[PPT] - New bioeconomics of fisheries and forestry Olli Tahvonen University PowerPoint Presentation

SLIDE 1

New bioeconomics of fisheries and forestry Olli Tahvonen University of Helsinki EAERE Venice Summer School 2011 Section 1, Fisheries Section 1, Fisheries

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SLIDE 2

The question of managing

1. Introduction

The question of managing biological resources Actual resource management Resource Applied Actual resource management is dominated by ecologists and MSY -type objectives both in forestry and fisheries economics Applied ecology Detailed Economic objectives with "oversimplified" y New bioeconomics: The aim is to integrate sound economics and realistic ecological models with MSY -type

bjectives

ecology economics and realistic models taken directly from ecology Applied h i

cf. economics of nonrenewable

mathematics resources

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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,...

rt

px(t )e w max

 

rt { h }

max U( h,x )e dt

   

1

rt { t }

max e 

s.t. x F( x ) h, x( ) x    

Some extensions/alternatives: Age-structured models Spatial models Some extensions/alternatives : Environmental values Market level age structured models p Multispecies models,... Market level age-structured models Stand level size-structured models,... Optimal rotation with optimal thinning, initial density

3

initial density,...

SLIDE 4

New bioeconomics of fisheries and forestry Content Content 0 Introduction 1 Fisheries 1 1 Age-structured population models in fisheries 1.1 Age-structured population models in fisheries 1.2 Generic age-structured optimization problem 1.3 Empirical example of an age-structured fishery model 1 4 On numerical optimization 1.4 On numerical optimization 2 Forestry 2.1 Market level age-structured model for timber/old growth/agriculture g g g 2.2 Stand level size-structured models 2.3 Generic size structured optimization problem 2.4 Empirical example of a size-structured model p p 3 Summary and discussion

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SLIDE 5

Memory refresh: optimal solution for the Schaefer-Gordon-Clark bi h ti d l biomass harvesting model

 

rt

max U( h ) C( h x ) e dt

 





25

Numerical example :

 

{ 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

( )

Yield

5 10 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 return equals interest"

Biomass, x

20 40 60 80 100 120

Optimal yield

x

2. Optimal yield is an increasing function of biomass
3. The optimal solution approaches the steady state

monotonically

4. If C =0 and F'(0)< it is optimal to deplete the



Optimal yield Growth

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

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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 ti

1. The classical Gordon-Schaefer-Clark biomass model describes a biological population

but 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 targeted harvesting 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 old people's residential area, the second is a population of young children, and the third is a population

f 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 ) th would 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

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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

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Remarks: Age-structured models are becoming important in general economics as well Age structured models are becoming important in general economics as well Instead 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 fundamental dd 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.

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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 populations

L. Lefkovich (1965) Matrix models for size structured populations

M.B. Usher (1966) Matrix models for tree populations =>Presently population studies in ecology rest heavily on age- or size structured models structured models and in fishery economics (or fishery ecology...difficult to make the distinction) Baranov (1918): The problem raised Beverton and Holt (1957): Famous "Dynamic pool model" Walters (1969): Pulse fishing solutions ( ) g Clark (1976, 1990): "The problem is incomprehensible" Hannesson (1975): Pulse fishing solutions Horwood (1987): Smooth harvesting solutions => almost all studies have used only numerical methods

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A life cycle graph for an age-classified population with density dependence in recruitment in recruitment

2 2

x  x  x 

 

4 1 s st s

x  





1 1

x 

1 1t

x 

2 2t

x 

3 3t

x 

2 2t

x 

3 3t

x 

4 4t

x 

1t

x

2t

x

3t

x

4t

x

1 1t

x 

2 2t 3 3t

1t

h

4t

h

2t

h

3t

h

1 2 3 4 1

st

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      1 1 1

s s

share of individuals that survive in age class s the share s ,...,n die due to natural reasons ( natural morta         1

s

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

therecruitment function :the number of offspring that survive to age class h the harvesting mortality (control variables) 

SLIDE 11

Examples of commonly used recruitment functions Examples of commonly used recruitment functions

1

bx

Beverton Holt recruitment function : ( x ) ax / ( bx ) Ri h i f i ( ) 



   10

bx

Richer recuitment function : ( x ) ax e   cruits 8

0 9 1 0 1 ( x ) . x / ( . x )   

mber of rec 4 6

0 05

0 9

. x

( x ) x e 



Num 2

0 9 ( x ) . x e  

N mber of eggs 20 40 60 80

N b f " "

11

Number of eggs

Number of "eggs"

SLIDE 12

The age class model can be written as a difference equation system g q y

1 1 1 n ,t s st

x x ,  



      



1 1 1 1 1 1 1

1 2

s 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 1 2 1 1 2 2 ,t t t t t t t

x x h ( x ) x x h  

 

                               

2 1 1 2 2 2 3 ,t t t t

h 



                                                                 

1 1 1 n ,t n,t n n nt nt

h x x h  

  

                                         

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Perfect selectivity vs. nonselective harvesting Perfect selectivity: it is possible to control the age specific harvest levels

1

st

h , s ,...,n 

separately. This is seldom possible in fishery.

Nonselective harvesting: “effort” is controlled and the catch per age class can be given as 1 1

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 :    1

st 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

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SLIDE 14

1 1 Let / ( b ) ( )

Some properties of the age-structured model (and the connection with the biomass model)

1 1 2 2 1 1 1 2 2

1 1 2 1 1 3 4

,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

4 5 2 1 2 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

2 1 2 1

st s st t st i

n , h q E x , s , , , q q q, ax / ( bx ) and that w den           

1 2

1

tes 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

1 2 1 2

1 1 1 ( E ) ation (3) implies x x . ( E )      

1 2 2 1

1 6 1 1 7 1 2 7 ( E ) Denote , implying ( ) ( E ) x x . ( ) E ti ( ) ( ) d ( ) i l         

1 1 1 1 1 1 1

1 2 7 1 1 1 1 1 1 Equations ( ),( )and ( ) imply a x a a x x b x a b x b x b x and that the steady state                             is given as 14 and that the steady state

1 2

1 1 8 is given as a a x , x . ( a,b ) b b        

SLIDE 15

1 2 1 1 2 1 2

1 1 When E the steady state becomes a x , x . b             

1 1 2 2 1 2 cc

Substituting 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 E T  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

1 a E= . Note that E<1. a         

1 1 1 2 2 2

5 The 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 inthesteady state harvest function we can write E x and x as functions of B implying that H becomes a funct  ion of B Write

1 2

E, 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 cc 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

x H>0.  

SLIDE 16

Numerical example 1:

1 2 1 2

1 3 19 49 6 1 2 2 5 20 6000 Assume the following parameter values: , , , a , b , w , w and the Beverton Holt recruitment function           

2

2 5 20 6000 19 6 20

t

Thus, the model can be written as x x 

 

1 1 2 2 1 1 2

49 1 6 6000 1 3 1 2 5

,t t ,t t t t

x , x x x E x

 

    

 

1

t

E 

 

2 5

,

 

1 2 1 2

1 3 2 2 5 2

t t t t t t t t

H x E x E , B x x .    

1 2 1 2

100 125 350

t t t

The carrying capacity population level becomes x , x and the carrying capacity biomass .  

1 2

y g p y The critical level of fishing mortality that implies B=x x

btains the val

  49 0 71 69 ue E .  ฀

16

SLIDE 17

In addition, we obtain:

duals in arvest 400 30 35 number of individ and 2, biomass, ha 200 300 ibrium Harvest 15 20 25 0 0 0 2 0 4 0 6 0 8 Steady state n age classes 1 a 100 Equili 5 10 Fishing effort E 0.0 0.2 0.4 0.6 0.8 Individuals in age class 1 Individuals in age class 2 Total population biomass Equilibrium Biomass 100 200 300 400 Total population biomass Harvest

H th d l d i diff f th bi d l? How the model dynamics differ from the biomass model?

17

SLIDE 18

Let us fix the total harvest level H i e

1 1 1 2 2 2 1 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 t t t

H H w x E w x E E . w x w x         

1 1 2 1 1 1 2 2 1 1 1 2 2 2 1 1 1 2 2 2

1 1

,t t ,t t t t t t t

x ( x ), H H x 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 same
2. Do the biomass and age structured models give the same

prediction?

18

SLIDE 19

Is the harvesing equal to H=30 sustainable, i d i B 187 h B 124?

1 2

1 124 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 i.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 higher dot the equilibrium B=187. 70 arvest 25 30 35 ass x2 50 60

1 5

Equilibrium Ha 10 15 20 Size of age cla 20 30 40

2

100 200 300 400 E 5 124 B  187 B  20 40 60 80 100 120 140 10

3 4

Equilibrium Biomass Size of age class x1 20 40 60 80 100 120 140

124 Computing the model forward yields the results: Initial states 1 and 2 have B and converge toward B  124 124 g Initial states 3 and 4 have B , but are unsustainable Initial 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. 

SLIDE 20

The equilibrium can be unstable for all initial states

utside the equilibrium implying unsustainability or unexpected

   

2

3 1 1 2 2 1 1 2

0 9 1 0 9 1

t

x ,t t ,t t t t t

x x e , x . x q . x q ,

  

    

8

q p y g y p fluctutions and no convergence toward the steady state

2 2

0 9

t t t t t

B x , H . x q  

6 8 class x1 4

1 2 H 

Age 2

1 2 H .  H  1 2 H . 

2 4 6 8 10

1 79 B . 

Age class x1 2 4 6 8 10

20

SLIDE 21

Summary this far: Steady state harvest becomes a function of the biomass as in the generic bi 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 harvest Crucial differences between the two approaches :

1. The dynamic behavior of models become different

2 I d d l h ilib i bi

2. In age-structured model the equilibrium biomass–

equilibrium harvest function depends on catchability coefficients, i.e. on harvesting technology Th MSY t b d t i d l i Thus, MSY cannot be determined applying biological information only. This cannot be understood in the biomass framework Equilibrium biomass

21

Equilibrium biomass

SLIDE 22

The generic nonlinear age-structured optimization model

1 1 1 n ,t s st s

x x  

 

      



     

1 1 1 1 1 1

1 1 2 1 1

s ,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-structured model, effort Et as control

1 n t s s st s t s s

H w x q ( E ) where q ( E ) are fishing mortality functions with the properties 



 

t

variable

1

s s s

q ( ) , q ( E ) , q '  

s

( E ) , and q ''( E ) .  

 

n t

max V( ) U w x q ( E ) C( E ) b 



 

 

x Objective function; ' '' U U  

 

1 1 0 1

t st

s s st s t t t s { E , x , s ,...,n, t , ,...}

max V( ) U w x q ( E ) C( E ) b . 

   

     

 

x

0,

1,..., , 1

s

x s n are given 

0, 0, ' 0, '' 0 (effort cost) U U C C    

Initialstate

0, 1,..., , 0.

st t

x s n E   

Nonnegativity constraints

22

SLIDE 23

Simplified two age classes version of the age-structured population model (Schooling fishery)

Let

1 1 2 2 t t t t t

H w x qE x E ,     where 1 w  is the weight of fish in age class 1 with respect to fish in age class 2 and 1 q  is catchability parameter in age class 1 (in age class 2 it is 1). Solving for

t

E yields

1 1 2 2

1

t t t t t

H E , where we assume interior solutions in the sence that E w 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 2 1 1 2 2 1 1 2 2 t t t t t t t t t t t t

q x q x x x H . w x q x            Denote x q x   

 

1 1 2 2 1 2 1 1 2 2

1 1 1

t t t t t t

x q x G 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.

SLIDE 24

The optimization problem can now be written as

 

     

1 2

0 1

t t t

t t t H ,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 2 t t t

x and x given, x , x , H ,    where where

 

1 1 2 2 1 2 1 1 2 2 t t t t t t

x q x G x ,x . w x q x        Assume that U ' ,U ''   and that the recruitment function is either Beverton-Holt

r Richer -type. Note: the fecundity parameter for age class 1 is zero.

24

SLIDE 25

Lagrangian and the necessary conditions for interior solutions 



1 2 t t t

x , x , H  can be given as

     

 

     

1 2 1 1 2 1 1 2 2 1 2 2 1 2 1 2

3

t t t t ,t t t t t t t ,t t t t t t t

L b U H x x x x H G x ,x x , L b U ' H G x ,x , H       

    

                   



 

1

1 2 1 1 1 1 1 2 1 1 1

4

t t t ,t t x ,t ,t ,t t

H L b b H G x ,x , x L   

     

            

 

 

 

1 1 2 1 2 1 2 1 2 1 t ,t ,t ,t t ,t

L b b ' x b H x      

     

    

 

2

1 1 2 1 2

5

x ,t ,t t

G 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):

 

1

1 2 1 x

b HG .      Substituting this into (5), dividing by

2

b , taking into account that

 

1 1 b / r   and rearranging terms yields the steady conditions in the form

             

1 2

2 1 2 1 2 2 1 2 1 2

1 6 1 1 7

x x

' x G x ,x '( x ) H G x ,x r, r r x x ,                        

25

     

1 2 2 1 1 2 2 1 2

8 x x x HG( x ,x ).     

SLIDE 26

Let us first study the case with "knife edge" fishing gear where q  , i.e. where harvest includes

nly fish from age class 2. In this case the steady state is implicitly given as

         

1 2 2 1 2

1 9 1 10 '( x ) r, r x x ,             ( ) H  Interpretation: at the steady state it holds that

 

2 1 1 2 2

11 x x x H ,      because q  implies

1 2

1

x x

G , 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 natural

mortality. Surplus production

can be harvested without consuming the "biological capital"

2 1 2

'( ) x H x x        Maximizing steady state surplus production with respect to requires

 

2

(1 ) 12    

g g p

2

x  Given r=0, equation (12) equals equation (9). Note that

1 2

'( ) x     is marginal effect of

2

x on surplus production via changed recruitment and 



2

1   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

SLIDE 27

Write condition (9) in the form:

1 2 2 1 2 2

'( ) ( , , , , ) (1 ) 1 x y x r r r               . We obtain

2 1 2 2 1 1 2 2

'' 0. . , 1 ' ( '' ' ' 1, , 1, . (1 ) 1 1 y Thus the steady state is unique In addition x r x y y y y r r r r                                      

1 2

(1 ) 1 1 r r r r           Thus, we can write:

2 2 1 2

( , , , ) x x r     , i.e.

2

x as a function of the given parameters. The comparative statics derivatives become

1 2 2 2 1 2 2 1 1 1

' 1 ' (1 ) 0, 0, '' '' y y x x r r y y r                                    

2 2 2 2 2 2 1 2 2 2 1 2 1

1 ( '' ' 1 0, . '' '' x x r y y x x x y y                                     

1 2 1 2 2

1 y y x x r              Thus, steady state level of

2

x 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.

SLIDE 28

To study the stability of the optimal steady state we use implicit function theorem and write

t

H as a function of

t

 using equation (3). This yields

2

( ), ' 1/ ''.

t t

H H H U    The necessary optimality conditions can now be written as the system

1, 1 2 2

( ),

t t

x x  

 

 

 

2, 1 1 1 2 2 2 2 2 1 1 1 1

( ), ,

t t t t t t t

x x x H        

 

    

 

 

1, 1 2 1 1 2 2 2 2 1 2, 1 1

, ' ( ) .

t t t t

b x x H b         

 

   The Jacobian matrix takes the form ' ' H        

1 2 2 2 2 2 1 2 1 2 1 2 1 2 2 2 2 2 2 1 1 2 1 2 2 2 1 2

' ( ) '' ( ) '' ' ( ) '' ' . ( ' ) ( ' ) ' ( ' ) H b b b b H J b b b                                                         

1 2 1

1 b                  28

SLIDE 29

This yields the characteristic equation:

4 3 2 4 3 2 1

( ) , u u u w u w uw w where       

2 4 2 1 2 2 2 2 3 1 2 2 2 2 2 1 2 1 1 1 2

, ' ' '' ' '' 1 ' , ' '' ' '' ' ' w b U U w b bU b U b                              

2 2 2 2 1 2 1 2

, ' 1 . w b b w b        

1 2

b The facts lim ( ) , lim ( ) , (0) 0, (1) 0, ( 1)

u 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

ver time.

29

SLIDE 30

Optimal

5

saddle point steady state

3 4

x2

2 1 2 3 4 5 1

x1

1 2 3 4 5

Figure 2. Cyclical population dynamics but sadd Figure 2 Cyclical development of unharvested g 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

SLIDE 31

Steady states in biomass vs. age-structured model (n=2 case)

 

1 2 2 1 1 2 2

, The steady state satisfies x 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 biomass just 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 is T

2 1 2 2 2

, '( ) '( ) (1 ) hus within the biomass framework the optimal sustainable biomass level is defined by F x x r        

1 2 2 2

: '( ) 1 1 This can be compared withthe steady state condition in the age structured framework x r r          1 r For Bever  & '' ' , 1 ton Holt and Richer recruitment functions when Thus given discounting and b the biomass model yields higher steady state biomass and yield compared to the age structured model      

31

state biomass and yield compared to the age structured model

SLIDE 32

Assume next that 0 1 1 q 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 2 1 2 2 1 2 1 2 2 1 1 2 2 1 2

1 6 1 1 7 8

x x

' 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

2

x on the level of H due to changes in yield composition between

1

x and

2

x . Proposition1 : Given nonselective gear and 0<q<1, the steady state levels of

1 2

x 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 2 2 2 2 2

: ( ). / / ( ) ( ) ' / Interpretation x 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 x share of steady state , implying that harvest includes higher share of

32

SLIDE 33

Number of fish in age class 2 Figure 4. The effects of harvesting cost on optimal solution Parameters: U=H0.5, q1=0, q2=1, ==0, ==1, r=0.01, 

Comparison of steady states: biomass model vs. age-structured models p y g "optimal extinction" results

2 5 3.0 3.5 (a) (b) yield 2.5 3.0 3.5 4 5 6 (c) Biomass 0.5 1.0 1.5 2.0 2.5 Equilibrium y 0.5 1.0 1.5 2.0 2.5 Biomass 1 2 3 4 Rate of interest 0.0 0.1 0.2 0.3 0.4 0.5 0.0 Figure 5a c Comparision of steady states of the age structured and the biomass models Biomass 10 20 30 40 0.0 Rate of interest 0.0 0.2 0.4 0.6 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

SLIDE 34

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 g in 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 change Clark (1976, 1990): Adding age structure to bioeconomic analysis of fisheries will hardly change any basic bioeconomic principles" Th lt h The result here: Adding age structure changes all the basic properties of optimal harvesting compared to the generic biomass model

34

SLIDE 35

Empirical example: Baltic sprat fishery (schooling fishery) Th t Wh t i th ti l h ti l ti f

 

1 t t

pH max b

  

 The model: The setup: What is the optimal harvesting solution for Baltic sprat given different natural mortality levels determined by Baltic cod (a predator of sprat)

   

0 1 1 1

1

t

t { 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):

1 1 1 1 1 1 1

1 2

t s s st s 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

s

w [kg] Catchability

s

q 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 1

st

x , 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 88 0 016 14 3

1 1 1

1 2

s s st st n s s st s n n ,t

q x G , s ,...,n , w q x q G  

  

   



1 1 1 n n ,t n n nt n s s st s

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 1 and t , ,....  8 1.0 0.0110 1.0 0.7189 0.7711 0.4025 0.716

 

104 2 0 5032 Recruitment function: where

t

ax x a b    

35  

6

104 2 0 5032 0 07 10 1000 where Price of fish net of unit harvesting cost tons

t t

x , a . , b . . b x : € . per      

SLIDE 36

#code for Tahvonen, Quaas,Schmidt and Voss (2011). "Effects of species interaction on

ptimal 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;

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;

ption 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;

ption knitro_options "maxit=2000 opttol=1.0e-9 multistart=1 ms_maxsolves=10";

solve;

ption display_width 2;

display H;

SLIDE 37

usand tonnes 200 250 300 350 Equilibrium yield, tho 50 100 150 200 Population biomass, thousand tonnes 500 1000 1500 2000 2500 3000 3500 E Reference Low cod High cod

Figure 1. Equilibrium biomass–harvest relationships

37

SLIDE 38

1200 1400

(a)

ss, thousand tonnes 600 800 1000 2010 2015 2020 2025 2030 2035 2040 Biomas 200 400 Years 800

(b)

thousand tonnes 400 600 2010 2015 2020 2025 2030 2035 2040 Yield, t 200 Years 2010 2015 2020 2025 2030 2035 2040 Interest rate 2% Interest rate 10%

38

Figures 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

SLIDE 39

tons)

2500 3000

D C B A biomass ('000

1500 2000

Total stock

500 1000 2010 2015 2020 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2010 2015 2020

Figures 3a,b,c. Population biomass of Baltic sprat. (A) Historic stock size 1974-2008 (ICES, 2009b), only for 2008 the distribution to age-classes is displayed; (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 management for high cod case. Stacked bars show distribution of biomass to age-classes from age 1 (bottom) to age 8 (top)

39

SLIDE 40

y 0 5 0.6 0.7 tonnes 600 800

(a) (b)

Fishing mortality 0 1 0.2 0.3 0.4 0.5 al catc, in thousand 200 400 600 Years 2008 2010 2012 2014 2016 2018 2020 0.0 0.1 Year 2008 2010 2012 2014 2016 2018 2020 Annua Reference case Low cod case High cod case

Figures 4a,b. 5 Optimal fishing mortalities and catch for 2008-2020 assuming r=2% and the predation reference case (solid lines), low cod case (long dash) and high cod case (short dash)

40

SLIDE 41

nnes 1200

(a)

120 nes

(b)

yield, thousand ton 600 800 1000 60 80 100 ield, thousand tonn 2 4 6 8 10 12 14 Biomass, Annual y 200 400 2 4 6 8 10 12 20 40 Biomass, Annual y Rate of interest 2 4 6 8 10 12 14 Biomass Annual yield Rate of interest 2 4 6 8 10 12 B 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 5a and high cod case, Figure 5b.

41

SLIDE 42

Summary

1. Important extensions of the classical biomass harvesting model:

A) Age-structured, B) Multispecies and C) Spatial models 2 Adding age structure changes all fundamental model properties

2. Adding age structure changes all fundamental model properties

A) Optimal steady state becomes different B) "Optimal extinction" results change C) Steady state stability results differ C) Steady state stability results differ D) Pulse fishing becomes possible in age-structured models E) Optimal yield may decrease in biomass F) MSY b d d t ti F) MSY becomes dependent on gear properties In addition, fishery regulation becomes different (It becomes reasonable to regulate fishing gear in addition to total catch)

42

SLIDE 43

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 d

R. 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
ptimal harvesting of an age-structured schooling fishery, manuscript.

C.J. Walters, A generalized computer simulation model for fish population studies,. Transactions

f 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.

SLIDE 44

New bioeconomics of fisheries and forestry Olli Tahvonen Olli Tahvonen University of Helsinki EAERE Venice Summer School 2011 Section 2, Forestry , y

"FORESTRY IS AMONG THE GREATEST FORESTRY IS AMONG THE GREATEST CHALLENGES IN APPLIED ECOLOGY SINCE IT IS LARGE SCALE ECONOMIC ACTIVITY THAT IS BASED ON UTILIZING LIVING BIOLOGICAL RESOURCES"

44

LIVING BIOLOGICAL RESOURCES" HANSKI ET AL. IN "EKOLOGIA 1998"

SLIDE 45

About 31% of earth total land About 31% of earth total land area is covered by forests This makes 0.6ha per capita

45

Source: FAO

SLIDE 46

Totally 7%

Value of global industrial roundwood removals about $100 billion annually $100 billion annually Trend toward plantations

46

Source: FAO

SLIDE 47

=> limiting the economic analysis to timber production only is a serious

47

Source: FAO limiting the economic analysis to timber production only is a serious restriction

SLIDE 48

Let:

Memory refresh: the classical economic approach to forest resources

Let: annual(market)interest rate planting cost (€) per hectare (ha) stand clearcut value (€) as a function of stand age (per ha) r w , 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 3 rt rt rt r t rt r t

J w e V(t ) e [ w e V(t )] e [ w e V(t )] e

     

       

rt

e

2 r t

e

3 r t

e

( )

discount factor cash flow

48

J w e V(t ) e [ w e V(t )] e [ w e V(t )] e ....           

SLIDE 49

2 3 rt rt rt r t rt r t

J 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

rit rt i

J(t) e w e V ( t )

   

       



By the theorem of geometric series:

1 , 1. 1

i i

q when q q

 

  



1





Let ( 1, 0)

rt

e q when r



  

1 1

rit rt i

e e

   

  



Bare land value can now be given as the Faustmann (1849) formula:

rt

w e V ( t )



 

t

w b V(t )  

r in discrete time:

1

rt

w e V ( t ) J ( t ) . e    

1

t

w b V(t ) J(t ) b   

49

where b=1/(1+r).

SLIDE 50

Some generalizations of the optimal rotation model

Generalized size and age-class models Uneven-aged models Generic rotation model

1

rt rt t

w e V ( t ) max J ( t ) e

 

    Faustmann 1849, Ohlin 1921, Samuelson 1976,... 1 e

Optimal stopping; Stochastic growth Stochastic price Market level age- structured models

Mitra and Wan 1985,...

Imperfect capital markets

Tahvonen et al. 2001,...

Econometrics

f timber supply

Kuuluvainen 1990,...

p

Reed and Clarke 1990,...

Environmental preferences

Hartmann 1976

Optimal rotation and thinnings

Martin and Ek 1981,...

Carbon sequestration

van Kooten et al.1995,... 50 Hartmann 1976,... ,

SLIDE 51

2.1 Market level age-structured models in forestry g 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 represent a 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

SLIDE 52

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

SLIDE 53

The age-structured model with land allocation between forestry, agriculture and old growth Notation and setup:

1 2 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

1

2 1 1 3

n t t st s 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 1

1 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



   

     

4

t

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

   

t

y t

ility from agricultural land is: W y Q y dy, where W'>0,W''  



53

SLIDE 54

Notation and setup, cont.:

 

6

nt 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 2

s 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 1

1 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 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 1 n ,t nt n ,t n,t

This yields : z x x x . Thus, total harvest per period equal

  

  

   

2 1 1 1 1 1 1 n t s st s ,t n nt n ,t n,t s

s: c f x x f x x x

      

    



54

SLIDE 55

The social planners optimization problem:

 

     

1

s ,t

t t t nt x ,s ,...,n,t ,... t

max b U c W y A x



   

     



subject to

   

2 1, 1 1 1, , 1 1 1

, 1 ,

n t s st s t n nt n t n t s n t st s

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

   

     



, 1 1 1

1, 0, 1..., , 0, 1,..., , 1.

s t s st n s s s

x x s n x s n given x

  

     

 

Note: The choice of

1 1

1 1 1 2

s ,t

x , 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.

SLIDE 56

     

 

   

2 1 1 1 1 1 1 1 1

0 1 1

n n t t t nt t s,t st st s ,t n ,t nt n ,t n,t t 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 1 1 1

1

t t t t ,t ,t

L b bf U ' c bW ' y b x  

    

      

       

2 1 2

t

, L b f U ' c bf U ' c bW ' y b s n   



   

               

 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 1 2 3

s t s t t t s ,t st s ,t t n 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 1 1

4 1 5

n,t s,t s,t s,t

L x , x , s ,...,n, x

   

     

   

1 2 

 

5

st st s

, x   

   

 

1 1 1 1 1 1 1 1

1 2 6 1

t s ,t n ,t n ,t nt n ,t n,t n t t s,t s

x , s ,...,n , , x x x , x ,    

       

          



1 1 0 1 0 1

st t

where , 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.

SLIDE 57

Let us restrict the analysis to interior steady states where

n

c , y , x .   

   

1 1 1

m m s s m s

Assumea unique Faustmann rotation m satisfying b f / b b f / b , for s ,...,n. Direct substition shows that    

       

1

7 1 2 1 2

s i s s i

W ' b f U ', s ,...,n solves and as equalities in the interior steady state . Since the rotation b  

  

    



 

1

m i m m

eriod is m z and Thus W ' b f U ' Multiplying by b / b yields 

 





Since the rotation b

 

     

1 1 8 1 1 1

m m m i m m n m m

eriod is m, z and . Thus W ' b f U ' . Multiplying by b / b yields W ' y b f b y x U ' f , b b m U ' f b bW ' bA' 



                



     

1 1 2 1 1

1 3 2 1

n n n n n

U ' f b bW ' bA' Next from : . Eliminating and from written for x yields after some b   

    

    

 

2

8 1 1

n

cancellation bA' W ' b . Applying allows to write this condition in the form b b



   

       

1 1

1 9 1 1 8 5

n m n m n m m n n

A' x b f y x U ' f . b m b Finally, use and eliminate bW ' from the solution of . By condition it must hold that . This  

  

             yields

   

1 10 1 1

m n m n m m

bA' x f b y x U ' f . b b m            

57

SLIDE 58

Together equations (8), (9) and (10) determine an optimal steady state continuum for the 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

   

1 8 1 1

m m n m m

W ' y b f b y x U ' f , b b m            

Present value of marginal ag land equals Faustmann bare land value when timber price equals U' and annual timber production from normal forest equals

1

n m

y x f m  

Faustmann land value for a land unit ready to be clearcut

   

1 9 1 1

n m n m n m m

A' x b f y x U' f , b m b



           

y must 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

   

1

1 10 1 1

m n m n n m m

bA' x f b y x f U ' f . b b m



                  

exceed the value of such land unit if clearcut Otherwise it is optimal to move land from preservation to timber production.

58

SLIDE 59

Numerical example:

 

5 0 5 0 5

0 9 0 5 0 5 3 1 21 0 0 10 15 22 30 40 51 65 82 101123 148 175 203 234 264 293 321 346 346

. . . n

b . ,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 101123 148 175 203 234 264 293 321 346 346 f , , , , , , , , , , , , , , , , , , , , 

Timber price 0.30 0.35 0.40 0.45 Land in agriculture, y(t) 0.10 0.15 0.20 Timber harvesting, c(t) 10 15 20 25 Time 20 40 60 0.20 0.25 Time 20 40 60 L 0.00 0.05 Time 20 40 60 T 5 Land initially old growth

L d i iti ll ld th f t

3.0 0.30

Land initially old growth Land initially in agricuture

Land initially as old growth forest Land initially as one period old forest

nd rent, w'(y) 2.0 2.5 d old growt land, xn 0.15 0.20 0.25 20 40 60 Lan 1.0 1.5 20 40 60 Preserved 0.00 0.05 0.10

59

Time Time

SLIDE 60

Remarks: 1 Detailed analysis of the model is somewhat more complex than shown here

1. Detailed analysis of the model is somewhat more complex than shown here
2. 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 cycle

3. 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

SLIDE 61

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 trees T b i f t t t Two 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 understorey Shade 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

SLIDE 62

The possibilities to apply different forest management systems like p pp y g y even-aged and uneven-aged management depend on biological/ecological factors, economic parameters, preferences and harvesting technology Often ecologists attempt to develop forest management systems based biological factors only =>maximizing volume yield, etc. Resource and environmental economists have studied almost entirely even-aged management =>economics of uneven-aged management is rather purely understood This is rather serious limitation because

1. In some cases uneven-aged management may be economically superior to

even-aged management 2 U d b f d d i l d bi di i

2. 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

SLIDE 63

Even aged management g

≈80 yrs

63

SLIDE 64

Mixed species uneven aged uneven-aged stand

64

SLIDE 65

A life cycle graph for a size-classified population:

2



3



4



( )

t

x 

1



2



3



3 4

1t

x

2t

x

3t

x

4t

x

1 2 3 4 F i l

1



2



3



4



1

h

2

h

3

h

4

h

1 2 3 4 1 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 class the share of trees that remain in size class s         

Remark: in addition of trees the size-structured model is suitable

1 1

s s s s

the share of trees that remain in size class s the share of trees that die in size class s numb          er of seedlings( or seeds ) per tree in size class s x total number of seeds or seedlings

for fish and in situations where the perfect selectivity assumption is possible

65

s

f g the recruitment or " ingrowth" function h the number of trees harvested from size class s 

SLIDE 66

   

1 x x h      x The sizeclass matrix model can be written as aset 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

   

, , 1

4 ,

n t st s s

H h f



 

t

where H is total harvest in wei , 1,...,

s

ght or volume units f s n denotes the size of individuals in size class s  , , , ( ).

s

f f in 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

SLIDE 67

Using matrix notation the model takes the form:

1 t t t t  

 x x G q or

 

1 1 1 1 1 2 1 ,t t t t

x x h x x h    



                            x

2 1 1 1 2 2 2 3 3 1 1 ,t t t t n n t

x x h h h     

 

                                                                   .

1 1 1 1 n n ,t n,t n n nt n

x x h   

   

                        

t

G

t

q

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 decline Model 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)

SLIDE 68

A generic model specification for optimal harvesting of size-structured population:

 

   

, 1,..., , 0,1,... 1 1 1 1 1

max ,

st

t t t h s n t t t t t

U H b subject to x x h  

   

  



x

 

1, 1 1 1 1 1, 1 1 1, 1 1

, , 1,..., 1, ,

t t t t s t s st s s t s t n t st s s

x x h x x x h s n H h f h    

      

        



x 0, 0, , 1,..., .

st st s

h x x s n are given    68

SLIDE 69

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 8 109 9 7 0 3 x x B N         

Note the density dependence;

1, 1 1 2, 1 2 3, 1 3

0.8 109 9.7 0.3 0.04 0.9 , 0.02 0.9

t t t t t t t t

x x B N x x x x

  

                                           

Note the density dependence; φ is decresing function of the 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 x and x x x B          area and to total number of trees

t

N

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 h x x x h

 

       

areas of the trees in the stand. Units: m3

3, 1 2 3 3

0.02 0.9 .

t t t t

x x x h

 

  Assuming no harvest, it is possible to solve the steady state by assuming all variables are constant in time in the differential system above. This yields the steady state:

1 2 3

400, 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       ฀ ฀ ฀

SLIDE 70

Stand development without harvest and two initial states

600 800

umber of trees

400 600

Nu

200

Time

20 40 60 80 100

Time

size class 1 size class 2 size class 3

70

SLIDE 71

Based on this growth model we obtain the following economic Based on this growth model we obtain the following economic

ptimization problem:

 

 

1 1 2 2 3 3 1 2 3 0 1

max

t t t t h s t

p h p h p h b



 



 

, 1,2,3, 0,1,... 1, 1 1 1 2 1 1 2 2

109 9.7 0.3 0.8 , 0.04 0.9 ,

st

h s t t t t t t t t t t t

subject to x B N x h x 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 h x x x h x x x B x x x

 

         

1 2 3 1

,

t t t t t t

N x  

2 3 ,

0, 1,2,3 0, 1,2,3.

t t st st

x x x s h s     

1 2 3

0.3, 8, 20. The market prices of trees are: p p p    71

SLIDE 72

#Bioeconomics 2011, Olli Tahvonen, #model file #B i d Mi hi (1980) d t

AMPL code for Buongiorno and Michie (1980)

#model file param T; param ac; param n; param p {s in 1..n}; #Buongiorno and Michie (1980) data. #data file param T:=100; param ac:=1;#0.1; param r:=0;#0.1; param y {s in 1..n}; #basal area per tree param α {s in 1..n}; param β {s in 1..n}; param r; param b=1/(1+r); p param n:=3; param y:= 1 0.02# 2 0.06 3 0 13; param b 1/(1 r); param x0 {s in 1..n}; #initial state var 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 0.04 2 0.02 3 0; β var X {t in 0..T}=sum{s in 1..n} x[s,t];#total no. of trees var Y {t in 0..T}=sum{s in 1..n} y[s]*x[s,t]; #total basal area var φ {t in 0..T}; maximize objective: param β:= 1 0.8 2 0.9 3 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 8 3 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}: 1 840 2 234 3 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];

SLIDE 73

500

The features of optimal solution

Revenues

200 300 400

assuming r=0: At optimal steady state all trees are cut when they reach size class 2 (at the end of

R

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

Harvest from size class 2

20 40 60 80

This implies by t

2 2 t t

x x  he equation for that at the steady state

20

rees in

1000 1200

Number of tr size class 1

200 400 600 800

Time in 5yrs periods

20 40 60 80

73

initial state: [100, 45, 5] Initial state: [840, 234, 14]

SLIDE 74

More about density dependence in forestry models

In Buongiorno and Michie (1980) density dependence exists only in the regeneration function However, the growth of larger trees may also depend on stand density This 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 d

2 4 1

( / 2) 10 , 3.1415926... ( / 2)

t st s s s

y x d d s x d   



 



where is the tree diameter insizeclass 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 4

10

n 



( / 2)

st s st

x d y  

2 4 1

10 ( / 2) 10 , 1,..., 2

st s k s

x 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 st

d y y s n d 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

SLIDE 75

A more general transition matrix model

1 { 1 0 1 }

max ( ) ( ) , (the objective function)

t t t h s n t

V R C b



 



x

A more general transition matrix model

{ , 1,..., , 0,1,...}

st

h s n t t   



1 1 2 2 1 1 1 2 2

( ) , (annual gross revenues, sawntimber price, sawntimber vol per tree, same for pulp)

n t st s s s s s

R h p p p p    



 



Objective function revenues, cost

1, 2 1 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 d h h h     h d d h t

1, 1 1 1 1 1

( ) [1 ( ) ( )] , (development of smallest size class, regeneration, t iti t l

t t t t t t

x x h    

 

    x x x t lit )

1

transition, natural m  

rtality)

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 t

x x x h s n   

     

       x x x

1 1 1

( ) [1 ( )] . (development of largest size class)

t t t t t t

x x x h       x x

Nonlinear size structured model

, 1 1 1,

( ) [1 ( )] . (development of largest size class)

n t n t n t n t nt nt

x x x h  

  

 x x 0, 0, 1,..., , 0,1,..., , 1,..., (nonnegativity constraints)

given. (initial state)

st st s

h x s n t x s n     

Technical

,2 ,3 ,..., when (additional restriction for taking into account that harvesting can be done every kth period only)

st

h t k k k   where the value of k is a positive integer.

constraints

75

SLIDE 76

Empirical estimation results for the transition matrix model Norway spruce, 93 sample plots, Central Finland, y p p p Oxalis-Myrtillus (OMT) and Myrtillus (MT) forest site types Time step three years

2.1368 0.104 0.107

1, (regeneration, total number of trees)

t t

N y t

e N 

 

 

1

 

1 3.752 2.560 0.296 0.849ln( ) 0.035

1 , 1,...,10 (transition)

s s t st

d d y y s

e s 

        

  

 

 

1 3.606 0.075 0.997ln( )

1 1 10 (mortality)

st s

y d   



 

 

( )

1 , 1,...,10 (mortality)

st s

y s

e s    

2 2

39691 , 1,...,10 1000 25683 37785 (length of trees, m)

s s

d h s d d    

76

SLIDE 77

diameter, cm 7 11 15 19 23 27 31 35 39 43 sawn timber 0.14136 0.29572 0.45456 0.66913 0.88761 1.12891 1.39180 pulp d 0.01189 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

3

m 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 pulpvol t t t

C H H     

1

22.386 0.50001 0.59 2.1001366 300, 1000 85.621

n st s t s s t

h vol N vol N



         



26 350495 2 82183045 25 701440 3 33144

cc sawvol pulpvol

C H H     

1

26.350495 2.82183045 25.701440 3.33144 146.17 0.44472 0.94 2.1001366 300, 1000 862.05

t t t n st s t s s t

C H H h vol N vol N



              



where denotes thinning cost and clearcut cost, and a

th cc sawvol pulpvol t t t t

C C H H re the total volumes of sawlogsand pulpwood yieldsper cutting and is thetotal (commercial) volume of a stem from size class .

s

vol s

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

SLIDE 78

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 g

3. How even-aged and uneven-aged management systems can be compared?

78

SLIDE 79

1. How volume maximization solution looks like?

40

ree years, m3

30

Cuttings per thr

10 20 8 10 12 14 16

C Basal area before cuttings, m2 Steady state Initial state/initial optimal cuttings Figure 1. Optimal development of basal area and cuttings toward the MSY steady state

79

SLIDE 80

150 200 class 100 150

f trees per size c

123456 10 20 30 40 50 50 Number 123456789 10 60 Time periods, in three years intervals Size classes

Figure 2. Development of the size class distribution over time Figure 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

SLIDE 81

2. How the economically optimal solution looks like?

60 120 180 240

Basal area after and before harvest, m2/ha

5 10 15 20 25 30 60 90 120 150 180 2 4 6 8 10 12 14 16 60 120 180 240

mber of trees after d before harvest per ha

300 400 500 600 700 800 30 60 90 120 150 180 200 300 400 500 600 700 800 60 120 180 240

Num and

300

lume after and

efore harvest, m3/ha

20 40 60 80 100 120 140 160 180 30 60 90 120 150 180 200 20 40 60 80 100 120 140 60 120 180 240

Vo be

tal yield, m3,

wlogs yield, m3 r 15 years/ha

40 60 80 100 120 140 30 60 90 120 150 180 50 60 70 80 90 60 120 180 240

To saw per

20 40

revenues, € ues net of cutting er 15 years/ha

2000 3000 4000 5000 6000 30 60 90 120 150 180 40 2000 2500 3000 3500 4000 60 120 180 240

Gross r revenu cost pe

1000 2000

h, number of r three years/ha

30 40 50 60 30 60 90 120 150 180 1500 2000 30 35 40 45 50 55

81

60 120 180 240

Ingrowth trees per

10 20

Time, years Time, years

30 60 90 120 150 180 20 25 30

Interest rate 0% Interest rate 3%

SLIDE 82

Optimal steady state size distribution and selection of harvested trees Interest rate 0 or 3%, cutting periods 15 and 12 years Interest rate 0 or 3%, cutting periods 15 and 12 years

120

(a) Zero interest rate

mber of trees/ha

40 60 80 100 120

Diameter class

7 11 15 19 23 27 31 35 39 43

Num

20

(b) Th t i t t t

r of trees/ha

40 60 80 100

(b) Three percent interest rate

Diameter classes

7 11 15 19 23 27 31 35 39 43

Number

20 40

Diameter classes Harvested trees

82

SLIDE 83

How even-aged and uneven-aged management systems can be compared?

3000 3500 rtificial n € 2000 2500 3000

Uneven-aged optimal

st from ar

egeneration 1000 1500 2000

Valid area

C re 500 1000

Even-aged optimal Break even curve

Interest % 1 2 3 4 5 6 Interpretation: Even-aged management requires the regeneration investment after the clearcut. This competes with the natural regeneration that may produce lower number of seedlings but is free of cost. When this

83

p g investment cost and the interest rate is high uneven-aged management becomes always optimal

SLIDE 84

Summary The generic Faustmann model is brilliant but as such too simple for almost any purposes Market 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 years This 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 management The economist's argument: the proof is missing New resource economic studies have shown that the silviculturalists view is unwarranted New resource economic studies have shown that the silviculturalists view is unwarranted =>The Finnish ministry of Agriculture and Forestry has initiated a change toward official acceptance of uneven-aged forestry and general liberalization of forest policy

84

SLIDE 85

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

SLIDE 86

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 86 513 530 Agricultural 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

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