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.

86