut model and
multiplying the latter by a factor of 1.15. The fixed harvesting cost equals 300€.
Optimization procedure
Taking into account the empirical specification (9)-(14b), the optimization problem (1)(6) is smooth in the sense that all functions are continuous and differentiable. Thus it is
8
possible to use efficient gradient-based optimization algorithms that are built on the
Karush-Kuhn-Tucker theorem of nonlinear programming. This study applies Knitro
optimization software that includes state-of-the-art interior point algorithms (Byrd et al.
2006). The difficulties follow from the rather large number of state variables (40-60) and
nonconvexities, i.e. the possibility of several locally optimal solutions. To find the global
optima it is necessary to use 15-100 randomly chosen initial guesses until further guesses
do not increase the objective function value.
3. Results
Maximizing volume yield
It is useful to first study the model properties in the simplest (theoretical) case where trees
are harvested every period (i.e., every five years) and the aim is to maximize volume
yield or the saw timber volume yield. In this setting, with no discounting, it is possible to
find the optimal steady state by solving a static optimization problem, i.e.,
max
{ hs ,s =1,...,n }
n
∑[v (d
s =1
1
s +1
) + p2 v2 ( d s +1 )] hs
x1 = φ ( x ,d ) − h1 ,
xs +1 = xsα s ( x ,d ) − hs , s = 1,...,n,
(15)
d1 = δ 0 ( x ,d ),
d s +1 = d s + δ s ( x ,d ), s = 1,...,n − 1.
For this computation 20-30 age classes are needed to guarantee that the solution is not
restricted by the number of age classes. The results are given in Table 1. Both total and
saw timber yield increase while the basal area and number of standing and harvested trees
tends to decrease with the temperature sum. The explanation is presented in the next two
lines: given the highest temperature sum, it takes 60 years after ingrowth until tree
9
diameter is about 29 cm and cutting is optimal. At the lowest temperature sum, the
diameter of harvested trees is about 23 cm when tree age is 90 years after ingrowth. Thus,
when tree growth is slower, the forest includes a higher number of age classes and trees
compared to when tree growth is faster. Although the basal area becomes higher, the
higher number of trees implies that ingrowth is higher with lower temperature sum (cf.
Equation 9). Comparing ingrowth with the numbers of harvested trees shows that with
these stand densities only 0-2 trees per cohort die. In all cases it is optimal to cut all trees
from the oldest age class and leave others untouched.
Table 1. Optimal steady states for maximizing total yield
Table 2. Optimal steady states for maximizing saw timber yield
Table 3. Maximizing total yield under even-aged management
Figures 1a and b. Optimal stand structure after harvest
Figures 2a and b. Steady-state total yield and the productivity of standing volume
Table 2 shows similar results when the aim is to maximize annual saw timber
yield. Total yield is now lower, standing volume increases, trees are cut later with larger
diameters and the numbers of harvested trees are lower. Together the results in Tables 1
and 2 suggest that the model output is coherent.
Figures 1a and b show the stand age structure at the steady state. In Figure 1a, the
number of trees decreases with size. This is in line with the classical inverted J-curve
model The stand structure follows from natural mortality and the fact that ingrowth
remains constant given constant harvest. Figure 1b shows the steady-state tree diameter
pattern and, equivalently, how tree diameters grow until they are cut.
Another view on the model properties and on the production capacity of the
uneven-aged Norway spruce stand can be obtained by computing the steady state “surplus
production.” This is the periodic yield net of natural mortality as a function of standing
10
Table 1. Optimal steady states for maximizing total yield
1300
annual total yield,m
5.1
annual saw timber yield m3
4.5
basal area before cut m2
12.3
basal area after cut m2
9.6
no. of standing trees before cut 552
no of harvested trees a-1
8.4
standing vol. before cut m3
98
age* of harvested trees, years
60
diameter of harvested trees cm 28.7
ingrowth, no. of trees a-1.
8.7
Tsum
3
1200
4.9
4.3
13.4
10.8
627
8.3
106
75
28.4
8.6
1100
4.7
4.0
13.4
10.9
708
9.3
103
75
26.5
9.8
1000
4.5
3.8
14.7
12.2
848
9.8
109
85
26.5
10.5
900
4.3
3.4
15.8
13.3
1017
11.6
111
90
23.3
12.7
Note: Harvest every five years, * age after ingrowth
Table 2. Optimal steady states for maximizing saw timber yield
1300
annual total yield,m
5.0
annual saw timber yield m3
4.6
2
basal area before cut m
14.0
basal area after cut m2
11.4
no. of standing trees before cut 570
no of harvested trees a-1
7.0
standing vol. before cut m3
115
age* of harvested trees, years
80
diameter of harvested trees cm 30.9
ingrowth, no. of trees a-1.
7.4
Tsum
3
1200
4.8
4.4
15.0
12.5
639
7.0
121
90
30.4
7.4
1100
4.6
4.1
15.9
13.5
727
7.1
127
100
29.6
7.8
1000
4.4
3.9
16.9
14.5
848
7.5
131
110
28.4
8.5
900
4.1
3.6
18.5
16.3
992
7.3
141
130
27.8
8.9
Note: Harvest every five years, * age after ingrowth
Table 3. Maximizing total yield under even-aged management
temperature sum, d.d.
initial diameters for 3 size classes, cm
saw timber yield m3a-1
total yield m3a-1
1300
5.6,7.1,8.8
6.7
7.6
1100
4.6, 6.1,7.7
5.8
6.6
900
4.6, 5.2,6.3
4.9
5.6
11
46
Diameter, cm
Number of trees per age class
48
44
42
40
38
36
34
5
10
15
20
25
35
30
25
20
15
10
5
0
0
30
25
Diameter cm
50
75
100 125 150
Age years
1300 d.d
1200 d.d
1100 d.d.
1000 d.d.
900 d.d.
0.6
28
26
24
22
20
18
16
14
12
10
Fraction of volme
harvested per five years
Total steady state
yield per 5 years
Figures 1a and b.
Optimal steady state structure after harvest
Objective: saw timber volume maximization,
Harvest every 5 periods
0.5
0.4
0.3
0.2
0.1
0.0
50
100
150
200
Standing volume before harvesting
50
100
150
200
Standing volume before harvesting
Temperature zone (d.d.)
1300
1100
900
Figure 2a and b. Steady state surplus production and the productivity of standing volume
12
tree volume. This function is obtained by maximizing yield and varying a
minimum constraint on standing volume. The outcomes are given in Figures 2a and b.
In the case of 1300 d.d., the highest total yield is obtained when the standing
volume is about 98m3 before cutting. At this steady state about 42 trees are harvested
every five years. Ingrowth per five years is about 43.5 trees, i.e. natural mortality
decreases the surplus production of each cohort by 1.5 trees.
In comparison, when the standing volume before the cut is kept at 152m3 the yield
per five years is 22.2m3. This is about 87% of the maximized yield. Three trees are
harvested from age class 120, and 19 trees from age class 125. Their diameters are 35 cm
and 37 cm respectively. Ingrowth is 25 trees per five years, i.e., three trees from each
cohort die. Figure 2b gives a rough view on “the productivity of standing volume” in fiveyear terms. For the 1300 d.d. temperature zone, it varies between 56% and 9%, implying
an annual productivity between 9% and 2%.
Because of the model’s complexity, it is not evident that the optimal solutions
converge toward steady states representing uneven-aged management with constant
variables over time. It this model even-aged management is an admissible solution type
instead of uneven-aged management (Tahvonen 2009). This question is studied in Figures
3a, b, and c. The optimal solution is computed from seven different initial states. In spite
of the rather strong initial variation in periodic harvesting, basal area, and number of
trees, all the optimal solutions converge toward the same steady-state solution. This
steady state represents uneven-aged management and equals that shown in Table 1,
column 3.
Figures 3a, b, and c. Volume maximization over time
The question of constant versus cyclical harvesting can be further analyzed by
computing optimal steady-state solutions under the constraint that harvesting is only
13
80
60
40
20
0
30
25
20
15
Number of trees after cuts
10
Basal area before cut m2
Total yield per 5 yrs
100
5
800
700
600
500
400
300
0
10
20
30
40
50
Time in 5 yrs
Figure 3a,b and c.
Volume maximization over time
Cutting every five years,
Temperature sum 1200 d.d.
14
Average annual ingrowth
Average annual yield m3
5.2
5.0
4.8
4.6
4.4
4.2
4.0
3.8
3.6
0
2
4
6
8
10
13
12
11
10
9
8
7
6
5
0
12
2
4
6
8
10
12
Harvesting interval in 5 yrs
Harvesting interval in 5 yrs
1200 d.d
1300 d.d
900 d.d
1000 d.d
1100 d.d
Figure 4a and b. Maximized volume yield and ingrowth as functions of harvesting
interval.
9000
Revenues per 5 yrs
8000
7000
6000
5000
4000
3000
2000
1000
0
10
20
30
Time
Figure 5. Convergence toward the steady state from various initial states
Tsum=1200, interest rate 3%.
15
Table 4. Optimal steady state solutions for maximizing present value
of roadside revenues net of harvesting cost
Interest
rate,
%
Tsum
d.d.
Total
yield
m3a-1
Saw
timber
yield
m3a-1
Basal area
before/after
harvest,
m2
Number of
trees
before/after
harvest
Diameter
range of
harvested
trees, cm
Net
revenues
€ (20a)-1
Present
value,
infinite
horizon€
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
900
900
900
900
1000
1000
1000
1000
1100
1100
1100
1100
1200
1200
1200
1200
1300
1300
1300
1300
4.20
3.75
3.75
3.59
4.40
4.22
4.03
4.03
4.58
4.47
4.30
4.02
4.76
4.70
4.54
4.41
4.92
4.88
4.65
4.54
3.30
2.90
2.58
2.42
3.60
3.27
2.99
2.66
3.94
3.64
3.37
3.02
4.19
3.97
3.71
3.52
4.38
4.21
3.84
3.68
18.4/8.7
15.6/6.0
12.9/3.8
12.5/3.0
17.6/7.8
14.95/5.18
13.22/3.71
11.33/2.33
17.63/7.62
15.15/5.08
13.51/3.63
11.71/2.28
17.35/7.10
15.38/5.01
13.81/3.58
12.87/2.8
16.8/6.2
15.9/4.5
14.2/2.8
13.76/2.2
1015/772
874/602
716/428
650/357
823/615
723/481
644/384
542/271
714/534
647/431
587/350
506/253
629/462
588/391
541/323
507/287
564/401
529/341
475/259
448/224
20.9-26.1
19.0-25.0
16.7-24.0
15.4-23.9
21.7-28.2
18.4-26.9
17.9-25.9
15.1-25.9
22.7-30.8
19.7-29.0
18.9-27.8
15.8-27.7
23.4-32.3
22.9-30.9
19.9-29.5
18.0-29.3
25.0-33.4
21.3-31.7
21.2-31.4
18.4-30.7
3162
2936
2631
2479
3421
3179
2952
2651
3648
3464
3247
2948
3845
3720
3519
3364
4019
3917
3620
3492
17522
8978
5895
4560
18958
9720
6614
4877
20216
10592
7276
5423
21307
11375
7884
6188
22271
11978
8109
6424
Notes: Harvesting interval 20 years (four 5 year periods).
16
allowed to be positive every k periods (cf. Equation 8). In addition, it must be required
that the steady-state stand structures (number of trees and tree diameters) are repeated at
the end of the harvesting interval after each cutting. The results in Figures 4a and b reveal
that both the average annual output and ingrowth decrease if harvesting does not take
place every period. This is in line with the dynamic solutions in Figures 3a-c and supports
the result that when maximizing volume yield and relying on natural regeneration, it is
optimal to harvest trees every period.
Figure 4. Maximized volume yield and ingrowth as functions of harvesting interval.
Table 4. Maximizing total yield under even-aged management
Ideally it should be possible to use the same forest growth model for both unevenand even-aged management. Results from such an experiment are given in Table 3. The
second row shows the thermal zone-specific initial states (25 years after clearcut)
assuming that the initial stand is represented by three size classes. In all cases, the initial
number of trees is 1700. Of these, 24.5% is in the largest size class, 49% in the middle
size class, and the rest is in the smallest size class. In maximizing total yield, the sizespecific intensity of two thinnings and their timing are optimized together with the
rotation period. Comparison with Table 1 suggests that artificial thinning with even-aged
management is capable of producing higher volume yield than uneven-aged management
based on natural regeneration.
Maximizing present value of net forestry income
Maximizing economic net revenues brings the price difference between saw
timber and pulpwood, the harvesting costs, and the interest rate into the computations.
Because harvesting costs include a fixed cost component equal to 300€, it is not optimal
to harvest the stand every period. For simplicity, the results will be computed assuming a
17
cutting interval equal to 4 periods (20 years). With a low interest rate (1%) a harvesting
Revenues net
of harvesting costs, €
Total revenues, €
8000
7000
6000
5000
4000
20
Basal area after
harvest
Basal area before
harvest
3000
15
10
5
Harvested trees
600
500
400
300
200
Pulp yield per
5 yrs, m3
100
140
120
100
80
60
40
20
0
70
60
50
40
30
20
10
Ingrowth, number
of trees per 5 yrs
Harvest, number
of trees per 5 yrs
Sawn timber yield
per 5 yrs, m3
Number of trees
after cut
0
0
20
40
60
80
100
Time in 5 yrs periods
Figure 6. Optimal solution for maximizing roadside revenues net of harvesting cost
Note: Tsum =1200, interest 1%
18
interval equal to 5 periods may yield a slightly higher net present value of
revenues, as will an interval equal to 3 periods in the case of a high interest rate.
Figure 5 shows periodic net revenues from seven optimal solutions starting from
different initial stand states. They all converge toward the same steady-state equilibrium.
Figure 6 shows a more detailed example for an optimal convergence toward the steadystate stand structure and harvesting. The basal area, number of trees and ingrowth
fluctuate because harvests only occur once every five periods. For the same reason
ingrowth is highest at the period following low basal area and low number of trees.
Figure 5. Convergence toward the steady state from various initial states
Figure 6. Dynamic solution for maximizing roadside revenues net of harvesting cost over time
Computing the optimal dynamic solutions1 for thermal zones between 1300-900
d.d. and for interest rates between 1-4% produces the steady states characterized in Table
4. Given any interest rate, a higher terminal zone always yields higher total and saw
timber yield as well as higher steady-state net revenues. Similarly, as in maximizing
volume yield, optimal steady-state stand density decreases with thermal zone. Increasing
the interest rate decreases saw timber yield and periodic revenues but may increase total
yield (and pulpwood). In addition, a higher interest rate implies lower steady-state density
as measured by basal area or number of trees. However, when stand density decreases,
trees grow faster in diameter. Thus, compared to steady states for volume output
maximization, the decrease in saw timber yield is quite low. With a 1% interest rate the
optimal solutions produce about 95-90% of the maximum volume output and with 4% the
output is about 80-70% of the maximum. The volume obtained per harvest varies between
76m3 and 105m3. The harvesting cost level (not shown) varies between 8.7€/m3 and
7.4€/m3. The figures for the basal area and number of trees do not contain trees below the
ingrowth diameter. This basal area may be expected to vary between 1-2m2.
Table 4. Optimal steady states for maximizing present value of roadside revenues
19
(b)
100
100
80
80
Number of trees
per size class
Number of trees
per size class
(a)
60
40
20
60
40
20
0
0
5
10 15 20
Diameter cm
5
25
10
15
20
Diameter cm
(d)
70
60
50
40
30
20
10
0
Number of trees
per size class
Number of trees
per size class
(c)
100
80
60
40
20
5
10
15
20
25
0
30
5
10
Diameter cm
50
80
Number of trees
per size class
100
30
20
10
0
5
10 15 20 25 30 35
20
25
30
(f)
60
40
15
Diameter cm
(e)
Number of trees
per size class
25
60
40
20
0
5
10 15 20 25 30
Diameter cm
Diameter cm
Remaining trees
Harvested trees
Figure 7. Optimal tree structure at the steady state
a) Tsum=900, r=0.01,
b) Tsum=900, r=4%
c) Tsum=1100, r=0.01,
d) Tsum=1100, r=4%
e) Tsum=1300, r=0.01,
f) Tsum=1300, r=0.04%
20
180
160
140
120
100
80
60
40
20
0
Number of trees
Number of trees
(a)
5-10
10-15 15-20 20-25 25-30 30-35
Diameter classes, cm
160
140
120
100
80
60
40
20
0
(b)
5-10 10-15 15-20 20-25 25-30 30-35
Diameter classes, cm
Remaining trees
Harvested trees
Figure 8. Stand structure in 5 cm size classes
1300 d.d. a) r=1%, b( r=2%
21
Figure 7a-f. Optimal tree structure at the steady state
Figures 7a-f show steady-state stand structures and harvested trees. Harvesting
removes the four largest age classes while others are left growing. When the number of
trees in the (endogenously-determined) size classes is depicted, as in Figures 7a-f, the
stand structures represent a serrate form. This structure is determined by the 20-year
harvesting interval and the associated variation in stand density and ingrowth. After each
harvest, stand density is low, implying that ingrowth is high. During the next four periods
density increases, implying lower ingrowth and smaller tree cohorts. Figures 7a-f show a
mirror image of this process because cohort age increases along the x-axes.
Figure 8. Stand structure in 5 cm size classes
1300 d.d. a) r=1%, b) r=2%.
To compare this structure with the traditional inverted -J structure in the unevenaged literature, it must be noticed that in Figures 7a-f the frequency of the endogenously
determined size classes decreases with diameter. Figures 8a and b depict the steady states
of 1300 d.d. for interest rates of 1% and 4% when trees are grouped into 5 cm diameter
classes. The number of trees becomes highest in the smallest size class, but in general the
size-class structure still clearly differs from the classic inverted J-curve. Comparing
Figures 7e and 7f to 8a and 8b reveals that the number of trees tends to decrease with size
not because of natural mortality but because the diameter differences between size classes
increase in time and size.
Comparing the economic performance of uneven- and even-aged management
Comparing the economic outcomes from even- and uneven-aged management requires
information on maximized bare land value under even-aged management. One possibility
is to take the bare land values from existing studies. The advantage of this choice is that
several existing studies apply stand growth models that have been designed for even-aged
22
Table 5. Comparing the even-and uneven-aged management
Tsum, Interest BLV€
d.d.
rate, % this
study
900 1
11092
900 2
2378
900 3
285
1100 1
14779
1100 2
3800
1100 3
1017
1300 1
18537
1300 2
5230
1300 3
1778
Total
yield
m3a-1
5.4
4.5
4.2
6.3
5.4
5.2
7.4
6.4
6.2
BLV
Pukkala
et al. 2010
10360
2146
353
15057
4532
1583
20673
7800
3695
BLV
Hyytiäinen
et al. 2010
2560
-55
-641
9875
2335
403
20448
6099
2208
clearcut Break even
value € BLV €
4828
4004
3189
5540
4514
3887
5639
4945
4122
12694
4974
1369
14676
6078
3389
16632
7033
3987
Table 6. Investment efficient steady states
Ts, d.d. Interest NPV* NRT
rate %
900
1100
1300
1
2
3
1
2
3
1
2
3
17066
7185
4082
19536
8397
4832
21669
9393
5462
4035
3863
3627
4523
4464
4215
4981
4890
4789
BLV** NPV
Pukkala
et al. 2010
11092 15283
2378
6180
285
3371
14779 18082
3800
7432
1017
4217
18375 19954
5230
8504
1778
4937
*Investment efficient steady state without
requiring the Weibull distribution
**BLV computed in this study
23
management and should offer reliable results. In addition, these studies describe stand
growth based on artificially planted seedlings and thus include the potential genetic
superiority compared to natural regeneration. The drawback is that various details like
prices and harvesting costs may deviate from the values used here. Another possibility is
to compute the bare land values applying the specifications (9)-(14) and exactly
comparable economic parameter values. Here, both of these approaches are used.
To compute the even-aged solution requires solving for the number and timing of
thinnings, the number of harvested trees in thinnings, and the rotation period. The initial
state is taken to be the same as in Table 3. Regeneration costs are given as
W ( N 0 ,r ) = 425€ + 1700 × 0.39€ b 2 + 292€ b11 ,
where the first cost item refers to clearing and mounding the regeneration area, the next to
planting costs (1700 seedlings) that materialize two years after the clearcut, and the last to
tending costs that materialize 11 years after the clearcut. Harvesting costs are taken from
Kuitto et al (1994). The optimized number of thinnings varies between zero and two.
Table. 5. Comparing even- and uneven-aged management
The results for maximizing bare land values are shown in Table 5, column 3.
Column 4 shows the associated average annual volume output per hectare. Bare land
values are somewhat lower than in Pukkala et al. (2010) where timber prices were higher
and regeneration costs lower. The differences from the bare land values of Hyytiäinen et
al. (2010) may follow from lower regeneration costs and considerably lower stand growth
in the 900 d.d. thermal zone.
To compare the economic superiority of even- and uneven-aged forestry, the task
is to solve the problem
24
max
{ hst ,t = 0 ,1,...,T , s =1,...,n ,T }
T

n
∑ ∑ [ p ν ( d
t =0
s =1
1 1
s ,t +1

) + p2 v2 ( d s ,t +1 )]hst − Cti ( ht ,dt ) bt + bTV *

(16)
subject to restrictions (1)-(6), where V* is the maximized bare land value. The essential
feature in this problem is the optimal choice of a possible switch from uneven-aged
management to even-aged management. The choice T = ∞ implies that uneven-aged
management is applied forever; choosing some finite T ≥ 0 implies a switch to even-aged
management.
Complete analysis of (16) is tedious since the optimal T may depend on the initial
state, and the set of possible initial states is infinite. A computationally simple possibility
is to assume that the initial state equals the optimal uneven-aged steady state at the
beginning of the period with harvest. Given this initial state the simplest possibility is to
compare the cases T = ∞ and T = 0 , i.e., the solution of following uneven-aged
management forever and the solution where the stand is clearcut immediately and
managed as an even-aged stand thereafter. Let RUEA denote the steady state net revenue
(from Table 4) that is obtained every k th period under optimal uneven-aged management.
In addition, let Rcc denote the net revenues from a clearcut of this same stand (applying
the clearcut harvesting cost function (Kuitto et al 1994). An immediate switch to evenaged management is at least as profitable as continuing uneven-aged management if
RUEA
≤ Rcc + V * .
1 − b5 k
(17)
When (17) is an equality, V* may be defined as the break-even bare land value.
Table 6. Stand clearcut value and break even bare land value
25
(a)
Present values, €
20000
18000
16000
10000
8000
6000
4000
2000
0
20
40
60
80
100
120
80
100
120
80
100
120
(b)
Present values, €
20500
20000
19500
19000
10000
8000
6000
4000
0
20
40
60
(c)
Present values, €
24000
22000
12000
10000
8000
6000
0
20
40
60
Transition period, in 5 yrs
Interest rate=0.01
Interest rate=0.02
Interest rate=0.03
The present value from
uneven-aged management
Figure 9. Uneven-aged management compared to switches to even-aged management
a) 900d.d.
b) 1100 d.d
c) 1300 d.d.
26
The results for equation (17) are given in Table 5, column 8. The bare land values
obtained in this study exceed the break-even bare land values in two cases: when the
temperature sum is 1100 d.d or 1300 d.d. and the interest rate is 1%. In addition, the bare
land values in the study by Pukkala et al. (2010) exceed the breakeven level when the
temperature sum is 1300 d.d. and the interest rate is 2%. The study by Hyytiäinen et al.
(2010) yields higher bare land values in one case: when the temperature sum is 1300 d.d.
and interest rate is 1%. These results are intuitive: even-aged management becomes
competitive with a lower interest because the interest costs of artificial regeneration
decrease. In addition, stand growth is faster with higher temperature sum and costly
artificial regeneration becomes profitable compared to free but scanty natural
regeneration.
Figure 9. Uneven-aged management compared to switches to even-aged management.
Figures 9a-c shows results on possible later switches to even-aged management.
The figures have been computed using the bare land values obtained in this study. Thus,
in line with Tables 5 and 6, the immediate switch to even-aged management is optimal
given the temperature sum is either 1100 d.d or 1300 d.d., and the interest rate equals 1%.
Finally, it must be emphasized that this comparison allows thinning from above in evenaged management. If only thinning from below were allowed, the bare land values would
be lower than the break-even bare land values in all cases.
5. Discussion
This study applies a single-tree model for analyzing the economics of uneven-aged
management. The ecological model has been applied without any simplifications or ad
hoc constraints. The objective function includes detailed, empirically-estimated
harvesting cost functions specified for the purposes of uneven-aged management. The
27
optimization problems (with discounting) are solved in their most general form, i.e. by
computing finite time solutions with a long enough planning horizon to obtain a close
approximation of infinite horizon solutions. The most important findings of the study are:
1. Maximization of volume output and reliance on natural regeneration yields
uneven-aged management as the optimal solution.
2. When the thermal zone varies between 900 and 1300 d.d., volume output varies
between 5.1 and 4.3 m3 a −1ha −1 , the diameter of harvested trees between 23 and 29
cm, and the pre-harvest basal area between 16 and 12 m 2 .
3. The dynamically optimal solutions converge toward the optimal uneven-aged
steady states independently on the initial stand structure.
4. Even-aged management and artificial regeneration (1700 seedlings) yield higher
volume output than uneven-aged management when both solutions are computed
by the same empirical single-tree growth model.
5. When the thermal zone varies between 900 and 1300 d.d. and the interest rate
varies between 1% and 4%, the economically optimal output varies between 4.9
and 3.6 m3a −1ha −1 , the diameter of harvested trees between 15-33 cm and the preharvest basal area between 18 and 13 m 2 .
6. In economically optimal solutions, the steady-state size class structure resembles
serrate form instead of the traditional inverted J-curve.
7. Given the optimal uneven-aged steady state as the initial state, it is optimal to
switch to even-aged management only if the interest rate is 1%-2% or lower and
the thermal zone 1100 d.d. or higher.
28
Until now, similar economic optimization based on a single-tree model has not been
performed for Norway spruce. However, although the model specifications differ, it is
possible to compare these results with earlier uneven-aged studies on Norway spruce.
Among the first attempts to specify a growth model for uneven-aged Norway
spruce is the (nonlinear) transition matrix model by Kolström (1993). In his model,
ingrowth depends on the gaps left by harvested trees (cf. Usher 1966). Assuming harvest
removes some fixed proportion of trees from each size class, Kolström shows that the
uneven-aged stand may produce up to 8m3ha-1a-1 (1200 d.d. sites between high and
medium fertility). When this model is applied (Tahvonen 2007, 2009) to maximize
present value stumpage revenues (3% interest rate, harvest every 5 years) the steady-state
volume output equals 7m3a-1ha-1. The diameter of harvested trees is 26 cm and the afterharvest basal area 17m2. In stylized comparison, the present value of stumpage revenues
from uneven-aged management was clearly higher than that of even-aged management. It
is important to recognize that Kolström’s (1993) specification yields excessive ingrowth,
and then it becomes optimal to thin the smallest size class rather heavily. In addition, the
transition from the smallest size class to the next is density independent. This explains the
high basal area levels compared to the results here.
Andreassen (1995) reports results on 16 long term experimental plots and
compared the volume output between even-and uneven-aged management. According to
his results uneven-aged management yields about 15-20% lower volume production.
When compared to the results based on artificial planting and systematic volume
maximization (Tables 1 and 3) these losses here are somewhat higher (25-32%).
Wikström (2000) uses six separate studies in constructing a single-tree model for
Norway spruce. He refers to Kolström (1993) and specifies ingrowth independently of
stand density, applying fixed ingrowth equal to 50 trees per five years with an average
29
diameter equal to 5 cm. To reduce the number of decision variables, cuttings are specified
within 6 cm size classes. Post-harvest standing volume is required to be at least 150m3.
Harvesting cost is independent of the management method. The interest rate equals 3%.
According to the results, the optimal even-aged solution yields an average volume
output of 6.3m3 a −1ha −1 while uneven-aged management yields 3.2m3 a −1ha −1 . Basal area
remains above 16m3 and trees are cut when they reach 19-25 cm in diameter. The optimal
uneven-aged solution is computed over 41 (5-year) periods and yields 96-88% of the net
present value of revenues for the even-aged solution.
When these results are compared with the results of the study at hand it is
essential to note that a lower bound constraint on standing volume (150m3 ha −1 ) must
decrease the economic outcome from uneven-aged management (cf. Tables 1, 2, and 5,
and Figure 2a). Even when this restriction is taken into account, the volume yield from
uneven-aged management is low compared to the results in this study. Wikström explains
the low yield by some special properties of the growth model used.
Pukkala et al. (2010) utilize the same growth data (i.e., Pukkala et al. 2009) as is
used here. Their aim is to present practical instructions for uneven-aged management of
Norway spruce and Scots pine. The economic approach is based on Duerr and Bond
(1952) and Bare and Opalach (1987). Thus they search post-thinning, steady-state stand
structures that maximize net present value (NPV) defined as
NPV =
NRT
− ( NRcc − NR pc ),
( 1 + r )T − 1
(18)
where NRT is the roadside revenue net of harvesting cost (or stumpage revenues) realized
every T periods, NRcc is the net revenue if the steady-state stand is clearcut and NR pc is
the net revenue if it is cut to the steady-state, post-harvesting structure. The interpretation
of this “investment efficient” steady state in Getz and Haight (1989: 269-272), Pukkala et
30
al. (2010), and others is that NRcc − NR pc is viewed as an opportunity cost or investment
cost of stocking needed to perform uneven-aged forestry2. Maximizing (18) requires
solving for the number of trees and their diameters over the cycle periods taking (1)-(6) as
constraints. To simplify this problem, Pukkala et al. (2010) follow Bare and Opalach
(1987) and assume that the post-harvesting stand structure is represented by a Weibull
distribution function. The problem is computed by optimizing the parameters of the
Weibull function instead of the number of trees directly.
Their method yields a steady-state stand structure where the number of trees
decreases with diameter and trees larger than 19 cm are cut. Typically, about 50% of the
removal is pulpwood. In comparing even- and uneven-aged management the latter turns
out to be superior both for Scots pine and Norway spruce, the only exception being a
Norway spruce medium fertility site, 1300 d.d. thermal zone, and 1% interest rate.
The study by Pukkala et al. (2010) includes two separate features that deserve
attention: the investment-efficient steady state, and the strategy of optimizing the
parameters of the Weibull distribution function. The investment-efficient steady state is
analyzed by Getz and Haight (1989, p. 269-272), where it is shown to lack of a sound
theoretical basis. One main problem is the ad hoc forest valuation represented by
NRcc − NR pc . As a consequence, both the steady-state stand structure and the economic
profitability results become questionable.
This study shows that, in general, the steady-state stand structure deviates from
the classic, inverse J-curve. Solving for the post-harvesting stand structure by optimizing
the Weibull distribution function parameters may not yield a stand structure depicted in
Figures 7a-f or 8a and b. Thus, this solution method produces an ad hoc constraint in
optimization and must decrease the economic outcome from uneven-aged management.
In addition, the use of the Weibull function may explain the high fraction of pulpwood
31
obtained by Pukkala et al. (2010), that is, 50% vs. 20-30% here. The remaining question
is how the investment-efficient steady state defined by (18) — but computed without the
Weibull simplification — distorts the profitability figures. The results for this question
are shown in Table 6. The cutting interval is four periods as in the comparisons by
Pukkala et al. (2010). The revenues per 20 years (NRT) are considerably higher than the
optimal steady-state revenues obtained here3 (see Table 4, column 8). In addition, the
investment-efficient NPVs (third column) clearly exceed the maximized bare land values
in all cases (given any estimate and independent of the interest rate and thermal zone).
This shows that the investment-efficient steady-state revenues and their comparison with
the bare land value will overestimate the relative economic performance of uneven-aged
management. In addition, comparing the third and the last columns in Table 6 shows that
simplifying the optimization using the Weibull distribution function decreases the NPVs
obtained without this simplification.
Table 6. Investment efficient steady states
The study by Tahvonen et al. (2010) applies a nonlinear transition matrix model
for studying uneven-aged management of Norway spruce. The harvesting costs and other
economic features are approximately the same as they are here. Getz and Haight (1989, p.
250) compare the performance of a single-tree model and a stage-structured model very
similar to the one used in Tahvonen et al. (2010). They find that both models yield very
similar projections, assuming exogenously given harvesting. Here, it is possible to
compare the optimal solutions obtained by these two types of models.
The model in Tahvonen et al. (2010) is suitable for only one site type that is
between Oxalis-Myrtillus and Myrtillus. The temperature sum is 1200 d.d. Thus, the
results of this earlier study may be approximately compared to the results obtained
assuming 1300 d.d. in this study. Their volume maximization results are very similar to
32
the results here, the only difference being that the transition matrix model yields slightly
larger tree diameters and volumes for harvested trees. Similarly as here artificial
regeneration with thinning from above produces higher volume than uneven-aged
management although the latter dominates assuming natural regeneration.
Economically optimal solutions can be compared assuming a 3% interest rate.
Tahvonen et al. (2010) used a 12-year harvesting interval. Before- and after-harvest
steady-state basal area levels were 10 and 4m2, total and saw timber yield 4.6 and 4.1m3a1
ha-1, and revenues at an annual level 178€ha-1. At the steady state the diameter of
harvested trees was between 23 and 39 cm. Again, the results are rather close to results in
Table 4. However, since the results in Table 4 are based on a 20-year harvesting interval,
the basal area and the number of trees reach higher levels before the harvest. The other
difference is that in Table 4 the diameter of harvested trees is lower, implying lower saw
timber yield. In addition, the diameter variation in the transition matrix model is larger
than the diameter variation in the single-tree model (16 vs. 10 cm) although in the later
case the harvesting interval is longer. This difference follows from the fact that the
transition matrix model yields too slow growth for some trees and too fast for others.
In Tahvonen et al. (2010), a harvesting interval of 12-15 years yields similarly
fluctuating ingrowth as here. However, at the steady state the number of trees is
monotonically decreasing, although with diameters between 15 cm and 27 cm, their
number remains almost constant. The transition matrix model is not capable of producing
a serrate stand structure because only a fraction of trees (depending on the stand density)
moves to the next size class, implying that the variation in ingrowth is smoothed out. In
contrast the single-tree model used here maintains the original cohorts and variation in
ingrowth.
33
In comparing the economic superiority of forest management forms, Tahvonen et
al. (2010) found that given a 3% interest rate, the break-even bare land value is 4450€.
This is somewhat higher than here (3989€) and may be a consequence of the transition
matrix model that overestimates the volumes of harvested trees. Overall, Tahvonen et al.
(2010) found that uneven-aged management was superior to even-aged management,
although with low interest rates the difference becomes small. In comparison, the study at
hand shows that the relative superiority of even-aged management increases with thermal
zone. An obvious explanation is that when forest growth is faster, the interest costs
related to artificial regeneration obtain faster payoff.
6 Conclusions
Finnish forest policy has strongly supported even-aged management and forest legislation
is designed to prevent selective cuttings. At the statute level, forest harvesting operations
are classified into two groups: stand improvement cuttings and regeneration cuttings. For
stand improvement cuttings, the forest statute (528/2006) specifies the lower bound basal
area level that must be met after each stand improvement cutting. The lower bounds
depend on the dominant height of the stand, and are site fertility and thermal zone
specific. If these restrictions are applied to the 1% interest rate cases and to the thermal
zones of 900 d.d., 1100 d.d., and 1300 d.d. in Table 4, the lower bounds for the basal
areas after cuttings become 10, 13, and 13m2. In comparison, the corresponding after-cut
basal areas in Table 4 are 8.7, 7.6, and 6.2m2. With higher interest rates, the after-cut
basal areas become lower, implying that all the solutions in Table 4 violate the legal
lower bound restrictions. By requiring higher than optimal stand density the restrictions
decrease ingrowth, and future harvesting possibilities. In addition, the statute states that
“in stand improvement cuttings the residual trees must be primarily the trees from the
34
highest canopy layers.” It is somewhat unclear how this statement should be understood,
but straightforward interpretation suggests that the steady-state cuttings described in
Figure 7a-f violate this statement.
It has been shown elsewhere (Hyytiäinen and Tahvonen 2003) that in the case of
even-aged management Finnish forest policy promotes maximum sustainable yield. The
results of this study suggest that strong promotion of even-aged management is in line
with this policy and that uneven-aged management may yield about 15-25% less timber.
The obvious question is why volume maximization is the primary goal of forestry and not
economic surplus. In addition to the direct losses to forest owners, it tends to decrease the
market price of timber, causing another income distribution effect between forest owners
and forest industry.
Future economic studies on uneven-aged management would greatly benefit from
further developments in growth and yield models. In particular, the ingrowth specification
should describe the earlier development of seedlings instead of implicitly assuming the
existence of a pre-ingrowth seedling stock independent of stand state, as in the present
specification. Economic models should also be extended to analyze mixed species stands
including Norway spruce, Scots pine, and silver birch (cf. Bollandsås et al. 2008). Such
extension will be necessary to increase understanding of the potential to apply unevenaged management in Nordic conditions.
Notes:
1. The dynamic solutions are computed using time horizons from 100 to 350 periods
depending on the interest rate. These horizon lengths are long enough to produce a close
approximation toward the optimal steady state.
35
2. Note that NRcc − NR pc equals the direct harvesting value of residual trees. In
computing (18) the value is obtained by multiplying timber volumes by their prices.
3. It should be understood that under discounting, a higher level of steady-state revenues
is no sign of optimality. The optimal steady state can be found only by solving the full
dynamic problem.
36
Appendix A
Kuitto et al. (1994) performed an experiment for estimating the hauling and logging cost
models for harvesting Norway spruce stands in Finnish conditions. In this study it is
assumed that logging costs rate per our equals 82.5€ and hauling cost rate per hour 59.5€
respectively. The costs for thinnings and clearcuts in the case of even-aged management
are then given as
Ctth = 21.906306 + 3.3457762 H tsawvol + 25.5831144 + 3.77754938H tpulpvol +
∑
n
s =1

hst 
22.386
0.50001 + 0.59vols −
2.1001366 N t + 300,
vols 1000 + 85.621 N t 

Ctcc = 26.350495 + 2.82183045H tsawvol + 25.701440 + 3.33144 H tpulpvol +
∑

hst 
146.17
+
vol
−
0.44472
0.94

2.1001366 N t + 300,
s
s =1
vols 1000 + 862.05 N t 

n
where H tsawvol and H tpulp are the total volumes of sawlogs and pulpwood yields per cutting and
vols is the total (commercial) volume of a stem from size class s.
37
References
Andreassen, K., 1995. Long term experiments in selectively cut Norway spruce (Picea
abies) forests. Water, Air and pollution 82, 97-105.
Andreassen, K., B.-H. Øyen, B-H., 2002. Economic consequences of three silvicultural
methods in uneven-aged mature coastal spruce forests of central Norway. Forestry
75, 483-488.
Bare, B.B., Opalach, D., 1987. Optimizing species composition in uneven-aged forest
stands. For. Sci. 33, 958-970.
Bollandsås, O.M., Buongiorno, J., Gobakken, T., 2008. Predicting the growth of stands of
trees of mixed species and size: A matrix model for Norway. Scandinavian Journal of
Forest Research 23, 167-178..
Duerr, W.A., Bond, W.E. 1952. Optimum stocking of a selection forest. J. of Forestry 50,
12-16.
Faustmann, M. 1849. Berechnung des Wertes welchen Waldboden, sowie noch nicht
haubare Holzbestände für die Waldwirtschaft besitzen. Allgemeine Forst- und JagdZeitung 25, 441-455.
Getz, W.M., Haight, R.G., 1989. Population harvesting: demographic models for fish,
forest and animal resources. Princeton University Press, New Jersey.
Gobakken, T., Lexerod, N.L., Eid, T., 2008. A forest simulator for bioeconomic analysis
based on models for individual trees. Scandinavian Journal of Forest Research 23,
250-265.
Haight, R.G., 1987. Evaluating the efficiency of even-aged and uneven-aged stand
management. For. Sci. 33, 116-134.
Haight, R.G., Monserud, R.A., 1990. Optimizing any-aged management of mixed-species
stands. II: Effects of decision criteria. For. Sci. 36, 125-144.
Haight, R.H.,. Monserud, R.A., 1991. Optimizing any-aged management of mixedspecies stands. I. Performance of a coordinate-search process. Can. J. of For. Res. 20:
15-25.
Heinonen, J., 1994. Koealojen puu- ja puustotunnusten laskentaohjelma KPL.
Käyttöohje.(In Finnish) Reseach Reports, Finnish Forest Research Institute, 504.
Hooke. R., Jeeves, T.A., 1961. Direct search" solution of numerical and empirical
problems. J. Assoc. Comput. Math. 8, 212-229.
Hyytiäinen, K., Tahvonen, O., 2001. Economics of forest thinnings and rotation periods
for Finnish conifer cultures. Scand. J. of For. Res. 17, 274-288.
Hyytiäinen, K., Tahvonen, O., 2003.Maximum sustainable yield, forest rent or
Faustmann, does it really mater? Scand. J. of For. Res. 18, 457-469.
Hyytiäinen, K., Tahvonen O., Valsta, L., 2010. On economically optimal rotation periods
and thinnings for Scots pine and Norway spruce (in Finnish), Working papers 143,
Finnish Forest Research Institute, Vantaa..
Kolström, T., 1993.Modelling the developmenet of an uneven-aged stand of Picea Abies.
Scand. J. For. Res. 8, 373-383.
Kuitto, J-P., Keskinen, S., Lindroos, J., Ojala, T., Räsänen, T., Terävä, J., 1994.
Mechanized cutting and forest haulage. Metsäteho. Report 410 (In Finnish).
Painovalmiste KY, Helsinki.
Pukkala, T., Lähde, E., Laiho, O., 2009. Growth and yield models for uneven-sized forest
stands in Finland. Forest Ecology and Management 258, 207-216.
Pukkala,T., Lähde, E., Laiho, O., 2010. Optimizing the structure and management of
uneven-sized stands in Finland. Forestry 83, 129-142.
Sarvas, R., 1944. Tukkipuun harsintojen vaikutus Etelä-Suomen yksityismetsiin (In
Finnish).Valtioneuvoston kirjapaino, Helsinki
39
Siiskonen, H., 2007. The conflict between traditional and scientific forest management in
the 20th century Finland. For. Ecol. Manage. 249, 125-133
Surakka, H., Siren, M., 2004. Selection harvesting: state of the art and future directions
(in Finnish), Metsätieteen Aikakauskirja 4:24, 373-390.
Tahvonen, O., 2009. Optimal choice between even-and uneven-aged forestry. Natural
Resource Modeling 22, 289-321.
Tahvonen, O., Pukkala, T., Laiho, O., Lähde E., Niinimäki, S., 2010. Optimal
management of uneven-aged Norway spruce stands, For. Ecol. Manage. 260, 106115.
Usher, M.B., 1966. A matrix approach to the management of renewable resources, with
special reference to selection forests-two extensions, J. of Applied Ecology 6, 347346.
Valkeapää, A., Paloniemi, A., Vainio, R., Vehkalahti, A., Helkama, K., Karppinen, H.,
Kuuluvainen, J., Ojala, J., Rantala, T., Rekola, M., 2009. Finnish forests and forest
policy: analysis of public opinions (in Finnish) Reports 55, Department of Forest
Economics, Helsinki University, Helsinki..
Wikström, P., 2000. A solution method for uneven-aged management applied to Norway
spruce. Forest Science 46, 452-463.
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