When is the Optimal Economic Rotation Longer than the Rotation of Maximum Sustained Yield?

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

WHEN IS THE OPTIMAL ECONOMIC ROTATION LONGER THAN THE ROTATION OF MAXIMUM SUSTAINED YIELD?

C l a r k S . Binkley

F e b r u a r y 1 9 8 5 WP-85-9

A s s o c i a t e P r o f e s s o r of F o r e s t r y

S c h o o l of F o r e s t r y a n d E n v i r o n m e n t a l S t u d i e s Yale U n i v e r s i t y

Eew H a v e n , CT 06511, a n d

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

KOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

WHEN IS THE OPTIMAL ECONOMIC ROTATION LONGER THAN THE ROTATION OF M~~ SUSTAINED YIELD?

Clark S . Binkley

Associate P r o f e s s o r of F o r e s t r y

School of F o r e s t r y a n d Environmental S t u d i e s Yale U n i v e r s i t y

N e w Haven, CT 06511, a n d a n d

R e s e a r c h S c h o l a r F o r e s t S e c t o r P r o j e c t IIASA

A-2361 L a x e n b u r g AUSTRIA

Working Papers 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 I n s t i t u t e f o r Applied Systems Analysis a n d h a v e r e c e i v e d only lim- ited 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 I n s t i t u t e o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2 3 6 1 L a x e n b u r g , Austria

ABSTRACT

C o n t r a r y t o t h e a s s e r t i o n s of many, t h e r o t a t i o n which maximizes t h e n e t p r e s e n t value of t i m b e r r e c e i p t s may b e Longer t h a n t h e r o t a t i o n which maximizes a v e r a g e annual physical yield. This c i r c u m s t a n c e may a r i s e e v e n in a v e r y simple economic model o n c e r e g e n e r a t i o n c o s t s a r e recognized.

Along with a t h e o r e t i c a l comparison of t h e two r o t a t i o n c r i t e r i a , a n example using Pinus patuLa plantations in Tanzania d e m o n s t r a t e s t h e p o t e n t i a l p r a c t i c a l i m p o r t a n c e of t h i s conclusion.

FOREWORD

The objective of t h e F o r e s t S e c t o r P r o j e c t at IIASA is t o study long- t e r m development a l t e r n a t i v e s f o r t h e f o r e s t s e c t o r on a global basis. The emphasis in t h e P r o j e c t i s on issues of major r e l e v a n c e t o industrial and governmental policy m a k e r s in d i f f e r e n t regions of t h e world who are responsible f o r f o r e s t policy, f o r e s t industrial s t r a t e g y , and r e l a t e d t r a d e policies.

The key elements of s t r u c t u r a l change in t h e f o r e s t industry are r e l a t e d t o a v a r i e t y of issues concerning demand, supply, and international t r a d e in wood p r o d u c t s . Such issues include t h e growth of t h e global econ- omy and population, development of new wood p r o d u c t s and of s u b s t i t u t e f o r wood p r o d u c t s , f u t u r e supply of roundwood and a l t e r n a t i v e f i b e r s o u r c e s , development of new technologies for f o r e s t r y and industry, pollution regu- lations, c o s t competitiveness, t a r i f f s and non-tariff t r a d e b a r r i e r s , etc.

The aim of t h e P r o j e c t i s t o analyze t h e consequence of f u t u r e expectations and assumptions concerning such substantive issues.

This a r t i c l e r e p r e s e n t s a background study on timber supply econom- ics. The optimal r o t a t i o n h a s been studied in t h e p r e s e n c e of costs which a r e nonlinear r e l a t i v e to timber removals. In p a r t i c u l a r , t h e effect of fixed r e g e n e r a t i o n c o s t s f o r s h o r t r o t a t i o n f o r e s t r y plantations is analyzed and illustrated with numerical examples.

Markku Kallio Leader

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

CONTENTS

I N T R O D U C T I O N

1. TWO ROTATION M O D E L S E c o n o m i c R o t a t i o n s M a x i m u m S u s t a i n e d Y i e l d

2. C O M P A R I S O N O F MSY A N D ECONOMIC ROTATIONS

3. A N E X A M P L E : PNUS PAWLA P L A N T A T I O N S IX T A N Z A N I A 4 . C O N C L U S I O N S

R E F E R E N C E S

WHEN IS THE OPTIMAL ECONOMIC ROTATION LONGER THAN THE ROTATION OF

MAXIMUM

SUSTAINED YKELD?

Clark S . Binkley

INTRODUCTION

Can t h e optimal economic r o t a t i o n e v e r e x c e e d t h e r o t a t i o n which max- imizes t h e a v e r a g e annual yield of t h e f o r e s t ? This question h a s b e e n dis- cussed f o r y e a r s . Many (e.g. Samuelson, 1976; C l a r k , 1976; Bentley a n d Teeguarden, 1964; Hyde, 1980; Chang, 1 9 8 3 a n d Andersson a n d Lesse, 1984) n a v e a r g u e d t h a t t h e optimal economic r o t a t i o n ( t h e r o t a t i o n which maxim- izes t h e n e t p r e s e n t vaiue of t i m b e r r e c e i p t s o v e r a n infinite planning h o r - izon) a p p r o a c h e s t h e culmination of mean annual increment ( o r t h e point of maximum sustained physical yield) only as t h e discount rate a p p r o a c h e s z e r o . This p a p e r shows t h a t t h i s conclusion i s valid only if r e g e n e r a t i o n a n d management c o s t s are ignored. Once even a v e r y simple c o s t formulation i s introduced i n t o t h e problem, economic r o t a t i o n s may equal o r e x c e e d t h e maximum sustained yield r o t a t i o n .

This r e s u l t i s i m p o r t a n t from s e v e r a l p e r s p e c t i v e s . On p u r e l y t h e o r e t - ical g r o u n d s i t i s i n t e r e s t i n g t o n o t e t h a t no firm conclusions c a n b e drawn a b o u t t h e b e s t r o t a t i o n without c a r e f u l c o n s i d e r a t i o n of p r e c i s e c o s t s and r e t u r n s a r i s i n g in a s p e c i f i c situation. From a somewhat m o r e p r a g m a t i c point of view, t h a t t h e economic r o t a t i o n may b e l o n g e r t h a n t h e maximum sustained yieid (MSY) r o t a t i o n gives r i s e t o t h e "backward bending" long r u n t i m b e r supply c u r v e

-

t h e supply c u r v e h a s n e g a t i v e r a t h e r t h a n posi- t i v e s l o p e .

This p a p e r f i r s t summarizes t h e economic a n d sustained yield optimiza- tion problems. Comparing t h e solutions t o t h e s e problems shows how t h e economic r o t a t i o n may b e l o n g e r t h a n t h e MSY r o t a t i o n . The most i n t e r e s t - ing s i t u a t i o n s a r i s e with fast growing, s h o r t r o t a t i o n s p e c i e s . Consequently w e consicier a n e m p i r i c a l example based on P i n u s p a t u l a plantations in Tanzania. The p a p e r concludes with some comments o n t h e significance of t h e t h e o r e t i c a l and e m p i r i c a l r e s u l t s .

1. TWO ROTATION MODELS

Consider two simple r o t a t i o n problems. The r o t a t i o n problems are p a r - t i c u l a r l y simple b e c a u s e we t a k e t h e p e r s p e c t i v e of s t a n d l e v e l optimiza- tion. Xence no forest-level c o n s t r a i n t s o r s t a n d i n t e r a c t i o n s will o b s c u r e t h e main point of t h e analysis.

E c o n o m i c R o t a t i o n s

The f i r s t model i s similar in s t r u c t u r e t o t h o s e used by Chang (1983), Hyde (1980), a n d J a c k s o n (1980) if management intensity i s t a k e n as f i x e d , and i s identical, in t e r m s of t h e r o t a t i o n a g e decision, t o t h e models of

Samueison (1976) a n 6 Clark (1976257-263).

The economic optimization probiem r e q u i r e s five important assump- tions.

i. Capital m a r ~ e t s a r e p e r f e c t s o t h e f o r e s t owner c a n lend and bor- row a t a known i n t e r e s t r a t e i which is constant through time.

ii. Demand f o r timber is c e r t a i n and constant s o p r i c e s equal p/unit f o r a l i periods.

iii. Timber yield v ( t ) p e r unit a r e a is a known function of stand a g e t which does not change o v e r time.

iv. The c o s t p e r unit a r e a of r e g e n e r a t i n g t h e stand is c which does not change o v e r time, and

v. The even-aged f o r e s t is r e g e n e r a t e d promptly after c l e a r c u t t i n g if i t is p r o f i t a b l e t o do so.

Other more complex (and p e r h a p s more r e a l i s t i c ) assumptions c a n p r o - duce r e s u l t s similar t o t h o s e d e r i v e d below. For example, Dykstra (1985) showed t h a t when unit logging c o s t s decline with tree size, t h e optimal economic r o t a t i o n c a n e x c e e d t h e MSY rotation. The p r e s e n c e of valuable nontimber f o r e s t p r o d u c t s c a n lengthen (Hartman, 1976) o r s h o r t e n (Bowes et a l . , 1984) t h e optimal economic r o t a t i o n compared t o t h a t when n e t timber r e c e i p t s alone are considered. A v e r y simple model w a s adopted h e r e t o show t h a t , even f o r v e r y simple c a s e s , t h e relationship between t h e economic and MSY r o t a t i o n a g e is ambiguous.

Our assumptions imply t h a t t h e problem is s t a t i o n a r y in t h e s e n s e t h a t t h e solution f o r t h e f i r s t r o t a t i o n will b e identical t o t h e solution f o r t h e second and subsequent periods. The economic r o t a t i o n problem is t h e n

max .rr(t)

= -

c + p v ( t ) e d t

+

. r r ( t ) e d t 1.1 t

R e a r r a n g i n g t e r m s g i v e s a continuous v e r s i o n of t h e familiar Faustmann o r land e x p e c t a t i o n model.

-

c + p v ( t ) e - i t max r ( t )

=

t 1 - e d t

The f i r s t o r d e r optimality condition f o r t * , t h e optimal economic r o t a t i o n , d x

c a n b e e a s i i y found by solving

- =

0 ( s e e J a c k s o n , 1980; Hyde, 1980, or d t

Cnang, 1 9 8 3 )

w h e r e

Maximum Sustained Yield

Consider a f o r e s t of unit a r e a . If all a g e c l a s s e s a r e equally r e p r e s e n t e d in t h e f o r e s t , e a c h will o c c u p y a n area of l / t w h e r e t i s t h e o l d e s t a g e c l a s s o r t h e r o t a t i o n a g e . The o l d e s t a g e c l a s s c o n t a i n s a volume of v ( t ) / u n i t area, a n d e a c h y e a r t h i s e n t i r e a g e c l a s s i s h a r v e s t e d . Conse- quently t h e a n n u a l yield of t h i s unit f o r e s t i s v ( t ) /

t .

The maximum sus- t a i n e d yield r o t a t i o n c a n t h e n b e found by solving t h e p r o b l e m

v ( t )

max - 1.4

t t

The f i r s t o r d e r condition f o r tm i s

2. COXPARISON OF

MSY

AND ECONOMIC ROTATIONS

To see t h e r e l a t i o n s h i p between t h e economic and MSY r o t a t i o n s , com- p a r e t h e f i r s t o r d e r conditions f o r t h e two cases, e q u a t i o n s 1.3 a n d 1 . 5 , r e s p e c t i v e l y . F i g u r e 1 shows t h e s e g r a p h i c a l l y . To see t h a t t h e v a r i o u s c u r v e s a r e d r a w n in p r o p e r r e l a t i o n s h i p , f i r s t o b s e r v e t h a t

i 1

2 - f o r a l l t

1 - e + t t

This c a n b e s e e n by c o n s i d e r i n g t h e s e r i e s e x p a n s i o n of e

(it )' ⁺

e 4 f = 1 - i t +- _2!

-

_3!

^{. . .}

Substituting i n t o t h e l e f t h a n d s i d e of 2 . 1 gives

t~~~ t * t (years)

FIGURE 1. Optimal economic and MSY rotations.

i

- -

¹ _{5 -} ¹

1 - 1 + i t -o(it) o(it) t t

--

i

From 2 . 3 , w e c a n a l s o s e e t h a t lim i

= -

1 i + o 1 - e - t t t

S e c o n d , if b o t h c a n d p are positive

From t h e s e a r g u m e n t s , we see t h a t F i g u r e 1 d e p i c t s a plausible set of r e l a t i o n s h i p s among t h e f i r s t o r d e r conditions 1.3 a n d 1 . 5 . A s d r a w n , c / p i s l a r g e enough so t h e optimal economic r o t a t i o n i s g r e a t e r t h a n t h e MSY r o t a t i o n .

Finally, n o t e t h a t

-

^v

iim

- -

_{, a n d}

P + - y

- -

C ^V

P

This latter c a s e , 2.6b, combined with t h e limiting b e h a v i o r n o t e d in 2.4, gives r i s e to t h e c o n t e n t i o n t h a t t h e optimal economic r o t a t i o n a p p r o a c h e s t h e MSY r o t a t i o n as t h e discount rate a p p r o a c h e s z e r o . However, t h i s a s s e r t i o n i s t r u e only if p r o d u c t i o n costs are z e r o , are in t h e limit as stum- p a g e p r i c e s grow v e r y l a r g e .

Having e s t a b l i s h e d g r a p h i c a l l y t h a t t h e MSY r o t a t i o n may b e s h o r t e r t h a n t h e economic r o t a t i o n , i t i s i n t e r e s t i n g to examine t h e p o i n t at which t h e y coincide. To d o so, i t i s useful to r e w r i t e 1.3 as

Equating t h e r i g h t hznd s i d e s of 1.5 a n d 2 . 7 gives t h e r e l a t i o n s h i p among c , p , i and v which must hold in o r d e r f o r t h e economic a n d MSY r o t a t i o n to b e i d e n t i c a l

In t h e case of a competitive economy, w e c a n a d d t h e p r o v i s o t h a t 7r 2 0 ( o t h e r w i s e t h e optimal p l a n i s to c u t w h a t e v e r trees are standing a n d a b a n - don t h e l a n d ) , which f r o m 1.1 implies t h a t

-

C

^<

^{v e I t 2.9}

P

The t e r m c / p c a n b e eliminated from t h e p r o b l e m by equating t h e r i g h t h a n d s i d e s of 2.8 a n d 2.9

.

This g i v e s t h e limiting v a l u e s of i a n d t w h e r e t h e economic r o t a t i o n e x c e e d s t h e MSY r o t a t i o n b u t .rr

>

0 . R e a r - r a n g i n g t h e r e s u l t i n g e q u a t i o n g i v e s

( I

-

^{e I t ) ( i t}

-

^{I )}

⁼

⁰

Which implies e i t h e r

In t h e f i r s t case, e i t h e r t

=

0 or i

=

0 . In t h e s e c o n d , m o r e i n t e r e s t i n g c a s e , i

=

l / t . Thus f o r economic r o t a t i o n s to b e g r e a t e r t h a n t h e MSY r o t a t i o n , t h e i n t e r e s t rate must b e less t h a n t h e i n v e r s e of t h e MSY rota- tion. F o r c o m p a r a t i v e l y s l o w growing s p e c i e s s u c h as Douglas-fir, t h e white

pines o r many t e m p e r a t e hardwoods, t h e KSY r o t a t i o n will b e on t h e o r d e r of 1 0 0 y e a r s . In t h i s case, Z . l l b implies a maximum discount rate of 1/100 o r i

=

3 . C 2 , and t h e situation d e s c r i b e d h e r e i s of l i t t l e p r a c t i c a l impor- t a n c e . On t h e o t h e r hand, f a s t growing s p e c i e s s u c h as t h e s o u t h e r n pines ( p a r t i c u l a r l y on hign s i t e s managed f o r c u b i c r a t h e r t h a n b o a r d foot p r o - duction) o r t r o p i c a l f o r e s t plantations of s p e c i e s such as Pinus patula, Pinus caribaea o r Gmelina arborea- may r e a c h t h e culmination of mean annual increment in l e s s t h a n 20 y e a r s . In s u c h s i t u a t i o n s t h e optimal economic r o t a t i o n might b e g r e a t e r t h a n MSY u n d e r a wide r a n g e of c o s t a n d p r i c e p a r a m e t e r s .

3.

AN EXAMPLE:

PINLrS PATLEA PLANTATIONS

IN

TANZANIA

To give a b e t t e r s e n s e of t h e p r a c t i c a l importance of t h i s situation, equations 2.8 a n d 2.9 w e r e computed f o r high s i t e (33 m at a g e 20) Pinus patula p i a n t a t i o n s at S a o Hill in s o u t h e r n Tanzania. This f a s t growing pine, n a t i v e t o Mexico, h a s b e e n widely planted t h r o u g h o u t E a s t Africa. Pianta- tion yields at S a o Hill a r e similar though n o t identical t o t h e yields of P . p a t u l a plantations e l s e w h e r e in Africa, including Kenya, Malawi a n d Uganda (Adegbehin, 1982). F o r a n a l y t i c a l convenience, Adegbehin's (1982) yield e s t i m a t e s w e r e f i t t e d to a two-parameter yield function:

ln[v ( t ) ]

=

7.42

-

^15.5/ t R~

=

0.989

(73.4) (-32.7) n

=

1 7 3.1

w h e r e v ( t ) re f e r s t o t h e t o t a l s t a n d volume, outside b a r k , in m3/ha. The numbers in p a r e n t h e s e s are t - s t a t i s t i c s f o r t h e null hypothesis t h a t t h e coefficient e q u a l s z e r o .

The Durbin-Watson s t a t i s t i c for t h e initial o r d i n a r y l e a s t s q u a r e s e s t i - mate of t h i s mociel s u g g e s t e d t h e p r e s e n c e of p o s i t i v e s e r i a l c o r r e l a t i o n among t h e r e s i d u a l s , a condition which r e n d e r s t h e e s t i m a t e s inefficient b u t unbiased. Consequently, t h e e s t i m a t e s in 3.1 w e r e o b t a i n e d by a n i t e r a t i v e g e n e r a l i z e d l e a s t s q u a r e s p r o c e d u r e w h e r e in e a c h s t e p t h e e s t i m a t e d lag- o n e s e r i a l c o r r e l a t i o n c o e f f i c i e n t i s u s e d to correct t h e v a r i a n c e / c o v a r i a n c e m a t r i x of t h e G L S e s t i m a t e s until s a t i s f a c t o r y c o n v e r - g e n c e of t h e e s t i m a t e d c o e f f i c i e n t s i s o b t a i n e d .

F i g u r e 2 g r a p h s 2.8 a n d 2 . 9 i s a function of t h e discount r a t e i f o r t h i s yield c u r v e . T h e t w o e q u a t i o n s divide t h e g r a p h i n t o f o u r areas. In r e g i o n s I a n d 11, t h e c / p r a t i o i s high enough so

t* > tm

In r e g i o n I1 c / p i s so high t h a t t i m b e r p r o d u c t i o n i s u n p r o f i t a b l e . Unless subsidized, we would e x p e c t n o p r o d u c t i o n at a l l f o r t h e s e combinations of c / p a n d i . In r e g i o n s I11 a n d IV, t h e c / p r a t i o i s l o w enough so t h a t

t* < tm

The t w o c u r v e s cross at i

=

1/

t

, o r at i

=

0.065. T h a t i s , at i n t e r e s t rates g r e a t e r t h a n i

=

0.065 t h e optimai economic r o t a t i o n wiIl always b e l e s s t h a n t h e MSY r o t a t i o n .

At a discount r a t e of i

=

0.04, c / p r a t i o s between 1 5 7 a n d 332 l e a d to

t* > tm

As a point of r e f e r e n c e , in 1 9 8 0 c

=

3 1 7 1 s h s / h a (Kowero, 1 9 8 4 ) , which g i v e s a p o s s i b l e r a n g e in p r i c e s of 9.6-20.2 shs/m3. In 1 9 8 0 , t h e r o y - a l t y f o r P. p a t u l a stumpage in t h i s s i z e r a n g e was 20 shs/m3 ( D y k s t r a , 1 9 8 5 ) . Consequently, t h i s simple economic model s u g g e s t s t h a t , at a d i s c o u n t rate of i

=

0.04, t h e optimal economic r o t a t i o n f o r t h e s e p l a n t a t i o n s would b e g r e a t e r t h a n t h e a g e of maximum s u s t a i n e d yield.

FIGURE 2. t*

>

tx,, Pznus patula.

4. CONCLUSIONS

In g e n e r a l , economic r o t a t i o n s may b e g r e a t e r t h a n , equal t o , o r l e s s t h a n t h e r o t a t i o n which maximizes t h e s u s t a i n e d physical o u t p u t of t h e f o r e s t . The r e l a t i o n s h i p between t h e MSY a n d economic r o t a t i o n s d e p e n d s on t h e yield function f o r t h e s p e c i e s in question, t h e management c o s t s , t h e stumpage p r i c e and t h e i n t e r e s t r a t e . In p r a c t i c e , economic r o t a t i o n s exceeding K S Y r o t a t i o n s are more likely t o a r i s e with fast t h a n with slow growing s p e c i e s .

Tne r e l a t i o n s h i p between t h e economic a n d MSY r o t a t i o n i s i m p o r t a n t for u n d e r s t a n d i n g lone r u n t i m b e r supply. I t i s well-known t h a t a n i n c r e a s e in stumpage p r i c e will l e a d to a d e c r e a s e in t h e optimal r o t a t i o n (differen- t i a t e 1.3 with r e s p e c t to t t o see t h i s , o r see Chang, 1983:271). If p r i c e s are high enough s o t h e economic r o t a t i o n i s s h o r t e r t h a t MSY, t h e n t n e r e d u c t i o n in r o t a t i o n a t t e n d i n g a p r i c e i n c r e a s e will l e a d to a Lower l e v e l of a v e r a g e o u t p u t . The long r u n t i m b e r supply c u r v e will t h u s bend backward.

If high p r o d u c t i o n costs, high i n t e r e s t rates a n d low growth c o n s p i r e t o make p r o f i t s n e g a t i v e at a n y r o t a t i o n l o n g e r t h a n MSY, n o p a r t of t h e long r u n s u p p l y c u r v e will h a v e a positive s l o p e . Binkley (1985) g i v e s a m o r e complete a c c o u n t of t h e long r u n t i m b e r s u p p l y model.

A m o r e r e a l i s t i c model of f o r e s t management would r e l a x t h e assump-

t i o n s of t h e p r e s e n t a n a l y s i s . F o r example, p r i c e s would r e s p o n d t o chang- ing supply/demana b a l a n c e s . Management c o s t s would c h a n g e in r e s p o n s e to c h a n g e s in l a b o r a n d o t h e r f a c t o r m a r k e t s . Yields might c h a n g e o v e r time as biological knowledge a c c u m u l a t e s o r as environmental d e g r a d a t i o n t a k e s i t s toll. The nontimber p r o d u c t s of t h e forest would b e r e c o g n i z e d in t h e economic optimization. T h e s e complications will s u r e l y u p s e t t h e simple comparisons between economic a n d MSY r o t a t i o n s p r e s e n t e d h e r e b u t are n o t likely to a l t e r t h e ambiguous r e l a t i o n s h i p between t h e t w o a p p r o a c h e s .

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