Steady State Analysis of the Finnish Forest Sector

W O R K I N G P A P E R

STEADY STATE ANALYSIS OF THE FINNISH FOREST SECTOR

Markku Kallio M a r g a r e t a S o i s m a a

J u n e 1983 UP-83-53

l n t e r n a l i o n a l l n s t ~ t u l e for Appl~ed Systems Analysis

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

STEADY STATE ANALYSIS OF THE FINNISH FOREST SECTOR

M a r k k u K a l l i o M a r g a r e t a S o i s m a a

J u n e 1983 WP-83-53

W o r k i n g P a p e r s a r e i n t e r i m r e p o r t s o n work o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s a n d h a v e r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o r

o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e - s e n t t h o s e o f t h e I n s t i t u t e o r o f i t s N a t i o n a l Member O r g a n i z a t i o n s .

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A - 2 3 6 1 L a x e n b u r g , A u s t r i a

FOREWORD

The o b j e c t i v e of t h e F o r e s t S e c t o r P r o j e c t a t IIASA i s t o s t u d y long- t e r m development a l t e r n a t i v e s f o r t h e f o r e s t s e c t o r on a g l o b a l b a s i s . The emphasis i n t h e P r o j e c t i s on i s s u e s o f major r e l e v a n c e t o i n d u s t r i a l and governmental p o l i c y makers i n d i f f e r e n t r e g i o n s of t h e world who a r e respon- s i b l e f o r f o r e s t r y p o l i c y , f o r e s t i n d u s t r i a l s t r a t e g y , and r e l a t e d t r a d e p o l i c i e s .

The key e l e m e n t s of s t r u c t u r a l change i n t h e f o r e s t i n d u s t r y a r e r e l a t e d t o a v a r i e t y o f i s s u e s concerning demand, s u p p l y , and i n t e r n a t i o n a l t r a d e of m o d p r o d u c t s . Such i s s u e s i n c l u d e t h e development of t h e g l o b a l economy and p o p u l a t i o n , new wood p r o d u c t s and s u b s t i t u t i o n f o r wood p r o d u c t s , f u t u r e s u p p l y of roundwood and a l t e r n a t i v e f i b e r s o u r c e s , t e c h n o l o g y develop- ment f o r f o r e s t r y and i n d u s t r y , p o l l u t i o n r e g u l a t i o n s , c o s t c o m p e t i t i v e n e s s ,

t a r i f f s and n o n - t a r i f f t r a d e b a r r i e r s , e t c . The aim of t h e P r o j e c t i s t o a n a l y z e t h e consequences of f u t u r e e x p e c t a t i o n s and assumptions c o n c e r n i n g such s u b s t a n t i v e i s s u e s .

The r e s e a r c h program of t h e P r o j e c t i n c l u d e s a n a g g r e g a t e d a n a l y s i s of long-term development of i n t e r n a t i o n a l t r a d e i n wood prodv-cts

,

and t h e r e b y a n a l y s i s o f t h e development of wood r e s o u r c e s , f o r e s ' t i n d u s t r i a l p r o d u c t i o n and demand i n d i f f e r e n t world r e g i o n s . The o t h e r main r e s e a r c h a c t i v i t y i s a d e t a i l e d a n a l y s i s of t h e f o r e s t s e c t o r i n i n d u s t r i a l c o u n t r i e s . Research on t h e s e m u t u a l l y s u p p o r t i n g t o p i c s i s c a r r i e d o u t s i m u l t a n e o u s l y i n c o l l a b o r a - t i o n between IIASA and t h e c o l l a b o r a t i n g i n s t i t u t i o n s of t h e P r o j e c t .

T h i s p a p e r i s a s p e c i f i c s t u d y of t h e F i n n i s h f o r e s t s e c t o r . I t s g o a l i s t o a n a l y z e one of t h e main c o s t f a c t o r s , t h e wood c o s t , and t h e e f f e c t o f t h i s i n t e r n a l l y p r i c e d f a c t o r on t h e c o m p e t i t i v e n e s s of t h e F i n n i s h f o r e s t i n d u s t r y .

Mar kku Kal 1 i o P r o j e c t Leader

F o r e s t S e c t o r P r o j e c t

ABSTRACT

D u r i n g t h e r e c e n t y e a r s t h e t o t a l c o s t of round wood f o r t h e Finnish f o r e s t industry has b e e n in t h e o r d e r of US$ 1.5 billion annually. T h e s h a r e of s t u m p a g e p r i c e r e p r e s e n t s roughly o n e half w h e r e a s harvesting, t r a n s p o r t a t i o n etc a c c o u n t f o r t h e r e s t . T h e purpose of t h i s s t u d y i s t o i n v e s t i g a t e long t e r m equilibrium p r i c e s f o r wood (and t h e r e b y t o t a l round wood c o s t s ) u n d e r various conditions of world m a r k e t of wood products.

In t h e f i r s t p a r t of t h i s p a p e r a ( d i s c r e t e t i m e ) d y n a m i c l i n e a r model f o r t h e f o r e s t s e c t o r is discussed. T h e s t e a d y state version of i t is a n a l y z e d in m o r e d e t a i l . A n a p p l i c a t i o n of t h e s t e a d y state f o r e s t r y m o d e l is c a r r i e d o u t f o r t h e F i n n i s h forests. As a r e s u l t , a l t e r n a t i v e sustained yield solutions f o r t h e Finnish f o r e s t s a r e obtained.

In t h e a n a l y s i s of t h e second p a r t , a s t e a d y state s e c t o r i a l model is a d o p t e d t o c a r r y o u t a S t a c k e l b e r g equilibrium analysis f o r t h e round wood m a r k e t . F u r t h e r e l a b o r a t i o n a p p e a r e d n e c e s s a r y until t h e s t e a d y state m o d e l b e c a m e s u i t a b l e f o r t h i s g a m e t h e o r e t i c analysis. This e l a b o r a t i o n involves definitions of o b j e c t i v e f u n c t i o n s of t h e key p a r t i e s ( t h e f o r e s t r y and t h e industry) in t h e f o r e s t s e c t o r g a m e . A d e m a n d f u n c t i o n of c o n s t a n t p r i c e e l a s t i c i t y i s a d o p t e d f o r wood pro- ducts.

Table of Contents

1 Introduction 1

2 A Dynamic Linear Model f o r t h e F o r e s t r y and Wood Processing Industry 2

2.1 F o r e s t r y 2

2.2 Wood Processing Industry 3

3 Sustained Yield in F o r e s t r y

3.1 T h e S t e a d y S t a t e Formulation 3.2 Application t o t h e Finnish F o r e s t r y

4 A Steady S t a t e Model of t h e F o r e s t Industries 13

5 A Steady S t a t e Model of t h e F o r e s t S e c t o r 15

6 A S t a c k e l b e r g G a m e 6.1 T h e P r o f i t Functions

6.2 Demand Functions and O p t i m a l P r i c e s f o r Wood P r o d u c t s 6.3 T h e P r o f i t Maximization Problem f o r F o r e s t r y

7 Equilibrium Solutions f o r Finland

7.1 T h e Single P r o d u c t

-

Single T i m b e r A s s o r t m e n t C a s e 7.2 T h e Seven P r o d u c t s

-

T w o T i m b e r A s s o r t m e n t s C a s e

8 Summary and Conclusions 2 8

APPENDIX 1. 30

APPENDIX 2. 3 1

REFERENCES 33

STEADY STATE ANALYSIS O F THE FINNISH F O R E S T S E C T O R Markku Kallio a n d M a r g a r e t a S o i s m a a

1 Introduction

D u r i n g t h e l a s t f e w y e a r s a growing r e s e a r c h e f f o r t h a s b e e n d i r e c t e d t o w a r d s ( r e n e w a b l e ) n a t u r a l resources. Since t h e prosperity of many nations i s d e p e n d e n t o n a sensible e x p l o i t a t i o n of t h e s e r e s o u r c e s t h e s i g n i f i c a n c e of studies dealing w i t h s u c h problems b e c o m e s g r e a t . F o r t h e Finnish e c o n o m y f o r e s t s r e p r e s e n t t h e m o s t i m p o r t a n t n a t i o n a l resource. N o t only as s u c h but b e c a u s e a n e n t i r e l i n e of production ranging f r o m pulp and sawn goods t o paper, f u r n i t u r e a n d p r e f a b r i c a t e d houses, i s b a s e d on wood n o t t o mention industry producing machi- nery for f o r e s t r y and wood processing. This also e m p h a s i z e s t h e i m p o r t a n c e of t h e f o r e s t s e c t o r , which includes b o t h f o r e s t r y and f o r e s t b a s e d industries, f o r emplo- y m e n t and foreign t r a d e .

In the past t h e t o t a l c o s t of round wood f o r t h e Finnish f o r e s t industry h a s b e e n of t h e o r d e r of US$ 1.5 billion annually. T h e s h a r e of s t u m p a g e p r i c e r e p r e s e n t s roughly o n e half w h e r e a s harvesting, t r a n s p o r t a t i o n , etc a c c o u n t f o r t h e r e s t . T h e p u r p o s e of t h i s s t u d y i s t o i n v e s t i g a t e long r a n g e equilibrium p r i c e s f o r w o o d ( a n d t h e r e b y t o t a l round wood c o s t ) under various conditions f o r world m a r k e t of wood products.

In Section 2 w e will p r e s e n t a d y n a m i c l i n e a r m o d e l f o r t h e e n t i r e f o r e s t s e c t o r . In S e c t i o n 3 w e t a k e f o r e s t r y s e p a r a t e l y a n d d e t e r m i n e t h e o p t i m a l h a r v e s t i n g p o l i c i e s in a s t e a d y state. S e c t i o n 4 d e a l s w i t h a s t e a d y state model f o r t h e forest industries. In S e c t i o n 5 w e c o m b i n e t h e s e t w o p a r t s a n d f o r m u l a t e a s t e a d y s t a t e model f o r t h e f o r e s t sector. In S e c t i o n 6 w e s u p p l e m e n t t h e model of S e c t i o n 5 t o m a k e i t applicable f o r solving t h e long r a n g e equilibrium wood p r i c e s as solution t o a S t a c k e l b e r g game. In S e c t i o n 7 w e p r e s e n t n u m e r i c a l results f r o m e x p e r i m e n t s with Finnish d a t a . Finally, S e c t i o n 8 s t a n d s f o r s u m m a r y a n d conclusions.

2 A Dynamic Linear Model for the Forestry and Wood Processing Industry

We s h a l l c o n s i d e r f i r s t t h e i n t e g r a t e d and d y n a m i c s y s t e m of wood supply and w o o d processing; i e f o r e s t r y a n d f o r e s t industry. T h e model h a s b e e n a d o p t e d from Kallio, Propoi, a n d S e p p a l a

121.

T h e discussion begins with t h e f o r e s t r y p a r t d e s c r i b i n g t h e g r o w t h of t h e f o r e s t given harvesting a n d planting a c t i v i t i e s , as w e l l as land availability o v e r time. T h e wood processing p a r t c o n s i s t s of a n input -output model describing t h e production process as well as of p r o d u c t i o n c a p a c i t y and f i n a n c i a l r e s o u r c e considerations. E a c h p a r t i s a d i s c r e t e t i m e l i n e a r m o d e l describing i t s o b j e c t o v e r a c h o s e n t i m e interval.

2.1 F o r e s t r y

L e t w ( t ) b e a v e c t o r d e t e r m i n i g t h e n u m b e r of various t r e e s p e c i e s (say pine, s p r u c e a n d birch) in d i f f e r e n t a g e c a t e g o r i e s at t h e beginning of t i m e period t.

We d e f i n e a s q u a r e t r a n s i t i o n (growth) m a t r i x Q s o t h a t Qw(t) i s t h e n u m b e r of t r e e s a t t h e beginning of period t + l given t h a t nothing i s h a r v e s t e d o r planted.

T h u s , m a t r i x Q d e s c r i b e s aging a n d n a t u r a l d e a t h of t h e trees. L e t p(t) a n d h(t) b e v e c t o r s f o r levels of d i f f e r e n t kinds of planting a n d h a r v e s t i n g a c t i v i t i e s , r e s p e c t i v e l y (eg planting of d i f f e r e n t species, t e r m i n a l harvesting, thinning, e t c ) , a n d l e t t h e m a t r i c e s P a n d H be d e f i n e d s o t h a t P p ( t ) a n d -Hh(t) a r e t h e i n c r e - m e n t a l i n c r e a s e in t h e t r e e q u a n t i n t y c a u s e d by t h e planting and h a r v e s t i n g a c t i v i - t i e s . Then, f o r t h e state v e c t o r w ( t ) of t h e n u m b e r of t r e e s in d i f f e r e n t a g e c a t e g o r i e s w e h a v e t h e following equation:

Planting is r e s t r i c t e d through l a n d availability. We m a y f o r m u l a t e t h e land c o n s t - r a i n t s o t h a t t h e t o t a l s t e m volume of t r e e s in f o r e s t s c a n n o t e x c e e d a given v o l u m e L ( t ) during t. Thus, if W i s a v e c t o r of s t e m volume p e r t r e e f o r d i f f e - r e n t s p e c i e s in various a g e groups, t h e n t h e land availability r e s t r i c t i o n may b e s t a t e d as

G i v e n t h e l e v e l of h a r v e s t i n g a c t i v i t y h(t), t h e r e i s a minimum r e q u i r e m e n t f o r t h e planting a c t i v i t y p(t) ( r e q u i r e d by t h e law, f o r i n s t a n c e ) as follows:

w h e r e N is t h e m a t r i x t r a n s f o r m i n g t h e l e v e l of h a r v e s t i n g a c t i v i t i e s t o planting r e q u i r e m e n t s .

In t h i s s i m p l e f o r m u l a t i o n w e shall l e a v e o u t o t h e r r e s t r i c t i o n s , s u c h as harves- t i n g l a b o r a n d c a p a c i t y . Finally, t h e wood supply y(t), given t h e l e v e l of harves- t i n g a c t i v i t i e s h(t), is given f o r period t as

I

H e r e t h e m a t r i x S = (Sij) t r a n s f o r m s a t r e e of a c e r t a i n s p e c i e s a n d a g e combi-

i

n a t i o n j i n t o a volume of t y p e i of t i m b e r a s s o r t m e n t ( e g pine log, s p r u c e pulp-

~

wood, etch I

2.2 Wood Processing Industry

F o r t h e industrial side, l e t x ( t ) b e t h e v e c t o r of production a c t i v i t i e s f o r period t (such as t h e production of sawn goods, panels, pulp, paper, a n d c o n v e r t e d wood products), a n d l e t U' b e t h e m a t r i x of wood u s a g e per u n i t of production a c t i v i -

ty.

The wood d e m a n d f o r period t is t h e n given by U1x(t). I t c a n n o t e x c e e d wood supply y(t):

N o t e t h a t t h e m a t r i x U' m a y a l s o h a v e n e g a t i v e e l e m e n t s . F o r i n s t a n c e , sawmill a c t i v i t y c o n s u m e s logs b u t produces pulpwood as a residual.

Let A be a n input-output m a t r i x s o t h a t (I

-

A)x(t) i s t h e v e c t o r of n e t producti- on. If D(t) i s t h e corresponding ( m a x i m u m ) e x t e r n a l d e m a n d , w e r e q u i r e

P r o d u c t i o n i s r e s t r i c t e d by t h e c a p a c i t y c ( t ) available:

T h e v e c t o r c ( t ) in t u r n h a s t o s a t i s f y t h e s t a t e e q u a t i o n

w h e r e g is a diagonal m a t r i x a c c o u n t i n g f o r (physical) d e p r e c i a t i o n and v(t) i s t h e i n c r e m e n t f r o m i n v e s t m e n t s during period t. T h e v e c t o r v(t) of i n v e s t m e n t a c t i v i t i e s i s r e s t r i c t e d through f i n a n c i a l considerations. T o s p e c i f y this, l e t m(t) b e t h e s t a t e v a r i a b l e f o r c a s h at t h e beginning of period t. L e t G(t) b e t h e v e c t o r of s a l e s r e v e n u e l e s s d i r e c t production c o s t s p e r unit of production inclu- ding, f o r i n s t a n c e , wood, energy, a n d d i r e c t labor costs. L e t F ( t ) b e t h e v e c t o r of m o n e t a r y fixed c o s t s p e r unit of c a p a c i t y , l e t I(t) b e t h e a m o u n t of e x t e r n a l f i n a n c i n g e m p l o y e d by t h e industry at t h e beginning of period t, l e t s b e t h e i n t e r e s t r a t e f o r e x t e r n a l financing per period, l e t I+(t) b e n e w loans t a k e n du- ring period t, l e t I-(t) b e l o a n r e p a y m e n t s during t, a n d l e t E(t) b e t h e v e c t o r of cash e x p e n d i t u r e p e r u n i t of i n c r e a s e in t h e production c a p a c i t y . Then, t h e state e q u a t i o n f o r c a s h may b e w r i t t e n as

Finally, f o r t h e industrial model, w e may w r i t e t h e state e q u a t i o n f o r e x t e r n a l financing as follows:

O D O O O O

IU Vt V I 11 LI I1

C

u

- - -

I ^C I

w

-

u Id I

h cn

I

u u L

I

-

I

-

³ _{I I a}

I u

w

p.

- ₊

C, w

-

h

- ₊

C, u

E

h

- ₊

4J w

U

h

4J w I

-

h C, w

+-

h C, w

-

h

C, w

E

h C, w >

h C, w

U

h C, w

X

-

h C, w

h

h

C, w

C

h C, u

P

h C, u

3

h

- ₊

C, w

3

0 0 - 1 0

II A! V I I!

u

I Z I V)

n I

-

0 3 I

u

( 2 . 1 0 ) l ( t + L > = l ( t )

-

I - ( t ) + l + ( t >

.

F i g u r e I p r e s e n t s t h e s t r u c t u r e of t h e c o n s t r a i n t m a t r i x of t h e f o r e s t s e c t o r model f o r period t.

3 Sustained Yield in Forestry

In t h e previous s e c t i o n w e p r e s e n t e d a d y n a m i c l i n e a r programming m o d e l e n c o m - p a s s i n g b o t h f o r e s t r y a n d f o r e s t b a s e d industries. In t h i s s e c t i o n w e f o c u s our a t t e n t i o n solely o n f o r e s t r y . W e p r e s e n t f o r e s t r y in a s t e a d y state by a s s u m i n g t h a t o n e period follows a n o t h e r o n e without changes. We shall i n v e s t i g a t e a l t e r n a - t i v e s u s t a i n e d a n d e f f i c i e n t t i m b e r yields in various t i m b e r a s s o r t m e n t s . We a l s o p r e s e n t a n a p p l i c a t i o n to t h e Finnish f o r e s t r y .

3.1 T h e S t e a d y S t a t e F o r m u l a t i o n

In S e c t i o n 2.1 w e p r e s e n t e d a g e n e r a l d y n a m i c f o r m u l a t i o n f o r forestry. In t h i s s e c t i o n w e d e a l with a s t e a d y state case of t h i s model.

We consider a f o r e s t land with a single t r e e s p e c i e s a n d with uniform soil, c l i m a - te, etc conditions. W e assign t h e t r e e s t o a g e groups a, f o r a = 1, 2,

...,

N. L e t d b e a t i m e i n t e r v a l ; e g 5 years. A t r e e belongs t o a g e g r o u p a if i t s a g e i s in t h e i n t e r v a l [ ( a - l ) d , a d ) f o r a l l a < N. T r e e s which h a v e a n a g e of at l e a s t (N-l)d belong t o a g e g r o u p N. We consider a d i s c r e t e t i m e s t e a d y s t a t e f o r m u l a - tion of t h e f o r e s t , w h e r e e a c h t i m e period i s also a n i n t e r v a l of d. L e t p b e t h e n u m b e r of t r e e s e n t e r i n g t h e f i r s t a g e g r o u p during e a c h period ( e g through p l a n t i n g o r n a t u r a l regeneration), a n d l e t w(a) b e t h e n u m b e r of t r e e s in a g e g r o u p a at t h e beginning of e a c h t i m e period, f o r I

5

a

-

< N (cf (2.1)). L e t h(a) b e t h e n u m b e r of t r e e s h a r v e s t e d during e a c h period f r o m a g e g r o u p a. In t h i s case, w e a s s u m e t h a t t h e h a r v e s t i n g a c t i v i t i e s e q u a l t h e number of t r e e s harvested f r o m e a c h a g e group during e a c h period. We d e n o t e by Qa t h e r a t i o of t r e e s p r o c e e d i n g f r o m a g e group a t o g r o u p a + l in o n e period given t h a t n o harvesting occurs. Without loss of g e n e r a l i t y w e a s s u m e 0

5

_Qa

5

^1, f o r a l l a.

F a c t o r s (1-Qa) a c c o u n t f o r n a t u r a l d e a t h of t r e e s , f o r e s t fires, etc as w e l l as f o r t h i n n i n g of f o r e s t s in a g e g r o u p a. T h e state e q u a t i o n s f o r f o r e s t r y in a s t e a d y state m a y t h e n b e w r i t t e n as follows (cf (2.1)):

w h e r e w e d e f i n e w ( N + l ) = 0.

The land c o n s t r a i n t p r e v e n t s e x c e s s i v e planting (cf (2.2)). L e t Wa b e t h e a m o u n t o f l a n d c o n s u m e d by e a c h t r e e in a g e g r o u p a, 1

5

a

5

N, a n d l e t L b e t h e total a m o u n t of land a v a i l a b l e in t h e f o r e s t s . A l t e r n a t i v e l y , t h e s p a c e l i m i t a t i o n s may b e t a k e n i n t o a c c o u n t d e n o t i n g by Wa t h e v o l u m e of wood p e r t r e e in a g e g r o u p a a n d by L t h e t o t a l possible v o l u m e of wood in t h e forests. In e i t h e r case t h e l a n d c o n s t r a i n t i s g i v e n as

As a p e r f o r m a n c e index f o r f o r e s t r y w e consider t h e physical wood supply. (Expe- r i e n c e shows t h a t when w e m a x i m i z e t h e physical wood supply w e usually g e t a policy which a l s o m e e t s o t h e r i m p o r t a n t r e q u i r e m e n t s , s u c h as p r e s e r v i n g t h e w a t e r s h e d . ) T h e t i m b e r a s s o r t m e n t s v a r y in v a l u e ( e g log, pulpwood, f u e l wood).

L e t j I , 2,

...I

r e f e r t o d i f f e r e n t t i m b e r a s s o r t m e n t s . Accordingly, l e t eaj b e t h e y i e l d (in m 3 / t r e e ) of t i m b e r a s s o r t m e n t j when a t r e e in a g e g r o u p a is h a r v e s t e d , a n d l e t gaj b e t h e yield p e r t r e e in a g e g r o u p a r e s u l t i n g f r o m thin-

- ning activities. A s s t a t e d e a r l i e r , o u r o b j e c t i v e i s to find a n e f f i c i e n t t i m b e r yield u s i n g t h e yields of t i m b e r a s s o r t m e n t s as c r i t e r i a . L e t ea a n d ga b e c o n v e x c o m b i n a t i o n s ( w e i g h t e d sums) of t h e c o e f f i c i e n t s eaj a n d gaj, r e s p e c t i v e l y . T h e objective i s t o m a x i m i z e t h e w e i g h t e d s u m of t h e yields of v a r i o u s t i m b e r a s s o r t - m e n t s a n d i t i s given as

T h e w e i g h t s t o b e used m a y b e proportional t o t h e m a r k e t p r i c e s of t h e t i m b e r a s s o r t m e n t s . Also o t h e r w e i g h t s m a y b e c o n s i d e r e d f o r s t u d y i n g e f f i c i e n t yields

(see Section 3.2 below). T h e f o r e s t r y problem, d e n o t e d by (F), i s t o find nonnega- tive scalars h(a) a n d w(a), f o r e a c h a, which m a x i m i z e (3.4) s u b j e c t t o (3.1)-(3.3).

The following r e s u l t is used t o d e r i v e a n o p t i m a l solution t o t h i s linear program:

P r o p o s i t i o n : F o r a n o p t i m a l solution of t h e f o r e s t r y problem (F) t h e r e is a n a g e g r o u p A s u c h t h a t h(a) = 0, f o r all a

1

A, a n d w(a) = 0, f o r a l l a > A.

Thus in t h e o p t i m a l h a r v e s t i n g schedule, a l l t r e e s a r e h a r v e s t e d , c l e a r c u t (besides t h i n n i n g a c t i v i t i e s ) if a n d only if t h e y r e a c h a g e g r o u p A. T h e r e f o r e , t h e r e a r e n o t r e e s in a g e groups higher t h a n A. Problem (F) m a y t h e n b e solved, f o r i n s t a n c e , c h e c k i n g a l l a l t e r n a t i v e h a r v e s t i n g policies of t h i s type.

-

For a proof of t h e Proposition, see Appendix I.

\

We c o n s i d e r now a p a r t i c u l a r policy a = A w h e r e t r e e s a r e h a r v e s t e d in an a g e g r o u p A. Then, a c c o r d i n g to (3.2),

f o r a

5

A

I

^{O f o r a > A.}

F o r t h e corresponding l e v e l of planting p~ t h e land c o n s t r a i n t (3.3) yields:

A

T h e number of t r e e s h a r v e s t e d , when policy A i s applied, i s given as

The e f f i c i e n t yield of t i m b e r a s s o r t m e n t j f r o m c l e a r c u t t i n g when policy A i s applied. is given as

As f o r c u t t i n g and thinning, t h e e f f i c i e n t yield of t i m b e r a s s o r t m e n t j under policy A is

A

3.2 Application t o t h e Finnish F o r e s t r y

W e will now apply this approach t o t h e f o r e s t r y in Finland. L e t t h e a g e group i n t e r v a l d be 5 y e a r s and N = 21 (so t h a t t h e o l d e s t group includes t r e e s of a t least 100 y e a r s old). W e consider t w o t i m b e r assortments: pulpwood ( j = l ) and log (j=2).

T a b l e 1 gives e s t i m a t e s f o r t h e transition probabilities Qa, t h e a v e r a g e volume of p u l p w o o d and log per t r e e in a g e group a eal a n d ea2, respectively, as well a s t h e t o t a l volume Wa. W e assume t h a t all losses indicated by t h e Qa coeffi- c i e n t s f o r a g e groups less than 20 a r e due t o thinning. Based on this, t h e yield c o e f f i c i e n t s gaj c a n be given as

for 4 - < a < 20. W e a s s u m e gaj = 0 for e a c h j, f o r a

2

20. The land c o n s t r a i n t (3.3) r e q u i r e s t h a t t h e t o t a l volume of log and pulpwood cannot exceed a n a m o u n t of L=1700 million m3, which is around t e n p e r c e n t above t h e a c t u a l c u r r e n t level in Finland. According t o t h e transition coefficients, 5.6 t r e e s h a v e t o b e p l a n t e d f o r e a c h grown t r e e harvested when policies A = 14, 15,

...,

²¹

a r e applied. This number i s roughly what is enforced by t h e Finnish law today.

F i g u r e 2 shows t h e annual yield of log and pulpwood when harvesting policies A

= 1 2 , 13,

...,

21 a r e applied. W e may n o t e t h a t a l t e r n a t i v e s A = 17, 18,

...,

²¹

a r e d o m i n a t e d by o t h e r alternatives; i e t h e r e i s a n o t h e r a l t e r n a t i v e whose yield i s b e t t e r f o r both of t h e t w o t i m b e r assortments. The o p t i m a l a l t e r n a t i v e de- pends on the weighting of log and pulpwood. If t h e weight f o r log is at l e a s t 150 p e r c e n t l a r g e r t h a n t h a t f o r pulpwood, t h e n A = 16 i s optimal; i e a t r e e g e t s h a r v e s t e d when i t grows 7 5 t o 80 y e a r s old. If this p e r c e n t a g e is 100 (which

r o u g h l y c o r r e s p o n d s t o t h e c u r r e n t p r i c e l e v e l s of log and pulpwood in Finland) then t h e a l t e r n a t i v e s A = 14 a n d A = 1 5 a r e a b o u t equally good; i e t r e e s in t h e age interval 6 5 t o 7 5 should b e harvested. When t h e weight f o r log drops t o only 75 p e r c e n t a b o v e t h a t f o r pulpwood, t h e h a r v e s t i n g a g e f a l l s t o 6 0 t o 6 5 years.

T a b l e I. Transition c o e f f i c i e n t s Qa, v o l u m e Wa, pulpwood yield e a l a n d g a l , a n d log yield ea2. Yield ga2 e q u a l s .O f o r a l l a g e groups e x c e p t f o r a = 1 3 f o r which ga2 = .003 (Volumes in m 3 / t r e e )

T h e yield a l o n g t h e l i n e s e g m e n t b e t w e e n t h e c o r n e r points in F i g u r e 2 may be o b t a i n e d when t w o policies a r e combined. Logs may a l s o be used as pulpwood.

This has b e e n i l l u s t r a t e d f o r A = 21 by points along t h e broken l i n e of F i g u r e 2.

N o t e t h a t e a c h s u c h point is inferior t o t h e e f f i c i e n t f r o n t i e r , a n d t h e s a m e i s t r u e f o r any o t h e r policy a l t e r n a t i v e A. Thus, in t h e s t a t i o n a r y state logs should not b e used as pulpwood r e g a r d l e s s of t h e p r i c e r a t i o of log and pulpwood. (In a t r a n s i t i o n period t h i s of c o u r s e may n o t hold.)

F i g u r e 2. A l t e r n a t i v e yield of log and pulpwood.

Yield of pulpwood (mill. m3lyear)

,

0 2 5 50

Yield of log (mill. m3/year)

T a b l e 2 s u m m a r i z e s t h e a l t e r n a t i v e s A = 12, 13,

...,

16. I t shows, f o r e a c h policy a l t e r n a t i v e A, t h e yields of log a n d pulpwood s e p a r a t e l y f r o m t h e harves- ting and t h e thinning a c t i v i t i e s when t h e t o t a l volume L of f o r e s t s is a s s u m e d t o be 1700 mill. m3. Also t h e n u m b e r of t r e e s t o be h a r v e s t e d a n d p l a n t e d annually i s shown in T a b l e 2.

T h e a g e distribution of t r e e s resulting f r o m policy a l t e r n a t i v e s A = 13,

...,

¹⁶

h a s b e e n i l l u s t a r t e d in F i g u r e 3. F o r comparison, t h e e s t i m a t e d a g e distribution i n 1 9 7 6 a d j u s t e d t o t h e s a m e t o t a l volume of f o r e s t s h a s been shown in F i g u r e

3. In Figure 4 w e have presented t h e distribution of t h e volume of t r e e s in different a g e groups for policies A=13, 14, 15, and 16. T h e e s t i m a t e d distribution of t h e volume f o r t h e year 1976 has also been presented.

Table 2. Yield by t i m b e r assortments, t r e e s harvested and t r e e s planted f o r harvesting policies A = 12,

...,

^16.

Harvesting policy A 12 13 14 15 16

Log yield, mill. m3/a

Harvesting 0 24.1 35.0 39.8 43.9

Thinning 0 3.7 2.7 2.3 2.0

Total 0 27.8 37.7 42.1 45.9

Pulpwood yield, mill. m3/a

Harvesting 76.2 39.5 26.5 19.7 13.0

Thinning 22.7 24.0 19.7 16.3 13.6

Total 98.9 63.5 46.2 36.0 26.6

T o t a l yield, mill. m3/a Harvesting, mill. t r e e s l a Planting, mill. t r e e s / a

Figure 3. Age distribution of t r e e s for policies A = 13,

...,

a n d comparison with t h e situation in 1976.

Trees in age group (bill, trees)

1

0.0 ',

0 20 40 60 80 100 Age (years)

F i g u r e 4. T h e v o l u m e distribution of t r e e s in a g e groups f o r policies A = 13, 14, 15, and 1 6 as c o m p a r e d t o t h e s i t u a t i o n of 1976.

Trees in age group (mill. m3) I

4 A Steady State Model of the Forest Industries

In this s e c t i o n w e shall c o n s i d e r t h e wood processing p a r t of t h e m o d e l of S e c t i o n 2 in a s t e a d y state.

Suppressing t h e t i m e index t in t h e industrial p a r t of t h e model: e q u a t i o n (2.5) yields

i e i n d u s t r i a l u s a g e of wood U'x c a n n o t e x c e e d wood supply y.

E q u a t i o n (2.6) r e q u i r e s t h a t n e t p r o d u c t i o n (I - A)x supplied t o t h e e x t e r n a l m a r k e t c a n n o t e x c e e d d e m a n d D:

A s in t h e d y n a m i c version (2.7), g r o s s production is l i m i t e d by c a p a c i t y c

T h e state e q u a t i o n (2.8) f o r c a p a c i t y yields

i e i n v e s t m e n t s v e q u a l (physical) d e p r e c i a t i o n gc. T h e state e q u a t i o n (2.10) f o r e x t e r n a l f i n a n c i n g i s r e w r i t t e n as

in o t h e r words, t h e l e v e l of e x t e r n a l financing r e m a i n s c o n s t a n t in t h e s t e a d y state formulation.

T a k i n g i n t o a c c o u n t (4.4) a n d (4.5) t h e m o d i f i c a t i o n of (2.9) r e s u l t s in t h e follo- wing f o r m u l a t i o n

E q u a t i o n (4.6) states t h a t t h e n e t i n c o m e f r o m s a l e s e q u a l s t h e e x p e n d i t u r e s caused by c a p a c i t y (fixed c o s t s a n d d e p r e c i a t i o n ) plus e x t e r n a l f i n a n c i n g ( i n t e r e s t payments). A l t e r n a t i v e l y w e m a y r e p l a c e e q u a l i t y in (4.6) by a n inequality. T h e s l a c k c a n t h e n b e i n t e r p r e t e d as a c o n s t a n t f l o w o u t f r o m t h e f o r e s t s e c t o r .

I t i s obvious t h a t in t h e o p t i m a l solution (4.3) holds as a n equality:

We d e f i n e a v e c t o r d f o r t h e e x t e r n a l d e m a n d which e q u a l s n e t production. Sol- ving x f r o m this, yields

In summary, f o r t h e s t e a d y state solution w e h a v e t o f i n d d which s a t i s f i e s (4.8),

a n d

5 A Steady State Model of the Forest Sector

A b o v e w e h a v e p r e s e n t e d t w o s t e a d y state models one, f o r f o r e s t r y a n d a n o t h e r for wood processing industries. In t h i s s e c t i o n . w e s h a l l m e r g e t h e s e t w o p a r t s t o o b t a i n a s t e a d y state model f o r t h e e n t i r e f o r e s t s e c t o r .

Efficient yields of pulpwood a n d log a r e shown by F i g u r e 2, in which t h e f e a s i b l e region of yields c a n b e defined by a set of linear inequalities:

w h e r e y i s a v e c t o r of m c o m p o n e n t s signifying t h e d i f f e r e n t t i m b e r assort- m e n t s , R i s a m a t r i x and s a vector. F o r t h e two-dimensional case of F i g u r e 2, t h e c o m p o n e n t s of R and s c a n be obtained i m m e d i a t e l y .

Thus, f o r U'x, t h e i n d u s t r i a l u s e of wood, w e r e q u i r e

R U ' x

5

^s

,

T h e s t e a d y state solution d f o r t h e e n t i r e s e c t o r i s t h e n o n e which s a t i s f i e s (4.9), (4.10) a n d (5.1).

So far w e h a v e n o t included i n t o t h e model any o b j e c t i v e functions. F o r f o r e s t r y w e might c h o o s e t o m a x i m i z e s t u m p a g e earnings; ie t h e i n c o m e f r o m selling w o o d t o industry less t h e production c o s t s f o r t h a t wood ( e g h a r v e s t i n g a n d transportation costs). As f o r industry, i n d u s t r i a l p r o f i t , t h e left-hand s i d e of e q u a - t i o n (4.9) o f f e r s o n e possible o b j e c t i v e f u n c t i o n t o be maximized. T h e s u m of t h e s e t w o could c o n s t i t u t e a joint o b j e c t i v e f u n c t i o n ( t h e joint profit) f o r t h e e n t i r e f o r e s t s e c t o r . We shall discuss t h i s s u b j e c t f u r t h e r in S e c t i o n 6.1.

6 A Stackelberg Game

In t h i s s e c t i o n w e s h a l l discuss a s p e c i f i c a t i o n of t h e s t e a d y s t a t e model t o be a p p l i e d f o r t i m b e r m a r k e t analysis. T h e model will be a u g m e n t e d with o b j e c t i v e f u n c t i o n s both f o r f o r e s t r y a n d wood processing industry. F u r t h e r m o r e , t h e d e - m a n d f o r final p r o d u c t s is r e p r e s e n t e d by a d e m a n d f u n c t i o n of c o n s t a n t p r i c e e l a s t i c i t y . T h e round wood m a r k e t i s v i e w e d as a S t a c k e l b e r g game.

I t i s a p p a r e n t t h a t t h e g a m e s i t u a t i o n in t h e f o r e s t s e c t o r involves t w o parties:

forestry and f o r e s t industry. S o f a r t h i s b i p a r t i t i o n has b e e n r e v e a l e d by s e p a r a t e m o d e l s f o r e a c h party. T h e s e models a r e i n t e r c o n n e c t e d through t h e a m o u n t of r o u n d wood supplied by f o r e s t r y t o t h e industry and through t h e prices of round wood.

T h e m a r k e t m e c h a n i s m which d e t e r m i n e s (round) wood p r i c e s may be d e s c r i b e d as follows: Given t h e p r i c e s a n d t h e availability of d i f f e r e n t t i m b e r a s s o r t m e n t s ( a t t h e s e prices) t h e industry c h o o s e s t h e q u a n t i t y i t will buy by maximizing i t s profit; t h e problem f o r f o r e s t r y i s t o c h o o s e prices t o m a x i m i z e i t s p r o f i t (given t h e resulting wood d e m a n d f o r t h a t price).

The decision p r o c e s s d e s c r i b e d a b o v e is c a l l e d a S t a c k e l b e r g g a m e 141. T h e p a r t y m a k i n g t h e f i r s t decision ( o n prices) i s c a l l e d t h e l e a d e r of t h e g a m e a n d t h e o t h e r party t h e follower. In our application, f o r e s t r y acts a s t h e l e a d e r and t h e

i n d u s t r y as t h e follower. We a s s u m e t h a t both t h e l e a d e r and t h e follower a r e p r o f i t m a x i m i z e r s and t h a t t h e y both h a v e p e r f e c t i n f o r m a t i o n on t h e g a m e ( e g on p r o f i t functions, supply and demand).

The c o m p e x i t y of t h e g a m e a r i s e s f r o m t h e f a c t t h a t t h e p r o f i t s of both p a r t i e s depend on r a w wood prices. T h e r e v e n u e s of f o r e s t r y i s d e t e r m i n e d by t h e p r i c e o f w o o d a n d t h e q u a n t i t y sold. In addition, t h e production c o s t f o r wood ( e g h a r v e s t i n g and t r a n s p o r t a t i o n c o s t s ) i n f l u e n c e t h e p r o f i t of f o r e s t r y . F o r t h e in- d u s t r y , t h e p r i c e of wood i n f l u e n c e s t h e c o s t of production. T h e s a l e s p r i c e of a n i n d u s t r i a l p r o d u c t influences i t s demand.

At t h e solution of t h e g a m e , i e at t h e S t a c k e l b e r g equilibrium, p r i c e s f o r t i m b e r a s s o r t m e n t s a r e at a l e v e l which m a x i m i z e s forestry's p r o f i t t a k i n g i n t o a c c o u n t t h e e f f e c t of t h i s p r i c e l e v e l o n wood demand.

6.1 T h e P r o f i t F u n c t i o n s

In o r d e r t o s o l v e t h e (Stackelberg) equilibrium p r i c e s w e shall a p p e n d t o t h e s t e a d y state model of S e c t i o n 5 p r o f i t f u n c t i o n s f o r b o t h parties.

L e t p = (pi) b e t h e v e c t o r of u n i t p r i c e s f o r industrial p r o d u c t s i on t h e i n t e r n a - t i o n a l m a r k e t , l e t v e c t o r c = (ci) s t a n d f o r t h e c o s t s of o n e u n i t of production i n c l u d i n g labor, e n e r g y and. f i x e d c o s t s , d e p r e c i a t i o n , a n d r e a l i n t e r e s t on t o t a l invested c a p i t a l b u t excluding wood cost. L e t z b e t h e v e c t o r of wood p r i c e s f o r t h e d i f f e r e n t t i m b e r a s s o r t m e n t s . D e n o t e

as t h e v e c t o r of t i m b e r a s s o r t m e n t s required f o r o n e u n i t of (industrial) producti- on. Industrial profit, d e n o t e d by PI, i s given by

w h e r e v e c t o r d s t a n d s f o r t h e v o l u m e of export.

A s f o r f o r e s t r y , d e n o t e by e t h e u n i t production c o s t f o r wood. F o r e s t r y profit, d e n o t e d by PF, i s given by

w h e r e y i s t h e q u a n t i t y of wood sold t o t h e industry.

6.2 D e m a n d F u n c t i o n s and O p t i m a l P r i c e s f o r Wood P r o d u c t s

In S e c t i o n 5, w e a s s u m e d t h a t t h e e x t e r n a l d e m a n d f o r wood products is l i m i t e d by a n (exogenous) u p p e r bound. However, f o r t h e S t a c k e l b e r g analysis i t is conve- n i e n t t o u s e a d e m a n d f u n c t i o n w i t h c o n s t a n t p r i c e e l a s t i c i t y

( f o r e a c h wood p r o d u c t i) w h e r e pi i s t h e price, k i i s a c o n s t a n t , a n d -bi i s t h e p r i c e e l a s t i c i t y o f demand. We m a y a s s u m e t h a t bi is g r e a t e r t h a n 1.

D e n o t e by p i t h e world m a r k e t p r i c e which r e s u l t s in t h e ( r e f e r e n c e level) of d e m a n d

zi.

F o r e x a m p l e , if

di

is t h e c u r r e n t ( e x t e r n a l ) d e m a n d , t h e n

pi

^shall

r e f e r t o t h e . c u r r e n t price. Using

pi

^{a n d}

ai

w e s o l v e f o r ki. S u b s t i t u t i n g i n t o (6.4) yields

I n s e r t i n g d = (di) f r o m (6.5) i n t o (6.21, w e c a n s o l v e t h e ( p r o f i t maximizing) o p t i m a l p r i c e

pr

f o r wood products. As a r e s u l t w e h a v e

6.3 T h e P r o f i t Maximization P r o b l e m f o r F o r e s t r y

In (6.6) w e o b t a i n t h e o p t i m a l p r i c e

pr

^{as a} f u n c t i o n of wood p r i c e z; in o t h e r words,

pr

^{= p$z).} Thus, e x t e r n a l (optimal) d e m a n d di i s a c t u a l l y a f u n c t i o n of w o o d p r i c e z. We s h a l l d e n o t e t h e v e c t o r of o p t i m a l d e m a n d q u a n t i t i d s as a f u n c t i o n o f z by d(z). T h e wood u s a g e y = y(z) corresponding t o t h e o p t i m a l wood product p r i c e s i s t h e n given as a f u n c t i o n of wood price:

According t o (5.11, t h e wood availability f r o m f o r e s t s r e s t r i c t s wood consumption as follows:

We h a v e c o m b i n e d t h e t w o models, o n e f o r f o r e s t r y a n d a n o t h e r o n e f o r indust- ry, t o yield t h e following o p t i m i z a t i o n problem f o r forestry:

( 6 . 9 ) m a x P F ( z ) = ( z

-

e ) y ( z ) z

s u b j e c t t o

T h e f o r e s t r y p r o f i t maximizing wood p r i c e v e c t o r , d e n o t e d by z*, i s t h e (Stackel- berg) equilibrium price.

7 Equilibrium Solutions for Finland

In t h i s s e c t i o n w e s h a l l p r e s e n t n u m e r i c a l r e s u l t s f o r t h e S t a c k e l b e r g g a m e w i t h F i n n i s h d a t a . We will c a r r y o u t t h e n u m e r c a l tests using a model d e a l i n g with t w o t i m b e r a s s o r t m e n t s (log and pulpwood) a n d w i t h s e v e n wood products: s a w n g o o d s , panels, o t h e r m e c h a n i c a l wood products, m e c h a n i c a l pulp, c h e m i c a l pulp, p a p e r , a n d c o n v e r t e d paper products.

F o r t h e f o r e s t r y s e c t o r w e employ t h e a l t e r n a t i v e s u s t a i n e d yield solutions deri- v e d in S e c t i o n 3. T h e set of sustained yield solutions of F i g u r e 3 i s used t o d e f i n e t h e c o n s t r a i n t s (6.10) defining t h e c o n v e x polyhedral set of f e a s i b l e round wood yield.

For the industrial model, w e a s s u m e d e m a n d f u n c t i o n s with p r i c e e l a s t i c i t y c o e f - f i c i e n t s bi=b being e q u a l f o r e a c h product. According t o t h e r e p r e s e n t a t i v e s of t h e Finnish f o r e s t industry, a r e a s o n a b l e a s s u m p t i o n c o n c e r n i n g t h e value of b i s t h e range b e t w e e n 1 0 a n d 30. However, sensitivity analysis shall b e p r e s e n t e d f o r t h e whole r a n g e of 1 < b - <

.

A n o t h e r highly s e n s i t i v e and u n c e r t a i n f i g u r e in t h e analysis i s t h e r e f e r e n c e level pi of t h e world m a r k e t price. F o r sensitivity analysis, t h r e e p r i c e s c e n a r i o s w e r e - c o n s t r u c t e d f o r e a c h f o r e s t product. S c e n a r i o 1: a n o p t i m i s t i c world m a r k e t p r i c e i s d e f i n e d as t o t a l production c o s t in Finland (including wood c o s t at pre- sent p r i c e s and a t e n p e r c e n t r e a l i n t e r e s t o n t o t a l i n v e s t e d capital). S c e n a r i o 3:

a peshistic p r i c e i s defined r e f l e c t i n g s u c h production c o s t s f o r t h e m a j o r f u t u r e s u p p l i e r s (such as N o r t h A m e r i c a n a n d L a t i n A m e r i c a n producers) in t h e world market 111. S c e n a r i o 2, a m o r e likely s c e n a r i o , i s t h e a v e r a g e of t h e t w o above.

A c c o r d i n g t o our d a t a , t h e p r i c e in S c e n a r i o 1 i s higher t h a n in S c e n a r i o 3 f o r e a c h wood product s e p a r a t e l y .

7.1 T h e Single P r o d u c t

-

Single T i m b e r A s s o r t m e n t C a s e

F o r q u a l i t a t i v e a n a l y s i s of t h e model w e shall f i r s t study t h e case of a single timber a s s o r t m e n t a n d a single product. In t h i s case, t h e equilibrium c a n a c t u a l l y b e solved analytically.

Depending o n t h e v a l u e of b t h e r e s u l t s shall b e s t u d i e d in t w o cases. We consi- der first t h e case when b i s s m a l l a n d when f o r e s t l a n d i s n o t fully exploited. T o s o l v e t h e equilibrium wood p r i c e z* w e m a x i m i z e f o r e s t r y p r o f i t as d e f i n e d in S e c t i o n 6.3. Taking i n t o a c c o u n t (6.5) a n d (6.6) a n d o m i t t i n g c o n s t a n t s w e h a v e

T h e equilibrium wood price z* f r o m (7.1) i s

N o t i c e t h a t z* i s independent of t h e world m a r k e t r e f e r e n c e p r i c e

p.

I t i s a d e c r e a s i n g f u n c t i o n of b, which a s y m p t o t i c a l l y a p p r o a c h e s wood production c o s t e (harvesting, t r a n s p o r t a t i o n , e t c ) as b increases.

Inserting z* i n t o (7.1) t h e m a x i m u m f o r e s t r y p r o f i t i s

A s f o r industrial p r o f i t given by (6.2), t h e following f o r m u l a r e s u l t s

Along w i t h b, f o r e s t u t i l i z a t i o n i n c r e a s e s u n t i l t h e t o t a l f o r e s t land a r e a i s exp- loited. In t h i s second case, when f o r e s t land i s fully exploited, w e s o l v e t h e e q u i l i b r i u m wood p r i c e z* a s s u m i n g t h a t t h e d e m a n d f o r round wood e q u a l s t h e maximum supply. T h e m a x i m u m p r o d u c t i o n i s d e n o t e d by d*. F r o m (6.5) a n d (6.6) w e g e t

Solving t h e equilibrium wood p r i c e z* f r o m (7.5) r e s u l t s in

In t h i s case, t h e equilibrium p r i c e z* i s a c o n c a v e f u n c t i o n of b which a s y m p t o t i - c a l l y a p p r o a c h e s (p-c) ( t h e u n i t p r o f i t when wood c o s t i s o m i t t e d ) as b i n c r e a s e s .

Inserting (7.6) i n t o (6.6) yields t h e o p t i m a l wood product p r i c e

which a s y m p t o t i c a l l y a p p r o a c h e s j5 ( t h e world m a r k e t price) as b a p p r o a c h e s infi- nity.

Using (7.4), (7.6), a n d (7.7) t h e i n d u s t r i a l p r o f i t i s defined as

A s b i n c r e a s e s t h e equilibrium p r i c e z* a s y m p t o t i c a l l y a p p r o a c h e s a l e v e l absor- bing a l l p r o f i t of t h e f o r e s t s e c t o r i n t o wood price.

As f o r f o r e s t r y profit, (6.3) gives us

( 7 . 9 ) P F = ( z *

-

e ) d *

which a s y m p t o t i c a l l y a p p r o a c h e s

(p

- c

-

e)d* ( t h e m a x i m u m p r o f i t of t h e e n t i r e f o r e s t s e c t o r ) as b a p p r o a c h e s infinity.

In F i g u r e 5 a w e p r e s e n t t h e equilibrium p r i c e z* of r a w wood a s a f u n c t i o n of b. F i g u r e s 5 b a n d 5 c show t h e behavior of f o r e s t r y p r o f i t P F a n d industrial p r o f i t P I a s f u n c t i o n s of b, respectively.

, F i g u r e 5. Equilibrium p r i c e s and p r o f i t s a s f u n c t i o n of t h e p r i c e e l a s t i c i t y c o e f f i c i e n t b f o r t h e single product

-

single t i m b e r a s s o r t m e n t case.

I

^{, b}

a) The equilibrium price z*.

Forestry profit PF

b b ) The forestry profit P,

Industrial profit PI

T

I I

C ) The industrial profit PI

7.2 T h e S e v e n P r o d u c t s - T w o T i m b e r A s s o r t m e n t s C a s e

F o r e a c h wood p r i c e v e c t o r z, t h e profit m a x i m i z i n g solution f o r industry, a n d t h e r e b y wood d e m a n d y(z), c a n be e x p r e s s e d analytically. Thus t h e problem of determining t h e equilibrium p r i c e z* c a n b e s t a t e d a s a n e x p l i c i t nonlinear prog- r a m m i n g p r o b l e m (6.9)

-

(6.10) w i t h n o n l i n e a r i t i e s b o t h in t h e o b j e c t i v e and in t h e c o n s t r a i n t s .

We s h a l l r e d e f i n e t h e v a r i a b l e s so t h a t t h e r e s u l t i n g problem has n o n l i n e a r i t i e s only in t h e o b j e c t i v e . L e t t h e i n v e r s e f u n c t i o n of y(z) be d e f i n e d a s

Substituting t h i s i n t o (6.9) - (6.10) yields t h e following problem with l i n e a r c o n s t - r a i n t s

s u b j e c t to

F o r m o d e r a t e v a l u e s of b w e c a n s o l v e t h i s p r o b l e m using s t a n d a r d nonlinear p r o g r a m m i n g codes. T h e MlNOS c o d e

131

w a s e m p l o y e d in t h i s study.

S i n c e w e only know g(y) t h r o u g h i t s i n v e r s e function, t h e following p r o c e d u r e w a s i m p l e m e n t e d f o r e v a l u a t i n g t h e gradient: (i) Employing i t e r a t i v e m e t h o d s , solve f o r t h e p r i c e v e c t o r z corresponding t o t h e c u r r e n t v a l u e f o r y; (ii) d e t e r - mine the J a c o b i a n m a t r i x E(z) = (ayi(z)

/

3 z j ) f o r c u r r e n t y a n d z, a n d finally, (iii) c a l c u l a t e t h e g r a d i e n t as

vy

PF(y) =

vz

PF(z) E - ~ ( z ) .

F o r l a r g e values of b, t h e problem i s illbehaved a n d t h e r e b y nonsolvable. Howe- v e r , f o r b = ^m w e o b t a i n t h e equilibrium p r i c e z* f r o m t h e d u a l solution of t h e following l i n e a r p r o g r a m maximizing joint p r o f i t f o r industry a n d f o r e s t r y a s fol- lows:

Figure 6. Equilibrium round wood prices a s functions of t h e price elasticity c o e f f i c i e n t b f o r world m a r k e t p r i c e Scenarios 1-3.

Equilibrium price of pulpwood ($/m3 1

Price level in 1978

7

a) The equilibrium price for pulpwood

price of log ($/m3

60

Scenario 1 Scenario 2 Scenario 3

Scenario 1

Scenario 2

Scenario 3

b ) The equilibrium price for log

( 7 . 1 3 ) m a x ( p

-

c

-

e U ) d

.

d , ~ s u b j e c t t o

Proposition: Assume problem (7.13-16) t o be n o n d e g e n e r a t e . If

$

i s t h e dual o p t i m a l solution corresponding t o c o n s t r a i n t (7.14), t h e n z* = ev* i s t h e S t a c k e l - b e r g equilibrium p r i c e f o r b = ol.

.

W h e n f o r e s t r y sets t h e s t u m p a g e p r i c e at p* a n d y - < y* ( t h e o p t i m a l wood c o n s u m p t i o n ) i t will m a x i m i z e i t s earnings, which, in t h i s case, a r e e q u a l t o t h e t o t a l p r o f i t f o r t h e e n t i r e f o r e s t s e c t o r .

-

^{F o r} a proof of t h e Proposition, see Appendix 2.

In F i g u r e s 6 a and 6 b w e h a v e t h e equilibrium wood p r i c e s as f u n c t i o n s of t h e e l a s t i c i t y p a r a m e t e r b f o r t h e t h r e e world m a r k e t p r i c e scenarios.

F i g u r e s 7 a and 7 b s h o w t h e p r o f i t s f o r f o r e s t r y and f o r industry at t h e equilib- rium. For l a r g e values of b (ie b = ol. ), f o r e s t r y , a b s o r b s t h e t o t a l p r o f i t of t h e s e c t o r . ( N o t e t h a t t h e n e c e s s a r y r e t u r n on c a p i t a l h a s b e e n t a k e n i n t o a c c o u n t a s a c o s t f a c t o r f o r t h e f o r e s t industry. Z e r o profit f o r industry m e a n s , t h e r e f o - r e , t h a t r e t u r n on c a p i t a l e q u a l s t h i s minimum.)

F o r b = 10, 20, 3 0 a n d ^ol.

,

t h e n u m e r i c a l r e s u l t s h a v e b e e n given in T a b l e 3.

F i g u r e 7. Equilibrium p r o f i t f o r t h e f o r e s t r y and t h e industry a s a f u n c t i o n

of t h e p r i c e e l a s t i c i t y c o e f f i c i e n t b f o r world m a r k e t p r i c e S c e n a r i o s 1-:

Forestry profit (bill. $/a)

, -

^{Scenario 1}

0

Scenario 2

Scenario 3

a) Forestry profit

Industrial profit (bill. $/a)

Scenario 1

Scenario 2

Scenario 3

0.25 0 .oo

0 10 20 30 m

b ) Industrial profit

Table 3. Equilibrium prices for pulpwood and log as compare d to current prices when b equals 10, 20, 30 and

m

.

Table 3a. The case of b

=

10.

b =

10

Table 3b. The case of b

=

20.

Current Price

($/m3) 4 8 38 30

15 -

Wood Price

I

Stumpage

1 ^Price

Log

Pu 1 pwood Log

Pu 1 pwood

b

=

2 0

Table 3c. The case of b

=

30.

-

Equilibrium Price ($/m3)

Cur rent Price

Wood P r i c e Stumpage Price

Sce 1

3 4 2 8 17

6

Log

Pu 1 pwood Log

Pu 1 pwood

b

=

3 0

(

$/m3

)

32 2 4

38 28 25

30

I

7

15 1 1 i

f

Equilibrium Price ($/m3)

Cur rent Price

($/m3)

-

Wood Price

I ^Stumpage

/ ^Price

Sce 2

2 8 2 8

~ c e 1 I ^{~ c e 2}

Log

Pu 1 pwood Log

PU 1 pwood

Sce3

2 3 2

6

Sce3

-

^--.I

Equilibrium Price ($/m3) ;

5 l 1 ⁶ I ^{, 3}

Sce 1 Sce 2

4 8 43

1

34

I

Sce3 j

?

j

25 :

38

i

34 29

1

24

i

30 26 16

I

1 I

15

^I

12

³

7 i

i 2

-

I

I- - --.

4 8 4 8 3 9

3 8 _i _ _ _ _ * _ I _ _ I _ - . . 3 8 i 3 3 ...^._.C - - . . 2 8

i

I

S t u m p a g e

1

L o g ! 3 0

I I 3 0

!

^{2 1 1 4 i}

1 !

1 5 i P u l p w o o d

1

P r i c e 1 1 5 1 0 5 j ^:

T a b l e 3d. T h e case of b =

= .

G e n e r a l l y , w e c o n c l u d e t h a t t h e c u r r e n t p r i c e l e v e l s f o r pulpwood a n d log a r e much higher t h a n t h e equilibrium p r i c e s resulting f r o m our analysis. On t h e o t h e r h a n d , t h e r e a r e s u b s t a n t i a l d i f f e r e n c e s b e t w e e n p r i c e s resulting f r o m t h e d i f f e - r e n t p r i c e scenarios.

8 Summary and Conclusions

In t h e f i r s t p a r t of t h i s p a p e r a ( d i s c r e t e t i m e ) d y n a m i c l i n e a r model f o r t h e forest s e c t o r was discussed. T h e s t e a d y state version of i t was a n a l y z e d in m o r e detail. An a p p l i c a t i o n of t h e s t e a d y state f o r e s t r y m o d e l w a s c a r r i e d o u t f o r t h e F i n n i s h forests. A s a result, a l t e r n a t i v e sustained yield solutions f o r t h e Finnish f o r e s t r y w e r e obtained.

In t h e s e c o n d p a r t of t h e paper, a s t e a d y state s e c t o r i a l model was a d o p t e d t o c a r r y o u t a S t a c k e l b e r g equilibrium analysis f o r t h e round wood m a r k e t of Fin- land. F u r t h e r e l a b o r a t i o n was n e e d e d f o r t h e s t e a d y state model u n t i l i t b e c a m e s u i t a b l e f o r t h i s g a m e t h e o r e t i c analysis. This e l a b o r a t i o n involved definitions of o b j e c t i v e f u n c t i o n s f o r t h e f o r e s t r y a n d f o r t h e industry.

F o r t h e industrial model, a d e m a n d f u n c t i o n with a c o n s t a n t p r i c e e l a s t i c i t y c o e f f i c i e n t b was c h o s e n f o r e a c h product. A r e a s o n a b l e a s s u m p t i o n conserning t h e v a l u e of b i s in t h e r a n g e b e t w e e n 1 0 a n d 30. If b i s g r e a t e r t h a n 3 0 w e p r i c e ourselves o u t of t h e m a r k e t with a 10 p e r c e n t i n c r e a s e in price. On t h e o t h e r h a n d , when b is u n d e r 1 0 t h e d e m a n d i s very rigid; in o t h e r words, c h a n - g e s in p r i c e d o n o t a f f e c t demand, which d o e s n o t c o r r e s p o n d t o t h e p r e s e n t market situation. However, sensitivity analysis was c a r r i e d o u t on t h e whole r a n g e

of l < b L

- ^.

T h e o t h e r highly u n c e r t a i n and sensitive f i g u r e in t h e analysis is t h e world m a r k e t p r i c e (defined as s a l e s p r i c e when b a p p r o a c h e s infinity). F o r s e n s i t i v i t y analysis, t h r e e p r i c e s c e n a r i o s w e r e c o n s t r u c t e d f o r e a c h f o r e s t pro- duct a s follows: (1) An o p t i m i s t i c world m a r k e t p r i c e i s d e f i n e d as t o t a l p r o d u c t i o n c o s t in Finland (including wood c o s t at p r e s e n t p r i c e s a n d a t e n p e r c e n t r e a l interest on t o t a l i n v e s t e d c a p i t a l ) , (3) a pessimistic world m a r k e t p r i c e i s defined as being roughly e q u a l t o t h e production c o s t of our m a j o r f u t u r e c o m p e t i t o r s in t h e w o r l d m a r k e t , a n d (2) a likely s c e n a r i o which i s t h e a v e r a g e of t h e t w o above.

As t h e n u m e r i c a l r e s u l t s p r e s e n t e d in S e c t i o n 7 show t h e c u r r e n t p r i c e levels f o r pulpwood a n d log a r e m u c h higher t h a n t h e equilibrium p r i c e s resulting f r o m our a n a l y s i s . On t h e o t h e r hand, t h e r e a r e s u b s t a n t i a l d i f f e r e n c e s b e t w e e n p r i c e s r e s u l t i n g f r o m t h e t h r e e p r i c e s c e n a r i o s f o r t h e world m a r k e t p r i c e s of wood products.

APPENDIX 1.

Proposition: F o r a n o p t i m a l solution of t h e f o r e s t r y problem ( F ) d e f i n e d on p a g e 6 t h e r e i s a n a g e g r o u p A such t h a t h(a) = 0, f o r a l l a

f

A, a n d w(a) = 0, f o r a l l a > A.

P r o o f : C l e a r l y , f o r a n o p t i m a l solution w(1) = p > 0. L e t a = A b e t h e s m a l l e s t a g e g r o u p f o r which w ( A + l ) = 0. T h e n h(A) > 0 a n d w(a) = h(a) = 0 f o r a l l a >

A. T o show t h a t h(a) = 0 f o r a < A, w e c o n s i d e r t h e o p t i m a l b a s i s f o r (F) p a r t i t i o n e d as follows:

p w ( 1 ) w(2)

....

w ( A ) h ( A ) other bacic variables

Figure: An o p t i m a l b a s i s m a t r i x f o r (F).

Constraints (3.1 )-(3.3) for all a 5 A

Other Constraints

H e r e B l I is s q u a r e a n d B21 = 0. Thus, B22 i s nonsingular a n d t h e r e f o r e , h(a) is nonbasic f o r a < A.l l