Renewable Resource Economics - Optimal Rules of Thumb

Working Paper

RENEWABLE RESOURCE ECONOMICS

-

OFl'IMAL RULES

OF

THUMB

A E.

Andersson (IIASA) P. Lesse (CSIRO)

October 1904 WP-84-84

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT M)R QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ENERABLE RESOURCE ECONOMICS

-

OPTIMAL RULES

OF

TI-MdI3

A

E. Andersson (IIASA) P. Lesse (CSIRO)

October 1984 WP-84-84

Working Papers a r e interim reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

The objective of t h e Forest Sector Project a t IIASA is t o study long- t e r m development alternatives for t h e forest sector on a global basis.

The emphasis in t h e Project is on issues of major relevance t o industrial and governmental policy m a k e r s in different regions of the world who a r e responsible for forestry policy, forest industrial strategy, a n d related trade policies.

The key elements of structural change in t h e forest industry a r e related t o a variety of issues concerning demand, supply, and interna- tional trade of wood products. Such issues include t h e development of t h e global economy a n d population, new wood products and substitution f o r wood products, future supply of roundwood and alternative fiber sources, technology development for forestry a n d industry, pollution regulations, cost competitiveness, tariffs a n d non-tariff t r a d e barriers, etc. The a i m of t h e Project is t o analyze t h e consequences of future expectations a n d assumptions concerning such substantive issues.

In this article t h e supply of roundwood is discussed within t h e framework of renewable resource economics. Quantitative guidelines for forestry a r e derived and tested against possible disturbances t o planta- tion managenent conditions. It is shown t h a t certain rules of t h u m b for renewable resource management a r e robust with respect t o a broad s e t of incidental disturbances, e.g., weather conditions, market fluctuations, etc.

Markku Kallio Project Leader

Forest Sector Project

CONTENTS

1. INTRODUCTION

2. A SIMPLE MODEL FOR OPTIMUM MANAGEMENT OF FORESTS 3. CONSEQLrENCES OF USING THE FOREST AS A POLLUTION SINK 4. THE "SAFE" OPTIMUM MANAGEMENT OF FORESTS

4.1 The Modified Growth Equations

4.2 Optimum Management in t h e Presence of Incidental Factors

5. OPTIMUM THINNING POLICIES MADE SIMPLE (CONCLUSIONS) REFERENCES

RENEWABLE RESOURCE ECONOMICS

-

OPTIWK

RULES

OF THUMB

A

E. Andersson (IIASA) and P. Lesse

(CSIRO)

In this paper we formulate a simple model for optimum manage- m e n t of large forests. When seeking t h e optimum management policy, we aim t o reconciliate t h e ever-present conflict between t h e forest a s a valuable capital resource t o be exploited, and as a valuable biotope t o be protected and preserved.

We a r e aware of previous work in this field, which already led t o valu- able results. Kilkki and Vaisanen (1969) developed an optimum thinning policy for Scotch pine stands in Finland. Their work has been Further refined by Clark (1976). In both cases, t h e problem has been treated a s t h a t of linear optimum control leading to a "bang-bang" cutting policy.

There is also a large g r o u p ' o f models using operations research approaches. These models can often be viewed as resource management

simulators. A r e c e n t example is t h e integrated s e t of models developed in New Zealand by Garcia (1981), Levack and Jennings (1981), and Lee (1981, 1982). Large scale (20,000 variables, 10.000 constraints) mathematical programming model for forestry management were developed by Whyte and Baird (1982, 1983) and Dykstra (1984). (See also Kallio, Andersson, a n d Seppal5 1984 and t h e forthcoming book on renew- able resource a n d forest economics by Lofgren and Johansson 1985).

In this paper, we concentrate on deriving qualitative rules of t h u m b plantation for management. This approach takes into a c c o u n t both economic and some simple ecological aspects of the problem. For an easy interpretation, we derive the qualitative rules using a very simple model a t ArsL Later, we investigate t h e robustness of t h e s e rules, i-e., we t e s t t h e i r validity by gradually relaxing some of t h e assumptions underlying t h e simple model.

2.

A

SIMPLE MODEL FOR OFl'IMUM MANAGEMJCNT OF

FORESTS

The aim of this section is t o formulate a simple model capable of generating a s clear-cut a n d hard-boiled conclusions a s possible. We a r e primarily interested in management of large areas, a n d h e n c e do not address t h e problem of t h e optimum period. The model i s based on t h e following simplifying assumptions:

(a) Harvested volume h is proportional t o t h e standing volume z, i.e. h ( t )

=

u ( t ) z ( t ) where u ( t ) E [O; u,,] is a control variable called thinnang e f f o r t .

(b) The n e t revenue derived f r o m t h e a m o u n t harvested p with prices a s s u m e d t o be known is independent of t h e scale of h a r - vesting a n d time. Accordingly, t h e n e t revenue per u n i t har- vested,

p,

is a constant.

(c) Growth of wood is given by t h e differential equation

i =

az

-

b z 2 , ₍₁₎

which is t r e a t e d a s t h e only c o n s t r a i n t relevant t o o u r problem.

(d) Parts or t h e whole of t h e plantation can be sold a t a n y t i m e in a market. This assumption implies t h a t t h e r e i s no t e r m i n a l d a t e a t which t h e optimization m u s t e n d

( e ) The plantation owner maximizes t h e profit discounted over time by

6,

which i s t h e maximal r e a l r a t e of i n t e r e s t in a n y seg- m e n t of t h e capital m a r k e t . Thus, the plantation owner s e e k s to

subject t o c o n s t r a i n t (1).

Following Clark (1976), we formally derive t h e Euler Lagrange equa- tion corresponding t o ( I ) , (2). By inserting for uz

=

rzz

-

b z 2

- ⁱ

^{i n t o}

(2). we obtain

The corresponding Euler Lagrange equation leads t o t h e expression for t h e optimal path z

a n d h e n c e

The equation (3) defines a singular solution to t h e l i n e a r optimum con- trol problem ( I ) , (2). The control r u l e i s t o use t h e maximum effort of harvesting u,,, whenever z

>

z *, a n d stop thinning, if z

<

^z*. Along t h e o p t i m u m path, t h e optimum policy should satisfy

u z *

=

^ux*

-

b ~ * ~ , a n d h e n c e if z * # 0

Note t h a t t h e optimal steady-state harvest, equal t o

2 u

=

( a 2

-

d2)/4b, is always Less t h a n the muximum sustainable yield. which equals t o a2/ 4 b . The relative difference compared to max- i m u m yield i s (6/ a)'. For example, if t h e p a r a m e t e r a corresponds t o 10% a n n u a l growth a n d t h e i n t e r e s t r a t e 6 is 5%. t h e n t h e difference is 25%.

The optimum thinning policy t h u s leads t o a stationary (equili- b r i u m ) s t a t e . This equilibrium s t a t e is stable, provided t h a t a

>

6, a s c a n b e shown by introducing a new variable X

=

2-z*. a n d by transforming (1) i n t o

The asymptotic stability of t h e solution X

=

0 c a n be proven, e.g., by using t h e positive definite Lyapunov function

The equation (3'). (4) t h u s define a stable equilibrium point, which may be called a b i o - e c o n o m i c e q u i l i b r i u m . We have just proven the following Proposition:

PROPOSITION 1: The m a z i m i z a t i o n of n e t p r o f i t (Z), s u b j e c t t o t h e g r o w t h e q u a t i o n

( I ) ,

l e a d s t o a s t a b l e b i o e c o n o m i c e q u i l i - b r i u m g i v e n by e q u a t i o n s (39, (4), i f

the

mazimum e f f o r t of t h i n n i n g e z c e e d s the a r i f h r n e t i c a v e r a g e o f t h e l i n e a r g r o w t h r a t e a n d t h e d i s c o u n t r a t e (4).

The original problem can be easily transformed into an equivalent Hamil- toniari maximization problem:

In equilibrium we expect

As 2 # 0, in any sustainable plantation, this implies t h a t -A = j T e d f . As 6.p a r e parameters, i t follows t h a t

The problem of A+O can easily be avoided by inserting a requirement t h a t t h e standing volume should be kept always above a certain positive value. Our second proposition car, thus be formulated.

PROPOSITION 2: U n d e r t h e c o n d i t i o n s p e c i f i e d in m o d e l (5), t h e r a t e o f m t e r e s t is e q u a l t o t h e r e l a t i v e r a t e of c h a n g e o f t h e p l a n t a t i o n ' s v d u e ( m e a s u r e d in t e r m s of t h e s h n d o w p r i c e 6).

The interpretation of this proposition is obvious. If t h e plantation value would change slower than the r a t e of interest requires, t h e r e i s an inducement for t h e owner t o invest in another object, carrying t h e increase of value 6. If t h e r a t e of value increase is larger t h a n t h e r a t e of interest, resources a r e withdrawn from t h e alternative uses, until t h e condition is fulfilled. Such an adaptation could occur both in t h e non- forestry and t h e forestry markets.

This adaptation is closely related t o t h e harvesting effort as c a n be observed by t h e following condition

or by substituting expression (6) into (7):

A

=

AIL

+

h a

-

Zhbz

-

h u or

Above i t has been shown t h a t ) ; / A m u s t equal t h e i n t e r e s t r a t e 6 a t an optimal trajectory. The only way of achieving this i s by adjusting t h e size of t h e standing volume t o this requirement. Such a procedure c a n be illustrated in a simple diagram (see Figure 1). The Figure shows how t h e steady s t a t e standing volume z * increases from z; t o

z i ,

when t h e r a t e of i n t e r e s t changes from

b1

t o d2.

COROLLARY

1 : There is a danger of extinction of t h e whole standing volunie if a

<

6; i.e., if t h e real r a t e of i n t e r e s t is larger t h a n t h e coefficient of linear growth in t h e biological equation (1).

Figure 1. The relative role of growth -as function of standing t volume Z . Z

This c a n easily happen in a society where t h e industrial growth r a t e is high while t h e ecological conditions p r e v e n t a high biological growth r a t e . P a r t s of japan a n d similar c o u n t r i e s with a high r a t e of growth (and a correspondingly high real i n t e r e s t r a t e in t h e i r industrial sectors) fall into t h i s category. These c o u n t r i e s could lose c e r t a i n biological species, presumably due t o violating t h e condition specified by

Corol-

lnry 1.

CoroUa7y 1 t h u s illustrates a g e n e r a l r u l e t h a t decision making in forestry m a n a g e m e n t h a s t o be based on both ecological a n d economic facts.

3. CONSEQUENCES OF USING

THE

FOREST AS A

POLLUTION

SINK

Forests a c t as powerful pollution sinks, especially in industrialized c o u n t r i e s with moderate t o warm a n d h u m i d climatic conditions. Thus

we would, for most situations, need t o modify o u r analysis t o include con- siderations of t h e forests being of value both as flow g e n e r a t o r s and as s t o c k s .

The simple model is t h e n reformulated in t h e following way:

maximize

J 6 . u

^-I + o . z ) e d t Ir,u1 ⁰

s u b j e c t to:

= az -

b z 2 - uz (10)

o i n t h i s case denotes t h e n e t ecological value of t h e forest stock, i.e., t h e value of t h e forest a s a pollution sink, which, f o r obvious reasons depends on t h e standing volume.

The corresponding Hamiltonian formulation is

m a x

f

~ d t

= f

@uz + o z ) e t

-

A(= -bz2-) luP1 0 0

Following t h e s a m e procedure a s in t h e l a s t section we obtain p e d t = - A a n d ) ; / ~ = d

The condition

leads t o

and this implies t h a t t h e steady s t a t e (optimal) standing forest volume is deterrnined by t h e expression

The determination of t h e r a t e of harvesting, corresponding to steady state, is a slight extension of t h e earlier results (cf. (4))

The interpretation of equation (14) can be formulated as a C o r o U a r y 2.

COROLLARY 2 : If t h e standing volume of forest is valued p e r se (i.e., as a pollution-reducing ecological asset), the steady s t a t e standing volume increases proportionally t o t h e fraction value of t h e forest p e r se, over n e t value of forest as a material har- vested resource.

The procedure for determining w / p will not be addressed in this paper. However, determination of t h e relative valuation of t h e standing volume is one of t h e most complicated issues within t h e framework of public goods theory.

The simple rules for optimum management of forests derived in the preceding sections were obtained on t h e basis of a mathematical model.

This model i s certainly not an exact representation of reality. The ques- tion arises, how sensitive a r e t h e management rules to changes of t h e model. For example, should t h e conclusions reached on the basis of the simple model remain valid for a whole class of plausible models, one would feel m u c h more confident, when implementing t h e model in prac- tice. Alternatively, t h e parameters of the simple model can be expected to depend on unpredictable exogenous factors (e.g., weather, insect

attacks, etc.) a n d on slowly evolving variables (like age a n d o t h e r biologi- cal distribution factors). It is i m p o r t a n t t o know how t h e p a r a m e t e r changes influence t h e m a n a g e m e n t policies. To improve o u r under- standing of t h e s e problems, t h e simple model of t h e preceding sections is modified a n d generalized a n d t h e conditions sufficient t o preserve t h e validity of t h e m a n a g e m e n t rules already derived a r e also investigated i n Section 4.

4.1 The Modified Growth Equations

We c o m m e n c e by replacing the equation (1) by a more g e n e r a l equa- t i o n , t h u s allowing for t h e influence of incidental disturbances. These disturbances m a y be d u e t o biological, meteorological a n d o t h e r factors.

Let t h e dynamics be given by a m o r e g e n e r a l equation

z

=

z F ( z , u , v ) , (18)

where F ( z ) is a function, which i s Lipschitz continuous i n z. a n d l e t urnax ² u ¹ 0 denote t h e harvesting effort r e l a t e d t o the r a t e of thinning by t h e equation h

=

u z . The variable v s u m m a r i z e s t h e influence of t h e disruptive factors. We shall a s s u m e t h a t e q u a t i o n (16) is integrable for u E

nu.

v E

R,,.

which a r e t h e appropriate c o n t r o l and disruption sets.

To e n s u r e t h a t t h e r a t e of thinning is proportional t o t h e biomass z, t h e function F ( z , u , v ) is a s s u m e d to have t h e f o r m

F(Z,U,V)

=

u F l ( v )

+

F ~ ( X , V ) .

The t e r m Fl(v) 2 0 c a n be i n t e r p r e t e d a s d i s t u r b a n c e of t h e thinning r a t e d u e t o incidental factors. The function z F Z ( z , v ) t h e n r e p r e s e n t s a gen- e r a l form of growth dynamics influenced by fluctuations of biological,

meteorological and other n a t u r e . The reader can verify t h a t ( 1 ) is a spe- cial case of (16). if F 1

=

l,Fz

=

a-bz. The equation (16) has a t least one equilibrium point z

=

0. Let some other equilibrium point be denoted z, , i.e., z,: F(z,,u,v)

=

0 for some values of uo,vo from

R U , R v

In t h e vicinity of such an equilibrium point, t h e r.h.s. of equation (16) can be linearized:

zF(z,uo,vo) = z o F ( z o . ~ O . v o ) + Z ~ ~ I , , ( Z - Z , )

+ . . .

az The equilibrium will be stable, if

or, more specifically, if

51 ^az

^{c o .}

for some u,,v,. The condition can be written

+&)I

0 co.

az z ,=,

a

^z

A dynamics satisfying +-) < 0 has been called c o m p e n s a t o r y (cf.

az z

Clark, 1976).

The equilibrium point z,, which depends on u E R,.v

ER,

is t h u s stable, if t h e dynamics is Locally c o m p e m a t o r y . The equilibrium point z,(u,v) is stable with respect t o t h e harvesting efforts from t h e s e t

A,.

and with respect t o t h e incidental factors from A,, if the dynamics is locally compensatory for all u E A,,

c R,.

a n d v E

A,

c

R,.

Obviously, a model with dynamics (16), satisfying

-

aF2

( z , v )

c 0 for z

>

0.v E A,. is

a~

compensatory for all harvesting efforts, a n d for all values of incidental

factors v from t h e s e t A,

The model which is not compensatory, is said to be depensatory.

The distinction between compensatory and depensatory dynamics, and t h e possibility of a transition from one t o another, is a m a t t e r of crucial importance (c.f. Holling 1973). Indeed, as long as t h e r e is a chance t h a t an environmental or economic factor could distort t h e dynamics t o make i t become depensatory. t h e optimal management policies could lead to environmental damage, and even to an extinction of t h e biotope through destabilization of t h e system (see Figure 2). Conversely, a gen- eral class of models (16), which is locz!ly compensatory with respect t o sufficiently 'large' sets A,, A,, is safe to use. In particular, models with

a r e safe for all harvesting efforts, and for all incidental disturbances from A, In t h e following, we shall limit our attention t o models of this type only.

4.2 Optimum Management in the Presence of Incidental Factors

In this paragraph, we shall combine t h e general dynamics of 4.1 with an optimality criterion including a more general 'cost' function. This function will be assumed e i t h e r positive, in which case i t will represent t h e harvesting cost, or negative, in which case it will be interpreted as a benefit derived from the forest in a non commercial way (e.g., acting as a pollution sink. cf. Section 3). We shall now proceed t o derive t h e formu- lae for t h e steady s t a t e standing volume under optimal management, and for t h e relationship between t h e shadow price and t h e r a t e of

a) Compensatory dynamics; equilibrium at x = xo is stable.

b) Dtpensatory case;the dynamics having been distorted the equilibrium first becomes unstable since the system heads for extinction.

i n t e r e s t .

The valuation equation for t h e model is now assumed t o be of t h e form

where C ( u , v , z ) is general cost function which depends both on t h e thin- ning effort, a n d on t h e incidental factors v.

The c o s t function is a s s u m e d t o satisfy the r e q u i r e m e n t

c =

7f.Zc1(v)

+

C 2 ( 2 , v )

The cost function is separable i n t o two parts: t h e first p a r t is a l i n e a r function of t h e r a t e of thinning, influenced by disruptive factors. It m a y be called p r o d u c t i o n c o s t . The second p a r t is not directly under t h e con- t r o l of t h e forester, a n d may be i n t e r p r e t e d as additional c o s t s or benefits, n o t immediately connected with t h e thinning operation, b u t a t t r i b u t a b l e t o t h e interaction between, for example, weather or biologi- cal calamities on t h e one hand, a n d t h e biomass on t h e other. This p a r t c a n be called e n v i r o n m e n f d c o s t (benefit).

Denoting ~ ( v ) = p

-

c l ( v ) a n d calling i t p r o d u c e r s p r i c e , we c a n write ( 1 7 )

This expression is t o be compared with (2) a n d ( 1 0 ) .

In c o n t r a s t t o t h e p r e s e n t model, t h e earlier model t h u s a s s u m e d a c o n s t a n t n e t revenue p e r u n i t harvested

6 ) .

This revenue was con- s i d e r e d independent of t h e disturbing factors, while the environmental

cost/benefit was neglected. Following t h e s t a n d a r d procedure, we form t h e Hamiltonian

aH ₌

0. we obtain for z 20 From t h e condition

-

au

P ( v ) exp(-6t)

=

-hF1(v) (18)

Under special circumstances, t h e equation (13) allows, t o i n t e r p r e t t h e discount r a t e a s t h e relative change of t h e shadow price A. To reproduce t h e equation (9)

i t is necessary t o assume e i t h e r

-=

dv ^0, i.e. slowly acting disruptive d t

forces or, alternatively,

d n P

which translates into

- =

1, i.e. t h e producers price should have dlnF,

u n i t elasticity with r e s p e c t t o t h e disruptive effect o n t h e thinning r a t e due t o incidental factors.

Summing up: The discounting r a t e 6 c a n be interpreted a s t h e relative change of t h e shadow price (cf. 9) e i t h e r if t h e disturbances of t h e

/

environment a r e extremely slow, o r if disturbances affect t h e p r o d u c e r s price a n d harvesting r a t e in a similar m a n n e r .

Returning t o the optimum m a n a g e m e n t problem, we c a n obtain a condition connecting i n t e r e s t r a t e , consumers price, and value of stock,

8 H

.

by using t h e equation --

=

h which yields after some algebra

a~

In a special case when

F 1 = 1, F 2 = a - b z ,

-

=

^0.

( 1 9 )

becomes

a 2

a -2bz = 6 ( 2 0 )

i.e., t h e expression

( 9 )

obtained in Section

2.

Similarly, t h e equation

( 1 4 )

of Section 3 i s obtained a s a special case, when

F 1 = -1. F2 = a - b z , C l ( v ) =

0,

Cz = -oz,.

We have t h u s arrived a t a gen- eralization of Proposition

1,

a n d of Corollary 2.

PROPOSITION 3:

h a s y s t e m with t h e g e n e r a l c o m p e n s a t o r y d y n a m i c s (16), t h e o p t i m u m m a n a g e m e n t l e a d s to a s t e a d y s t a t e d e t e r m i n e d b y t h e d i s c o u n t i n g r a t e a c c o ~ d i n g t o e q u a t i o n (19). Other t h i n g s being e q u a l ,

an

i n c r e a s e d d i s c o u n t i n g .rate b r i n g s a b o u t a s e d u c t i o n

of

t h e s t a n d i n g v o l u m e

if

a6 - ^26F2 a 2 ~ , ^{F ,} a2c2

--

-

+z---- <

⁰

az az2 az2 P az2

f o r all v

^E

Av.

Proposition 4 t h u s extends a n d confirms t h e qualitative conclusions r e a c h e d on t h e basis of t h e simple model, which remain valid irrespec- tive of t h e influence of incidental factors in a wide range of situations; i n particular, it is sufficient, if t h e function

~ ~ ( z , v )

is positive, decreasing a n d nonconvex in t h e standing volume, and if t h e environmental cost (benefit) function

c z ( z , v )

is nonconcave in the s a m e variable.

5. OPIlMUM

THINNING

POLICIES

MADE

SIldPLE; (CONCLUSIONS)

On t h e basis of o u r considerations, we c a n draw t h e following conclu- sions:

( a ) The growth of wood depends o n t h e quality and m a t u r i t y s t r u c - t u r e s of t h e t r e e s , a n d often i n a secondary way on a n u m b e r of incidental factors including weather, pollution a n d interaction (competition, symbiosis) with o t h e r organisms.

(b) The thinning r a t e is in a first approximation proportional t o t h e standing volume of wood; t h e thinning r a t e p e r u n i t standing volume is d e t e r m i n e d by t h e forester according t o t h e prevail- ing economic conditions, a n d i t is also influenced by t h e incidental factors.

( c ) An optimal thinning policy a i m s a t maximizing discounted n e t profit over a period of time. Such policy leads in t h e long r u n to a steady s t a t e representing a bioeconomic equilibrium, i.e., a s t a t e a t which t h e biological growth is balanced by t h e thinning operations.

(d) The bioeconomic equilibrium is determined, in the first a p p o z -

imation, by t h e equation (3') (cf. Proposition 1). The equation (3') c a n be given a very simple meaning, if one takes into account t h a t b

= -

a ^where

K

is a carrying capacity of t h e habi-

K'

r K 6

tat. The s t e a d y s t a t e standing volume is t h e n z

=

$I-;) i.e., i t is e q u a l t o the standing volume corresponding to t h e maximum r a t e of growth (according t o the simple growth model this happens when t h e standing volume r e a c h e s 50% of carrying

6 .

capacity), r e d u c e d by t h e fraction -, 1.e.. by the r a t i o of t h e a

r a t e of i n t e r e s t t o t h e initial ( c o n s t a n t ) r a t e of growth of wood.

This simple formula becomes m o r e complicated, when t h e forest i s considered not only a s a s o u r c e of wood, b u t also as a public good indispensable a s a t m o s p h e r i c filter, reservoir of water, etc.

In this case, t h e steady s t a t e s t a n d i n g volume is increased, a n d t h e a m o u n t of i n c r e a s e depends on t h e ratio of ecological a n d commercial values of forest (cf. equation (14), Corollary 2). In general, t h e s t e a d y s t a t e depends not only on t h e r a t e s of growth, i n t e r e s t , a n d on price, b u t also on t h e f o r m of t h e cost/benefit function, a n d on t h e incidental factors. The gen- e r a l equation, from which t h e s t e a d y s t a t e standing volume c a n be obtained, is given in paragraph 4.2 (equation (19)).

( e ) Having d e t e r m i n e d the optimal s t e a d y s t a t e , t h e optimal thin- ning policy i s easy t o formulate: Whenever the a c t u a l standing volume exceeds t h e s t e a d y s t a t e standing volume, u s e max- h u m thinning effort. As soon a s t h e steady s t a t e standing volume i s r e a c h e d , stop thinning.

( f ) It i s i m p o r t a n t to know whether t h e unknown (incidental) fac- t o r s c a n spoil t h e effect of t h e theoretically optimal t h i n n i n g policy, for example by reducing irreversibly t h e standing volume below t h e steady s t a t e value. In an e x t r e m e case, t h i s reduction c a n lead to a c a t a s t r o p h i c destruction of forest. How- ever, Proposition 3 specifies t h e conditions, under which t h e

bioeconomic equilibrium is stable. The stability guarantees t h a t t h e influence of incidental factors (e.g. draft, pest calam- ity, i n c r e a s e d pollution, e t c . ) , cannot lead to a n irreversible damage so long a s t h e influence is kept within predetermined limits. We h a s t e n to add t h a t this conclusion is not valid abso- lutely, b u t only in t h e context of what appears t o be a large class of models. In o t h e r words, should reality be so compli- cated t h a t its modeling required relaxation of t h e assumptions specified in Proposition 3, then the optimal thinning policy need n o t be safe for implementation. In t h e final analysis, t h e practitioner's experience m u s t decide whether the theory and its underlying m a t h e m a t i c a l assumptions a r e of sufficient gen- erality t o accommodate t h e behavior of t h e particular ecologi- cal s y s t e m a t hand. If depensation phenomena s e e m t o be of g r e a t potential importance there is probably a strong case for viability a n d p e r m a n e n c e analysis using differential inclusion theory a s proposed by Aubin and Sigmund (1984).

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