new bioeconomics of fisheries and forestry olli tahvonen ... · msy -type objectives both in...
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New bioeconomics of fisheries and forestryOlli TahvonenUniversity of Helsinki EAERE Venice Summer School 2011Section 1, FisheriesSection 1, Fisheries
1
The question of managing
1. IntroductionThe question of managingbiological resources
Actual resource management
ResourceApplied
Actual resource managementis dominated by ecologists andMSY -type objectives both in forestry and fisheries
economicsApplied ecology
DetailedEconomic objectiveswith "oversimplified"
y
New bioeconomics:The aim is to integrate sound economics and realistic
ecological modelswith MSY -typeobjectives
ecologyeconomics and realistic models taken directly from ecology
Applied h i
cf. economics of nonrenewable mathematics resources
2
1. Introduction, cont.
Two generic models in resource economics:
Optimal rotation model (forestry)Faustmann 1849, Ohlin 1928, Samuelson 1976,...
Biomass harvesting model (fisheries)Schaefer 1954, Gordon 1957, Plourde 1972, Clark 1976,...
rtpx( t )e wmax 00
rt
{ h }max U( h,x )e dt
1 rt{ t }
maxe
00s.t. x F( x ) h,
x( ) x
Some extensions/alternatives:Age-structured modelsSpatial models
Some extensions/alternatives :Environmental valuesMarket level age structured models p
Multispecies models,...Market level age-structured modelsStand level size-structured models,...Optimal rotation with optimal thinning,initial density
3
initial density,...
New bioeconomics of fisheries and forestryContentContent0 Introduction1 Fisheries1 1 Age-structured population models in fisheries1.1 Age-structured population models in fisheries1.2 Generic age-structured optimization problem1.3 Empirical example of an age-structured fishery model1 4 On numerical optimization1.4 On numerical optimization
2 Forestry2.1 Market level age-structured model for timber/old growth/agricultureg g g2.2 Stand level size-structured models 2.3 Generic size structured optimization problem2.4 Empirical example of a size-structured modelp p3 Summary and discussion
4
Memory refresh: optimal solution for the Schaefer-Gordon-Clark bi h ti d lbiomass harvesting model
rtmax U( h ) C( h x ) e dt
25
Numerical example :
00
00
{ h }max U( h ) C( h,x ) e dt
s.t. x F( x ) h,x( ) x .
d, h 15
20
Some generic features:1. The optimal steady state h*,x* is defined by
0( )
Yie
ld
5
10
0
x
h
c ( h*,x*) F '( x*) ,U '( h*) C ( h*,x*)
F( x*) h* .
2 O ti l i ld i i i f ti f bi
"marginal rate of returnequals interest"
Biomass, x0 20 40 60 80 100 120
0
Optimal yield x
2. Optimal yield is an increasing function of biomass3. The optimal solution approaches the steady state monotonically4. If C =0 and F'(0)< it is optimal to deplete the
Optimal yieldGrowth
F(x)=0.5x(1-x/100)U(h)=h1-0.95 , C(h,x)=15h,x-1.5, r=0.02
U=u(h)-c(h,x)
population (Clark 1973)5. MSY solution is determined by biological factors only
5
Some problems related to the biomass models
1 Th l i l G d S h f Cl k bi d l d ib bi l i l l ti1. The classical Gordon-Schaefer-Clark biomass model describes a biological populationbut simplifies the population as a homogenous biomass with no age or size structure
2. The classical Gordon-Schaefer-Clark model cannot specify to which age classes har esting sho ld be targetedharvesting should be targeted
3. Harvesting activity may change the population age structure, regeneration level but these effects are not possible to be included in the biomass framework
4 These and other age truncation effects are intensively studied by ecologists 4. These and other age-truncation effects are intensively studied by ecologists
"Picture three human populations containing identical number of individuals. One of these is an oldpeople's residential area, the second is a population of young children, and the third is a population of mixed age and sex. No amount of attempted correlation with factors outside the population
ld l th t th fi t d d t ti ti ( l i t i d b i i ti ) thwould reveal that the first was doomed to extinction (unless maintained by immigration), the second would grow fast but after a delay, and the third would continue to grow steadily."From Begon et al. (2011, p. 401) "Ecology". (perhaps the globally most widely used ecology textbook)
Obviously something similar holds true in the cases of fish trees etc
6
Obviously something similar holds true in the cases of fish, trees etc
Discussion on adding age structure to economic models:
Wilen (1985, 2000): the biomass approach may at its best serve as a pedagogical tool
Clark (1976 1990): Unfortunately the dynamics of many important biological resources Clark (1976, 1990): Unfortunately, the dynamics of many important biological resources cannot be realistically described by means of simple biomass models
Hilborn and Walters 1992:. The biomass model is seen as a poor cousin of the age-structured analysis and is used only if age-structured data is unavailable
Clark (1990, 2006), Hilborn and Walters (1992) and Wilen (1985): age-structured models are analytically incomprehensible
However, this statement has turned out to be overly pessimistic
7
Remarks:Age-structured models are becoming important in general economics as wellAge structured models are becoming important in general economics as wellInstead of aggregate production functions with "capital stock"
it is possible to specify capital as "vintages" (e.g. Boucekkine et al JET 1997,...)Adding internal structure to capital stock or labor will change many fundamentaldd g e a s uc u e o cap a s oc o abo c a ge a y u da e a
properties in models on economic growth and business cycles, for example.
8
Some history of age- and size-structured population models in biology
P.H. Leslie (1945). Matrix models for age-structured populations L Lefkovich (1965) Matrix models for size structured populationsL. Lefkovich (1965) Matrix models for size structured populationsM.B. Usher (1966) Matrix models for tree populations=>Presently population studies in ecology rest heavily on age- or size
structured modelsstructured models
and in fishery economics (or fishery ecology...difficult to make the distinction)
Baranov (1918): The problem raisedBeverton and Holt (1957): Famous "Dynamic pool model"Walters (1969): Pulse fishing solutions( ) gClark (1976, 1990): "The problem is incomprehensible"Hannesson (1975): Pulse fishing solutionsHorwood (1987): Smooth harvesting solutions
=> almost all studies have used only numerical methods
9
A life cycle graph for an age-classified population with density dependencein recruitment in recruitment
2 2x x x
4
1 s stsx
1 1x
1 1tx2 2tx 3 3tx
2 2tx3 3tx 4 4tx
1tx 2tx 3tx 4tx
1 1tx
2 2t 3 3t
1th 4th2th 3th
1 2 3 40
1st
Four age classes, s , , ,x number of individuals in age class s in the beginnig of period t (state variables)
share of individuals that survive in age class s
11 1
s
s
share of individuals that survive in age class sthe share s ,...,n die due to natural reasons ( natural morta
0
1s
lity )number of offspring per individual in age class s
h i f i h b f ff i h i l
10
1
st
the recruitment function :the number of offspring that survive to age classh the harvesting mortality (control variables)
Examples of commonly used recruitment functionsExamples of commonly used recruitment functions
0
0 0 01bx
Beverton Holt recruitment function : ( x ) ax / ( bx )
Ri h i f i ( )
10
00 0
bxRicher recuitment function : ( x ) ax e
crui
ts
80 0 00 9 1 0 1( x ) . x / ( . x )
mbe
r of r
ec
4
6
00 050 9 . x( x ) x e
Num
2
00 00 9( x ) . x e
N mber of eggs
0 20 40 60 800
N b f " "11
Number of eggsNumber of "eggs"
The age class model can be written as a difference equation systemg q y
1 11
n
,t s stx x ,
1
1 1
1 1 1 1
1 2s
s ,t s st st
n ,t n n n ,t n nt nt
x x h , s ,...,n ,x x h x h .
Assumption: after age n-1 individuals , ,
Or in matrix form
remain approximately similar
1 1 1 10
2 1 1 2 2
0 0 0 0 00 0 0 0 0
,t t tt
t t t
x x h( x )x x h
2 1 1 2 2
2 30 0 0 0 0,t t t
th
1
1 1
0 0 0 0 0 00 0 0 0
n ,t
n,t n n nt nt
hx x h
12
Perfect selectivity vs. nonselective harvesting
Perfect selectivity: it is possible to control the age specific harvest levels
1sth , s ,...,n
separately. This is seldom possible in fishery.
Nonselective harvesting: “effort” is controlled and the catch per age class can be given as
11
st s t st
t s
h q ( E ,x ), s ,...,n,where E is effort and catcability functions q ,s ,...,n are nondecreasing in E.Commonly used example :
1st s t st
s
Commonly used example :h =q E x ,where q ,s ,...,n are constants and called as catchability coefficents.
“Effort” refers to number of nets, vessels weeks etc
13
1 1Let
/ ( b ) ( )
Some properties of the age-structured model (and the connection with the biomass model)
1 1 0 0
0 2
2 1 1 1 2 2
1 12
1 1 34
,t t t
t t
,t t t t t
x ax / ( bx ), ( )x x , ( )x x ( E ) x ( E ), ( )B ( )
1 1 2
1 1 1 2 2 2
45
2 1 2 0 1
t t t
t t t t t
B w x w x , ( )H w x E w x E , ( )where it is assumed that :n h q E x s q q q ax / ( bx ) and that
1 1 2 0 02 1 2 0 1st s st t st
i
n , h q E x , s , , , q q q, ax / ( bx ) and thatw den
1 2 1otes the weight of fish in age class i
Since q =q , effort can be taken directly as the control var iable and q ( )neclected:
Given a steady state, the variables are constant and the time subscrpts can be cancelled.
Thus, equ 12 1
2
11 1
( E )ation (3) implies x x .( E )
1
2
2 1
1 61 1
71 2 7
( E )Denote , implying ( )( E )
x x . ( )E ti ( ) ( ) d ( ) i l
11 1 1
1 1 1
1 2 7
1 0 1 0 1 01 1 1
Equations ( ),( )and ( ) imply
a x a ax x b x ab x b x b x
and that the steady state
is given as
14
and that the steady state
1 21 1 8
is given asa ax , x . ( a,b )
b b
1 2 11 2
1 2
01
1
When E the steady state becomesax , x .
b
1 1 2 2
1 2
ccSubstituting these into B=w x w x yields the carrying capacity biomass level , B .Both x and x decrease in E (from 6, 8a,b) implying that B decreases in ET
he level of E implying B=0 satisfies a =1 (from 8a,b). Applying (6) we obtain this critical E as
1 2
1 2
1aE= . Note that E<1.a
1 1 1 2 2 2
5The steady state harvest level ( equation ) was given as
H w Ex w Ex .
Since the steady state biomass is a decreasing function of E, we obtain the inverse of this function, i.e.E as a functionof B. Write E E( B ). Next in the steady state harvest function we can writeE x and x as functions of B implying that H becomes a funct
ion of B Write1 2E, x and x as functions of B implying that H becomes a funct
1 1 1 2 2 2 9
ion of B. Write
H w E( B )x ( B ) w E( B )x ( B ). ( )
Equilibrium biomass- Equilibrium harvest fu
This can be called as
nction.
15
1 2
1
0
0 0cc
cc
When B B it holds that E H . When B=0, and E E it hold that x =x =H=0.
When B ,B 0<E<E and x and
2 0x H>0.
Numerical example 1:
1 2 1 21 3 19 496 1 22 5 20 6000
Assume the following parameter values:
, , , a , b , w , w and the Beverton Holt recruitment function
2
2 5 20 6000
19 620 t
Thus, the model can be written as
xx
1 1
2
2 1 1 2
491 66000
1 312 5
,t
t
,t t t t
x ,x
x x E x
1 tE 2 5,
1 2
1 2
1 3 22 5
2
t t t t t
t t t
H x E x E ,
B x x .
1 2
1 2100 125350
t t t
The carrying capacity population level becomes x , xand the carrying capacity biomass .
1 2 0y g p y
The critical level of fishing mortality that implies B=x x obtains the val 49 0 7169
ue
E .
16
In addition, we obtain:
dual
s in
arve
st
400
30
35nu
mbe
r of i
ndiv
idan
d 2,
bio
mas
s, ha
200
300
ibriu
m H
arve
st
15
20
25
0 0 0 2 0 4 0 6 0 8
Stea
dy st
ate
nag
e cl
asse
s 1 a
0
100 Equi
li
0
5
10
Fishing effort E
0.0 0.2 0.4 0.6 0.8
Individuals in age class 1Individuals in age class 2Total population biomass
Equilibrium Biomass
0 100 200 300 400
Total population biomassHarvest
H th d l d i diff f th bi d l?How the model dynamics differ from the biomass model?
17
Let us fix the total harvest level H i e
1 1 1 2 2 21 1 1 2 2 2
Let us fix the total harvest level H, i.e.
Now the model takes the form:
t t t t tt t
HH w x E w x E E .w x w x
1 1 0
2 1 1 1 2 21 1 1 2 2 2 1 1 1 2 2 2
1 1
,t t
,t t tt t t t
x ( x ),
H Hx x x .w x w x w x w x
Fix H to some level that is lower than maximum sustainable yield.
Questions: 1. Given some initial state, does the solution converge toward a steady state, i.e. is the harvest level sustainable?
2. Do the biomass and age-structured models give the same2. Do the biomass and age structured models give the same prediction?
18
Is the harvesing equal to H=30 sustainable, i d i B 187 h B 124?
1 2
0
1124
Since w and w =2, all initial age class combinations above the dashed line have B and vise versa. The
Numerical example 1, cont.
35
0i.e. does it converge to B=187 when B >124?This is what the biomass approach suggests.
lower dot corresponds the equilibrium B=124 and the higherdot the equilibrium B=187.
70
arve
st 25
30
35
ass x
2 50
60
15
Equi
libriu
m H
a
10
15
20
Size
of a
ge c
la
20
30
40
2
0 100 200 300 400
E
0
5124B 187B
0 20 40 60 80 100 120 1400
10 34
Equilibrium Biomass Size of age class x1
0 20 40 60 80 100 120 140
0 124Computing the model forward yields the results:Initial states 1 and 2 have B and converge toward B0
0
0
124124
gInitial states 3 and 4 have B , but are unsustainableInitial state 5 have B , but is sustai
nable
S i bili f h h h i l l b
19
Sustainability of the chosen harvesting level cannot be deducted from the biomass information.
The equilibrium can be unstable for all initial statesoutside the equilibrium implying unsustainability or unexpected
231 1 2
2 1 1 20 9 1 0 9 1
tx,t t
,t t t t t
x x e ,
x . x q . x q ,
8
q p y g y pfluctutions and no convergence toward the steady state 2
20 9t t
t t t
B x ,H . x q
6
8
cla
ss x
1
41 2H
Age
2 1 2H . 0H
1 2H .
0 2 4 6 8 100
1 79B .
Age class x1
0 2 4 6 8 10
20
Summary this far:Steady state harvest becomes a function of the biomass as in the genericbi d l biomass model => biomass model is a simplification of the age-structured model=>Age class framework reveals that biomass model is based on strong simplifications
equilibrium harvestCrucial differences between the two approaches :1. The dynamic behavior of models become different2 I d d l h ilib i bi2. In age-structured model the equilibrium biomass–
equilibrium harvest function depends on catchability coefficients, i.e. on harvesting technologyTh MSY t b d t i d l iThus, MSY cannot be determined applyingbiological information only. This cannot be understood in the biomass framework
Equilibrium biomass021
Equilibrium biomass0
The generic nonlinear age-structured optimization model
1 11
n
,t s sts
x x
1 1
1 1 1 1
1 1 2
1 1s ,t s st s t
n ,t n n n t n nt n t
n
x x q E , s ,...,n ,
x x q E x q E ,
The nonlinear age-structuredmodel, effort Et as control
1
nt s s st s ts
s
H w x q ( E )
whereq ( E ) are fishing mortality functions with the properties
t
variable
0 0 1s s sq ( ) , q ( E ) , q ' 0 0s( E ) , and q ''( E ) .
n tmax V( ) U w x q ( E ) C( E ) b xObjective function;
' 0 '' 0U U 0 0 11 0 1t sts s st s t tt s{ E , x , s ,...,n , t , ,...}
max V( ) U w x q ( E ) C( E ) b .
x
0 , 1,..., ,0 1
sx s n are given
0, 0,' 0, '' 0 (effort cost)
U UC C
Initialstate0, 1,..., ,0.
st
t
x s nE
Nonnegativity constraints
22
Simplified two age classes version of the age-structured population model (Schooling fishery) Let 1 1 2 2t t t t tH w x qE x E , where 1w is the weight of fish in age class 1 with respect to fish in age class 2 and 1q is catchability parameter in age class 1 (in age class 2 it is 1). Solving for tE yields
1 1 2 2
1tt t
t t
HE , where we assume interior solutions in the sence that Ew x q x
The development of age class 2 can now be given as:
2 1 1 1 2 2
1 1 2 2 1 1 2 2
1 1,t t t t t
t t t t t
x x qE x E
x x E x q x
1 1 2 2 1 1 2 2
1 1 2 21 1 2 2
1 1 2 2
t t t t t
t tt t t
t t
qx q xx x H .
w x q x
Denote
x q x 1 1 2 21 2
1 1 2 2
1 1 0 1t tt t
t t
x q xG x ,x when w and q .w x q x
Note that the unit of G is numbers per weight and it transforms the total yield to numbers of harvested individuals in age class 2
23
class 2.
The optimization problem can now be written as
1 200 1t t t
tttH ,x ,x , t , ,...
max U H b
subject to
1 1 2
2 1 1 1 2 2 1 2
10 20
1
2,t t
,t t t t t t
x x ,
x x x H G x ,x ,x and x given
10 20
1 20 0 0t t t
x and x given,x , x , H , where where
1 1 2 21 2
1 1 2 2
t tt t
t t
x q xG x ,x .w x q x
Assume that 0 0U ' , U '' and that the recruitment function is either Beverton-Holt or Richer -type. Note: the fecundity parameter for age class 1 is zero.
24
Lagrangian and the necessary conditions for interior solutions 1 2 0t t tx , x , H can be given as
1 2 1 1 2 1 1 2 2 1 2 2 10
2 1 2 0 3
tt t t ,t t t t t t t ,tt
tt t t t
L b U H x x x x H G x ,x x ,
L b U ' H G x ,x ,H
11 2 1 1 1 1 1 2 11 1
0 4
t
tt ,t t x ,t ,t
,t
t
HL b b H G x ,x ,
xL
1 1 2 1 2 1 2 12 1
t,t ,t ,t t
,t
L b b ' x b Hx
2 1 1 2 1 2 0 5x ,t ,t tG x ,x .
For studying the steady state drop time subscripts and assume that variables are constant over ti Thi i ld f (4) time. This yields from (4):
11 2 1 xb HG . Substituting this into (5), dividing by 2b , taking into account that 1 1b / r and rearranging terms yields the steady conditions in the form
1
2
2 1 21 22 1 2
1 2
1 61 1
7
xx
' x G x ,x'( x ) H G x ,x r,r r
x x ,
25
1 2
2 1 1 2 2 1 2 8x x x HG( x ,x ).
Let us first study the case with "knife edge" fishing gear where 0q , i.e. where harvest includes only fish from age class 2. In this case the steady state is implicitly given as
1 22
1 2
1 91
10
'( x ) r ,r
x x ,
( ) HInterpretation: at the steady state it holds that
2 1 1 2 2 11x x x H , because 0q implies
1 21 0x xG , G G .
Surplus production:
2 1 2 2 2
1 2 2 2
( ) .
( ) 1 .
x x x H
H x x
Interpretation: at the steady state it holds that
We can write the steady state surplus production as
Growth net of naturalmortality. Surplus productioncan be harvested withoutconsuming the "biological capital"
2
1 2'( )
x
H xx
Maximizing steady state surplus production with respect to requires
2(1 ) 0 12
g g p
2x
Given r=0, equation (12) equals equation (9). Note that 1 2'( )x is marginal effect of 2x on surplus production via changed recruitment and 21 is the marginal increase in mortality. Thus, (12) states that in MSY the marginal surplus production, i.e. marginal growth is zero growth is zero. While (12) requires that marginal steady state surplus production is zero, (9) requires that marginal present value steady state surplus production must equal the rate of discount. Note in particular that the term 1 2'( )x must be discounted because it takes one period until the recruits can be harvested as two periods old fish.
26
Write condition (9) in the form:
1 22 1 2 2
'( )( , , , , ) (1 ) 01
xy x r rr
. We obtain 2
1
22
1 1 22
'' 0. . ,1
' ( '' ''1, , 1, .(1 ) 1 1
y Thus the steady state is unique In additionx r
xy y y yr r r r
1 2(1 ) 1 1r r r r Thus, we can write: 2 2 1 2( , , , )x x r , i.e. 2x as a function of the given parameters. The comparative statics derivatives become
12
2 2 12 2
1 1 1
' 1'(1 ) 0, 0,
'' ''
yyx xrr
y yr
2 2
22 2 2 1 2
2 212 1
1
( '' '1 0, .'' ''
x xry y
x x xy y
12 1
2 21y yx xr
Thus, steady state level of 2x is a decreasing function of the interest rate and increasing function of the survivability parameters while the effect of the fecundity parameter is a priory indeterminate
27
the survivability parameters while the effect of the fecundity parameter is a priory indeterminate.
To study the stability of the optimal steady state we use implicit function theorem and write tH as a function of
t using equation (3). This yields 2( ), ' 1/ ''.t tH H H U The necessary optimality conditions can now be written as the system
1, 1 2 2( ),t tx x
2, 1 1 1 2 2 2
22 1
11 1
( ),
,
t t t t
t t
t
x x x H
1, 1
2 1 1 2 2 2 2
12, 1
1
,' ( )
.
tt
tt
b x x H
b
The Jacobian matrix takes the form
0 ' 0 00 'H
1 2
2 2 22 1 2 1 2 12 1 2 2 2 2 2 2
1 1 2 12 2 2
1 2
0 '
( ) '' ( ) '' ' ( ) '' '.
( ' ) ( ' ) ' ( ' )
H
b b b b HJ
b b b
1 2
1
10 0 0b
28
This yields the characteristic equation:
4 3 24 3 2 1( ) ,u u u w u w uw w
where
24 2
1 222 2
3 1 2 2 2 2 21 2 1 1 1 2
,'
' '' ' '' 1' ,' '' ' '' ' '
wb
U Uwb bU b U b
2 22 2
1 2
1 2
,'
1 .
wb b
wb
1 2b The facts lim ( ) , lim ( ) , (0) 0, (1) 0, ( 1) 0
u uu u
imply that the absolute
values of two roots are above 1 and absolute values of two roots are below one. Thus, the steady state is a local saddle point. This implies that optimal solution is a path toward an equilibrium where all variables are constant over time.
29
Optimal
5
saddle pointsteady state
3
4
x 2
2
0 1 2 3 4 50
1
x1
0 1 2 3 4 5
Figure 2. Cyclical population dynamics but saddFigure 2 Cyclical development of unharvestedg y p p y point stability for the optimally harvest population
Figure 2. Cyclical development of unharvested population but saddle point stability for the optimally harvested population
30
Steady states in biomass vs. age-structured model (n=2 case)
1 2 2 1 1 2 2,The steady state satisfiesx x x x x H
1 2 2 2 2 2( )
. ,
H= x x x
This equation gives sustainable harvest as a function of harvestable biomassjust as in the biomass framework Thus we can write
2 1 2 2 2 2 2
,
) ( )
j f
H=F(x x x x
T
,hus within the biomass framework the optimal sustainable biomass level isT
2 1 2 2 2
,
'( ) '( ) (1 )
hus within the biomass framework the optimal sustainable biomass level isdefined by
F x x r
1 2 22
:
'( ) 11
This can be compared with the steady state condition in the age structured framework
x rr
1 r
For Bever
& '' 0 ' 0, 1
ton Holt and Richer recruitment functions whenThus given discounting and b the biomass model yields higher steadystate biomass and yield compared to the age structured model
31
state biomass and yield compared to the age structured model
Assume next that 0 1 0 1q w and , i.e. harvesting gear in nonselective and the weight of fish in age class 1 equals w (and the weight of age class 2 fish equals 1). The steady state is defied by the three equations
1
2
2 1 21 22 1 2
1 2 2 1 1 2 2 1 2
1 61 1
7 8
xx
' x G x ,x'( x ) H G x ,x r,r r
x x , x x x HG( x ,x ). ( ),
Interpretation: The term H reflects the effect of increasing 2x on the level of H due to changes in yield composition between 1x and 2x . Proposition1 : Given nonselective gear and 0<q<1, the steady state levels of 1 2x and x are higher compared 1 2
to their levels under the knife edge selectivity assumption q=0. Proof. Appendix 1.
:Interpretation
1 2 2 1 2 2
22 2 2 2
:( ). / / ( )
( ) ' / 0
Interpretationx x x x x x
x x x x
At the steady state it holds that Thus the share equals and / by the properties of . Thus, increases in steady state increases the
2 2.x xshare of steady state , implying that harvest includes higher share of
32
Number of fish in age class 2
Figure 4. The effects of harvesting cost on optimal solutionParameters: U=H0.5, q1=0, q2=1, ==0, ==1, r=0.01,
Comparison of steady states: biomass model vs. age-structured modelsp y g"optimal extinction" results
2 53.03.5
(a) (b)
yiel
d2.53.03.5
456
(c)
Bio
mas
s
0.51.01.52.02.5
Equi
libriu
m y
0.51.01.52.02.5
Bio
mas
s
1234
Rate of interest
0.0 0.1 0.2 0.3 0.4 0.50.0
Figure 5a c Comparision of steady states of the age structured and the biomass models
Biomass0 10 20 30 40
0.0
Rate of interest
0.0 0.2 0.4 0.60
Figure 5a-c. Comparision of steady states of the age-structured and the biomass models a) Equilibrium yield biomass relationships;
Solid line: Dotted line: Dased line: b) Selective gear C=0, Solid line: biomass model; Dashed line age-structured model
c) Nonselective gear x2)=x2/(1+0.4x2), 0.8, C=0,
33
x2) x2/(1 0.4x2), 0.8, C 0,
Solid line: biomass model; Dashed line: age-structured model
Remarks/summary: 1. When 0<q<1, the optimal solution may converge toward a limit cycle. This cycle may represent pulse fishing q p y g y y y p p gin the sense that every second year optimal harvest is zero. 2. When 0<q<1, the optimal steady state population level in the age-structured model may be higher
compared to the biomass model.3 Since the steady state is different compared to the biomass model the "optimal extinction" 3. Since the steady state is different compared to the biomass model the optimal extinction
results differ (cf. Clark 1973)4. It is possible to have examples where optimal yield is a decreasing function of biomass (when the population
consists a large fraction of young age class; "growth overfishing" situation)g y g g g g )5. The analysis can be generalized to any number of age classes (for details Tahvonen 2009a,b)
Clark (1976 1990): "Adding age structure to bioeconomic analysis of fisheries will hardly changeClark (1976, 1990): Adding age structure to bioeconomic analysis of fisheries will hardly changeany basic bioeconomic principles"
Th lt hThe result here:Adding age structure changes all the basic properties of optimal harvesting compared to the generic biomass model
34
Empirical example: Baltic sprat fishery (schooling fishery)
Th t Wh t i th ti l h ti l ti f
1t tpHmax b
The model:The setup: What is the optimal harvesting solution for
Baltic sprat given different natural mortalitylevels determined by Baltic cod (a predator of sprat)
00 0 1
1 1 0
1tt{ H , t , ,...}
,t t
n
max b ,
x x ,
x w x
Table 1. Parameters used in the economic-ecological model.
The data (ICES 2009):
01
1 1
1 1 1 1
1 2
t s s sts
s ,t s st t st
n,t n n ,t n nt t n ,t
x w x ,
x x H G ,s ,...,n ,
x x x H G ,
Age-class
Maturity s
Weight sw
[kg]
Catchability sq
Survival rate s reference
Survival rate s low cod
Survival rate s high cod
Numbers 1st Apr 2008
[109] 1 0.17 0.0053 0.31 0.6703 0.7118 0.3012 43.895
0 1stx , s ,...,n,
where
q x
2 0.93 0.0085 0.54 0.7261 0.7711 0.4360 56.741 3 1.0 0.0097 0.76 0.7483 0.7788 0.5066 19.540 4 1.0 0.0103 1.0 0.7558 0.7866 0.5434 3.952
1 0 0 0108 1 0 0 408 0 88 0 016 14 3 1
11
1 2s s stst n
s s sts
nn ,t
q xG , s ,...,n ,w q xq
G
1 1
1
n n ,t n n ntn
s s sts
x q x,
w q x
5 1.0 0.0108 1.0 0.7408 0.7788 0.5016 14.377 6 1.0 0.0112 1.0 0.7408 0.7788 0.4916 3.846 7 1.0 0.0113 1.0 0.7189 0.7711 0.4025 0.600 8 1 0 0 0110 1 0 0 7189 0 7711 0 4025 0 716
0 1and t , ,....8 1.0 0.0110 1.0 0.7189 0.7711 0.4025 0.716
0 104 2 0 5032
Recruitment function:wheretaxx a b
35
00
6
104 2 0 5032
0 07 10 1000
where
Price of fish net of unit harvesting cost tons
tt
x , a . , b . .b x
: € . per
#code for Tahvonen, Quaas,Schmidt and Voss (2011). "Effects of species interaction on optimal harvesting of an age-structured schooling fishery", manuscript
#data file (Balticsprat.dat.txt) param T := 100; param n := 8; param r := 0.02; param p:=0.07;
0
AMPL optimization code
1.
2.
p g g g y p#model file (Balticsprat.mod.txt) param T; param n; param r; param p; param w {s in 1..n}; param g {s in 1..n};
param ac :=0; param w:= 1 0.0053 2 0.0085 3 0.0097 4 0.0103 5 0.0108 6 0 0112 g { }
param q {s in 1..n}; param a {s in 1..n}; param x0 {s in 1..n}; param ac; var H {t in 0..T-1} >=0; #total harvest; unit 10^3 tonns var x {s in 1..n,t in 0..T} >= 0; #number of individuals; unit 10^9
6 0.0112 7 0.0113 8 0.0110; param q:= 1 0.31 2 0.54
var B {t in 0..T-1}=sum{s in 1..n} w[s]*x[s,t]*1000; #biomass; unit 10^3 tons var Xo{t in 0..T-1}=sum{s in 1..n} w[s]*g[s]*x[s,t]*1000; #spawning stock; unit 10^3 tonns var G {s in 1..n-1, t in 0..T}; #transformation function; unit number of #individuals in 10^9 per 10^6 tons maximize objective_function: sum{t in 0..T-1} (1/(1+r))^t*(((if H[t]=0 then 0 else p*H[t]^(1-ac)))/(1-ac));
3 0.76 4 1 5 1 6 1 7 1 8 1; param g:= subject to constraint1 {t in 0..T-1}: x[1,t+1]=(0.1042*Xo[t]/(0.5032+Xo[t]/1000));
subject to constraint2 {t in 0..T, s in 1..n-2}: G[s,t]=a[s]*q[s]*x[s,t]/(sum{i in 1..n} w[i]*q[i]*x[i,t]); subject to constraint2b {t in 0..T}: G[n-1,t]=(a[n-1]*q[n-1]*x[n-1,t]+a[n]*q[n]*x[n,t])/(sum{i in 1..n} w[i]*q[i]*x[i,t]); subject to constraint3 {s in 1..n-2, t in 0..T-1}: x[s+1,t+1]=a[s]*x[s,t]-H[t]*G[s,t]/1000;
param g:= 1 0.17 2 0.93 3 1 4 1 5 1 6 1 7 1
subject to constraint4 {t in 0..T-1}: x[n,t+1]=a[n-1]*x[n-1,t]+a[n]*x[n,t]-H[t]*G[n-1,t]/1000; subject to initial_condition {s in 1..n}: x[s,0] = x0[s];
8 1; param a:= #reference case 1 0.6703 2 0.7261 3 0.7483 4 0.7558 5 0 7408 5 0.7408 6 0.7408 7 0.7189 8 0.7189; param x0:= 1 43.895 2 56.741
#Run file reset; model Balticsprat.mod.txt; data Balticsprat.dat.txt; option solver knitro-ampl;
3.
36
2 56.741 3 19.540 4 3.952 5 14.377 6 3.846 7 0.5 8 0.716;
option knitro_options "maxit=2000 opttol=1.0e-9 multistart=1 ms_maxsolves=10"; solve; option display_width 2; display H;
usan
d ton
nes
200
250
300
350
Equil
ibrium
yield
, tho
50
100
150
200
Population biomass, thousand tonnes
0 500 1000 1500 2000 2500 3000 3500
E
0
Reference Low cod High cod
Figure 1. Equilibrium biomass–harvest relationships
37
1200
1400
(a)
ss, th
ousa
nd to
nnes
600
800
1000
2010 2015 2020 2025 2030 2035 2040
Biom
as
0
200
400
Years
800(b)
thous
and t
onne
s
400
600
2010 2015 2020 2025 2030 2035 2040
Yield
, t
0
200
Years
2010 2015 2020 2025 2030 2035 2040
Interest rate 2% Interest rate 10%
38Figures 2a,b. Optimal solutions assuming the maximization of the present value resource rent (η =0), r=2% or r=10% and the predation reference case
tons
)2500
3000 DCBAbi
omas
s ('0
00
1500
2000
Tota
l sto
ck
500
1000
2010 2015 20201975 1980 1985 1990 1995 2000 20050
2010 2015 20202010 2015 2020
Figures 3a,b,c. Population biomass of Baltic sprat. (A) Historic stock size1974-2008 (ICES, 2009b), only for 2008 the distribution to age-classes isdisplayed; (B) Optimal sprat management for cod stock size as in 2008 (reference case);(C) O f ( ) O(C) Optimal management for low cod case; (D) Optimal managementfor high cod case. Stacked bars show distribution of biomass to age-classesfrom age 1 (bottom) to age 8 (top)
39
y 0 5
0.6
0.7
tonn
es
600
800
(a) (b)
Fishin
g mor
tality
0 1
0.2
0.3
0.4
0.5
al ca
tc, in
thou
sand
200
400
600
Years
2008 2010 2012 2014 2016 2018 20200.0
0.1
Year
2008 2010 2012 2014 2016 2018 2020
Annu
a
0
Reference case Low cod caseHigh cod case
Figures 4a,b. 5 Optimal fishing mortalities and catch for 2008-2020assuming r=2% and the predation reference case (solid lines),low cod case (long dash) and high cod case (short dash)
40
nnes 1200
(a)120ne
s
(b)
yield,
thou
sand
ton
600
800
1000
60
80
100
ield,
thous
and t
onn
0 2 4 6 8 10 12 14
Biom
ass,
Annu
al y
0
200
400
0 2 4 6 8 10 120
20
40
Biom
ass,
Annu
al y
Rate of interest
0 2 4 6 8 10 12 14
Biomass Annual yield
Rate of interest
0 2 4 6 8 10 12B
y
Figures 5a,b. Dependence of the steady state biomass (solid line) and yield(short dash) on the interest rate in Baltic sprat – reference case Figure 5a(short dash) on the interest rate in Baltic sprat – reference case, Figure 5aand high cod case, Figure 5b.
41
Summaryy1. Important extensions of the classical biomass harvesting model:
A) Age-structured, B) Multispecies and C) Spatial models2 Adding age structure changes all fundamental model properties2. Adding age structure changes all fundamental model properties
A) Optimal steady state becomes differentB) "Optimal extinction" results changeC) Steady state stability results differC) Steady state stability results differD) Pulse fishing becomes possible in age-structured modelsE) Optimal yield may decrease in biomassF) MSY b d d t tiF) MSY becomes dependent on gear properties
In addition, fishery regulation becomes different (It becomes reasonable to regulatefishing gear in addition to total catch)
42
References Baranov, T.I. (1918) On the question of the biological basis of fisheries, Nauch. issledov. iktiol. Inst. Izv., I, (1), 81-128. Moscow. Rep. Div. Fish Management and Scientific Study of the Fishing Industry, I (1). Begon, M., Townsend, C.R. and Harper, J.L. (2011) Ecology, Blackwell, MA.Begon, M., Townsend, C.R. and Harper, J.L. (2011) Ecology, Blackwell, MA.R.J. H. Beverton, and S.J. Holt, On the dynamics of exploited fish populations. Fish Invest. Ser. II, Mar. Fish G.B. Minist. Agric. Fish. Food 19 (1957). Boucekkine, R, M. Germain, and A. Licandro (1997), Replacement echoes in the vintage capital growth model, J. Econ. Theory 74, 333-348. H. Caswell, Matrix population models, Sinauer Associates, Inc., Massachusetts 2001. C.W. Clark, Profit maximization and extinction of animal species, (1973) J of Polit. Econ. 81 (1973) 950 961(1973) 950-961.C.W. Clark, Mathematical bioeconomics: the optimal management of renewable resources, John Wiley & Sons, Inc. New York, 1990 (first edition 1976). W.M. Getz, and R.G. Haight, Population harvesting: demographic models for fish, forest and animal resources, Princeton University Press, N.J. 1989. R. Hannesson, Fishery dynamics: a North Atlantic cod fishery, Canadian J.of Econ. 8 (1975) 151-173. R Hilb R d C J W lt Q tit ti Fi h i t k t h i d i dR. Hilborn, R. and C.J. Walters, Quantitative Fisheries stock assessment: choice, dynamics and uncertainty. Chapman & Hall, Inc. London, 2001. J.W. Horwood, A calculation of optimal fishing mortalities, J. Cons. Int. Explor. Mer. 43 (1987) 199-208. Leslie, P.H. (1945) On the use of matrices in certain population mathematics, Biometrica 33: 183-212. G.C. Plourde, A simple model of replenishable resource exploitation, American Economic Review 60 (1970) 518-522. W.E.Ricker, Stock and recruitment, J. of Fisheries Resource Board Canada 11 (1954) 559-623. M.B. Schaefer, Some aspects of the dynamics of populations important to the management of commercial marine fisheries, Bull. Inter Am. Tropical Tuna Commission 1 (1954) 25-56. O. Tahvonen, (2008) Harvesting age-structured populations as a biomass: Does it work? Nat. Res. Mod. 21, 525-550. O. Tahvonen, (2009a) Optimal harvesting of age-structured fish populations, Marine Resources , ( ) p g g p p ,Economics, 24 147-169.. O. Tahvonen, (2009b) Economics of harvesting of age-structured fish populations, Journal of Environmental Economics and Management, 58, 281-299. Tahvonen O. (2010) Age-structured optimization models in fisheries economics: a survey, Optimal Control of Age-structured Populations in Economy, Demography, and the Environment” in R. Boucekkine, N. Hritonenko, and Y. Yatsenko, (eds.), Series “E i t l E i ” R tl d (T l & F i UK)“Environmental Economics”, Routledge (Taylor & Francis, UK).O. Tahvonen, M. Quaas, J.O. Schmidt and R. Voss (2011), Effects of species interaction on optimal harvesting of an age-structured schooling fishery, manuscript. C.J. Walters, A generalized computer simulation model for fish population studies,. Transactions of the Am. Fisheries Society 98 (1969) 505-512. C.J. Walters, and S.J.D. Martell, Fisheries ecology and management, Princeton University Press, Princeton, 2004..
43
J.E. Wilen (1985), Bioeconomics of renewable resource use, In A.V. Kneese, J.L. Sweeney (Eds.) Handbook of Natural Resource and Energy Economics, vol 1. Elsever Amsterdam. J.E. Wilen (2000), Renewable resource economics and policy. what differences we have made? Journal of Environmental Economics and Management 39, 306-327.
New bioeconomics of fisheries and forestryOlli TahvonenOlli TahvonenUniversity of Helsinki EAERE Venice Summer School 2011Section 2, Forestry, y
"FORESTRY IS AMONG THE GREATESTFORESTRY IS AMONG THE GREATESTCHALLENGES IN APPLIED ECOLOGYSINCE IT IS LARGE SCALE ECONOMICACTIVITY THAT IS BASED ON UTILIZING LIVING BIOLOGICAL RESOURCES"
44LIVING BIOLOGICAL RESOURCES"HANSKI ET AL. IN "EKOLOGIA 1998"
About 31% of earth total land About 31% of earth total land area is covered by forestsThis makes 0.6ha per capita
45Source: FAO
Totally 7%
Value of global industrial roundwood removals about $100 billion annually$100 billion annually
Trend toward plantations
46Source: FAO
=> limiting the economic analysis to timber production only is a serious
47Source: FAO
limiting the economic analysis to timber production only is a seriousrestriction
Let:
Memory refresh: the classical economic approach to forest resources
Let:annual (market) interest rateplanting cost (€) per hectare (ha)
stand clearcut value (€) as a function of stand age (per ha)
rw ,V( t ) ,( ) g (p )
value of bare land (€) (per ha)( ) ,
J
Assumption: all growing (or rotation) periods are of equal length
V(t)
2t 3t
t 2t 3t
...time
V(t)‐w V(t)‐w‐w V(t)‐w “cash flow”( ) ( )
2 3rt rt rt r t rt r tJ w e V( t ) e [ w e V( t )] e [ w e V( t )] e
rte 2r te 3r te
( )
discount factorcash flow
48
J w e V( t ) e [ w e V( t )] e [ w e V( t )] e ....
2 3rt rt rt r t rt r tJ w e V( t ) e [ w e V( t )] e [ w e V( t )] e J w e V( t ) e [ w e V( t )] e [ w e V( t )] e ....
1. rotation 2. rotation 3. rotation
0
rit rt
iJ(t) e w e V ( t )
By the theorem of geometric series: 0
1 , 1.1
ii
q when qq
1Let ( 1, 0)rte q when r 0
11
ritrti
ee
Bare land value can now be given as the Faustmann (1849) formula:
rtw e V ( t ) tw b V( t )
or in discrete time:
1 rt
w e V ( t )J ( t ) .e
1 t
w b V( t )J( t )b
49where b=1/(1+r).
Some generalizations of the optimal rotation model
Generalized size and age-class modelsUneven-aged models
Generic rotation model
1
rt
rtt
w e V ( t )m ax J ( t )e
Faustmann 1849, Ohlin 1921, Samuelson 1976,...
1 e
Optimal stopping;Stochastic growthStochastic price
Market level age-structured modelsMitra and Wan 1985,...
Imperfectcapital marketsTahvonen et al. 2001,...
Econometricsof timber supplyKuuluvainen 1990,...p
Reed and Clarke 1990,...
Environmental preferencesHartmann 1976
Optimal rotation andthinningsMartin and Ek 1981,...
Carbon sequestrationvan Kooten et al.1995,...
50
Hartmann 1976,... ,
2.1 Market level age-structured models in forestryg y
Some history: A classical forestry problem that dates back over several centuries: A classical forestry problem that dates back over several centuries:
"How to manage a large forest area in order to guarantee sustainable and smooth timber supply over time"timber supply over time
The classical answer by silviculturalists: "Develop the forest age structure to representa normal or regulated age structure* and clearcut the oldest age class every period "a normal or regulated age structure and clearcut the oldest age class every period.Forest scientists have presented 40-50 different formulas for transforming the age-structure toward the normal forest
However, these formulas are totally ad-hoc.
Economists remark to silviculturalists: where is the proof that the normal forest is optimal?p p
*Normal or regulated forest: The land area is evenly distributed over existing age classes=>every year clearcut the land with the oldest age class then regenerate the bare land
51
=>every year clearcut the land with the oldest age class, then regenerate the bare land=>timber supply will be smooth and sustainable over time
The corresponding questions from the economic point of view:
How is timber price determined in the optimal rotation framework?How is timber price determined in the optimal rotation framework?Does well functioning market equilibrium guarantee smooth timber supply over time?Is normal forest an optimal steady state with saddle point stability?
Dasgupta (1982): "These problems have turned out to be very difficult and still unsolved".
Mitra and Wan 1985 JET 1986 RES Wan 1994 IER: given zero interest rate Mitra and Wan 1985 JET, 1986 RES, Wan 1994 IER: given zero interest rate normal forest is the optimal steady state but numerical examples suggests that with discounting the steady state is cyclical; cycles are a very generic feature in forestry
Salo and Tahvonen (2002a,b, 2003, 2004a,b): analytical proof for the optimality of cycles under discounting but remark that cycles exists because of discrete time; =>cycles are not generic in forestry; y g y;=>model generalizations remove the cycles=>in generalized models normal forest is the optimal steady state with saddle point
stability properties give any number of age classes
52
The age-structured model with land allocation between forestry, agriculture and old growth
Notation and setup:
1
2 0 1 1
st
n
. Let x denote theland area allocated to stands of age s in the beginning of period t
L d h l d l h l l d l
p
12 0 1 1
3
nt t sts
s
. Let y denote theland area in agriculture y x when total land area equals
. Let the total timber content per land unit be given as : f , s
1 11 0
4
n n,...,n, assume : f ... f f
D t th i d d f ti b b P D h i th i di t t l ti b h ti d
0
4
0 0t
t t t
c
t t
. Denote the inverse demand for timber by P D c , where c is the periodic total timber harvesting and
consumption. The social utility from timber is : U c D c dc, where U ' , U ''
5. The social ut 0
ty
tility from agricultural land is : W y Q y dy, where W'>0, W'' 0
53
Notation and setup, cont.:
6 0 0nt nt. The old growth forest area equals x . The social utility from old growth is A x , where A' , A''
d l f h l h f f l d l h b f
7 1. Time development of the age class structure : the area of forest land in age class s in the beginning of next
period equals the area in ag
1 1 1 2s t st st st
e class s in the beginning of this period minus the area that is harvested , i.e.
x x z , s ,...,n , where z denotes theclearcutted land area from age class s. 1 1
1 1 11 2
s ,t st st st
st st s ,t st s ,t
f g
This yields : z x x , s ,...,n , where x x
1
1 1 1 1 1 1
0
n,t nt n ,t n ,t n ,t nt n ,t n n
("the cross vintage bound")
In addition, x x x z , where z denotes the harvest from both x and x . We assumed f f .
1 1 1n ,t nt n ,t n ,tThis yields : z x x x .
Thus, total harvest per period equal
21 1 1 1 11
nt s st s ,t n nt n ,t n ,ts
s: c f x x f x x x
54
The social planners optimization problem:
1 1 0 0s ,t
tt t ntx ,s ,...,n ,t ,... t
max b U c W y A x
subject to
21, 1 1 1, , 11
1
,
1 ,
nt s st s t n nt n t n ts
nt sts
c f x x f x x x
y x
1, 1
, 1 1,
1
, 1,..., 2,
,
1
s t st
n t nt n t
n
x x s n
x x x
x
, 11
0 01
1,
0, 1..., ,
0, 1,..., , 1.
s ts
stn
s ss
x
x s n
x s n given x
Note: The choice of 1 1 1 1 1 2s ,tx , s ,...,n , t , ,.... determines harvest levels as well as the the level of agricultural land and land area for old growth preservation
55
the level of agricultural land and land area for old growth preservation.
21 1 1 1 1 10 1 1
0 1
1 n ntt t nt t s ,t st st s ,t n ,t nt n ,t n ,tt s s
The Lagrangian and the Karush-Kuhn-Tucker conditions for all t , ,... are
L b U c W y A x x x x x x x ,
1 1 1 1 11 1
1 0tt t t ,t
,t
L b bf U ' c bW ' y bx
2 0 1 2t
,
L b f U ' c bf U ' c bW ' y b s n
1 1 1 1 11 1
1 1 1 1 1 1 1 11
2 0 1 2
3 0
s t s t t t s ,t sts ,t
tn t n t t n,t t n ,t n ,t
n ,t
b f U ' c bf U ' c bW ' y b , s ,...,n ,x
L b f U ' c bf U ' c bW ' y bA x ,x
1
1 11
4 0 0 1
5 0
n,t
s ,t s ,ts ,t
Lx , x , s ,...,n,x
0 1 2 0 0 5 0st st s, x
1 1 1 1 1 1
11
0 1 2 0 0
6 0 1 0
t s ,t n ,t n ,t nt n ,t n ,t
nt t s ,ts
x , s ,...,n , , x x x
, x ,
1 1 0 1 0 1st twhere , s ,...n , t , ,... and , t , ,... are Lagrangian multipliers.
Given bounded utility and b<1, the optimal solution exists by the theorem 4.6. in
56
Given bounded utility and b 1, the optimal solution exists by the theorem 4.6. in Stokey et al. 1989, p. 79.
Let us restrict the analysis to interior steady states where 0 0 0nc , y , x .
1 1 1m m s sm sAssume a unique Faustmann rotation m satisfying b f / b b f / b , for s ,...,n.
Direct substition shows that
1
07 1 2
1 2 0
s is si
W ' b f U ' , s ,...,n
solves and as equalities in the interior steady state .
Since the rotation b
10 0 0 1m i m meriod is m z and Thus W ' b f U ' Multiplying by b / b yields Since the rotation b
00 0 0 1
18 01 1
1
m m mi
mm n
mm
eriod is m, z and . Thus W ' b f U ' . Multiplying by b / b yields
W ' y b f b y xU ' f ,b b m
U ' f b bW ' bA'
11 2 1 1
13 2
1n
n n n n
U ' f b bW ' bA'Next from : . Eliminating and from written for x yields after some
b
2
0 81 1
n
cancellation
bA' W ' b . Applying allows to write this condition in the formb b
1 1
19 01 1
8 5 0
n mnm n
mm
n n
A' x bf y xU ' f .b m b
Finally, use and eliminate bW ' from the solution of . By condition it must hold that . This
yields
110 01 1
mn m n
mm
bA' x f b y xU ' f .b b m
57
Together equations (8), (9) and (10) determine an optimal steady state continuum forthe land allocation between forestry, agriculture and old growth preservation
The continuum exists because the cost-benefit consequences of adding a land unit to preservation differ from the consequences of decreasing preserved land
18 01 1
mm n
mm
W ' y b f b y xU ' f ,b b m
Present value of marginal ag land equals Faustmannbare land value when timber price equals U' and annual timber production from normal forest equals1 n
my x fm
Faustmann land value for a land unit ready to be clearcut
19 01 1
n mnm n
mm
A' x bf y xU ' f ,b m b
ymust be higher or equal to the present value of marginal preserved land unit when discounted over n-m periods(since it takes time until the land represents an old growth)
Otherwise it would be optimal not to clearcut a land unit
The present value of a preserved marginal land unit must
1110 0
1 1
mn m n
n mm
bA' x f b y xf U ' f .b b m
exceed the value of such land unit if clearcutOtherwise it is optimal to move land from preservationto timber production.
58
Numerical example:
5 0 5 0 50 9 0 5 0 5 3 1 21
0 0 10 15 22 30 40 51 65 82 101 123 148 175 203 234 264 293 321 346 346. . .
nb . , U . c , W . y , A . x , n ,f
0.50
0.25
0.30
25
30
0 0 10 15 22 30 40 51 65 82 101 123 148 175 203 234 264 293 321 346 346f , , , , , , , , , , , , , , , , , , , ,
Tim
ber p
rice
0.30
0.35
0.40
0.45
Land
in a
gric
ultu
re, y
(t)
0.10
0.15
0.20
Tim
ber h
arve
stin
g, c
(t)
10
15
20
25
Time
0 20 40 600.20
0.25
Time
0 20 40 60
L
0.00
0.05
Time
0 20 40 60
T
0
5
Land initially old growthL d i iti ll ld th f t
3.0 0.30
Land initially old growthLand initially in agricutureLand initially as old growth forest
Land initially as one period old forest
nd re
nt, w
'(y)
2.0
2.5
d ol
d gr
owt l
and,
xn
0.15
0.20
0.25
0 20 40 60
Lan
1.0
1.5
0 20 40 60
Pres
erve
d
0.00
0.05
0.10
59
Time Time
Remarks:1 Detailed analysis of the model is somewhat more complex than shown here1. Detailed analysis of the model is somewhat more complex than shown here2. In Salo and Tahvonen 2004 it is proved (without old growth) that the steady state is a local
saddle point 3 In optimal steady state with no agricultural land the steady state is a stationary cycle3. In optimal steady state with no agricultural land the steady state is a stationary cycle
Cycle is reasonable if there are periodic features in forestry operations (harvesting only during winter)4. Many extensions possible:A Multiple land types: normal forest feature vanishes –– in a model with many land types each A. Multiple land types: normal forest feature vanishes in a model with many land types each
having their own age structure timber supply may become smooth without the normal forest feature B. Forests and carbon sequestration (single stand models restrictive)
60
2.2 Stand* level size-structured models
Some concepts:Even-aged stand: at any moment all trees in the stand are of equal age (but not
necessarily of equal size, cf. plantations)Uneven-aged stand: at any moment the stand may contain some heterogeneous
age distribution of treesT b i f t t tTwo basic forest management systems:
Even aged management: artificial regeneration=>thinnings=>clercut=>artificial.regeneration...Uneven-aged management: trees are cut selectively every 15 yrs for example, no clearcuts
Shade tolerant trees: tree species that regenerate and grow as understoreyShade tolerant trees: tree species that regenerate and grow as understorey(e.g. Norway spruce, beech, sugar maple)
Shade intolerant trees: trees that do not tolerate shading and regenerate and grow slowly as understorey (e g silver birch and Scots pine)slowly as understorey (e.g. silver birch and Scots pine)
* A stand may be defined as a group of trees that can be managed as a unit.
61
The possibilities to apply different forest management systems like p pp y g yeven-aged and uneven-aged management depend on biological/ecologicalfactors, economic parameters, preferences and harvesting technology
Often ecologists attempt to develop forest management systems basedbiological factors only =>maximizing volume yield, etc.
Resource and environmental economists have studied almost entirely even-agedmanagement =>economics of uneven-aged management is rather purely understood
This is rather serious limitation because1. In some cases uneven-aged management may be economically superior to
even-aged management2 U d b f d d i l d bi di i2. Uneven-aged management may be preferred due to environmental and biodiversity
reasons
Initially optimal uneven-aged management models were developed by forest scientists Initially optimal uneven-aged management models were developed by forest scientists and economists (e.g. Adams and Ek 1974, Haight 1987, Getz and Haight 1989).Further economic analysis can be found from Tahvonen 2009, Tahvonen et al. 2010
62
Even aged managementg
≈80 yrs
63
Mixed species uneven ageduneven-agedstand
64
A life cycle graph for a size-classified population:
23 4
0( )tx
1 2 3
3 4
1tx 2tx 3tx 4tx
1 2 3 4F i l
1 2 3 4
1h 2h 3h4h
1 2 3 40
10 1
st
s
Four size classes, s , , ,x number of trees in size class s in period t
share of trees that grow to the next size classthe share of trees that remain in size class s
Remark: in addition of trees thesize-structured model is suitable
0 11
0
s
s s
s
the share of trees that remain in size class sthe share of trees that die in size class s
numb
0
er of seedlings( or seeds ) per tree in size class sx total number of seeds or seedlings
for fish and in situations wherethe perfect selectivity assumptionis possible
65
0
s
f gthe recruitment or " ingrowth" function
h the number of trees harvested from size class s
1 x x h x
The size class matrix model can be written as a set of difference equations:
1, 1 1 1 1
1, 1 1 1, 1
, 1 1 1,
1 ,
2 , 1,..., 2,
3 ,
t t t t
s t s st s s t s t
n t n n t n nt nt
x x h
x x x h s n
x x x h
x
, ,
14 ,n
t st ssH h f
t
whereH is total harvest in wei
, 1,...,s
ght or volume unitsf s n denotes the size of individuals in size class s, , ,( ).
sf fin volumeor weight units
Remark:According to equation (2) the number individuals in the beginning of next period in size class s+1 equals the number of individuals that will reach this size in size class s within period t plus the individuals in size class s+1 that are still in this size class minus the number of individuals th t h t d f thi i l t th d f th i d
66
that are harvested from this size class at the end of the period
Using matrix notation the model takes the form:
1t t t t x x G q or
1 1 1 1 1
2 1
0 0 0 00 0 0
,t t t tx x hx x h
x
2 1 1 1 2 2
2 3 3
1 1
0 0 00 0 0
0 0 0 0
,t t t
t
n n t
x x hh
h
.
1 1
1 10 0 0n n ,t
n ,t n n nt nx x h
tG tq
If the recruitment function φ is linear and increasing, the model is called as the "generic size classified model" in population ecology (Caswell 2001)"generic size- classified model" in population ecology (Caswell 2001)Linear model is simpler but yields exponential growth or declineModel with density dependence, i.e. with linear and decreasing or nonlinear φ is more interesting for economic purposes and well known in population ecology (Getz and Haight 1989)interesting for economic purposes and well known in population ecology (Getz and Haight 1989)
Recall: Density dependence is discovered by T. Malthus (1798).This is acknowledged in population ecology (Caswell 2001 p 504)
67
This is acknowledged in population ecology (Caswell 2001,p. 504)
A generic model specification for optimal harvesting of size-structured population:
0, 1,..., , 0,1,...
1 1 1 1 1
max
,
st
ttth s n t
t t t t
U H b
subject tox x h
x 1, 1 1 1 1
1, 1 1 1, 1
1
,, 1,..., 1,
,
0 0
t t t t
s t s st s s t s t
nt st ss
x x hx x x h s n
H h f
h
x
0
0, 0,, 1,..., .
st st
s
h xx s n are given
68
Empirical example 1:
Buongiorno and Michie (1980) estimated a size structured growth model for sugar maple forests. In this model time step is 5 yrs. The estimation yielded the following size structured model.:
0 8 0 0 109 9 7 0 3x x B N Note the density dependence;1, 1 1
2, 1 2
3, 1 3
0.8 0 0 109 9.7 0.30.04 0.9 0 0 ,
0 0.02 0.9 0
t t t t
t t
t t
x x B Nx xx x
Note the density dependence;φ is decresing function ofthe number of trees
1 2 3 1 2 3
10 20 30
0.02 0.06 0.13 , ,840, 234, 14.
refers to basal
t t t t t t t t
t
where B x x x N x x xand x x xB
area and to total number of treestN
Definition:Basal area is the sum of the cross section
Or if written as a set of difference equations:
1, 1 1 1
2, 1 1 2 2
109 9.7 0.3 0.8 ,0.04 0.9 ,
t t t t t
t t t t
x B N x hx x x h
areas of the trees in the stand. Units: m3
3, 1 2 3 30.02 0.9 .t t t tx x x h Assuming no harvest, it is possible to solve the steady state by assuming all variables areconstant in time in the differential system above. This yields the steady state:
1 2 3400, 160, 32.x x x Solving the characteristic roots for the dynamic system (without harvesting) yields:
69
1 2 3
2 2
0.847, 0.930 0.116 , 0.930 0.116 .
0.93 0.116 0.937 1.The steady state is stable because
r r i r i
R
Stand development without harvest and two initial states
600
800
umbe
r of t
rees
400
600
Nu
200
Time
0 20 40 60 80 1000
Timesize class 1size class 2size class 3
70
Based on this growth model we obtain the following economicBased on this growth model we obtain the following economic optimization problem:
1 1 2 2 3 31 2 3 0 1
max tt t th s t
p h p h p h b
, 1,2,3, 0,1,... 0
1, 1 1 1
2 1 1 2 2
109 9.7 0.3 0.8 ,0.04 0.9 ,
sth s t t
t t t t t
t t t t
subject tox B N x hx x x h
2, 1 1 2 2
3, 1 2 3 3
10 20 30
1 2 3
0.04 0.9 ,0.02 0.9 ,
840, 234, 14,0.02 0.06 0.13 ,
t t t t
t t t t
t t t t
x x x hx x x hx x xB x x x
1 2 3
1
,t t t t
t tN x 2 3 ,0, 1, 2,30, 1,2,3.
t t
st
st
x xx sh s
1 2 30.3, 8, 20.The market prices of trees are: p p p
71
#Bioeconomics 2011, Olli Tahvonen, #model file #B i d Mi hi (1980) d t
AMPL code for Buongiorno and Michie (1980)
#model fileparam T;param ac;param n;param p {s in 1..n};
#Buongiorno and Michie (1980) data.#data fileparam T:=100;param ac:=1;#0.1;param r:=0;#0.1;
param y {s in 1..n}; #basal area per treeparam α {s in 1..n};param β {s in 1..n};param r;param b=1/(1+r);
pparam n:=3;param y:=
1 0.02#2 0.063 0 13;param b 1/(1 r);
param x0 {s in 1..n}; #initial statevar x {s in 1..n, t in 0..T} >= 0;var h {s in 1..n, t in 0..T} >= 0; var H {t in 0..T-1}>=0;var X {t in 0 T}=sum{s in 1 n} x[s t];#total no of trees
3 0.13;param α:=
1 0.042 0.023 0;
βvar X {t in 0..T}=sum{s in 1..n} x[s,t];#total no. of treesvar Y {t in 0..T}=sum{s in 1..n} y[s]*x[s,t]; #total basal areavar φ {t in 0..T};
maximize objective:
param β:=1 0.82 0.93 0.9;
param p:=sum {t in 0..T-1} b^t*(H[t])^ac;
subject to restriction_1 {t in 0..T}:φ[t]=109-9.7*Y[t]+0.3*(sum{s in 1..n} x[s,t]); subject to restriction_2 {t in 0..T-1}:x[1,t+1] = φ[t]+β[1]*x[1,t]-h[1,t];
1 0.3 2 83 20;
param x0:=1 840[ , ] φ[ ] β[ ] [ , ] [ , ];
subject to restriction_3 {s in 1..n-2,t in 0..T-1}:x[s+1,t+1]=α[s]*x[s,t]+β[s+1]*x[s+1,t]-h[s+1,t];
subject to restriction_4 {t in 0..T-1}:x[n,t+1]=α[n-1]*x[n-1,t]+β[n]*x[n,t]-h[n,t];
subject to restriction 5 {t in 0 T 1}:
1 8402 2343 14;
72
subject to restriction_5 {t in 0..T-1}:H[t]=sum{s in 1..n} p[s]*h[s,t];
subject to restriction_6 {s in 1..n}:x[s,0]=x0[s];
500 The features of optimal solutionR
even
ues
200
300
400 assuming r=0:At optimal steady state all trees are cutwhen they reach size class 2 (at the end of R
0
100
80
when they reach size class 2 (at the end of each period) and no trees are cut from size class 1. Thi i li b th ti f
Har
vest
from
si
ze c
lass
2
20
40
60
80 This implies by t 2
2 0t
t
xx
he equation for that at the steady state
0
20
rees
in
1000
1200
Num
ber o
f tr
size
cla
ss 1
200
400
600
800
Time in 5yrs periods
0 20 40 60 800
73initial state: [100, 45, 5]Initial state: [840, 234, 14]
More about density dependence in forestry models
In Buongiorno and Michie (1980) density dependence exists only in the regeneration functionHowever, the growth of larger trees may also depend on stand densityThis can be taken into account by specifying transition coefficents as functions of stand density
Total stand basal area as a density measure: 2 4( / 2) 10 3 1415926
n
d d h i th t di t i i l d2 4
1
( / 2) 10 , 3.1415926...
( / 2)
t st s ss
y x d d s
x d
where is the tree diameter insize class and
The basal area of trees larger than size class s trees (and half of size class s trees) as a density measure :2 4
2 410 n
( / 2)st sst
x dy 2 4
1
10 ( / 2) 10 , 1,...,2 st s
k sx d s n
The transition of trees between the size classes become functions of basal area (in addition of diameter)
( , , ), 1,..., 1,( ) 1 ( ) ( ) 0 ( ) 1 1
st s s t std y y s nd d d
( , , ) 1 ( , , ) ( , , ), 0 ( ) 1, 1,..., ,
( , , )
st s s t st s s t st s s t st s t
s s t st
d y y d y y d y y y s n
d y y
where denotes natural mortality.
74
A more general transition matrix model
1 0{ 1 0 1 }max ( ) ( ) , (the objective function)t
t th s n tV R C b
x
A more general transition matrix model
{ , 1,..., , 0,1,...} 0sth s n t t
1 1 2 2 11
1 2 2
( ),
(annual gross revenues, sawntimber price, sawntimber vol per tree, same for pulp)
n
t st s ss
s s
R h p p pp
Objective functionrevenues, cost
1, 21 2
( , ) , [ ,..., ][ , ,..., ]
(harvesting cost per operation, fixed cost, tree diameters (cm) harvested trees per size class in period t t f f n
t t nt
C C C C d d dh h h
h d dh t
1, 1 1 1 1 1( ) [1 ( ) ( )] , (development of smallest size class, regeneration, t iti t l t t t t t tx x h x x x
t lit )1 transition, natural m ortality)
1, 1 1 1 1, 1,( ) [1 ( ) ( )] , 1,..., 2 (development of size classes 2,...,n-1)s t s t st s t s t s t s tx x x h s n x x x
1 1 1( ) [1 ( )] . (development of largest size class)t t t t t tx x x h x x
Nonlinearsize structuredmodel
, 1 1 1,( ) [1 ( )] . (development of largest size class)n t n t n t n t nt ntx x x h x x
0
0, 0, 1,..., , 0,1,...,, 1,...,
(nonnegativity constraints)given. (initial state)
st st
s
h x s n tx s n
Technical
0 ,2 ,3 ,...,when (additional restriction for taking into account that harvesting can be done every kth period only) sth t k k k
where the value of k is a positive integer.
constraints
75
Empirical estimation results for the transition matrix modelNorway spruce, 93 sample plots, Central Finland, y p p pOxalis-Myrtillus (OMT) and Myrtillus (MT) forest site typesTime step three years
2.1368 0.104 0.107 1, (regeneration, total number of trees)t tN yte N
1 13.752 2.560 0.296 0.849ln( ) 0.0351 , 1,...,10 (transition)s s t std d y y
s e s
13.606 0.075 0.997ln( )1 1 10 (mortality)st sy d ( )1 , 1,...,10 (mortality)st sys e s
2
2
39691 , 1,...,101000 25683 37785
(length of trees, m)ss
dh sd d
76
diameter, cm 7 11 15 19 23 27 31 35 39 43 sawn timber 0 0 0 0.14136 0.29572 0.45456 0.66913 0.88761 1.12891 1.39180 pulp
d0.01189 0.05138 0.12136 0.08262 0.06083 0.06703 0.04773 0.04596 0.04672 0.04119
wood Table 1. Sawn timber and pulpwood volumes 3m per size classes
The roadside price for saw logs equals 51.7€m-3 and pulp logs 25€ m-3. p g q p p g
21.906306 3.3457762 25.5831144 3.77754938
The harvesting cost functions are (Kuitto et al. 1994):
th sawvol pulpvolt t tC H H
1
22.3860.50001 0.59 2.1001366 300,1000 85.621
n sts ts
s t
hvol Nvol N
26 350495 2 82183045 25 701440 3 33144cc sawvol pulpvolC H H
1
26.350495 2.82183045 25.701440 3.33144
146.170.44472 0.94 2.1001366 300,1000 862.05
t t t
n sts ts
s t
C H H
hvol Nvol N
where denotes thinning cost and clearcut cost, and ath cc sawvol pulpvolt t t tC C H H re the
total volumes of sawlogs and pulpwood yields per cutting and is the total (commercial)volume of a stem from size class .
svols
The linear parts in both cost functions denote the hauling costs and the two nonlinear components the logging cost. In the case of uneven-aged management the cost function is formed by taking the hauling cost components from the thinning cost function and the logging costs using the logging cost component from the clearcut cost function multiplied by a factor equal to 1.15. Fixed harvesting cost equals 300€.
77
q
Note: All the model components are based on empirically estimated parameters
Questions to be analyzed: 1. How volume maximization solution looks like?2. How the economically optimal uneven aged solution looks like?y g3. How even-aged and uneven-aged management systems can be compared?
78
1. How volume maximization solution looks like?
40
ree
year
s, m
3
30C
uttin
gs p
er th
r
10
20
8 10 12 14 16
C
0
Basal area before cuttings, m2
Steady stateInitial state/initial optimal cuttings
Figure 1. Optimal development of basal area and cuttings toward the MSY steady state
79
150
200
clas
s
100
150
of tr
ees p
er si
ze c
123456
010
2030
4050
0
50
Num
ber
12345678910 60 Time periods, in three years intervals
Size classes
Figure 2. Development of the size class distribution over timeFigure 2. Development of the size class distribution over time Number of trees before cuttings
In optimal solutions the forest is harvested continuously without clearcuts. Thus, given
80
natural regeneration it is optimal to apply uneven-aged management
2. How the economically optimal solution looks like?
0 60 120 180 240
Bas
al a
rea
afte
r and
be
fore
har
vest
, m2 /h
a
0
5
10
15
20
25
0 30 60 90 120 150 1802468
10121416
0 60 120 180 240
mbe
r of t
rees
afte
r d
befo
re h
arve
st p
er h
a300
400
500
600
700
800
0 30 60 90 120 150 180
200300400500600700800
0 60 120 180 240Num
and 300
olum
e af
ter a
ndef
ore
harv
est,
m3 /h
a
020406080
100120140160180
0 30 60 90 120 150 180200
020406080
100120140
0 60 120 180 240
Vo
be
otal
yie
ld, m
3 ,w
logs
yie
ld, m
3r 1
5 y
ears
/ha
406080
100120140
0 30 60 90 120 150 1800
50
60
70
80
90
0 60 120 180 240
To saw
per
2040
reve
nues
, €ue
s net
of c
uttin
g er
15
year
s/ha
2000
3000
4000
5000
6000
0 30 60 90 120 150 18040
2000
2500
3000
3500
4000
0 60 120 180 240Gro
ss r
reve
nuco
st p
e
1000
2000
h, n
umbe
r of
r thr
ee y
ears
/ha
30
40
50
60
0 30 60 90 120 150 1801500
2000
303540455055
810 60 120 180 240In
grow
thtre
es p
er
10
20
Time, years Time, years0 30 60 90 120 150 180
202530
Interest rate 0% Interest rate 3%
Optimal steady state size distribution and selection of harvested treesInterest rate 0 or 3%, cutting periods 15 and 12 yearsInterest rate 0 or 3%, cutting periods 15 and 12 years
120
(a) Zero interest ratem
ber o
f tre
es/h
a
40
60
80
100
120
Diameter class
7 11 15 19 23 27 31 35 39 43
Num
0
20
(b) Th t i t t t
r of t
rees
/ha
40
60
80
100(b) Three percent interest rate
Diameter classes
7 11 15 19 23 27 31 35 39 43
Num
ber
0
20
40
Diameter classes
Harvested trees
82
How even-aged and uneven-aged management systems can be compared?
3000
3500
rtific
ial
n € 2000
2500
3000
Uneven-aged optimal
ost f
rom
ar
egen
erat
ion
1000
1500
2000
Valid area
C re
0
500
1000
Even-aged optimal Break even curve
Interest %
1 2 3 4 5 60
Interpretation: Even-aged management requires the regeneration investment after the clearcut. This competes with the natural regeneration that mayproduce lower number of seedlings but is free of cost. When this
83
p ginvestment cost and the interest rate is high uneven-aged managementbecomes always optimal
SummaryThe generic Faustmann model is brilliant but as such too simple for almost any purposesMarket level problem consistent with even-aged management leads to an any number of
l bl th t b t d d t i l d l d ll ti b t ti b age classes problem that can be extended to include land allocation between timber production, old growth conservation and agriculture
Economic analysis for forest resources have concentrated to even-aged management=>restrictive due to pure economic and environmental reasons=>restrictive due to pure economic and environmental reasons
Uneven-aged management problem leads to size-structured optimization problems
Policy remark: Policy remark: In Finland (and Sweden) uneven-aged management has been practically illegal over last 60 yearsThis has been based on the silviculturalists view that uneven-aged management is This has been based on the silviculturalists view that uneven aged management is economically inferior compared to even-aged managementThe economist's argument: the proof is missingNew resource economic studies have shown that the silviculturalists view is unwarrantedNew resource economic studies have shown that the silviculturalists view is unwarranted
=>The Finnish ministry of Agriculture and Forestry has initiated a change toward officialacceptance of uneven-aged forestry and general liberalization of forest policy
84
Emerson, Lake and Palmer (1991) Romeo and Juliet
The idea: ELP takes Sergei Progofiev (1935) and add their own ideas and produce something new and interesting
In resource economics we take Faustmann (1849), Ramsey (1928), Hotelling (1931) etc and add our own ides and attempt to produce something new and interesting
85
References D M Adams and A R Ek Optimizing the management of uneven aged forest stands Can J of For Res 4 274 287 (1974)D. M. Adams and A.R. Ek, Optimizing the management of uneven-aged forest stands, Can. J. of For. Res. 4, 274-287 (1974).J. Buongiorno and B. Michie, A matrix model for uneven-aged forest management, For. Sci. 26(4), 609-625 (1980). M. Faustmann, Berechnung des Wertes welchen Waldboden, sowie noch nicht haubare Holzbestände für die Waldwirtschaft besitzen, Allgemeine Forst- und Jagd-Zeitung 25, 441--455 (1849). R.G. Haight, Evaluating the efficiency of even-aged and uneven-aged stand management, For. Sci. 33(1), 116-134 (1987). R. Hartman, (1976), The harvesting decision when a standing forest has value, Econom. Inquiry 4, 52-58. L.P. Lefkovitch (1965), The study of population growth in organisms grouped by stages, Biometrics 21, 1-18. T. Mitra, and H.Y. Wan, Some theoretical results on the economics of forestry. Rev. of Econ. Studies LII, 263-282 (1985). Mitra T and H Y Wan (1986) On the Faustmann solution to the forest management problem J Econ Theory 40 229 249 Mitra, T. and H.Y. Wan (1986), On the Faustmann solution to the forest management problem, J Econ. Theory 40, 229-249.
S. Salo and O. Tahvonen, On the Economics of forest vintages, J. of Econ. Dyn. Control 27, 1411-1435 (2003). P.A. Samuelson, (1976), Economics of forestry in an evolving society, Econ. Inquiry 14, 466--492. Salo, S. and O. Tahvonen, (2002a) On equilibrium cycles and normal forests in optimal harvesting or tree age classes. Journal of Environmental Economics and Management 4, 1-22. Salo, S. and O. Tahvonen, (2002b), On the optimality of a normal forest with multiple land classes, Forest Science 48, 530-542. Salo,S and O. Tahvonen, (2003), On the economics of forest vintages, Journal of Economic Dynamics and Control 27, 1411-1435. Salo, S. and O. Tahvonen, (2004) Renewable resources with endogenous age classes and allocation of land, Americal Journal of Agricultural Economics 86 513 530Agricultural Economics, 86 513-530. Stokey, N.L. and R.E. Lucas (1989), Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, Massachusetts. O. Tahvonen, S. Salo and J. Kuuluvainen,(2001) Optimal forest rotation and land values under a borrowing constraint, J. of Econ. Dyn. Control. 25l, 1595-1627. O. Tahvonen, (2004), Timber production vs. environmental values with endogenous prices and forest land classes, Canadian Journal of Forest Research (34) 1296-1310. O. Tahvonen,(2004) Optimal harvesting of forest age classes: a survey of some recent results, Mathematical Population Studies, 11, 205-232. O T h P kk l T L ih O Lähd E d Nii i äki S (2010) O ti l t f d N f tO. Tahvonen, Pukkala, T., Laiho, O., Lähde, E., and Niinimäki, S (2010), Optimal management of uneven-aged Norway spruce forests, Forest Ecology and Management 260, 106-115, 2010. M.B. Usher, (1966) A matrix approach to the management of renewable resources, with special reference to selection forests-two extensions, J. of Applied Ecology 6, 347-346. Wan, Y.H. (1994), Revisiting the Mitra-Wan tree farm, International Econ. Rev. 35, 193-198.
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