Pollution-Induced Forest Damage, Optimal Harvest Age and Timber Supply: Some Theoretical Considerations

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

POLLUTION-INDUCED FOREST DAMAGE. OPTIMAL HARVEST AGE AND TIMBER SUPPLY: SOME

THEORFI?CAL CONSIDERATIONS

1

E f f e O v a s k u i n e n

May 1987 WP-87-37

PUBLICATION

NUMBER

36 of the project:

Ecologically S u s t a i n a b l e Development of t h e B i o s p h e r e

l n t e r n a t l o n a l I n s t i t u t e for Applied Systems Analysis

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POLLUTION-INDUCED FOREST

DAMAGE,

OF'TIMAL

HARVEST AGE AND

TIMBER SUPPLY: SOME THEORETICAL CONSIDERATIONS

K l l e O v a s k a i n e n

May 1987 WP-87-37

PUBLICATION NUMBER 36 of t h e p r o j e c t :

EcoLogicaLly SustainabLe Development of t h e B i o s p h e r e

Working P a p e r s are interim r e p o r t s on work of t h e I n t e r n a t i o n a l Institute f o r Applied Systems Analysis and h a v e r e c e i v e d only limited review. Views o r opinions 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 of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , Austria

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ABOUT THE AUTHOR

Ville Ovaskainen i s a R e s e a r c h Fellow at t h e Department of F o r e s t Economics in t h e Finnish F o r e s t R e s e a r c h I n s t i t u t e in Helsinki. His r e s e a r c h i n t e r e s t s are mainly in t h e economic evaluation of silvicultural regimes in f o r e s t management.

C o r r e s p o n d e n c e should b e a d d r e s s e d to:

Ville Ovaskainen

Department of F o r e s t Economics The Finnish F o r e s t R e s e a r c h I n s t i t u t e

P.O. Box 3 7 SF-00381 Helsinki

Finland

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FOREWORD

This p a p e r , b y Ville Ovaskainen of t h e Finnish F o r e s t R e s e a r c h I n s t i t u t e , e x - amines t h e r e l a t i o n s h i p s of pollution-induced f o r e s t d a m a g e , optimal a g e of s t a n d h a r v e s t , a n d t i m b e r s u p p l y f r o m a t h e o r e t i c a l p e r s p e c t i v e . S i n c e f u t u r e l e v e l s of f o r e s t d a m a g e will b e i m p o r t a n t d e t e r m i n a n t s of h a r v e s t i n g a n d s i l v i c u l t u r a l p r a c - t i c e s a n d t h u s of wood s u p p l y , t h e y must b e e x p l i c i t l y t a k e n i n t o a c c o u n t in r e a l i s t i c a n a l y s i s of a l t e r n a t i v e s c e n a r i o s of f u t u r e f o r e s t d e c l i n e a n d a p p r o p r i a t e p o l i c y r e s p o n s e s .

T h e p a p e r i s a n o u t p u t of t h e IIASA F o r e s t S t u d y , which e x a m i n e s t h e c o n s e - q u e n c e s f o r t h e w o r l d ' s f o r e s t s a n d t h e f o r e s t p r o d u c t s s e c t o r of c h a n g e s in f o r e s t p a t t e r n s a n d g r o w t h d u e t o a i r b o r n e p o l l u t a n t s . T h e e m p h a s i s in t h e f i r s t p h a s e of t h e s t u d y i s o n i s s u e s of m a j o r r e l e v a n c e t o i n d u s t r i a l a n d g o v e r n m e n t policy- m a k e r s in E u r o p e . T h e r e s e a r c h p r o g r a m i n c l u d e s a n a n a l y s i s of f u t u r e wood supply in E u r o p e u n d e r d i f f e r e n t a s s u m p t i o n s a b o u t t h e rate a n d e x t e n t of f o r e s t de- c l i n e . In a d d i t i o n , a n u m b e r of p a p e r s a r e b e i n g p r o d u c e d t o a d d r e s s v a r i o u s to- p i c s r e l a t e d t o f o r e s t d e c l i n e a n d t h e E u r o p e a n f o r e s t s e c t o r in g e n e r a l .

T h e F o r e s t S t u d y i s p a r t of IIASA's P r o j e c t o n Ecologically S u s t a i n a b l e Development of t h e B i o s p h e r e , which s e e k s t o c l a r i f y t h e policy implications of long-term, l a r g e - s c a l e i n t e r a c t i o n s b e t w e e n t h e w o r l d ' s economy a n d i t s e n v i r o n - ment. T h e P r o j e c t c o n d u c t s i t s work t h r o u g h a v a r i e t y of b a s i c r e s e a r c h e f f o r t s a n d a p p l i e d case s t u d i e s . A l i s t of t h e P r o j e c t ' s p u b l i c a t i o n s a p p e a r s at t h e end of t h i s d o c u m e n t .

R.E. Munn L e a d e r

E n v i r o n m e n t P r o g r a m

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I would like t o t h a n k Dr. H . A . Loikkanen f o r t h e stimulus t o i n i t i a t e t h i s p a p e r and f o r comments o n t h e Finnish d r a f t , as well as J. Kuuluvainen. Thanks are a l s o d u e t o Dr. P. Kauppi f o r discussions a n d t h e idea of c o l l a b o r a t i o n with IIASA, a n d t o Dr. S. Nilsson f o r h i s c o n t r i b u t i o n t o t h i s collaboration.

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^vii

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CONTENTS

1. INTRODUCTION 1

2. THE ASSUMPTIONS OF THE ANALYSIS 2

3. THE EFFECT OF CHANGES IN THE GROWTH FUNCTION ON THE OPTIMAL

HARVEST AGE 5

3.1 Once-and-for-all changes with identical rotations 5

3.1.1 Damage in t h e mature f o r e s t 6 3.1.2 T o o r e r s i t e " and 'lower input efficiency" e f f e c t s 7 3.2 The optimal h a r v e s t a g e f o r standing timber under expected

changes in f u t u r e rotations 1 2

3.2.1 Standing timber unaffected 1 2

3.2.2 Observed damage in standing timber 1 3

3.3 Concluding r e m a r k s 1 5

4. THE EFFECT ON EQUILIBRIUM IN THE ROUNDWOOD MARKET 1 8

4.1 Short-term e f f e c t s 4.2 Long-run e f f e c t s 5. DISCUSSION

References Appendices

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POLLLTION-INDUCED FOREST DAMAGE, OPTIKAL HARVE;ST AGE AND TIMBER SUPPLY: S O M THEORETICAL CONSIDERATIONS

ELLe O v a s k a i n e n

1. INTRODUCTION

Recently, a c o n c e r n f o r t h e e f f e c t s of c h a n g e s in t h e a t m o s p h e r e o n forests h a s emerged. F o r e s t damage d u e t o a i r pollutants h a s b e e n r e p o r t e d in C e n t r a l Eu- r o p e a n c o u n t r i e s , accompanied by a n anticipation of sanitation fellings a n d t h e subsequent a d v e r s e e f f e c t s o n t h e roundwood m a r k e t . Less d r a s t i c c h a n g e s in t h e productivity of f o r e s t ecosystems might even h a v e long-term impacts t h r o u g h a de- c l i n e in growth a n d s u s t a i n a b l e h a r v e s t .

T h e r e i s p r e s e n t l y no r e a s o n t o o v e r e s t i m a t e t h e s c a l e a n d s e r i o u s n e s s of t h e damage (e.g., t h e r e are no indications of i n c r e a s e d sanitation fellings (Kauppi, 1987)). However, if some damage h a s been o b s e r v e d a n d c h a n g e s may b e anticipat- e d in t h e f u t u r e , t h e p o t e n t i a l consequences are p r o b a b l y i m p o r t a n t enough t o b e worth a t h e o r e t i c a l discussion, i r r e s p e c t i v e of t h e s c a l e of t h e a c t u a l damage. F o r t h e moment, not t o o much i s known a b o u t how pollution and i t s e f f e c t s will p r o c e e d , o r how pollutants a c t u a l l y a f f e c t tree v i g o r a n d growth. I t i s n a t u r a l enough, t h e n , t h a t l i t t l e analysis o n t h e economic e f f e c t s e x i s t s .

The p r e s e n t p a p e r will n o t aim at f o r e c a s t s . I n s t e a d , i t i n q u i r e s a b o u t t h e po- t e n t i a l e f f e c t s u n d e r a l t e r n a t i v e assumptions: given c e r t a i n h y p o t h e t i c a l damage levels in standing t i m b e r and c h a n g e s in f o r e s t growth, what are t h e likely e f f e c t s on h a r v e s t i n g decisions a n d t h e roundwood m a r k e t equilibrium? A few h y p o t h e t i c a l , intuitively plausible t y p e s of e f f e c t s are c o n s i d e r e d to gain insight i n t o t h e quali- t a t i v e a s p e c t s of t h e problem. The t e c h n i q u e employed is c o m p a r a t i v e s t a t i c s , t h a t is, w e c o n s i d e r mainly t h e d i r e c t i o n of changes.

F i r s t , t h e e f f e c t of c h a n g e s in t h e growth c u r v e on t h e optimal h a r v e s t a g e of f o r e s t s i s studied using t h e t r a d i t i o n a l Faustmann model. The d r a s t i c damage in older s t a n d s i s by no means t h e only possibility, p a r t i c u l a r l y when d i f f e r e n t time s c a l e s are recognized. The assumptions will t h e n b e d e s c r i b e d in more d e t a i l . Be- s i d e s once-and-for-all c h a n g e s assuming identical r o t a t i o n s , t h e h a r v e s t a g e f o r

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t h e standing timber will b e analyzed by a modification t h a t allows t h e f u t u r e r o t a - tions and t h e c u r r e n t stands t o possess different growth p a t t e r n s .

The p a r t i a l equilibrium analysis of t h e optimum r o t a t i o n is limited by i t s res- t r i c t i v e assumptions, especially t h a t of a n exogenously fixed, constant p r i c e level.

To recognize t h e p r i c e effects, a market equilibrium setting i s introduced. Assum- ing t h a t just a p a r t of t h e exploitable f o r e s t s is damaged, t h e r e will b e r e s p o n s e s o t h e r t h a n t h e pulse of timber from sanitation fellings. The p a p e r concludes with a discussion of t h e r e l e v a n c e and limitations of t h e r e s u l t s .

2. THE ASSUMPTIONS OF THE ANALYSIS

In t h e following analysis, t h e most traditional a p p r o a c h t o f o r e s t utilization, t h e problem of optimum r o t a t i o n , will b e employed. The steady-state solution f o r t h e problem is well-known as t h e Faustmann r u l e (e.g., see Samuelson, 1976;

Johansson and Loefgren, 1985). The model assumes p e r f e c t capital, timber and land markets. P r i c e s , c o s t s and i n t e r e s t rates are assumed t o b e known constants in t h e p a r t i c u l a r steady-state solution. Given t h e assumptions, t h e solution h a s been shown t o b e socially optimal (Johansson and Loefgren, 1985).

Implicitly, t h e model a l s o assumes constant r e t u r n s t o s c a l e in harvesting technology, so t h e optimal cutting decision can b e made at t h e s t a n d level (Johans- son and Loefgren, 1985).

The time profile of fellings or t h e age-class composition of t h e e n t i r e f o r e s t i s not constrained. F u r t h e r m o r e , t h e choice of optimal cutting time is assumed not t o b e r e s t r i c t e d by t h e capacity of timber harvesting (cf. Heaps and Neher, 1979).

The assumptions are s t r o n g , hence simplifying. In t h e p r e s e n t c o n t e x t , t h e assumption of a constant p r i c e level i s especially r e s t r i c t i v e , because non-marginal s h i f t s in supply (e.g., due t o sanitation fellings) imply p r i c e changes. In o t h e r words, t h e p a r t i a l analysis ignores t h e p r i c e effects.

The s t a n d a r d Faustmann model aims at maximizing t h e p r e s e n t n e t value of a n infinite chain of subsequent r o t a t i o n s , o r land expectation value. This r e q u i r e s , besides constant economic v a r i a b l e s , t h a t t h e growth p a t t e r n i s identical f o r a l l rotations. Thus, biotechnology (e.g., t h e quality of plants, seeds, and management intensity) as well as f o r e s t soil productivity are assumed t o b e unchanged. Then, all r o t a t i o n s will b e identical, and t h e problem converges t o t h e well-known com- p a c t formulation.

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In a way, t h e simplification i s overly r e s t r i c t i v e . For example, t h e choice of tree species i s not f o r e v e r constant, since f o r e s t tree breeding changes t h e ge- netic p r o p e r t i e s of plants and seeds and t h e rate of t r e e growth. In analysing t h e e f f e c t of biotechnological improvements on optimum rotation, Johansson and Loef- g r e n (1985) show how t h e assumptions of t h e model can b e r e l a x e d to allow f o r a n explicit treatment of differences between c h a r a c t e r i s t i c s of t h e standing timber and f u t u r e rotations. Walter (1980) refers t o t h e problem of maintaining f o r e s t soil productivity and how t h i s influences t h e formulation of t h e problem. To t h e b e s t of my knowledge, t h e r e h a s been little f u r t h e r study on t h i s m a t t e r , r e p o r t e d in t h e optimum-rotation literature1. However, t h e discussion below on changes in t h e composition of t h e atmosphere, and soil acidification, and t h e i r e f f e c t s on f o r e s t s , suggests t h a t t h e issue of soil productivity and t h e uncertainty of f u t u r e growth conditions may become more relevant.

There i s no straightforward answer t o t h e question of how a i r pollutants af- f e c t forests. The r e a c t i o n s undoubtedly v a r y a c r o s s a wide r a n g e of s t a g e s and de- g r e e s of pollution, as w e l l as a c r o s s various t r e e s p e c i e s and t y p e s of f o r e s t soil.

Furthermore, in t h e atmospheric changes a number of d i f f e r e n t chemical com- pounds a r e involved (e.g., s u l f u r dioxide, nitrogen oxides, ozone and carbon dioxide).

Here i t will suffice t o refer t o Kauppi's (1987) s u r v e y of study r e s u l t s thus far available. One conclusion is t h a t old t r e e s are particularly susceptible t o damage. The sulfur content and t h e a v e r a g e a g e of foliage on needle-leaved trees a r e r e l a t e d . However, besides t h e hypothesis of immediate damage through toxic ef- f e c t s on t h e needles, t h e hypothesis of soil acidification as a link t o delayed im- p a c t s on crown density and t r e e vigor h a s become more important. Thus, t h e dam- a g e may w e l l b e caused by accumulating soil acidification, t h e consequent changes in t h e balance of calcium, magnesium and aluminium, and t h e i r e f f e c t on plant r o o t s along with heavy metal stress. On t h e o t h e r hand, increased carbon dioxide con- t e n t s of t h e a i r may induce a so-called C02 fertilization effect.

In t h e Faustmann r u l e of t h e optimum rotation f o r a n even-aged f o r e s t (stand), t h e e f f e c t of changing growth conditions e n t e r s in terms of shifts o r o t h e r changes in t h e growth c u r v e expressing standing volume as a function of stand age. The ef- 'ln some versions of the Faustmann model (Hyde, 1980; Jackson, 1980; Chang, 1983), biotechnology i s represented a s a choice variable through endogenous silvicultural effort. The choice of a higher level of inputs results in a s h i f t t o a higher growth curve, and the optimal level, once determined, i s maintained through the future. Here we are dealing with an exogenous change, or deterioration of the growth function due t o pollution.

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f e c t can not b e unambiguously c h a r a c t e r i z e d , and i t may also be different in dif- f e r e n t time scales. T h e r e f o r e t h e p r e s e n t p a p e r will consider t h r e e different possibilities. The c a s e s , although hypothetical, s e e m t o have some r e l e v a n c e in t h a t a meaningful i n t e r p r e t a t i o n can b e given t o each of them from a n intuitive point of view. They are squeezed into t h e following t h r e e assumptions. The e f f e c t of e a c h t y p e of onceand-for-all changes (i.e., assuming all rotations are identical) on t h e optimum rotation will b e studied in t u r n by comparing t h e 'new' optimum with t h a t of t h e r e f e r e n c e conditions (the norm), o r undamaged f o r e s t .

(1) "Forest damage". Young stands, being practically unaffected, grow normally, while growth is reduced later on in mature stands by various d e g r e e s of loss of vi- g o r and increased tree mortality.

(2) " P o o r e r site". The trees do not drastically lose t h e i r vigor a s in (1) above, but t h e productivity of f o r e s t soil declines. The standing volume i s assumed t o decline proportionally, i.e. by t h e s a m e p e r c e n t a g e at e a c h age. The maximum potential volume p e r h e c t a r e d e c r e a s e s , as if t h e trees were effectively growing on a poorer s i t e t h a n t h e original one.

(3) "Lower input efficiency". The maximum potential volume p e r h e c t a r e is unaffected but s t a n d growth i s delayed, which implies a non-proportional decline in standing volume. The damaged stand resembles one growing under a lower management intensity, s o one c a n say t h a t t h e effectiveness of silvicultural inputs h a s de- c r e a s e d .

Next, t h e e f f e c t of observed and/or expected growth changes (damage) on t h e optimal h a r v e s t a g e of c u r r e n t standing timber i s analyzed more explicitly. The growth c u r v e s of f u t u r e r o t a t i o n s are assumed t o b e identical with e a c h o t h e r but d i f f e r from t h e growth p a t t e r n of t h e standing timber. The standing timber is f i r s t assumed t o b e unaffected, and subsequently also damaged, while in both c a s e s t h e f u t u r e stands possess p o o r e r growth p a t t e r n s in comparison t o t h e norm.

In t h e p a r t i a l equilibrium setting of t h e optimum rotation, a constant parametric p r i c e f o r wood i s assumed. However, t h e pulse of supply from damaged f o r e s t s i s not without implications t o timber p r i c e whenever t h e demand f o r timber i s not perfectly elastic. Furthermore, i t i s reasonable t o assume t h a t just a p a r t of t h e f o r e s t s in t h e r e l e v a n t market a r e a i s damaged. A conventional market equilibrium setting will be employed t o i l l u s t r a t e t h e point t h a t , through t h e involved p r i c e changes, t h e r e will b e a f u r t h e r indirect e f f e c t on supply from t h e undamaged f o r e s t s .

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3. THE EFFECT OF CHANGES IN THE GROWTH FUNCTION ON THE OPTIMAL HARVEST AGE

3.1. Once-and- f or-all changes w i t h i d e n t i c a l rotations

In t h e t r a d i t i o n a l Faustmann model t h e problem i s to c h o o s e t h e r o t a t i o n length t s o as to maximize t h e n e t p r e s e n t value V of r e c e i p t s from a n infinite c h a i n of s u b s e q u e n t r o t a t i o n s (land e x p e c t a t i o n value):

where: p

=

stumpage p r i c e ,

f ( t )

=

t h e growth function u n d e r r e f e r e n c e conditions

( t h e norm); f '(t )

>

0 , f "(t)

<

0 o v e r t h e r e l e v a n t r a n g e , c

=

r e g e n e r a t i o n c o s t , a n d

r

=

m a r k e t rate of i n t e r e s t .

By d i f f e r e n t i a t i n g (3.1) with r e s p e c t to t a n d s e t t i n g t h e d e r i v a t i v e e q u a l to z e r o , t h e f i r s t - o r d e r condition f o r a maximum i s obtained:

The condition i s t h e Faustmann r u l e which s a y s t h a t a f o r e s t i s p r o f i t a b l y c u t when i t s value i n c r e m e n t is l e s s t h a n or equal to t h e o p p o r t u n i t y cost ( i n t e r e s t ) on t h e c a p i t a l tied up in t h e standing s t o c k plus t h e value of f o r e s t land ( f o r f u r t h e r analysis, see: Samuelson, 1976; Johansson a n d Loefgren, 1985). lgnoring r e g e n e r a - tion c o s t ( c

=

O), t h e condition simplifies to:

which s a y s t h a t at t h e optimum, t h e r e l a t i v e rate of growth of t h e s t a n d must b e e q u a l to t h e m a r k e t rate of i n t e r e s t c o r r e c t e d f o r t h e land r e n t element. Note t h a t f u t u r e r o t a t i o n s are r e c o g n i z e d in t h e h a r v e s t i n g decision. To study t h e ef- f e c t of c h a n g e s in t h e growth function on t h e optimum r o t a t i o n , we r e w r i t e t h e

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function in t h e form

Q

= f ( t r a ) ; a E ( 0 , l )

-

Thus, w e have simply made explicit a p a r a m e t e r t h a t changes t h e growth c u r v e as compared t o t h e norm without defining t h e form of t h e e f f e c t more specifically. The problem in (3.1), ignoring r e g e n e r a t i o n cost, now r e a d s

3.1.1. Damage in the mature forest

Old trees are p a r t i c u l a r l y susceptible t o forest damage due to pollutants (Kauppi, 1987). I t may t h e r e f o r e b e reasonable t o assume t h a t growth in t h e young stand i s not disproportionately a f f e c t e d but declines in comparison to t h e norm in t h e mature forest. The d e g r e e of damage may r a n g e f r o m minor growth reduction to s e r i o u s damage through t h e loss of tree vigor. In t h e e x t r e m e case where t h e volume lost through mortality e x c e e d s t h e increment of remaining live trees, implying a negative n e t volume increment, t h e t e r m ' f o r e s t dieback' may b e used and a sanitation felling may b e e x p e c t e d .

One possible case of t h i s t y p e is shown in Figure 1. Regarding t h e formal analysis, t h e r e i s no simple, s p e c i a l parametrization for t h i s c a s e , and t h e in- t e r p r e t a t i o n of t h e g e n e r a l f o r m may become difficult. T h e r e f o r e a simplifying re- gularity assumption will b e made, t h u s r e s o r t i n g t o a special yet still plausible case. A s depicted in Figure 1, for a l l a g e s t E (to,tmax) t h e growth c u r v e of t h e damaged stand is less s t e e p t h a n t h e norm. Thus t h e c u r r e n t growth or t h e deriva- tive of t h e growth c u r v e i s l e s s t h a n t h a t of t h e norm for all r e l e v a n t ages. Also, t h e growth c u r v e remains s t r i c t l y concave:

ft'(t , a )

>

0, ft;'(t , a )

<

0 a l l t E ( t O , t max)

.

^a^E^(0.1)

^.

This s a y s simply t h a t no i r r e g u l a r i t i e s in t h e sense of convex portions in t h e growth function are assumed t o emerge. The f i r s t - o r d e r condition for a n optimum now r e a d s

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Compared t o t h e s t a n d a r d Faustmann r u l e (3.3), t h e left-hand s i d e i s changed.

Under t h e r e g u l a r i t y assumption w e may conclude t h a t t h e optimum r o t a t i o n in t h e damaged f o r e s t will b e s h o r t e n e d in comparison t o t h e norm. This i s c l e a r from Fig- u r e 2, b e c a u s e

as B o

<

B 1 ( f o r e f e r s t o t h e norm a n d f l t o t h e damaged f o r e s t ) . From t h e g r a p h i - c a l i n t e r p r e t a t i o n of t h e d e r i v a t i v e ( t h e s l o p e of a c u r v e ) , i t i s s e e n t h a t t h e r e l a - t i v e rate of growth in t h e damaged s t a n d f o r a n y r e l e v a n t a g e T i s lower t h a n t h a t of t h e norm, s o t h e s t a n d will b e h a r v e s t e d at a lower a g e . The qualitative r e s u l t holds t r u e i r r e s p e c t i v e of t h e d e g r e e of damage whenever t h e r e g u l a r i t y assumption holds.

3.1.2. "Poorer site" and "lower input efficiency" effects

Intuitively, t h e a b o v e case p e r h a p s b e s t r e p r e s e n t s t h e a c u t e e f f e c t s of a i r pollutants o n t h e h e a l t h of trees d u e t o d i r e c t t o x i c e f f e c t s through t h e foliage.

However, r e g a r d i n g soil acidification, l e s s e x t r e m e and qualitatively d i f f e r e n t long-term e f f e c t s might a l s o b e possible ( t h a t is, t h e productivity of e a c h t y p e of soil o r s t a n d i s r e d u c e d , y e t without d r a s t i c 'dieback' e f f e c t s ) . Two s p e c i a l p a r a m e t r i z a t i o n s r e p r e s e n t i n g t h i s possibility are c o n s i d e r e d h e r e .

F i r s t , assume t h e e f f e c t i s p r o p o r t i o n a l . Thus, standing volume i s r e d u c e d by t h e same p e r c e n t a g e at e a c h a g e , s o t h e p o t e n t i a l maximum volume a l s o declines, a n d t h e trees are effectively growing on a p o o r e r s i t e . The growth c u r v e i s depict- e d in F i g u r e 3 , a n d t h e volume i s given by t h e equation:

Note t h a t t h e case i s formally identical with t h e time-neutral biotechnological improvement c o n s i d e r e d by Johansson a n d Loefgren (1985). The only d i f f e r e n c e i s t h a t t h e p a r a m e t e r a i s i n t e r p r e t e d as a 'pollution p a r a m e t e r ' , h e n c e taking on v a l u e s l e s s t h a n one.

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Volume rn3/ha

Current growth

F i g u r e 1. The c h a n g e in t h e g r o w t h f u n c t i o n with d a m a g e in t h e m a t u r e f o r e s t .

F i g u r e 2. Diagrammatic p r e s e n t a t i o n of t h e c h a n g e in r e l a t i v e g r o w t h rate in F i g u r e 1.

(15)

If r e g e n e r a t i o n c o s t i s ignored f o r t h e moment, t h e problem c a n b e w r i t t e n as

- p

a f ( t ) (3.7) M a x

v -

t e f t -1

The f i r s t - o r d e r condition f o r a n optimum, when a r r a n g e d , is

Clearly, t h e s t a n d a r d Faustmann r u l e (cf. (3.3)) i s obtained. Thus, a p r o p o r - tional c h a n g e in t h e growth c u r v e will not c h a n g e t h e optimum r o t a t i o n when r e g e n - e r a t i o n cost i s n o t t a k e n i n t o a c c o u n t (c

=

0). The r e s u l t i s obvious b e c a u s e t h e optimum condition (3.7) d o e s not depend o n a. Intuitively, a p r o p o r t i o n a l c h a n g e l e a v e s t h e r e l a t i v e growth rate unchanged.

I t i s a l s o c l e a r t h a t t h e c h a n g e in t h e growth c u r v e will d e c r e a s e t h e value of s e e d e d ( b a r e ) land, b e c a u s e t h e r o t a t i o n p e r i o d remains unchanged b u t t h e volume of t i m b e r h a r v e s t e d declines.

By recognizing explicitly t h e r e g e n e r a t i o n c o s t (c

>

0) t h e problem i s r e w r i t - t e n as:

The f i r s t - o r d e r condition t a k e s t h e form

By r e a r r a n g i n g to:

i t becomes c l e a r t h a t a p r o p o r t i o n a l downward s h i f t in t h e growth c u r v e ( a c ( 0 , l ) ) i n c r e a s e s t h e r e l a t i v e rate of value growth in comparison t o t h e norm ( a = l ) , and h e n c e l e n g t h e n s t h e optimum r o t a t i o n . Because a

<

1 , t h e e f f e c t of p r o p o r t i o n a l d e t e r i o r a t i o n of t h e growth function i s effectively t h e same as a once-and-for-all r e l a t i v e i n c r e a s e of t h e r e g e n e r a t i o n c o s t with a n u n a l t e r e d growth c u r v e ( o r a d e c l i n e in p r i c e ; cf. Johansson a n d Loefgren, 1985). F o r t h i s i t holds t h a t

at / ac

>

0, which e s t a b l i s h e s t h e claim.

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Secondly, suppose t h e growth c u r v e i s given by t h e equation

A c u r v e of t h i s t y p e i s p r e s e n t e d in Figure 4. The change of t h e c u r v e i s c l e a r l y non-proportional.

The distinguishing f e a t u r e of t h i s case i s t h a t given enough time, t h e stand will potentially r e a c h t h e s a m e maximum volume as t h e norm. This p r o p e r t y resembles t h e assumption employed by Chang (1983) where s t a n d s established a t d i f f e r e n t densities on a given s i t e eventually converge t o t h e same volume (via a random pat- t e r n ) . In o t h e r words, w e assume t h a t t h e damaged soil c a n r a i s e t h e s a m e volume of trees b u t with a delay in comparison t o t h e norm, as if t h e stand were growing under a lower management intensity. The r e s p o n s e of timber yield t o a given level of management intensity is lower t h a n t h e norm. The o t h e r way round, then, t h e case i s formally identical with a d e c r e a s e d efficiency of silvicultural inputs.

Assuming f o r simplicity t h a t c

=

0, t h e problem r e a d s

The f i r s t - o r d e r condition f o r a n optimum c a n b e r e a r r a n g e d t o

Taking into account t h a t land value is maximized at t h e optimum t* ^,aV/ a t

=

0 , s o t h e second-order condition i s

(3.15) a 2 f " ( a t )

-

^raf ' ( a t )

<

^0,

which holds by assumptions ( a f O ( a t )

>

0 , a 2 f " ( a t )

<

0 in t h e r e l e v a n t range).

Looking at (3.14), t h e right-hand side of t h e equation i n c r e a s e s as compared t o t h e norm because a

<

1. But t h e left-hand s i d e a l s o changes in a way t h a t i s not readily c l e a r . I t c a n b e shown mathematically t h a t a non-proportional change in t h e growth c u r v e of t h e form Q

=

f ( a t ) ; a € (0.1) will lengthen t h e optimum r o t a - tion in comparison to t h e norm. The formal proof i s technical, and i s p r e s e n t e d in Appendix 1. I t i s obvious again t h a t t h e d e t e r i o r a t i o n of t h e growth c u r v e de-

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Volume d / h a

Current growth

Figure 3 . A p r o p o r t i o n a l c h a n g e in t h e growth c u r v e ( t h e " p o o r e r s i t e " e f f e c t ) . Volume &/ha

Current growth

F i g u r e 4. A non-proportional c h a n g e in t h e growth c u r v e ( t h e c a s e of "lower input efficiency ").

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creases t h e land value, because t h e rotation a g e is lengthened and t h e volume of h a r v e s t is lower than t h e norm.

3.2. The optimal harvest age for standing timber under expected changes in future stands

3.2.1. Standing timber unaffected

Next, assume t h a t c u r r e n t mature stands are not y e t seriously affected by a i r pollutants, but t h e r e i s r e a s o n t o believe t h a t f u t u r e r o t a t i o n s will show a reduced growth. The assumption intuitively r e f e r s to areas like t h e Nordic c o u n t r i e s where no a c u t e f o r e s t damage h a s actually been observed, but still a c o n c e r n f o r t h e fu- t u r e growing conditions exists. The problem is, then, whether standing timber should b e h a r v e s t e d m o r e o r less rapidly than t h e "planned1' rotations ( f o r r e f e r - e n c e conditions) imply.

In what follows w e explicitly recognize t h e standing timber, aged b y e a r s , as t h e s t a r t i n g point instead of b a r e land. Regarding realism, t h e initial a g e b should r e f e r to mature s t a n d s at a n e a r l y s t a g e of maturity. The standing timber is assumed t o follow t h e norm (original f o r e c a s t growth c u r v e for t h e s i t e ) while t h e fu- t u r e rotations, being identical t o e a c h o t h e r , follow a lower growth p a t t e r n . The problem is t o maximize t h e p r e s e n t value of t h e standing timber plus t h e seeded f o r e s t land ( t h e p r e s e n t value of t h e s e r i e s of rotations s t a r t i n g on t h e newly seeded land) (cf

.

Johansson and Loef g r e n , 1985):

where: b

=

t h e a g e of t h e standing timber;

T

=

optimal h a r v e s t a g e f o r t h e standing timber; and t

=

optimum r o t a t i o n f o r t h e f u t u r e stands.

By differentiating (3.16) f i r s t with r e s p e c t to t

.

t h e f i r s t - o r d e r condition f o r t h e optimum rotation in t h e f u t u r e s t a n d s i s obtained:

which holds if and only if

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Thus, t h e optimum r o t a t i o n t f o r t h e f u t u r e stands is independent of t h e h a r v e s t a g e T of t h e standing timber. Thus t h e problem c a n b e solved by f i r s t solving independently f o r t h e optimum t % and substituting t h e corresponding optimal land value into equation (3.16) (cf. Johansson and Loefgren, 1985). For t h e optimal h a r - v e s t a g e of t h e standing timber, t h e following condition is obtained:

which can b e p r e s e n t e d in t h e form

From (3.20) it i s easy t o s e e t h a t t h e expected change of t h e growth function (decline in soil productivity) in t h e f u t u r e rotations a f f e c t s t h e h a r v e s t a g e of t h e standing timber through t h e value of seeded f o r e s t land, i.e. through a change in land r e n t o r shadow c o s t of land use. From (3.19) o r (3.20) i t is obvious t h a t t h e expected d e t e r i o r a t i o n in t h e growth of f u t u r e stands will lengthen t h e optimal h a r v e s t a g e f o r t h e standing timber. Because all kinds of growth reduction de- c r e a s e t h e value of b a r e land, t h e i n t e r e s t on f o r e s t land a l s o declines, and t h e r i g h t hand side in (3.20) declines. The equilibrium i s re-established by letting t h e trees grow older.

The r e s u l t is intuitive in t h a t a reduced growth of f u t u r e stands naturally r e d u c e s t h e implicit loss owing t o t h e delay of r e c e i p t s from f u t u r e h a r v e s t s o r land r e n t a t t r i b u t e d t o growing t h e c u r r e n t stand f u r t h e r . A s long as no damage i s observed in t h e standing timber, i t is a l s o n a t u r a l t o utilize t h e growth potential of existing stands longer, ceteris paribus.

3.2.2. Observed damage in the standing timber

Empirically, t h e most r e l e v a n t c a s e may b e t h e observed a c u t e damage (deterioration of t h e health of t r e e s ) in mature stands with t h e consequent reduced r a t e of growth (this i s what i s said t o happen in Central Europe). In terms of o u r setting t h e phenomenon is most a p p r o p r i a t e l y presented as a growth change in t h e standing timber; on t h e o t h e r hand, i t i s natural in t h i s c a s e t o e x p e c t t h a t f u t u r e r o t a t i o n s a l s o d i f f e r from t h e norm (the predicted growth p a t t e r n of t h e relevant

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s i t e under r e f e r e n c e conditions).

To i l l u s t r a t e t h e situation w e write a slight modification of t h e formulation in (3.16). It is assumed t h a t t h e standing timber is s u b j e c t t o a change of t h e "forest dieback" t y p e (Section 3.1.1 above; Figure 1, satisfying t h e r e g u l a r i t y assumption), while t h e specific t y p e of d e t e r i o r a t i o n in t h e f u t u r e stands does not matter:

Again, t h e optimum r o t a t i o n in t h e f u t u r e stands t

*

is solved independently.

Then i t holds f o r t h e optimal h a r v e s t a g e T* of t h e standing timber t h a t

In comparison t o t h e norm o r t h e s t a n d a r d Faustmann r u l e , t h e equations have t w o differences t h a t a f f e c t t h e outcome in opposite directions. F i r s t , t h e e x p e c t e d lower growth c u r v e of t h e f u t u r e stands d e c r e a s e s t h e value of land, h e n c e land r e n t , and ceteris p a r i b u s tends to lengthen t h e optimal h a r v e s t age. On t h e o t h e r hand, t h e observed damage in t h e standing timber a f f e c t s t h e left-hand side of t h e equation. According to discussions above, t h e r e l e v a n t change implies a decreasing r e l a t i v e rate of growth, and hence a s h o r t e r optimal h a r v e s t a g e in comparison to t h e norm. Also, t h e second term on t h e right-hand side of (3.22) i s a f f e c t e d ; t h e decline in t h e standing volume j'(T) r a i s e s , loosely speaking, t h e "effective guiding rate of i n t e r e s t " and ceteris p a r i b u s s h o r t e n s t h e optimal h a r v e s t age.

Owing to t h e s e changes of opposite direction, t h e net effect i s ambiguous, depending on quantitative r e l a t i o n s of d i f f e r e n t e f f e c t s (e.g., t h e r e l a t i v e s t r e n g t h of pollution e f f e c t s in t h e standing timber u s . in t h e f u t u r e stands). However, intuitively w e may e x p e c t t h a t t h e e f f e c t through t h e standing timber dominates. The ef- f e c t through f u t u r e r o t a t i o n s , even if unrealistically considered as known with c e r t a i n t y , tends to be small, not to s a y discounted out of existence, in "European"

f o r e s t r y with long r o t a t i o n periods. Hence w e may conclude t h a t o b s e r v e d damage in t h e standing timber will typically s h o r t e n i t s optimal h a r v e s t a g e as compared t o t h e planned rotation even u n d e r expectations of reduced growth in t h e f u t u r e rotations.

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3.3. Concluding remarks

The r e s u l t s c o n c e r n i n g t h e e f f e c t of pollution-induced c h a n g e s in t h e growth function on t h e optimal h a r v e s t a g e are summarized below. The p o t e n t i a l implications t o t i m b e r supply will b e b r i e f l y discussed.

F i r s t , t h e s t a n d a r d Faustmann model was employed assuming t h a t a l l r o t a t i o n s will b e identical. Regarding t h e d i r e c t i o n of t h e e f f e c t o n optimum r o t a t i o n , t h e s p e c i f i c t y p e of growth c h a n g e , i.e. t h e r e l a t i v e e f f e c t s at v a r i o u s tree a g e s , t u r n e d o u t t o b e decisive:

(1) Damage o b s e r v e d f i r s t in m a t u r e s t a n d s ( t h a t "locally" r e d u c e s t h e r e l a t i v e rate of growth) will s h o r t e n t h e optimum r o t a t i o n as compared t o t h e norm, while

(2) both p r o p o r t i o n a l a n d non-proportional c h a n g e s a f f e c t i n g e n t i r e r o t a t i o n s (causing a " p o o r e r s i t e " o r a delay of growth) will lengthen i t .

Second, t h e e f f e c t s on t h e optimal h a r v e s t a g e f o r existing m a t u r e s t a n d s w e r e examined with allowance f o r t h e growth p a t t e r n of f u t u r e r o t a t i o n s t o d i f f e r from t h e standing timber. Then,

(3) u n d e r e x p e c t e d growth r e d u c t i o n s in t h e f u t u r e r o t a t i o n s , t h e optimal h a r v e s t a g e f o r u n a f f e c t e d standing t i m b e r will b e lengthened.

This case might h a v e some r e l e v a n c e in, f o r example, t h e Nordic c o u n t r i e s , w h e r e n o c o n s i d e r a b l e damage h a s b e e n o b s e r v e d f o r t h e moment ( t h e r e may e v e n h a v e been signs of i n c r e a s e d f o r e s t growth). On t h e o t h e r hand,

(4) a c u t e damage o b s e r v e d in standing timber will typically s h o r t e n i t s optimal h a r v e s t a g e .

By "typically" I r e f e r t o t h e f a c t t h a t f o r e s t damage may r a n g e f r o m minor growth r e d u c t i o n t o s e r i o u s damage (labelled " f o r e s t dieback") t h a t implies immediate s a n i t a t i o n fellings. In t h e usual case w e assume t h a t t h e u n e x p e c t e d r a p i d d e c l i n e in t h e r e l a t i v e rate of s t a n d growth i s marked enough t o outweigh t h e counter-affecting c h a n g e in land r e n t . Intuitively, t h e case might r e s e m b l e t h e situation in C e n t r a l E u r o p e with r e p o r t e d decline in crown densities a n d i n c r e a s e d mortality of trees.

If t h e s h i f t p a r a m e t e r of t h e growth curve would, instead, b e given values a b o v e unity, t h e a n a l y s i s could b e e a s i l y r e i n t e r p r e t e d as t h e e f f e c t of t h e so- c a l l e d '%Oz fertilization" e f f e c t ( s e e Kauppi, 1987). Naturally, t h e impacts of a n exogenous improvement in f o r e s t growth would b e r e v e r s e d in comparison with t h e a b o v e analysis ( s e e Johansson a n d Loef g r e n , 1985).

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Sometimes, t h e maximization of sustained yield could be t a k e n as t h e objective of f o r e s t management. If t h e a v e r a g e output o v e r t h e rotation period were t o b e maximized, t h e qualitative r e s u l t s would b e identical t o those p r e s e n t e d h e r e . This, in f a c t , is no s u r p r i s e because i t is well-known t h a t t h e Faustmann r o t a t i o n period a p p r o a c h e s t h e o n e maximizing sustained yield in t h e case where t h e rate of in- terest tends t o z e r o .

Turning t o t h e implications of t h e changes in f o r e s t r o t a t i o n s t o roundwood supply, t h e limitations of t h e model should b e recognized. The r e l e v a n c e of t h e r e s u l t s is p a r t l y hampered by t h e underlying c e t e r i s p a r i b u s assumption concerning t h e constant level of timber p r i c e . Ignoring t h i s f o r t h e moment, w e may rea- sonably summarize t h e e f f e c t s as "potential" supply implications. These may b e considered at two levels. First, t h e effect on short-term supply refers t o t h e r e s p o n s e during t h e adjustment period, i.e. when t h e changes have been o b s e r v e d and t h e adjustment to t h e new optimum rotation starts by cutting stands t h a t have t u r n e d o v e r m a t u r e or by letting trees grow o l d e r (cf. Clark, 1976). The long-term effect, on t h e o t h e r hand, refers t o "supply" in t h e s e n s e of maximum sustained h a r v e s t levels ( t h e quantity f ( t ) / t ), i.e. t h e supply potential as a flow concept.

The qualitative supply implications, again in comparison t o t h e norm o r "fore- cast" without f o r e s t damage, are summarized in Table 1 f o r t h e cases with identical rotations. Table 1 shows t h e intuitive expectation of increased short-term supply d u e t o forest damage, and t h e decline in long-term supplies in all c a s e s , while t h e l a s t two short-term r e a c t i o n s are h a r d t o g r a s p intuitively. Note t h a t besides t h e r o t a t i o n period, t h e growth c u r v e or t h e volume h a r v e s t e d at r o t a t i o n a g e also changes. So, f o r example, long-term supply in t h e proportional case would even b e r e d u c e d without r e g e n e r a t i o n c o s t (i.e. if t h e rotation period remains unchanged).

The cases with non-uniform growth c u r v e s are m o r e interesting. The s h o r t - term r e s p o n s e , again, refers to adjustment e f f e c t s due t o r o t a t i o n changes, but t h e long-term r e s p o n s e may b e examined at two time spans. In Table 2, t h e quantity j ' ( T ) / T refers t o t h e p r e s e n t stands (standing timber), and j ' ( t ) / t t o t h e flow

f r o m f u t u r e stands.

In areas with no damage in standing timber, t h e expectation of growth reduction in f u t u r e stands would imply a decline in short-term supply through a lengthening h a r v e s t age. On t h e o t h e r hand, observed damage will i n c r e a s e s h o r t - t e r m supply whenever t h e shortening of t h e rotation period dominates o v e r t h e change in land r e n t .

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T a b l e 1. Potential supply r e s p o n s e s f r o m once-and-for-all changes in t h e forest growth c u r v e ("+" means positive e f f e c t ;

"-"

means negative effect).

Case Effect on Supply Response

Rotation

Period S h o r t - Long- term t e r m

Damage in t h e

- + -

Mature F o r e s t

" P o o r e r Site"

+ - -

(proportional)

"Lower Intensity"

+ - -

(non-orooortional)

T a b l e 2. Potential supply r e s p o n s e s when standing timber and f u t u r e stands d i f f e r

Case Effects on Supply r e s p o n s e s

h a r v e s t a g e

of standing From Standing From Future

timber Timber Stands

Short- f ( T ) / T y ( t ) / t Term

Expected Growth

Reduction in

+ - + -

Future Stands Only Observed Damage

in Standing

- + - -

Timber

In t h e above p a r t i a l equilibrium analysis of t h e optimal h a r v e s t a g e , a p a r a m e t r i c p r i c e w a s assumed. However, owing to s h i f t s in timber supply both in t h e s h o r t t e r m (changes in t h e timing of fellings of standing timber) and t h e long t e r m (changing flows of wood), t h e equilibrium position of t h e roundwood market, and hence timber p r i c e , will b e a f f e c t e d whenever t h e demand f o r wood is not p e r - fectly elastic. Timber p r i c e , o n t h e o t h e r hand, matters f o r t h e choice of optimum r o t a t i o n (e.g. Johansson and Loefgren, 1985)'. T h e r e f o r e , if all t h e exploitable

%he formulation in Section 3.2 allows the treatment of price a s a function of time (age), i.e. exo- genous price variation (cf. Lohmander, 1984; Comolli, 1984). This will not do here because there i s a two-way causation between price and harvest age.

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forests are not equally s u b j e c t to damage, t h e r e will b e a two-fold effect on aggre- g a t e supply. In what follows, a conventional supply-demand framework will b e introduced to make t h i s point more c l e a r l y .

The s t a n d a r d f o r e c a s t concerning t h e economic consequences of f o r e s t dam- a g e is a considerable pulse of high timber supply in t h e s h o r t t e r m , based o n anti- cipations of sanitation fellings. Increased damage would r e s u l t in a lower p r i c e for timber (e.g. Kroth and Bartelheimer, 1984). On t h e o t h e r hand, t h e reduced productivity of forest ecosystems would mean a decline in t h e sustainable h a r v e s t (potential supply) in t h e long t e r m . Intuitive as t h e above views are, t h e y consider sanitation fellings and t h e decline in p r i c e as t h e o n e and only response. This, in t u r n , seems to r e q u i r e a n implicit underlying assumption t h a t all f o r e s t s in t h e economy under consideration will b e damaged, s o t h e question of i n d i r e c t e f f e c t s does not a r i s e .

Because t h e e f f e c t s of pollution v a r y widely with r e s p e c t to soil types, tree species, etc., i t might b e r e a s o n a b l e to analyze t h e problem theoretically, assuming t h a t just one p a r t of t h e economy's f o r e s t area i s damaged while t h e o t h e r remains unaffected. Under t h i s assumption timber supply consists of t w o sub- s e c t o r s t h a t i n t e r a c t t h r o u g h t h e market. An exogenous s h i f t of supply from t h e damaged f o r e s t s will now have f u r t h e r effects on t h e undamaged f o r e s t s (cf. Dyks- tra and Kallio, 1987).

4.1. Short-term effects

The m a r k e t for roundwood is formulated by making some v e r y g e n e r a l and conventional assumptions concerning t h e supply and demand functions. Actually, w e only know t h a t t h e short-term supply of timber depends positively on i t s p r i c e . Such a n assumption may b e theoretically justified by various models (e.g., s e e : Johansson and Loefgren, 1985; Binkley, 1987), including t h e r e s p o n s e of t h e above Faustmann model in t h e adjustment period, but only u n d e r t h e p a r t i c u l a r assumption of s t a t i c , unitary e l a s t i c p r i c e expectations. No matter what t h e underlying t h e o r e t i c a l model is, empirical r e s u l t s quite consistently s u p p o r t t h e normal, positively sloped supply c u r v e .

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The model is written as follows (subscripts are used throughout t o denote p a r - tial o r o r d i n a r y derivatives of functions):

(4.1) Q=: s D @ , 8 ) $ 2 0 , s : > O Q:

= s u b ) S;>O QD

=

d b )

%

< O

Q

= @ +

QJ (equilibrium condition).

The f i r s t equation defines supply from damaged forests as a non-decreasing function of t h e c u r r e n t p r i c e f o r timber and increasing in

8,

a p a r a m e t e r describ- ing t h e level of pollution. Increasing

8

means increased pollution and f o r e s t dam- a g e , hence increasing supply in t h e s h o r t t e r m (cf. Sections 3.1.1 and 3.2.2 above).

Second, t h e supply of wood from undamaged f o r e s t s i s a n increasing function of p r i c e only, while t h e demand f o r timber ( t h i r d equation) i s d e c r e a s i n g in p. The equilibrium is given as a n equality between demand and a g g r e g a t e supply, t h e l a t t e r being a horizontal summation of supplies from both sub-sectors.

In an initial equilibrium t h e following conditions hold:

To point out t h e qualitative effects of increased pollution on t h e equilibrium p r i c e and quantities in t h e market, suppose a change in

8.

Then t h e following c o m - p a r a t i v e s t a t i c r e s u l t s hold t r u e ( f o r proof, see Appendix 2):

s;(-($ + s p )

u

d Q D / d @

=

I J I >

⁰ ^, ^and

where

I

J

I

=s:-dp

+

S;>O.

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The r e s u l t s in (4.3) unambiguously t e l l t h a t i n c r e a s e d pollution r e s u l t s in in- c r e a s e d sales (fellings) from t h e damaged f o r e s t s a n d in a lower p r i c e f o r wood.

What i s i n t e r e s t i n g i s t h e f a c t t h a t t h e s a l e s from undamaged f o r e s t s at t h e same time decline, which i s d u e t o t h e i n d i r e c t e f f e c t of a c h a n g e in p r i c e . The t o t a l quantity t r a d e d being t h e sum of t h e t w o s u b - s e c t o r s , its c h a n g e i s t h e n e t e f f e c t (sum) of t h e two changes:

The n e t e f f e c t of t h e o p p o s i t e c h a n g e s o n t h e quantity t r a d e d i s unambiguously positive. However, n o t e t h a t t h e n e t e f f e c t i s l e s s t h a n t h e mere i n c r e a s e of supply from t h e damaged f o r e s t s . Thus, t h e e f f e c t of f o r e s t damage o n t h e equilibrium i s p a r t l y c a n c e l l e d by t h e simultaneous r e s p o n s e of t h e undamaged f o r e s t s to t h e p r i c e change. The magnitude of c h a n g e s in p r i c e a n d quantity d e p e n d s on t h e s l o p e s of supply a n d demand c u r v e s .

W e n e x t modify t h e model slightly by introducing t i m b e r p r i c e e x p e c t a t i o n s i n t o t h e supply function of undamaged f o r e s t s . In c o n t r a s t with t h e a b o v e myopic supply b e h a v i o r , r e a c t i n g only t o c u r r e n t p r i c e changes. i t i s assumed t h a t t i m b e r s u p p l i e r s make f o r e c a s t s of f u t u r e s h o r t - t e r m p r i c e fluctuations. The e x p e c t a t i o n s are a d j u s t e d ( a l b e i t imperfectly) on t h e b a s i s of o b s e r v e d p r i c e changes. Expec- t a t i o n s are assumed not t o e n t e r t h e supply function of t h e f o r e s t s s u b j e c t t o dam- a g e . The intuitive r e a s o n i n g i s t h a t along with o b s e r v e d damage, t h e h e a l t h a s p e c t s of t h e f o r e s t t e n d t o dominate, a n d " t h e r e i s n o s p a c e f o r p r i c e speculation".

The model remains t h e same as in (4.1) e x c e p t f o r t h e supply equation f o r t h e undamaged f o r e s t s , which now r e a d s :

Thus, supply from unaffected f o r e s t s i s a d e c r e a s i n g function of p r i c e e x p e c t a t i o n s p e , which in t u r n are a n i n c r e a s i n g function of t h e c u r r e n t p r i c e . Because t h e equilibrium p r i c e d e p e n d s negatively o n t h e pollution p a r a m e t e r @, t h e p r i c e e x - p e c t a t i o n p e a l s o becomes a d e c r e a s i n g function of t h e l e v e l of f o r e s t damage.

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The r e s u l t is t h a t t h e e f f e c t s a r e not unambiguous, because t h e signs of t h e comparative s t a t i c derivatives are not unambiguously known without r e f e r e n c e t o t h e r e l a t i v e magnitudes of t h e d i f f e r e n t p r i c e effects involved:

-Sb D

>

⁰

(4.6) d p / d @

= - -

I J I <

^'

d ~ ~

=

/ d ~

- >

⁰ , and

I J I <

' ( d p e / d p ) . where

1

J

1 =

s:

-

dp

+

sp

+

s

P

"

The ambiguity r e s u l t s from t h e change in p r i c e expectations and t h e i r e f f e c t on timber supply f r o m undamaged f o r e s t s . However, note t h a t by assumption, t h e elasticity of expectations i s between z e r o (expectations independent of c u r r e n t p r i c e changes) and unity ( p e r f e c t adjustment of expectations), i.e.

0

<

d p e / d p

<

1. Then i t is sufficient (but not n e c e s s a r y ) f o r

I

J

1 >

⁰and t h e r e s u l t s t o b e unambiguous t h a t t h e effect of t h e e x p e c t e d p r i c e on supply f r o m undamaged f o r e s t s , s U i s not g r e a t e r in absolute value t h a n t h a t of t h e c u r r e n t

P " '

p r i c e . S$ In t h e long-term, p r i c e i s at t h e long-run equilibrium level (path). and expectations tend t o b e c o r r e c t , s o t h e e x p e c t e d p r i c e i s equal to t h e c u r r e n t (actual) p r i c e . Then p e and p collapse t o t h e same variable, and t h e e f f e c t of t h e p r i c e v a r i a b l e is t h e sum of t h e two opposite effects.

Hence, a n i n t e r p r e t a t i o n f o r t h e r e q u i r e d non-negativity of t h e sum s:

+

s$

i s t h a t t h e long-term supply c u r v e of t h e undamaged f o r e s t s is not backward bending.3 If t h i s i s t r u e t h e n t h e signs in (4.6) are t h e same as in (4.3). and t h e changes in p r i c e and t h e t o t a l quantity t r a d e d are l a r g e r t h a n without decreasing p r i c e expectations. Otherwise t h e situation is ambiguous depending on r e l a t i v e magnitudes, and unfortunately, t h e possibility of a backward bending long-term supply c u r v e ' ~ u ~ ~ o s e the market i s in an equilibrium where timber price p i s unchanging over a considerable time. With all previous values of p constant (p

=

p ) , adaptive expectations imply that eventu- a l p the exeectation of price in period t

+

1 , held in period

t ,

equals the actual price or p t ,t +1

=

p

=

p t

.

^Withp e

=

p , t h e current and expected price e f f e c t i v e l y collapse t o one variable, and the net e f f e c t i s the sum of the two opposite e f f e c t s .

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may exist (see next section).

4.2. Long-run effects

Conceptually, short-term supply means a supply from a given stock resulting from t h e adjustment of t h e standing volume t o a new equilibrium. On t h e o t h e r hand, we may b e i n t e r e s t e d in t h e impact on long-run supply. This c a n b e taken as a flow concept, t h e sustainable h a r v e s t level, which r e q u i r e s a time span long enough f o r t h e growing stock t o adjust t o t h e new equilibrium level. In what follows, t h e above model i s straightforwardly used by making some modifications t o t h e sign assumptions.

In t h e long run, pollution implies a lower sustainable h a r v e s t due t o t h e reduced productivity of (a p a r t of) f o r e s t soils. Hence, t h e pollution p a r a m e t e r @ now a f f e c t s t h e supply from damaged f o r e s t s c o n t r a r y t o t h e above case, and we have s;

<

0.

Regarding t h e e f f e c t of p r i c e on long-term supply, two possibilities will b e considered. Thinking about long-run supply in terms of t h e Faustmann model, it is well-known t h a t t h e r u l e implies a backward-bending supply c u r v e if rotation a g e i s t h e only decision v a r i a b l e (Hyde, 1980; Johansson and Loefgren, 1985; Binkley, 1985). An economic i n t e r p r e t a t i o n of t h e situation is t h a t silvicultural e f f o r t and land use fail t o r e a c t t o t h e (expected) p r i c e , s o t h e only effect of a n increased p r i c e level is a s h o r t e r rotation, and subsequently lower sustainable h a r v e s t f ( t ' ) / t '

.

The supply equations under t h e s e assumptions are as follows:

The negative slopes of supply c u r v e s make t h e situation a bit complicated.

Namely, t h e r e i s a possibility of market instability (an unstable p r i c e adjustment process) t h a t emerges when t h e a g g r e g a t e supply c u r v e i s f l a t t e r t h a n t h e demand c u r v e (e.g., see Johansson and Loefgren, 1903). Because t h e r e have been no signs of such a phenomenon in t h e real world, w e r e s t r i c t t h e analysis t o t h e normal case with a s t a b l e adjustment. Hence, t h e following r e s u l t s are derived by assuming t h a t a g g r e g a t e supply i s s t e e p e r than t h e demand curve; i.e.,

$, <

:s

+

.:s Then t h e Jacobian determinant i s positive, and t h e following r e s u l t s are obtained:

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The derivations are not r e p e a t e d because t h e comparative s t a t i c derivatives given in (4.3) and Appendix 2 hold e x c e p t f o r changed signs of t h e derivatives of t h e r e l e v a n t functions.

A s could b e expected, decreasing sustainable h a r v e s t due t o r e d u c e d soil productivity in damaged areas will induce a h i g h e r p r i c e level. But again, t h e r e is a simultaneous r e a c t i o n from undamaged areas. Because t h e p r i c e level r i s e s , stands will b e c u t at a younger a g e , implying lower sustainable h a r v e s t s a l s o in undamaged a r e a s . This s t r e n g t h e n s t h e decline in a g g r e g a t e supply.

Declining supply from undamaged f o r e s t s , however, depends critically on t h e assumption of negative p r i c e elasticities. Suppose n e x t t h a t long-run supply from both sub-sectors reacts positively t o timber p r i c e . Then w e have supply equations

The more conventional effect of p r i c e on long-run supply i s obtained from t h e Faustmann model by assuming a n endogenous silvicultural e f f o r t (e.g

.

, Hyde, 1980).

The intuitive reasoning behind t h e positive effect i s t h a t , in undamaged a r e a s , a h i g h e r p r i c e a t t r a c t s more advanced silvicultural p r a c t i c e s (more efficient regen- e r a t i o n , fertilization, e t c ) , and land use might a l s o react. On t h e o t h e r hand, in damaged o r susceptible areas a higher p r i c e might justify investments in t h e pro- tection of f o r e s t soils against t h e impacts of acidification (improving t h e Ca and Mg balances, p e r h a p s ) o r more a p p r o p r i a t e management p r a c t i c e s (felling o r regen- e r a t i o n methods, f o r example). Under t h e s e assumptions, t h e r e s u l t s are unambiguously as follows:

(4.10) d p / d B > O , d @ / d / 9 < 0 , d Q U / d

6 >

0 , and d Q / d B

< o .

The r e s u l t s indicate t h a t timber p r i c e would r i s e , y e t much l e s s t h a n under negatively sloped supply c u r v e s . The t o t a l quantity t r a d e d would a l s o decline as b e f o r e but l e s s t h a n above

( 1

J

1

in t h e denominator i s g r e a t e r ) . A qualitative d i f f e r e n c e i s t h a t supply from undamaged f o r e s t areas would i n c r e a s e in t h e long

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r u n due t o a positive change in p r i c e level. Hence, t h e p r i c e e f f e c t would offset p a r t of t h e decline in supplies from damaged areas, t h a t is, t h e neutralizing e f f e c t is t h e m i r r o r image of t h a t in t h e s h o r t term.

5.

DISCUSSION

The e f f e c t s of f o r e s t damage a t t r i b u t e d t o a i r pollutants on optimum f o r e s t rotations and t h e equilibrium position of t h e roundwood market were considered above in t h e o r e t i c a l terms. The main r e s u l t s and conclusions c a n b e summarized as follows.

(1) Once-and-for-all changes of t h e growth function may e i t h e r s h o r t e n o r lengthen t h e optimum rotation, depending on t h e specific t y p e of change.

(2) Concerning t h e e f f e c t of a c u t e f o r e s t damage observed in c u r r e n t mature stands, t h e phenomenon i s to b e viewed as a n unexpected decline of t h e rela- tive growth r a t e of t h e f o r e s t . This will imply a s h o r t e n e d optimal h a r v e s t a g e f o r t h e standing timber, hence i n c r e a s e d supply in t h e s h o r t term. Theoreti- cally, however, e x p e c t e d growth changes in f u t u r e r o t a t i o n s are a l s o impor- t a n t h e r e according t o t h e Faustmann r u l e .

(3) In t h e s h o r t term, considerable f o r e s t damage would r e s u l t in a pulse of in- c r e a s e d supply of wood t o t h e m a r k e t , hence h i g h e r quantity t r a d e d and a lower p r i c e for t i m b e r a 4 However, t h e indirect e f f e c t s through t h e p r i c e change on supply from t h e undamaged f o r e s t s should b e recognized. Thus, t h e e f f e c t of damage on a g g r e g a t e supply may b e p a r t l y neutralized by a p r i c e - induced reallocation of fellings. This r e s u l t c a n b e t a k e n as a simple formal justification f o r t h e consideration of t h e neutralizing e f f e c t in quantitative s c e n a r i o s made with t h e IIASA f o r e s t s e c t o r model (Dykstra and Kallio, 1987).

The situation i s complicated by short-term p r i c e expectations held by timber suppliers.

(4) In t h e long r u n , declining productivity of (a p a r t of) t h e f o r e s t soils implies a reduced sustainable h a r v e s t , i.e. d e c r e a s e d supply and a h i g h e r p r i c e f o r timber. However, t h e magnitude of t h e quantitative e f f e c t depends on whether ' ~ r o t h and B a r t e l h e i m e r (1984) s u g g e s t e d t h a t along w i t h i n c r e a s e d damage, t i m b e r s u p p l y becomes more p r i c e i n e l a s t i c . If t h i s is t h e c a s e , a l o w e r p r i c e l e v e l is n o t t h e o n l y change. Assuming v a r i a - t i o n s i n t h e demand f o r t i m b e r ( s h i f t i n g demand schedule), a s t e e p e r s u p p l y c u r v e w i l l r e q u i r e l a r g e r p r i c e c h a n g e s f o r a g i v e n i n c r e a s e i n q u a n t i t y traded. Thus, g i v e n s h i f t i n g demand, p r i c e v a r i a t i o n s t e n d t o be s t r e n g t h e n e d .

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t h e l e v e l of silvicultural e f f o r t r e s p o n d s t o t h e p r i c e c h a n g e s , a n d w h e t h e r t h e long-run supply from undamaged f o r e s t s is positively o r negatively p r i c e e l a s t i c .

The r e l e v a n c e of t h e r e s u l t s , as a t h e o r e t i c a l b a s i s f o r quantitative s c e n a r i o s , f o r example, d e p e n d s on t h e empirical r e l e v a n c e of both growth and b e h a v i o r a l assumptions. In t h e analysis of optimum r o t a t i o n s , t h e assumption of present-value-maximizing b e h a v i o r i s made. However, qualitatively identical r e s u l t s will b e o b t a i n e d if t i m b e r management is volume-oriented (sustainable yield-maximizing). Regarding a c u t e damage in c u r r e n t s t a n d s , more r a p i d fellings a n d i n c r e a s e d s h o r t - t e r m supply are c l e a r from p u r e l y biological considerations.

I t should b e noted h e r e t h a t t h e optimal h a r v e s t a g e r e f e r s to t h e final c u t in even-aged management ( s t r i c t l y speaking, d e a r - c u t t i n g ) , being oversimplified in t h e s e n s e t h a t s p e c i a l management p r a c t i c e s , e.g. s e l e c t i v e h a r v e s t i n g of damaged t r e e s , may become more important in p r a c t i c a l terms as f o r e s t damage p r o g r e s s e s . ⁵

On t h e o t h e r hand, t h e assumptions behind t h e s h o r t - t e r m c h a n g e s in t h e roundwood m a r k e t equilibrium are v e r y g e n e r a l a n d n o n - r e s t r i c t i v e ( t h e usual s l o p e s of supply a n d demand c u r v e s ) . The price-induced neutralizing e f f e c t d u e to t h e r e s p o n s e f r o m undamaged f o r e s t s i s a l s o likely t o h a v e i t s c o u n t e r p a r t in f o r e s t r y u n d e r a non-price-oriented or price-insensitive management. A r e a l l o c a - tion of fellings may b e e x p e c t e d t o t a k e p l a c e u n d e r t h e institutional p r a c t i c e s of public f o r e s t s , or in planned economies as well (Dykstra a n d Kallio, 1987). The p r i c e sensitivity of long-run t i m b e r supply, o n t h e o t h e r hand, i s a n o p e n issue.

Although t h e intuitive r e a s o n i n g behind t h e analysis, h e n c e t h e qualitative implications, s e e m f a i r l y r e a l i s t i c , nothing c a n b e concluded a b o u t t h e q u a n t i t a t i v e significance of t h e economic consequences. O t h e r limitations a l s o e x i s t , s o t h e pa- p e r i s at b e s t a f i r s t approximation of t h e problem. Some of t h e s e limitations are pointed o u t below.

The analysis i g n o r e s t h e i n h e r e n t u n c e r t a i n t y involved in t h e problem. R a t h e r t h a n as c h a n g e s p e p r e s e n t e d deterministically, t h e e f f e c t of pollution could b e r e p r e s e n t e d as c h a n g e s in t h e s t o c h a s t i c p r o p e r t i e s of t h e growth p r o c e s s ( f o r some r e s u l t s on optimum c u t t i n g r u l e s with random elements, see Johansson and 5 ~ n mountainous r e g i o n s w h e r e t h e m o s t a e r i o u s damage has been r e p o r t e d , no l a r g e - s c a l e c l e a r - c u t t i n g i s possible. However, i n t h e l i m i t t h e arguments here apply t o individual t r e e s ; s o t h e

" f o r e s t stand" may r e f e r t o v e r y small plots o f f o r e a t land.