UBC Theses and Dissertations

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UBC Theses and Dissertations

Essays on the qualitative theory of forest economics Heaps, Terry 1981

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ESSAYS ON THE QUALITATIVE THEORY OF FOREST ECONOMICS by TERRY HEAPS B.Sc. U n i v e r s i t y of B r i t i s h Columbia Ph.D. U n i v e r s i t y of C a l i f o r n i a , Berkeley M.A. - Simon Fraser U n i v e r s i t y A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1980 DOCTOR OF PHILOSOPHY m Terry Heaps, 1980 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department O f E c o n o m i c s The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 - 1 -ABSTRACT T h i s t h e s i s i s c o n c e r n e d w i t h q u e s t i o n s r e l a t i n g t o t h e o p t i m a l r e g u l a t i o n o f l o g g i n g i n a s o l e l y owned f o r e s t . O p t i m a l i s t a k e n t o mean m a x i m i z i n g t h e p r e s e n t v a l u e o f n e t r e v e n u e s o b t a i n a b l e f r o m g r o w i n g and h a r v e s t i n g an i n f i n i t e s e q u e n c e o f c r o p s on a p i e c e o f f o r e s t l a n d . D i s c u s s i o n s o f o p t i m a l h a r v e s t i n g u s u a l l y assume t h e r e i s no a d v a n t a g e t o c h a n g i n g t h e age d i s t r i b u t i o n o f a f o r e s t . T h i s i s a l s o t h e a s s u m p t i o n o f C h a p t e r I . In t h i s c a s e , t r e e s a r e c u t when t h e i r age r e a c h e s t h e r o t a t i o n p e r i o d w h i c h i s d e t e r m i n e d f r o m what i s c a l l e d t h e Faustmann f o r m u l a . C h a p t e r I l o o k s a t t h e c o m p a r a t i v e s t a t i c s o f t h e s e r o t a t i o n p e r i o d s . The e f f e c t o f a change i n an exogenous p a r a m e t e r on t h e r o t a t i o n p e r i o d i s shown t o depend on how c e r t a i n e l a s t i c i t i e s a r e c h a nged. I t i s t h e n shown t h a t t h e r e a r e c o n d i t i o n s u n d e r w h i c h t h e s e r e s u l t s e x t e n d t o a more complex f o r e s t r y model where t h e manager c h o o s e s t h e l e v e l o f e f f o r t t o be expended on r e g e n e r a t i o n and s i l v i c u l t u r a l a c t i v i t i e s . The t e c h n i q u e s u s e d a r e drawn f r o m o p t i m a l c o n t r o l t h e o r y . C h a p t e r I I i n t r o d u c e s c o n s i d e r a t i o n s w h i c h may make i t a d v a n t a g e o u s t o a l t e r t h e age s t r u c t u r e o f a f o r e s t w h i l e l o g g i n g i t . I n p a r t i c u l a r , a v a r i a b l e a v e r a g e c o s t o f h a r v e s t i n g f u n c t i o n i s a l l o w e d f o r . A f o r e s t r y maximum p r i n c i p l e i s d e r i v e d w h i c h d e t e r m i n e s t h e d y n a m i c s o f o p t i m a l h a r v e s t i n g . T h i s i s s i m i l a r t o t h e maximum p r i n c i p l e f o r p r o c e s s e s i n c o r p o r a t i n g a d e l a y ( t h e t i m e between p l a n t i n g and h a r v e s t i n g ) . The u s u a l g r o w t h t h e o r y q u e s t i o n s a r e t h e n a s k e d . In t h e v a r i a b l e a v e r a g e c o s t c a s e , t h e " s t e a d y s t a t e " i i -age d i s t r i b u t i o n s t u r n out to be "normal" f o r e s t s w i t h the time between h a r v e s t s b e i n g d e t e r m i n e d by a Faustmann f o r m u l a G l o b a l a s y m p t o t i c s t a b i l i t y i s not proven but i s shown to be l i k e l y . F i n a l l y , Chapter I I I a p p l i e s the f o r e s t r y maximum p r i n c i p l e t o a problem o f d e t e r m i n i n g an o p t i m a l h a r v e s t i n g p o l i c y f o r a group o f f o r e s t s s u b j e c t to a s u s t a i n e d y i e l d c o n s t r a i n t . Assuming s t a b i l i t y and w i t h a few a d d i t i o n a l r e s t r i c t i o n s , i t i s shown t h a t the o p t i m a l l o n g r u n p o l i c y i s to c o n v e r t each s e p a r a t e f o r e s t t o a "normal" f o r e s t . The Faustmann f o r m u l a d e t e r m i n e s the number o f age c l a s s e s i n each f o r e s t . - i i i -C O N T E N T S A b s t r a c t i C o n t e n t s i i i L i s t o f F i g u r e s v A c k n o w l e d g e m e n t v i I N T R O D U C T I O N 1 C H A P T E R I : 8 T h e Q u a l i t a t i v e E f f e c t o f C o s t V a r i a t i o n s o n O p t i m a l R o t a t i o n P e r i o d s i n F o r e s t r y 1 . V a r i a t i o n s i n R o t a t i o n P e r i o d s 8 2 . T h e G e n e r a l S t e a d y S t a t e F o r e s t r y M o d e l . . 1 2 3 . V a r i a b l e R e p l a n t i n g a n d S i l v i c u l t u r a l E x p e n d i t u r e 26 C H A P T E R I I : 4 4 T h e F o r e s t r y M a x i m u m P r i n c i p l e 1 . I n t r o d u c t i o n 4 4 2 . T h e G e n e r a l i z e d F o r e s t r y M a n a g e m e n t P r o b l e m 4 8 3 . T h e F o r e s t r y M a x i m u m P r i n c i p l e 55 4 . S u f f i c i e n c y 6 1 5 . S t e a d y S t a t e s 66 6 . A s y m p t o t i c S t a b i l i t y 70 7 . C o n c l u s i o n 74 C H A P T E R I I I : 7 5 O n t h e O p t i m a l H a r v e s t i n g o f a R e g i o n a l F o r e s t 1 . I n t r o d u c t i o n 7.6 2 . T h e R e g i o n a l F o r e s t M a n a g e m e n t M o d e l 8 0 3 . T h e S i n g l e F o r e s t A r e a 82 - i v -4. M u l t i p l e F o r e s t Areas 86 5. C o n c l u s i o n s 91 APPENDIX 1 9 2 S u f f i c i e n t C o n d i t i o n s f o r R e p l a n t i n g or S i l v i c u l t u r e t o be P r o f i t a b l e APPENDIX 2 94 Second Order C o n d i t i o n s f o r the S i l v i c u l t u r e Problem APPENDIX 3 97 Envelope Theorem f o r O p t i m a l C o n t r o l Problems APPENDIX 4 98 The S i g n o f 8A(T)/8x APPENDIX 5 101 An Example o f the F i r s t V a r i a t i o n APPENDIX 6 103 S i m p l i f i c a t i o n o f P r e s e n t V a l u e APPENDIX 7 104 R e l a t i o n s h i p o f the F o r e s t r y Maximum P r i n c i p l e t o "The Economics o f F o r e s t r y when the Rate o f H a r v e s t i s C o n s t r a i n e d " APPENDIX 8 105 Continuous Nonmeshing o f R o t a t i o n P e r i o d s NOTES 106 BIBLIOGRAPHY I l l - v -LIST OF FIGURES F i g u r e 1 . . . . The O p t i m a l R o t a t i o n P e r i o d . . . . 16 F i g u r e 2 . . . . Change i n S i l v i c u l t u r a l E f f o r t . . . 31 F i g u r e 3 . . . . Va l u e Growth Curve . . 49 F i g u r e 4 . . . . A Logging P o l i c y 50 F i g u r e 5 . . . . The F i r s t H a r v e s t P a t t e r n 83 F i g u r e 6 . . . . The T r a j e c t o r i e s 99 - v i -ACKNOWLEDGEMENT I would l i k e t o thank my s u p e r v i s o r s , P h i l i p Neher and Y o s h i t s u g u Kanemoto f o r h a v i n g s t i m u l a t e d t h i s r e s e a r c h and f o r the v e r y u s e f u l d i s c u s s i o n s I have had w i t h them c o n c e r n i n g the c o n t e n t s o f t h i s t h e s i s . I am a l s o v e r y g r a t e f u l f o r the s u p p o r t I have r e c e i v e d d u r i n g my s t a y a t U.B.C. from the U.B.C. Programme i n N a t u r a l Resource Economics. - 1 -INTRODUCTION T h i s t h e s i s i s concerned w i t h q u e s t i o n s r e l a t i n g t o the o p t i m a l r e g u l a t i o n o f l o g g i n g i n a s o l e l y owned f o r e s t . I t may be n o t e d , f i r s t .of a l l ' , t h a t t h e r e have been wide d i v e r g e n -ces o f o p i n i o n over what o b j e c t i v e s s h o u l d be pursued i n d e t e r m i n i n g the o p t i m a l y i e l d from a f o r e s t . The d i f f e r e n t o b j e c t i v e s which have been proposed are d i s c u s s e d i n Gregory (1972, Ch. 14) and Samuelson (1976). Samuelson argues t h a t c o r r e c t c a p i t a l t h e o r e t i c a n a l y s i s r e q u i r e s t h a t the o b j e c t i v e to be pursued s h o u l d be t o maximize the p r e s e n t v a l u e o f net revenues o b t a i n a b l e from a l l the i n f i n i t e sequence o f h a r v e s t s t h a t can be o b t a i n e d from the f o r e s t l a n d . T h i s v i e w , which i s known i n the f o r e s t r y l i t e r a t u r e as the s o i l e x p e c t a t i o n v a l u e (Se) approach, i s the view which w i l l be used i n t h i s t h e s i s . The Se approach,.which i s a t t r i b u t e d - t o Faustmann (1849) 1 , r e s u l t s i n a y i e l d p l a n which c u t s each t r e e when i t reaches some o p t i m a l r o t a t i o n age. T h i s age i s det e r m i n e d from what i s c a l l e d the Faustmann f o r m u l a - a f o r m u l a which i n c o r p o r a t e s . b o t h economic and b i o l o g i c a l v a l u e s . These v a l u e s are e x t r e m e l y v a r i a b l e from f o r e s t s i t e t o f o r e s t s i t e even f o r the same s p e c i e s . Logging c o s t s can v a r y from s i t e t o s i t e due to d i f f e r e n c e s i n h a u l i n g time to the n e a r e s t wood u t i l i z i n g p l a n t and d i f f e r e n c e s i n type o f t e r r a i n . Growing c o n d i t i o n s can v a r y due t o d i f f e r e n c e s i n s o i l f e r t i l i t y or d i f f e r e n c e s i n c l i m a t e . The f i r s t c h a p t e r o f t h i s t h e s i s s t u d i e s the q u a l i t a t i v e impact o f v a r i a t i o n s i n the s e f a c t o r s on the o p t i m a l - 2 -r o t a t i o n a g e s . The b a s i c q u a l i t a t i v e r e s u l t i s t h a t t h e o p t i m a l r o t a t i o n age i n c r e a s e s w i t h i n c r e a s e s i n h a u l i n g t i m e f r o m f o r e s t s i t e t o m i l l . T h i s r e s u l t was o b t a i n e d by L e d y a r d - M o s e s (1976) f o r a model i n w h i c h t h e i n c l u s i o n o f r e p l a n t i n g was n e c e s s a r y i n o r d e r t o o b t a i n t h e r e s u l t . I n C h a p t e r I , however, i t w i l l be shown t h a t t h e r e s u l t c a n be d e r i v e d f r o m a few v e r y g e n e r a l a s s u m p t i o n s a b o u t the f o r e s t p r o f i t f u n c t i o n w h i c h do n o t r e q u i r e t h a t r e p l a n t i n g be a p r o f i t a b l e o p e r a t i o n . A n o t h e r r e s u l t , w h i c h c a n be f o u n d i n t h e e m p i r i c a l l i t e r a t u r e , i s t h a t b e t t e r g r o w i n g c o n d i t i o n s a r e a s s o c i a t e d w i t h s h o r t e r o p t i m a l r o t a t i o n p e r i o d s . T h i s r e s u l t w i l l be v e r i f i e d t h e o r e t i c a l l y , p r o v i d e d t h e f o r e s t g r o w t h f u n c t i o n s a t i s f i e s a c e r t a i n e l a s t i c i t y c o n d i t i o n . An i n t e r e s t i n g a s p e c t o f t h e work o f L e d y a r d - M o s e s i s t h a t t h e q u a l i t a t i v e r e s u l t c o n c e r n i n g h a u l i n g t i m e c o n t i n u e s t o h o l d when t h e more complex f o r e s t r y o p e r a t i o n o f a v a r i a b l e r e g e n e r a t i o n e f f o r t i s i n c o r p o r a t e d i n t o t h e m o d e l . They c l a i m , however, t h a t t h i s r e s u l t may n o t h o l d when s i l v i -c u l t u r e i s c o n s i d e r e d . C h a p t e r I c o n c l u d e s by e x a m i n i n g t h e s e c l a i m s w i t h i n a more g e n e r a l m o d el, and f i n d s t h a t t h e h a u l i n g t i m e r e s u l t a c t u a l l y c o n t i n u e s t o h o l d i n t h e more complex m o d e l s ; p r o v i d e d t h a t t h e r e g e n e r a t i o n and s i l v i c u l t u r a l c o s t f u n c t i o n s . . s a t i s f y some f a i r l y r e a s o n a b l e c o n d i t i o n s w i t h r e s p e c t t o t h e i r v a r i a t i o n w i t h r e s p e c t t o h a u l i n g t i m e . L e d y a r d - a n d Moses' c l a i m c o n c e r n i n g s i l v i c u l t u r e r e s u l t s f r o m t h e f a c t t h a t t h e y h o l d t h e l e v e l o f s i l v i c u l t u r a l e f f o r t - 3 -c o n s t a n t t h r o u g h o u t t h e g r o w i n g c y c l e o f t h e f o r e s t . However, t h e t e c h n i q u e s o f o p t i m a l c o n t r o l t h e o r y c a n be u s e d t o a n a l y s e t h e c a s e o f v a r i a b l e s i l v i c u l t u r a l i n p u t , and t h i s i s t h e a p p r o a c h o f t h i s t h e s i s . The r e l a t i o n s h i p o f o p t i m a l c o n t r o l t h e o r y t o y i e l d p l a n n i n g i n g e n e r a l i s t h e s u b j e c t o f C h a p t e r I I . T h e r e a p p e a r t o have been f e w e r a p p l i c a t i o n s o f P o n t r y a g i n ' s maximum p r i n c i p l e t o f o r e s t r y e c o n o m i c s t h a n t o o t h e r a r e a s i n n a t u r a l r e s o u r c e e c o n o m i c s . T h i s i s p e r h a p s b e c a u s e t h e " s t a t e " o f a f o r e s t o r t h e i n f o r m a t i o n n e e d e d by t h e y i e l d p l a n n e r i s more complex t h a n i n t h e c a s e o f o t h e r r e s o u r c e s . T h i s c h a p t e r t a k e s t h e age d i s t r i b u t i o n o f t h e t r e e s i n t h e f o r e s t t o be t h e a p p r o p r i a t e s t a t e v a r i a b l e f o r a c o n t r o l t h e o r e t i c a p p r o a c h t o y i e l d p l a n n i n g . The c o n t r o l v a r i a b l e i s t h e h a r v e s t i n g r a t e and t h e d y n a m i c s o f h a r v e s t i n g d e s c r i b e how a h a r v e s t i n g p a t t e r n changes t h e age d i s t r i b u t i o n . T h i s v i s i o n o f t h e h a r v e s t i n g p r o c e s s becomes i m p o r t a n t when the a v e r a g e c o s t s o f l o g g i n g a r e assumed t o v a r y i n t h e e c o n o m i s t ' s t r a d i t i o n a l U -shaped way. Heaps and Neher (1979) seem t o have b e e n t h e f i r s t t o a t t e m p t t o a n a l y s e o p t i m a l y i e l d p l a n n i n g u n d e r t h e s e c o n d i t i o n s . I t was shown t h e r e t h a t i f t h e age d i s t r i b u t i o n o f t h e f o r e s t was n o t n o r m a l t o b e g i n w i t h , t h e n o p t i m a l y i e l d p l a n n i n g w o u l d c o n t i n u a l l y change t h e age d i s t r i b u t i o n o f t h e f o r e s t f r o m h a r v e s t t o h a r v e s t . I t w i l l now be shown t h a t t h i s p r o c e s s c a n be c a p t u r e d by a f o r e s t r y maximum p r i n c i p l e w h i c h i s n o t q u i t e t h e same as t h e u s u a l maximum p r i n c i p l e b e c a u s e o f t h e i n f i n i t e d i m e n s i o n a l s t a t e - 4 -v a r i a b l e . N e v e r t h e l e s s , the f o r e s t r y maximum p r i n c i p l e i s not e n t i r e l y new because i t i s v e r y s i m i l a r t o what P o n t r y a g i n c a l l s the maximum p r i n c i p l e f o r p r o c e s s e s w i t h a d e l a y . The d e l a y i n f o r e s t r y i s the time between p l a n t i n g and c u t t i n g . The f o r e s t r y maximum p r i n c i p l e c h a r a c t e r i z e s the o p t i m a l c u t t i n g program as the s o l u t i o n o f a c e r t a i n ( f u n c t i o n a l ) d y n a m i c a l system, s u b j e c t t o the i n i t i a l s t a t e o f the f o r e s t . The u s u a l q u e s t i o n s o f growth t h e o r y then a r i s e . F i r s t , w hich i n i t i a l age d i s t r i b u t i o n s are st e a d y s t a t e s i n the sense t h a t the o p t i m a l c u t t i n g program w i l l r e c o n s t i t u t e these age d i s t r i -b u t i o n s a t r e g u l a r i n t e r v a l s . The answer t u r n s out t o be t h a t any age d i s t r i b u t i o n i s a st e a d y s t a t e i f average h a r v e s t i n g c o s t s a re c o n s t a n t and o n l y normal age d i s t r i b u t i o n s are s t e a d y s t a t e s i f average h a r v e s t i n g c o s t s a re v a r i a b l e . I n e i t h e r c a s e , the l e n g t h o f the i n t e r v a l s between h a r v e s t s i s g i v e n by a Faustmann f o r m u l a . Thus, Faustmann r o t a t i o n s are st e a d y s t a t e s f o r a more complex f o r e s t r y w hich admits t h a t t h e r e may be economic b e n e f i t s o b t a i n a b l e from a l t e r i n g the age d i s t r i -b u t i o n o f a f o r e s t . The o t h e r i m p o r t a n t q u e s t i o n o f growth t h e o r y i s the q u e s t i o n o f s t a b i l i t y . T h i s i s the q u e s t i o n o f whether any growth p a t h i n the system must converge towards a s t e a d y s t a t e , or i n the f o r e s t r y case whether the o p t i m a l age d i s t r i b u t i o n must e v o l v e towards a st e a d y s t a t e age d i s t r i b u t i o n . I n the l i t e r a t u r e , t h i s type o f s t a b i l i t y i s c a l l e d g l o b a l a s y m p t o t i c 3 s t a b i l i t y . I t has not been p o s s i b l e t o prove here t h a t o p t i m a l h a r v e s t i n g p o l i c i e s are g l o b a l l y a s y m p t o t i c a l l y s t a b l e . However, - 5 -i t i s shown t h a t i f t h e age d i s t r i b u t i o n o f t h e f o r e s t c o n v e r g e s a s y m p t o t i c a l l y t h e n i t c o n v e r g e s t o a s t e a d y s t a t e age d i s t r i b u -t i o n . M o r e o v e r , some e v i d e n c e i s g i v e n w h i c h s u g g e s t s t h a t a s y m p t o t i c c o n v e r g e n c e i s q u i t e l i k e l y . The f i n a l c h a p t e r o f t h i s t h e s i s , C h a p t e r I I I , a p p l i e s t h e f o r e s t r y maximum p r i n c i p l e t o a p r o b l e m o f s u s t a i n e d y i e l d h a r v e s t i n g f r o m a r e g i o n a l f o r e s t . A r e g i o n a l f o r e s t i s v i e w e d as b e i n g a c o l l e c t i o n o f d i s t i n c t f o r e s t a r e a s e a c h o f w h i c h a c t s as a s o u r c e o f s u p p l y o f raw m a t e r i a l f o r a r e g i o n a l m a n u f a c t u r i n g c e n t e r . These a r e a s w i l l d i f f e r i n t h e i r e c o n o m i c and b i o l o g i c a l c h a r a c t e r i s t i c s - c e r t a i n l y i n r e s p e c t t o h a u l i n g t i m e t o t h e c e n t e r . Thus, as a n a l y s e d i n C h a p t e r I , e a c h a r e a w i l l have a d i f f e r e n t o p t i m a l r o t a t i o n p e r i o d . I f e a c h a r e a i s l o g g e d a c c o r d i n g t o t h e s e r o t a t i o n p e r i o d s , t h e r e w i l l l i k e l y be extreme f l u c t u a t i o n s i n r e g i o n a l l o g s u p p l y . T h i s c h a p t e r assumes t h a t a s u s t a i n e d y i e l d r e s t r i c t i o n i s imposed on t h e r e g i o n a l l o g s u p p l y i n o r d e r t o a v o i d p o s s i b l e f l u c t u a t i o n s i n t h e a n n u a l c u t . T h i s r e s t r i c t i o n i s t h a t t h e sum o f t h e a n n u a l c u t s f r o m t h e v a r i o u s a r e a s must e x c e e d some minimum i n e a c h y e a r . The q u e s t i o n t h e n a d d r e s s e d i s what l o n g r u n l o g g i n g p o l i c y m a x i m i z e s t h e sum o f d i s c o u n t e d n e t r e v e n u e s f r o m t h e r e g i o n ' s f o r e s t s ? T h e r e a p p e a r s t o be j u s t two p o s s i b i l i t i e s . One p o s s i b i l i t y i s t o impose some common r o t a t i o n p e r i o d on a l l t h e d i s s i m i l a r a r e a s . The o t h e r i s t o c o n v e r t t h e f o r e s t i n e a c h a r e a i n t o a n o r m a l f o r e s t and t o c u t t h e o l d e s t age c l a s s i n e v e r y f o r e s t e v e r y y e a r . The - 6 -r o t a t i o n periods used here w i l l be d i f f e r e n t f o r the various areas as they w i l l be the Faustmann r o t a t i o n p e r i o d s . I t i s shown i n Chapter I I I that provided the optimal r e g i o n a l logging s t r a t e g y converges, i t converges to the second strategy mentioned above (provided a few r e s t r i c t i o n s h o l d ) . I t i s i n t e r e s t i n g that t h i s i s e s s e n t i a l l y the p o l i c y that many f o r e s t e r s have advocated, although they would have argued 4 f o r longer r o t a t i o n periods . Their reasons f o r advocating t h i s p o l i c y were somewhat d i f f e r e n t , however. In the past, due to high t r a n s p o r t a t i o n c o s t s , one f o r e s t area tended to supply one community and so sustained y i e l d was r e q u i r e d from the s i n g l e f o r e s t areas i n order to maintain the r e l a t i v e l y s t a t i c community. The p r a c t i c a l importance of the r e s u l t s of Chapter I I I i s as f o l l o w s . The models of t h i s t h e s i s have assumed that f o r e s t managers can guess c o r r e c t l y a l l fu t u r e p r i c e s and future rates of time preference and so on. C l e a r l y , t h i s i s u n l i k e l y to be the case i n p r a c t i c e . P r a c t i c a l planning may thus be r e s t r i c t e d to some f i n i t e time span and the o b j e c t i v e may be r e s t r i c t e d to maximizing the present value of net revenues over t h i s time span subject to some p h y s i c a l c o n s t r a i n t s of what the age d i s t r i b u t i o n of the f o r e s t can be at the end of the time span^. The r e s u l t s of Chapter I I I suggest that p h y s i c a l t a r g e t s should be to create normal f o r e s t s i n each f o r e s t area. Then, the planning process based on f i n i t e time spans and expectations about future economic parameters w i l l produce a pla n which may be a f a i r l y good approximation to the p l a n which would be made i f fut u r e economic parameters were known with - 7 -c e r t a i n t y f o r a l l t i m e . F i n a l l y , i t s h o u l d be n o t e d t h a t t h e r e a r e many c a v e a t s t h a t must be a p p l i e d t o t h e c o n c l u s i o n o f t h i s c h a p t e r . L o g g i n g a c t i v i t i e s i n any a r e a a r e n o t c o n d u c t e d i n d e p e n d e n t l y o f l o g g i n g e l s e w h e r e . An example i s a r o a d w h i c h i s u s e d f o r h a u l i n g l o g s f r o m s e v e r a l d i f f e r e n t a r e a s . These i n t e r -d e p e n d e n c i e s may make i t t o o c o s t l y t o l o g a p o r t i o n o f e v e r y f o r e s t a r e a i n e v e r y t i m e p e r i o d . - 8 -CHAPTER I THE QUALITATIVE EFFECT OF COST VARIATIONS ON OPTIMAL ROTATION PERIODS IN FORESTRY 1. V a r i a t i o n s i n R o t a t i o n P e r i o d s Anyone f a m i l i a r w i t h f o r e s t r y i s aware o f the m y r i a d o f p h y s i c a l and b i o l o g i c a l c h a r a c t e r i s t i c s o f f o r e s t s w hich have an e f f e c t on l o g g i n g costs"'". I t i s l e s s w e l l r e a l i z e d t h a t these c o s t v a r i a t i o n s a f f e c t the o p t i m a l r o t a t i o n p e r i o d f o r d i f f e r e n t f o r e s t a r e a s . There appears to be l i t t l e i n the f o r e s t r y l i t e r a t u r e which d i s c u s s e s the s e n s i t i v i t y o f r o t a t i o n p e r i o d s t o v a r i a t i o n s i n parameters a f f e c t i n g l o g g i n g c o s t s . E x c e p t i o n s are Haley and Smith (1964, 2 8 ) , Dobie (1966, 90-94) and K i l k k i and V a i s a n e n (1969, 12-18). The c o n c l u s i o n s o f the s e a u t h o r s a re based on e x p l i c i t c a l c u l a -t i o n s o f r o t a t i o n ages under v a r y i n g c o n d i t i o n s . I t i s , however, p o s s i b l e t o conduct a t h e o r e t i c a l a n a l y s i s o f the q u a l i t a t i v e e f f e c t s o f v a r i a t i o n s i n parameters a f f e c t i n g c o s t s on o p t i m a l r o t a t i o n p e r i o d s . Recent examples are Ledy a r d and Moses (1976), C l a r k (1976, 262-3) and Howe (1979, 230-2). The p r e s e n t c h a p t e r i s i n t e n d e d as an e x t e n s i o n o f t h i s work. Ledyard and Moses (1976) d e r i v e the r e s u l t t h a t the o p t i m a l r o t a t i o n p e r i o d i n c r e a s e s the f u r t h e r the l o g s must be t r a n s p o r t e d t o a p r o c e s s i n g p o i n t . T h e i r model i s somewhat p e c u l i a r i n t h a t t h e i r r e s u l t depends on making the l e v e l o f e x p e n d i t u r e on r e g e n e r a t i o n a c h o i c e v a r i a b l e . On the o t h e r - 9 -hand, i n the model used by C l a r k (1976, 259), one can o b t a i n the above r e s u l t w i t h o u t any r e f e r e n c e to r e g e n e r a t i o n . The d i f f e r e n c e between th e s e two models t u r n s out to be the way i n w hich they account f o r c o s t s . Ledyard and Moses (1976, 145) e m p h a s i z i n g t r a n s p o r t a t i o n c o s t s , account f o r c o s t s as average c o s t s p e r " b o a r d - f e e t or some o t h e r such measure". C l a r k (1976, 257-8) e m p h a s i z i n g f a l l i n g c o s t s , accounts f o r c o s t s as average c o s t s p e r t r e e . Both these and o t h e r types o f c o s t s such as b u c k i n g , y a r d i n g , l o a d i n g , overhead and o t h e r types o f c o s t s have a l e g i t i m a t e c l a i m to be i n c l u d e d i n a model of s t e a d y s t a t e f o r e s t r y . The f i r s t purpose o f t h i s c h a p t e r i s to l a y out a model of s t e a d y s t a t e f o r e s t r y which i s g e n e r a l enough to i n c o r p o r a t e a l l these c o s t s . Then i t w i l l be p o s s i b l e to i d e n t i f y the b a s i c p r o p e r t y t h a t l o g g i n g n e t revenue f u n c t i o n s must s a t i s f y i n o r d e r f o r the Ledyard-Moses r e s u l t to h o l d ; t h a t i s f o r the o p t i m a l r o t a t i o n p e r i o d to be l e n g t h e n e d by an i n c r e a s e i n some parameter a f f e c t i n g l o g g i n g c o s t s o r revenues. A l s o , an attempt w i l l be made to show t h a t the l o g g i n g net revenue f u n c t i o n has t h i s b a s i c p r o p e r t y when the v a r i a b l e parameter i s r o u n d - t r i p h a u l i n g t i m e . I t i s not o n l y revenue and c o s t c o n s i d e r a t i o n s t h a t determine o p t i m a l r o t a t i o n p e r i o d s but the growing c o n d i t i o n s f o r the f o r e s t - f o r example, the s o i l t y p e . T h i s paper a l s o a n a l y s e s the impact o f an improvement i n growing c o n d i t i o n s on the o p t i m a l r o t a t i o n p e r i o d . I t w i l l be shown t h a t , s u b j e c t to some a d d i t i o n a l c o n d i t i o n s , such an - 10 -improvement s h o r t e n s t h i s p e r i o d . The i n t e r e s t i n g f e a t u r e o f the Ledyard-Moses paper i s t h a t the r e s u l t s above c o n t i n u e to h o l d i n a model which i n c l u d e s a more complex f o r e s t r y o p e r a t i o n , namely a v a r i a b l e l e v e l o f e f f o r t expended on r e g e n e r a t i o n . However, they s t a t e (1976, 151, f n j ) t h a t t h e i r r e s u l t may no l o n g e r h o l d i f s i l v i c u l t u r a l o p e r a t i o n s are a l l o w e d f o r i n t h e i r model. The second p a r t o f t h i s c h a p t e r w i l l i n t r o d u c e b o t h v a r i a b l e r e g e n e r a t i o n and s i l v i c u l t u r e ( o f the n o n - t h i n n i n g k i n d ) i n t o the g e n e r a l model. I t w i l l be shown f i r s t o f a l l t h a t a l l o w i n g f o r these o p e r a t i o n s w i l l s h o r t e n the o p t i m a l r o t a t i o n p e r i o d - p r o v i d e d such o p e r a t i o n s are p r o f i t a b l e . T h i s agrees w i t h the e m p i r i c a l r e s u l t s o f Dobie (1966, 96-98) a l t h o u g h he found t h i s e f f e c t to be not v e r y s i g n i f i c a n t . S e c o n d l y , i t w i l l be shown t h a t t h e r e are c o n d i t i o n s under w h i c h the o p t i m a l r o t a t i o n p e r i o d v a r i e s p o s i t i v e l y w i t h the parameter x. These c o n d i t i o n s r e f e r not merely to the i n f l u e n c e s o f changes i n x, however, but r e f e r a l s o t o the n a t u r e o f the o p t i m a l s i l v i c u l t u r a l p a t t e r n . T h i s must be zero a t b o t h the b e g i n n i n g and end o f t h e h a r v e s t c y c l e . Some assumption about the o p t i m a l s i l v i c u l t u r a l p a t t e r n i s u n a v o i d a b l e when t r y i n g t o g e n e r a l i z e the Ledyard-Moses r e s u l t . I t w i l l be shown t h a t i f the o p t i m a l s i l v i c u l t u r a l e f f o r t i s n o n d e c r e a s i n g over the h a r v e s t c y c l e - then the 2 Ledyard-Moses r e s u l t may be r e v e r s e d . F i n a l l y , c o n d i t i o n s under w h i c h a g r e a t e r than m i n i m a l - 1 1 -r e g e n e r a t i o n o r s i l v i c u l t u r a l e f f o r t a r e p r o f i t a b l e w i l l b e i d e n t i f i e d . T h e r e s u l t s o f t h i s c h a p t e r p r o v i d e t h e b e g i n n i n g f o r a s e t o f g u i d e l i n e s f o r w h a t s h o u l d b e e x p e c t e d i n e m p i r i c a l s e n s i t i v i t y a n a l y s e s o f o p t i m a l r o t a t i o n p e r i o d s . T h e b a s i c c o n d i t i o n i n v o l v e s h o w a c e r t a i n e l a s t i c i t y v a r i e s w i t h c h a n g e s i n p a r a m e t e r s a f f e c t i n g l o g g i n g n e t r e v e n u e f u n c t i o n s . D e s i g n e r s o f l o g g i n g r e v e n u e - c o s t s t u d i e s s h o u l d t r y t o c a p t u r e t h e s e e f f e c t s i n c h o o s i n g t h e i r f u n c t i o n a l f o r m s . I t m a y , o f c o u r s e , t u r n o u t e m p i r i c a l l y t h a t r o t a t i o n p e r i o d s a r e n o t s i g n i f i c a n t l y s e n s i t i v e t o c h a n g e s i n a n y p a r a m e t e r a f f e c t i n g l o g g i n g n e t r e v e n u e s . T h e o n l y s t u d y w h i c h m a y s h e d s o m e l i g h t o n t h i s m a t t e r i s D o b i e ( 1 9 6 6 ) . D o b i e g i v e s a n a v e r a g e c o n v e r s i o n r e t u r n o f 42<f p e r c u b i c f t . ( p . 6 5 ) w i t h r e p r e s e n t a t i v e h a u l i n g c o s t o f 17$ p e r c u b i c f t . ( p . 1 5 ) f o r a 2 0 m i l e h a u l . D o u b l i n g t h e h a u l d i s t a n c e w o u l d g r e a t l y i n c r e a s e t h e h a u l i n g c o s t a l t h o u g h l e s s t h a n p r o p o r t i o n a l l y ( S m i t h a n d T s e , 1 9 7 7 , 5 3 ) . T h u s d o u b l i n g t h e h a u l d i s t a n c e m i g h t l e a d t o a 75% i n c r e a s e i n h a u l i n g c o s t s a n d a 30% r e d u c t i o n i n a v e r a g e c o n v e r s i o n r e t u r n s . D o b i e ' s c a l c u l a t i o n s ( 1 9 6 6 , 9 6 - 9 8 ) i n d i c a t e t h a t t h i s w o u l d l e a d t o a s u b s t a n t i a l i n c r e a s e i n t h e r o t a t i o n p e r i o d . A m o r e p r e c i s e s t a t e m e n t i s n o t p o s s i b l e b e c a u s e t h e n e w r o t a t i o n p e r i o d s f a l l o u t s i d e t h e r a n g e o f y e a r s c o n s i d e r e d b y D o b i e . - 1 2 -2 . T h e G e n e r a l S t e a d y S t a t e F o r e s t r y M o d e l T h e f r a m e o f r e f e r e n c e f o r t h i s s e c t i o n i s a c o l l e c t i o n o f f o r e s t s i t e s w h o s e b i o l o g i c a l a n d p h y s i c a l d i f f e r e n c e s c a n b e c h a r a c t e r i z e d b y v a r i a t i o n s i n t h e v a l u e s o f a n u m b e r o f p a r a m e t e r s . F w i l l d e n o t e t h e v o l u m e o f w o o d p e r h e c t a r e o n o n e o f t h e s i t e s . E a c h s i t e w i l l b e a s s u m e d t o h a v e a n e v e n -a g e d s t a n d o f t r e e s o n i t . T h e g r o w t h p r o c e s s e s o f l i v i n g o r g a n i s m s a r e e x t r e m e l y c o m p l e x a n d n o t w e l l u n d e r s t o o d . N e v e r t h e l e s s , e x p e r i m e n t a l b i o l o g i s t s h a v e s o u g h t , w i t h s o m e s u c c e s s , t o f i n d s i m p l e s t a t i s t i c a l r e p r e s e n t a t i o n s o f t h e g r o w t h r a t e o f o r g a n i s m s s u c h a s t r e e s . G o i n g b a c k a t l e a s t t o v o n B e r t a l a n f l y ( 1 9 3 8 ) t h e m o s t i m p o r t a n t e x p l a n a t o r y v a r i a b l e s i n t h e s e r e p r e s e n t a -t i o n s h a v e b e e n a g e a n d s i z e . T h u s i t w i l l b e a s s u m e d h e r e t h a t t h e g r o w t h p r o c e s s o n o u r f o r e s t s i t e s c a n a d e q u a t e l y b e r e p r e s e n t e d b y a b i o l o g i c a l g r o w t h f u n c t i o n o f t h e f o r m • F = f ( F , t ; y ) w h e r e t i s t h e a g e o f t h e t r e e s a n d y i s a v e c t o r o f p a r a m e t e r s c a p t u r i n g o t h e r d e t e r m i n a n t s o f g r o w t h w h i c h m i g h t v a r y b e t w e e n s i t e s . F o r e x a m p l e , o n e c o m p o n e n t o f y m i g h t b e a n i n d e x o f s o i l f e r t i l i t y w h i l e a n o t h e r m i g h t b e a m e a s u r e o f s t a n d d e n s i t y o r n u m b e r o f t r e e s p e r h e c t a r e ( s e e H a l e y a n d S m i t h ( 1 9 6 4 ) ) . A n e x a m p l e o f s u c h a g r o w t h f u n c t i o n w h i c h h a s a p p e a r e d i n t h e f o r e s t r y l i t e r a t u r e i s F - a t " b F e - p F 3 - 13 -F o r t h e t i m e b e i n g , t h i n n i n g w i l l n o t be a l l o w e d i n t h e model below so s t a n d volume and s t a n d age w i l l be c o r r e l a t e d and t h u s t h e g r o w t h law c a n be r e p r e s e n t e d as S e c o n d l y , t h e t o t a l n e t r e v e n u e o b t a i n a b l e f r o m l o g g i n g and d e l i v e r i n g volume F o f l o g s t o a p r o c e s s i n g p l a n t w i l l be d e n o t e d by R ( F ; x ) . H e r e x i s a v e c t o r o f p a r a m e t e r s c h a r a c t e r i z i n g d i f f e r e n c e s between s i t e s w h i c h a f f e c t e i t h e r l o g g i n g c o s t s o r r e v e n u e s . F o r example, a component o f x c o u l d r e p r e s e n t r o u n d - t r i p h a u l i n g t i m e t o the p r o c e s s i n g p l a n t o r t h e a v e r a g e s l o p e o f t h e s i t e ( s e e A n d e r s o n ( 1 9 7 6 ) ) . L e d y a r d and Moses (1976) w r i t e t h i s r e v e n u e f u n c t i o n as p ( x ) F . T h i s l i n e a r i t y i n F i s m i s l e a d i n g , however, b e c a u s e i t i s w e l l - k n o w n t h a t t h e r e a r e i m p o r t a n t e c onomies o f s c a l e i n l o g g i n g (Wackerman (1949, 3 6 1 - 2 ) ) . One r e a s o n f o r t h i s i s t h a t t h e c o s t s o f some l o g g i n g o p e r a t i o n s s u c h as f e l l i n g , y a r d i n g , l o a d i n g depend m a i n l y on t h e number o f p i e c e s r a t h e r t h a n t h e p i e c e s i z e so t h a t c o s t s p e r u n i t volume f a l l r a p i d l y 4 w i t h i n c r e a s i n g p i e c e s i z e . Thus i t w i l l be assumed h e r e t h a t a v e r a g e n e t r e v e n u e p e r u n i t volume r i s e s w i t h t h e s i z e o f t h e t r e e s o r (1) F = f ( F ; y ) (2) FRp - R > 0 - 14 -T h i s i s e q u i v a l e n t to s a y i n g t h a t the e l a s t i c i t y o f the net revenue f u n c t i o n w i t h r e s p e c t to volume exceeds 1. That i s (3) E R p = (FR-p) | R > 1 T h i s n e t revenue f u n c t i o n may now be i n s e r t e d i n a s t e a d y s t a t e f o r e s t r y model. I t i s assumed t h a t the s i t e was bare a t time 0 a t which time a crop o f t r e e s began growing on i t . T h i s crop i s l o g g e d a t time T, a second crop i s then a l l o w e d to grow on the s i t e w hich w i l l be l o g g e d a t time 2T and t h i s p r o c e s s o f l o g g i n g s u c c e s s i v e crops o f T y e a r o l d t r e e s i s c a r r i e d on i n d e f i n i t e l y . The p r e s e n t v a l u e o f n e t revenues from such a h a r v e s t i n g p o l i c y w i l l be (4) V(T;x,y) = R ( F ( T ) ; x ) e " r T ( 1 - e " r T ) " 1 where r i s the a p p r o p r i a t e m a r g i n a l r a t e o f time p r e f e r e n c e . The f o r e s t manager's problem i s to choose the r o t a t i o n p e r i o d T which w i l l maximize V s u b j e c t to the d i f f e r e n t i a l c o n s t r a i n t (1) and the i n i t i a l c o n d i t i o n F ( 0 ) = F Q , g i v e n . These c o n s t r a i n t s i m p l y that** (5) dF(T)/dT = f ( F ( T ) ) The f i r s t o r d e r c o n d i t i o n f o r the manager's problem i s then - 15 -(6) V T = [-rV + R F £ - r R ] [ e r T - l ] " 1 = 0 where the terms are e v a l u a t e d a t F(T) and T u n l e s s o t h e r w i s e s p e c i f i e d . A s t r a i g h t f o r w a r d c a l c u l a t i o n now y i e l d s the Faustmann f o r m u l a (see C l a r k (1976, 259)),. (7) R F • f = r R [ l - e " r T ] _ 1 w h i c h t o g e t h e r w i t h the d i f f e r e n t i a l e q u a t i o n F = f ( F ) determines the o p t i m a l r o t a t i o n p e r i o d . I t i s i n s t r u c t i v e t o r e w r i t e the Faustmann f o r m u l a i n the form (7) E R F • gp = r [ l - e _ r T ] _ 1 where gp = F/F = f ( F ) / F i s the growth r a t e o f volume per h e c t a r e . The term E^p . gp i s then the growth r a t e o f the v a l u e p e r h e c t a r e o f t h i s f o r e s t . One does not c u t the f o r e s t as l o n g as t h i s growth r a t e exceeds the r e t u r n on c a p i t a l r , a d j u s t e d t o take account o f the i n f i n i t e s t r i n g o f h a r v e s t s . A nother way t o l o o k a t the m a t t e r i s t h i s . R°° = R ( F ) ( 1 - e " r T ) - 1 = V ( T ) e r T i s the p r e s e n t v a l u e o f an i n f i n i t e s t r i n g o f h a r v e s t s , based on a r o t a t i o n p e r i o d T, a t time T. The growth r a t e o f t h i s 16 rT -1 i s the growth r a t e o f R, E R p • gp, minus r ( e -1) E q u a t i n g t h i s to the r a t e o f r e t u r n r on c a p i t a l g i v e s the Faustmann f o r m u l a . The o p t i m a l r o t a t i o n p e r i o d T* i s i l l u s t r a t e d i n F i g . 1 below. The growth r a t e of v a l u e per h e c t a r e E R p • gp - rT -1 must exceed r ( l - e ) f o r T < T* i n o r d e r f o r r o t a t i o n p e r i o d s l o n g e r t h a n T t o be more p r o f i t a b l e than the r e t u r n p o s s i b l e from f i n a n c i a l markets. 1/T E R F ( F ) . g p ( F ) r [ l - e _ r t ] -1 T or F(T) F i g u r e 1 The O p t i m a l R o t a t i o n P e r i o d - 17 -Our i n t e r e s t i n t h i s s e c t i o n i s w i t h how the o p t i m a l r o t a t i o n p e r i o d v a r i e s w i t h a change i n one o f the s i t e p arameters i n the v e c t o r s x and y. For n o t a t i o n a l s i m p l i c i t y , x or y w i l l be t r e a t e d as a s i n g l e parameter below. I n a p p l i c a t i o n , t h e n , the r e s u l t s below h o l d p r o v i d e d the s i t e s are s i m i l a r i n a l l c h a r a c t e r i s t i c s e x cept the one b e i n g c o n s i d e r e d . Below the r e s u l t s o f t h i s s e c t i o n are s t a t e d and i n t e r p r e t e d . F o l l o w i n g t h i s , p r o o f s o f the r e s u l t s are g i v e n and the s e c t i o n c o n c l u d e s w i t h a d i s c u s s i o n o f the case where x i s r o u n d - t r i p h a u l i n g t i m e . The main r e s u l t o f t h i s s e c t i o n i s Prop. 1: I f dE Rp/dx > 0 (< 0 ) , then the o p t i m a l r o t a t i o n p e r i o d T l ( x ) * i n c r e a s e s ( d e c r e a s e s ) w i t h i n c r e a s e s ( d e c r e a s e s ) i n x. The i n t u i t i o n o f t h i s r e s u l t may be seen by r e f e r r i n g to F i g . 1 above. I f x i n c r e a s e s t o x + Ax, then f o r a g i v e n F, E R p i s i n c r e a s e d and gp i s unchanged. Thus the growth curve s h i f t s up, f u r t h e r growth becomes p r o f i t a b l e f o r t r e e s o f age T ( x ) * , and hence T(x + A x ) * exceeds T ( x ) * . I t i s i n t e r e s t i n g a t t h i s p o i n t to l o o k a t the model o f Ledyard and Moses (1976). There R(F;x) = p ( x ) F so E R p = 1 and dE Rp/dx = 0. Thus i t was n e c e s s a r y f o r them to put r e g e n e r a t i o n c o s t s i n t o t h e i r model i n o r d e r t o get the r e s u l t t h a t the o p t i m a l r o t a t i o n p e r i o d i n c r e a s e d w i t h d i s t a n c e from the s a w m i l l . On the o t h e r hand i n C l a r k (1976), R(F;x) = pF - c ( x ) and d E R p / d x = c» ( x ) E R p R _ 1 so the o p t i m a l r o t a t i o n p e r i o d i n c r e a s e s w i t h d i s t a n c e from the - 18 s a w m i l l . Some f u r t h e r c h a r a c t e r i s t i c s of the o p t i m a l h a r v e s t are g i v e n by the nex t two p r o p o s i t i o n s . P r o p . 2: I f d E R F / d x > 0 (< 0) then ( i ) dF(T)/dx = f • (dT/dx) > 0 ( < 0) ( i i ) d(R/T)/dx > 0 ( < 0) i f R Y > 0 ( < 0) P a r t ( i ) says the l o n g e r the o p t i m a l r o t a t i o n p e r i o d , then the l a r g e r the volume o f wood i n the f o r e s t when h a r v e s t e d . P a r t ( i i ) r e f e r s t o average o r s u s t a i n e d n e t revenue r e c e i v e d from the l o g g i n g o p e r a t i o n . A t h a r v e s t - rT -1 ti m e , the growth r a t e o f net revenue i s r [ l - e ] which exceeds T ^ the growth r a t e o f t i m e ^ . Thus the growth r a t e o f s u s t a i n e d revenue i s p o s i t i v e . An i n c r e a s e i n x a f f e c t s the s u s t a i n e d n e t revenue d i r e c t l y i n R and i n d i r e c t l y by l e n g t h e n i n g the r o t a t i o n p e r i o d i f d E R p / d x > 0. The i n d i r e c t e f f e c t i s p o s i t i v e i f the growth r a t e o f s u s t a i n e d revenue i s p o s i t i v e . P r o p . 5: ( i ) d ( r V ) / d x > 0 ( < 0) i f R > 0 ( < 0) ( i i ) d 2 ( r V ) / d x 2 > o i f R v > 0 A A T h i s p r o p o s i t i o n r e f e r s t o the maximum b i d - r e n t ( rV) a l o g g i n g o p e r a t i o n would be w i l l i n g to pay f o r the use o f the l a n d . T h i s b i d - r e n t has s i m i l a r p r o p e r t i e s w i t h r e s p e c t to x as the net revenue f u n c t i o n . - 1 9 -T h e a b o v e t h r e e p r o p o s i t i o n s g i v e t h e s i t u a t i o n w h e n t h e v a r i a b l e p a r a m e t e r i n f l u e n c e s o n l y t h e n e t r e v e n u e f u n c t i o n . I t i s a l s o p o s s i b l e t o g e t s o m e c o m p a r a t i v e s t a t i c r e s u l t s w h e n t h e v a r i a b l e p a r a m e t e r y a f f e c t s t h e g r o w t h f u n c t i o n i n t h e f o r m F = f ( F , y ) . T h e s e r e s u l t s d e p e n d o n t h e e l a s t i c i t y o f t h e g r o w t h f u n c t i o n w i t h r e s p e c t t o v o l u m e p e r h e c t a r e ( E £ p = F f p / f ) a n d h o w t h i s e l a s t i c i t y v a r i e s w i t h y . T h e r e s u l t i s P r o p . 4 : I f f > 0 ( < 0 ) a n d 8 E £ p / 3 y < 0 ( > 0) f o r a l l f o r e s t s i z e s F , a n d i f R p p < 0 a n d f p < 0 a t t h e o p t i m u m t h e n t h e o p t i m a l r o t a t i o n p e r i o d T ( y ) * d e c r e a s e s ( i n c r e a s e s ) w i t h i n c r e a s e s i n y . T h i s r e s u l t i s l e s s a p p a r e n t i n t u i t i v e l y t h a n u i t s a n a l o g u e P r o p . 1 . R e f e r r i n g t o F i g . 1 , a n i n c r e a s e i n y a f f e c t s t h e g r o w t h r a t e o f n e t r e v e n u e s E R p • g p i n t w o w a y s . T h e r e i s a d i r e c t i n c r e a s e i n g p , e . g . a n i n c r e a s e d r a t e o f g r o w t h o f v o l u m e a t a n y v o l u m e p e r h e c t a r e s i n c e f > 0 . T h i s s h i f t s t h e c u r v e E R p • g p u p . H o w e v e r , t h e f a c t t h a t a t a n y a g e v o l u m e p e r h e c t a r e i s h i g h e r a l s o m e a n s t h a t t h e r a t e o f n e t r e v e n u e g r o w t h i s l o w e r . T h i s i n d i r e c t e f f e c t s h i f t s t h e c u r v e E R p • g p b a c k d o w n . T h e a d d i t i o n a l c o n d i t i o n s i n P r o p . 4 w i l l e n s u r e t h a t t h e i n d i r e c t e f f e c t o u t w e i g h s t h e d i r e c t e f f e c t . T h e c o n d i t i o n s R p p < 0 a n d f p < 0 a t t h e o p t i m u m d o n o t c o m e c o m p l e t e l y o u t o f t h e b l u e . I n f a c t , t h e y a r e s u f f i c i e n t c o n d i t i o n s f o r V t o h a v e a m a x i m u m ( e . g . V ^ T < 0 ) . N o e c o n o m i c - 20 -o r b i o l o g i c a l e x p l a n a t i o n o f the o t h e r c o n d i t i o n 8E^p/3y < 0 has been f o u n d , however. M o r e o v e r , i t does n o t seem t o be p o s s i b l e t o p i n down any o t h e r c h a r a c t e r i s t i c s o f the o p t i m a l h a r v e s t s u c h as t h e s i g n o f d F ( T ) / d y . F ( T ) i s i n c r e a s e d by the b e t t e r g r o w i n g c o n d i t i o n s ( f > 0) b u t d e c r e a s e d by t h e s h o r t e r r o t a t i o n p e r i o d . The b a l a n c e does n o t seem t o be d e t e r m i n a b l e . The n e x t p o r t i o n o f t h i s s e c t i o n w i l l f o r m a l l y p r o v e P r o p s . 1 - 4 . To b e g i n , d i f f e r e n t i a t e t h e f i r s t o r d e r c o n d i t i o n (6) t o o b t a i n (9) dT/dx = -V T xV~l A l s o f r o m ( 6 ) , V T = R [ E R p • g p - r [ l - e " r T ] _ 1 ] [ e r T - l ] " 1 so (10) V T x = ( 8 E R F / 8 x ) • g F • [ e r T - l ] _ 1 Vjrj, i s n e g a t i v e by t h e s e c o n d o r d e r c o n d i t i o n f o r a maximum T h e r e f o r e by (9) dT/dx has t h e same s i g n as Vrr, and by (10) V has t h e same s i g n as 8 E R p / 8 x . T h i s p r o v e s P r o p . 1. P r o p . 2 ( i ) i s immediate and f o r 2 ( i i ) d ( R / T ) / d x = T " 2 [ T R c f - R ] ( d T / d x ) + T _ 1 R r X S i n c e dT/dx > 0 and R > 0 the p r o p o s i t i o n f o l l o w s i f - A . TRpf - R > 0. From the f i r s t o r d e r c o n d i t i o n (7) 21 -TRpf - R = R [ r T [ l - e " r T ] _ 1 - 1] > 0. 7 F i n a l l y , Prop. 3 ( i ) i s an envelope theorem since d(rV)/dx = rV + rV T(dT/dx) rR r e r T - l ] " 1 + 0 > 0 D i f f e r e n t i a t i n g again d 2 ( r V ) / d x 2 = r V x x + rV x T(dT/dx) = r R x x t e r T " " r V x T VTT > 0 The proof of Prop. 4 w i l l r e q u i r e more e f f o r t . F i r s t the f o l l o w i n g lemma w i l l be needed Lemma f ( T ) " 1 ( 8 F ( T ) / 8 y ) = C f . ( s ) f ( s ) ' X d s Jo  y fT fF(T) • _ ± Proof T = I dt = [ f ( s ) dF changing v a r i a b l e s o D i f f e r e n t i a t e t h i s w i t h respect to y to get 0 = f ( T ) _ 1 ( 9 F ( T ) / 9 y ) - f f v ( s ) f ( s ) " 2 d F ./p y or changing v a r i a b l e s again ° /T f v ( s ) f ( s ) _ 1 d s o y Turn now to the proof of Prop. 4 . As usual (11) dT/dy = - V T y V ^ 22 -M o r e o v e r d i f f e r e n t i a t i n g (6) (12) V y = R p • ( 3 F ( T ) / 3 y ) • [ e r T - l ] " 1 (13) V T = [ R F • f - r R [ l - e " r T ] " 1 ] [ e r T - l ] - 1 (14) V = [ ( R F F • f + R £ . f p - r R p [ l - e " r T ] " 1 ) ( 9 F ( T ) / 9 y ) + RF £y ] [ e r T " H " 1 Now f r o m the lemma, 9 F ( T ) / 9 y = T> f(T)•£ ( s ) • f ( s ) " 1 f o r some s M o r e o v e r , d(£ (s) f ( s ) ~ 1J)/ds = f y f - f y * f p f " 1 ^ 0 s i n c e 9 E £ p / 9 y < 0, and so 3 F ( T ) / 9 y > T«£ ( T ) . C o n s e q u e n t l y (15) V T y < [ ( R p p - f + Rp-fp + Rp/T - r R p d - e ^ V 1 ) ( 9 F ( T ) / 3 y ) ] [ e r T - l ] " 1 A g a i n , by t h e lemma, 3 F ( T ) / 3 y > 0. Then s i n c e R p p < 0, f p < 0 and T _ 1 - r [1 - e " r T ] _ 1 < 0 ( s e e f n . 6) i t f o l l o w s t h a t V < 0 and t h e n by (11) dT/dy < 0. y ^ An example o f t h e use o f t h e above r e s u l t s i s p r o v i d e d by the s i t u a t i o n c o n s i d e r e d by L e d y a r d and Moses (1976) where the e f f e c t o f v a r i a b l e d i s t a n c e s f r o m the p r o c e s s i n g c e n t e r on o p t i m a l r o t a t i o n p e r i o d s i s c o n s i d e r e d . The a c t u a l p a r a m e t e r u s e d s h o u l d be a v e r a g e r o u n d - t r i p h a u l i n g time - 23 -i n c l u d i n g average d e l a y time a t the l o a d i n g and u n l o a d i n g g l a n d i n g s r a t h e r than m i l e s from the s a w m i l l . V a r i a t i o n s i n t h i s parameter a f f e c t o n l y the c o s t o f l o g g i n g . I t w i l l be assumed here t h a t these e f f e c t s are s e p a r a b l e from the e f f e c t on c o s t s o f v a r i a t i o n s i n volume t r a n s p o r t e d . Thus i t i s supposed t h a t Here S(F) r e p r e s e n t s a l l revenues and c o s t s which are indepen-dent o f the h a u l i n g o p e r a t i o n . T h i s n e t revenue f u n c t i o n w i l l be assumed to e x h i b i t n o n ^ d e c r e a s i n g r e t u r n s to s c a l e f o r the same reason R was assumed to have t h i s p r o p e r t y . The term t ( x ) C ( F ) r e p r e s e n t s h a u l i n g c o s t s and the term d(x) any o t h e r c o s t s dependent on d i s t a n c e such as r o a d maintenance c o s t s and the c o s t s o f t r a n s p o r t i n g men and equipment to the l o g g i n g s i t e . H a u l i n g c o s t s i n c r e a s e w i t h i n c r e a s e s i n r o u n d - t r i p h a u l i n g time. However, the i n c r e a s e i n c o s t i s not i n f u l l p r o p o r t i o n to the i n c r e a s e i n time (Smith and Tse, (1977, 53)) . Thus (16) R(F;x) = S(F) - t ( x ) C ( F ) - d(x) Thus (17) FS„ - S > 0 t (18) t* (x) > 0 x t ' ( x ) - t ( x ) < 0 24 -Simi rlar€-y thexQ a r e economies o f s c a l e i n h a u l i n g l a r g e r l o g s b e c a u s e more volume can be c a r r i e d p e r t r u c k l o a d (Wackerman (1949, 3 6 1 - 2 ) ) . Thus (19) C p >: 0 , FCp - C < -0 A l s o i t w i l l be assumed t h a t (20) d * ( x ) > 0 o r t h a t r o a d m a i n t e n a n c e and o t h e r c o s t s d e p e n d e n t on d i s t a n c e do n o t d e c r e a s e w i t h d i s t a n c e . G i v e n t h a t R(F;x) has t h e f o r m shown i n (16) E R p = FRpR" 1 = [FSp - t ( x ) F C F ] [ S - t ( x ) C ( F ) - d ( x ) ] " 1 and a s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t 3 E R p / 9 x has the same s i g n as ' , . . [ t ' : ( x ) C > : f l « I ( ^ [ F S p - S] + [ f ( x ) S .+ t ( x ) d ' ( x ) ] [ C - FCp] -+ d ' ( x ) [ S - t ( x ) C ] + " d ( x ) t ' ( x ) F C p From the a s s u m p t i o n s above t h i s i s non-'negative s i n c e S - t ( x ) C > d ( x ) i f t h e l o g g i n g o p e r a t i o n i s t o be p r o f i t a b l e . Thus P r o p . 5: The o p t i m a l r o t a t i o n p e r i o d i n c r e a s e s w i t h i n c r e a s e s i n r o u n d - t r i p h a u l i n g time i f d ' ( x ) f 0 o r i f 25 t ' ( x ) f 0 and.one o f FSp - S, C - FCp or d(x) i s s t r i c t l y p o s i t i v e . - 26 -3. V a r i a b l e R e p l a n t i n g and S i l v i c u l t u r a l E x p e n d i t u r e I n t h i s s e c t i o n , the g e n e r a l s t e a d y s t a t e f o r e s t r y model i s b r o a d e n e d t o g i v e t h e f o r e s t manager c o n t r o l o v e r h i s e x p e n d i t u r e s on r e g e n e r a t i o n and s i l v i c u l t u r e o f t h e non-t h i n n i n g k i n d . I t w i l l be s e e n t h a t t h e r e a r e c i r c u m s t a n c e s i n w h i c h the r e s u l t s o f t h e p r e v i o u s s e c t i o n c o n t i n u e to h o l d i n t h i s b r o a d e r framework. However, i t w i l l a l s o be shown t h a t t h e r e a r e c i r c u m s t a n c e s i n w h i c h t h e s e r e s u l t s a r e r e v e r s e d . V a r i a b l e r e g e n e r a t i o n can be a l l o w e d f o r by a l l o w i n g f o r v a r i a b l e s i z e s e e d l i n g s t o be u s e d . Thus F q becomes a c h o i c e v a r i a b l e . The c o s t s o f r e p l a n t i n g a t l e v e l F Q w i l l be d e n o t e d by D ( F Q ; x ) . The manager w i l l a l s o be a l l o w e d t o choose the time p r o f i l e < L ( t ) > o f s i l v i c u l t u r a l a c t i v i t y ( s u c h as p r u n i n g , f e r t i l i z a t i o n ) o v e r t h e r o t a t i o n p e r i o d w h i c h w i l l m aximize the p r e s e n t v a l u e o f n e t r e v e n u e s f r o m t h i s f o r e s t . The e f f e c t o f t h i s a c t i v i t y i s r e c o g n i z e d i n a b i o l o g i c a l 9 p r o d u c t i o n f u n c t i o n o f the f o r m (1) F = f ( F , L ; y ) where f ^ > 0 and < 0- The c o s t o f t h i s s i l v i c u l t u r a l a c t i v i t y on t y e a r o l d t r e e s w i l l be d e n o t e d W ( L ( t ) ; x ) . W i t h t h e s e two m o d i f i c a t i o n s , the p r e s e n t v a l u e o f n e t r e v e n u e s f r o m the f o r e s t becomes - 27 -(2) V ( T , F o , < L > , x,y) = [ R e ~ r T - D ( F Q ) - r T W ( L ) e ~ r t d t ] [1 - e " r T j _ 1 The f o r e s t manager's problem i s to f i n d the r o t a t i o n p e r i o d T and r e g e n e r a t i o n and s i l v i c u l t u r a l e f f o r t s F Q and <L> /which w i l l maximize t h i s p r e s e n t v a l u e . The s o l u t i o n o f t h i s p roblem can f i r s t o f a l l be compared to the s o l u t i o n o f the model of the p r e v i o u s s e c t i o n . I n c r e a s i n g e x p e n d i t u r e on r e g e n e r a t i o n e f f o r t s above m i n i m a l r e q u i r e m e n t s i s i n the f i r s t i n s t a n c e l i k e the h a u l i n g example o f the p r e v i o u s s e c t i o n ( w i t h t ( x ) = 0 ) . The d e s i r e to reduce the p r e s e n t v a l u e o f these added c o s t s tends to l e n g t h e n the r o t a t i o n p e r i o d . On the o t h e r hand, the volume of wood i n the f o r e s t w i l l be i n c r e a s e d a t any age of the f o r e s t . T h i s tends to decrease the growth r a t e o f the net v a l u e o f the f o r e s t (see F i g . 1) and, hence, to reduce the i n t e r v a l o f ages over which c o n t i n u e d growth i s p r o f i t a b l e . I t w i l l be shown below t h a t t h i s e f f e c t p r e d o m i n a t e s . I f a g r e a t e r than m i n i m a l e x p e n d i t u r e on r e g e n e r a t i o n i s p r o f i t a b l e , t hen the o p t i m a l r o t a t i o n p e r i o d i s s h o r t e n e d . T h i s agrees w i t h the e m p i r i c a l r e s u l t s o f Dobie (1966, 96-98), a l t h o u g h he found t h i s e f f e c t not to be v e r y s i g n i f i c a n t . S i m i l a r remarks a p p l y i f i t i s p r o f i t a b l e to a p p l y some p o s i t i v e l e v e l o f s i l v i c u l t u r e a t some time d u r i n g the r o t a t i o n p e r i o d . The dominant e f f e c t i s the d e s i r e to c a p i t a l i z e on the e x t r a growth g e n e r a t e d by the s i l v i c u l t u r e , - 28 -and thus the o p t i m a l r o t a t i o n p e r i o d i s s h o r t e n e d . These r e s u l t s are summarized i n Prop. 8: I f non-minimal e x p e n d i t u r e s on e i t h e r r e g e n e r a t i o n or s i l v i c u l t u r e are p r o f i t a b l e (and c e r t a i n r e g u l a r i t y c o n d i t i o n s a re t r u e ) , then the o p t i m a l r o t a t i o n p e r i o d w i t h these e x p e n d i t u r e s i s s h o r t e r than the o p t i m a l r o t a t i o n p e r i o d when these e x p e n d i t u r e s are not made. More-o v e r , the volume o f wood h a r v e s t e d per r o t a t i o n w i l l be reduced. Some f e e l i n g f o r when non-minimal e x p e n d i t u r e s on r e g e n e r a t i o n o r s i l v i c u l t u r e are p r o f i t a b l e i s g i v e n i n the next p r o p o s i t i o n . P r o p . 9: ( i ) A non-minimal e x p e n d i t u r e on r e g e n e r a t i o n i s p r o f i t a b l e i f where T i s the o p t i m a l r o t a t i o n p e r i o d g i v e n a m i n i m a l (fJ) l e v e l o f e f f o r t devoted t o r e g e n e r a t i o n . R F ( F ( T ) ) f ( F ( T ) ) f " 1 ( _ 0 ) e -rT - D F (0) > 0 o ( i i ) I f f o r the o p t i m a l p a t h w i t h o u t s i l v i -c u l t u r e R F ( F ( T ) ) f ( F ( T ) ) f L ( F ( t ) , 0 ) e -rT - f ( F ( t ) ) W L ( 0 ) e -rT > 0 f o r some age t , then a p o s i t i v e l e v e l o f s i l v i c u l t u r e i s 29 -p r o f i t a b l e . The i n t e r p r e t a t i o n o f t h e s e r e s u l t s r e q u i r e s t h e f o l l o w i n g f a c t s a b o u t how t h e g r o w t h o f t h e f o r e s t depends on i t s age and t h e f o r e s t ' s i n i t i a l s t a t e . These f a c t s a r e 10 (3) 9 F ( T ) / 9 T = f (F (T) ) , 9F tt')V'3 F q . f (-F ( t ) ) f ( F ^ ) The e x p r e s s i o n i n P r o p . 9 ( i ) thus can be s e e n t o be (up t o s i g n ) the m a r g i n a l p r e s e n t v a l u e p r o d u c t o f t h e f i r s t i n c r e m e n t t o the m i n i m a l r e g e n e r a t i o n e f f o r t . T h i s s h o u l d be p o s i t i v e and i n d e e d r e g e n e r a t i o n e f f o r t s h o u l d be c a r r i e d o u t up t o t h e p o i n t a t w h i c h th e m a r g i n a l p r e s e n t v a l u e p r o d u c t becomes 0. P a r t ( i i ) has a s i m i l a r i n t e r p r e t a t i o n . R p ( F ( T ) ) f ( F ( T ) ) f ( F ( ' t ) ) " 1 f L ( F ( t ) , 0 ) e " r T i s . e s s e n t i a l l y t h e m a r g i n a l p r e s e n t v a l u e p r o d u c t o f a p p l y i n g a s m a l l i n c r e m e n t o f s i l v i c u l t u r e t o an u n t o u c h e d t - y e a r o l d f o r e s t w h i l e - r t ' W^(0)e i s t h e d i s c o u n t e d m a r g i n a l c o s t o f d o i n g s o . A g e n e r a l i z a t i o n o f the r e s u l t s o f L e d y a r d and Moses can now be p r e s e n t e d . F i r s t o f a l l i t i s n e c e s s a r y t o be more s p e c i f i c a b o u t the p r o p e r t i e s o f the c o s t f u n c t i o n s D ( F Q ; X ) and W(L;x). I n c r e a s i n g e f f o r t s h o u l d i n v o l v e i n c r e a s i n g c o s t and d i m i n i s h i n g r e t u r n s t o s c a l e so ( 4 ) D F > 0 o O O > 0 w LL > 0 30 -As w e l l , i n c r e a s e s i n x have been r e p r e s e n t e d as c o s t i n c r e a s i n g changes so i t w i l l be assumed t h a t (5) D x > 0 D F x > 0 W > 0 WT > 0 x L x -R < 0 x A l s o = LW^/W w i l l d e n o t e the e l a s t i c i t y o f s i l v i c u l t u r e c o s t w i t h r e s p e c t t o s i l v i c u l t u r a l i n p u t . I t w i l l s u b s e q u e n t l y be p r o v e n t h a t P r o p . 10: Under the a s s u m p t i o n s l i s t e d b elow an i n c r e a s e i n x w i l l i n c r e a s e t h e o p t i m a l r o t a t i o n t i m e and t h e volume o f wood p e r h a r v e s t as w e l l as d e c r e a s i n g r e g e n e r a t i o n e f f o r t . T h a t i s dT/dx > 0, d F ( T ) / d x > 0, d F Q / d x < 0. The a s s u m p t i o n s a r e ( i ) t h e c o s t and r e v e n u e f u n c t i o n s R, W and D s a t i s f y (4) and--(5); ( i i ) d E R p / d x > 0 and d E ^ / d x > 0; ( i i i ) the o p t i m a l s i l v i c u l t u r a l e f f o r t i s 0 a t b o t h the b e g i n n i n g and the end o f the h a r v e s t . I t w o u l d be n i c e t o add s o m e t h i n g a b o u t th e change i n s i l v i c u l t u r a l e f f o r t due t o an i n c r e a s e i n x. S i n c e s i l v i -c u l t u r a l c o s t s a r e i n c r e a s i n g one w o u l d e x p e c t s i l v i c u l t u r a l e f f o r t t o be r e a r r a n g e d i n the d i r e c t i o n o f a v o i d i n g t h e s e c o s t s . T h i s can be done b o t h by an a b s o l u t e r e d u c t i o n and by - 31 -d e l a y i n g the a p p l i c a t i o n o f s i l v i c u l t u r e . F i g . 2 below i l l u s t r a t e s these p o s s i b i l i t i e s where Ax > 0. I t i s not p o s s i b l e to a n a l y s e t h i s s i t u a t i o n i n more d e t a i l , however, w i t h o u t b e i n g more s p e c i f i c about the c o s t f u n c t i o n W(L) so t h i s m a t t e r w i l l n o t be d i s c u s s e d any f u r t h e r . Assumption ( i i i ) about the o p t i m a l s i l v i c u l t u r a l e f f o r t may not be c o m p l e t e l y e s s e n t i a l i n Prop. 10. N e v e r t h e l e s s , i t appears t h a t some a s s e r t i o n about the o p t i m a l s i l v i c u l t u r a l - 32 -e f f o r t i s n e c e s s a r y i n o r d e r t o get comp a r a t i v e s t a t i c r e s u l t s . I t w i l l be shown below t h a t t h e r e are assumptions about the f o r e s t growth f u n c t i o n w h i c h l e a d to the c o n c l u s i o n t h a t the o p t i m a l s i l v i c u l t u r a l e f f o r t i s n o n - d e c r e a s i n g over the growth c y c l e . F u r t h e r m o r e , i t then becomes p o s s i b l e t h a t dT/dx < 0 -t h a t i s t h a t the Ledyard-Moses r e s u l t i s r e v e r s e d . F i n a l l y , no p r o p o s i t i o n l i k e Prop. 10 has been o b t a i n e d i n the case o f the parameter y. The remainder o f t h i s paper g i v e s f o r m a l p r o o f s o f Pro p s . 8-10. B e g i n n i n g w i t h Prop. 8 l e t T*, F J , F ( T ) * , L ( t ) * be the o p t i m a l s o l u t i o n o f the f o r e s t manager's problem. The r e g u l a r -i t y c o n d i t i o n s r e q u i r e d are somewhat l i k e a g l o b a l c o n c a v i t y r e q u i r e m e n t . There s h o u l d e x i s t f e a s i b l e < L ( t , 8 ) > and F0 ( e ) > 0 < 9 < 1 such t h a t d L ( t ) / d 9 > 0 , dF Q (e)/d9 >0, and i f ( 6 ) V(6) = MAX [ R ( F ( T ) e " r T - D(F (8)) m 0 - f T W ( L ( 9 ) ) e " r t d t ] [1 - e " r T ] s u b j e c t t o F = f ( F , L ( 9 ) ) F(0) = F o ( 8 ) , then V i n c r e a s e s w i t h 9. Moreover L ( t , 0 ) = 0, F o ( 0 ) = P. a nd L ( t , l ) = L ( t ) * , F Q ( 1 ) = F* so w i t h 9 = 0 , (6) r e p r e s e n t s the f o r e s t r y problem w i t h no e x p e n d i t u r e on s i l v i c u l t u r e or e x t r a r e g e n e r a t i o n e f f o r t w h i l e w i t h 8 = 1 the s o l u t i o n to (6) must be T* . Prop. 8 w i l l f o l l o w i f i t can be shown t h a t dT(8)/d9 < 0 and d F ( T ) ( 9 ) / d 9 < 0 f o r a l l 9 where T ( 9 ) , F ( T ) ( 8 ) are the s o l u t i o n o f ( 6 ) . D i f f e r e n t i a t i n g (6) 33 -(7) V ( 6 ) T = [-r(V+R)+R F:.:f(T) W ( L ( T , 0 ) ) ] [ e r T - l ] _ 1 = 0 (8) V ( 9 ) T T = [ R F F . f ( T ) + R F . ( f F - r ) ] £ ( T ) [ e r T - l ] _ 1 <0 (9) V > 0 by the r e g u l a r i t y a s s u m p t i o n (10) v ^ ) T e = c " r V e " w L C d L ( T ) / d e ) + ( R F F . £ ( T ) + R F ( f F - r ) ) ( d F ( T ) / d 0 ) ] r r T , , -1 [e - 1] [ r V Q + W L ( d L ( T ) / d e ) ] [ e r T - l ] _ 1 + £(T) _ 1V T T(dF(T)/de) I t w i l l f o l l o w t h e n t h a t d i ) T ( e ) = - v T e v - i [ r V Q + W L ( d L ( T ) / d e ) ] V ^ [ e r T - l ] _ 1 - f ( T ) _ 1 ( d F ( T ) / d 0 ) < 0 p r o v i d e d t h a t d F ( T ) / d 6 > 0. B u t t h i s f o l l o w s s i n c e t h i s d e r i v a t i v e i s a c o m p o s i t e o f two e f f e c t s . One e f f e c t i s t h r o u g h t h e d i f f e r e n t i a l e q u a t i o n F = f ( F , L ( e ) ) and i s p o s i t i v e a c c o r d i n g t o t h e lemma u s e d t o p r o v e P r o p . 4 s i n c e - 34 -d f ( F , L ( 9 ) ) / d 0 = f L - L ' ( 0 ) > 0. The o t h e r e f f e c t i s through the change i n F q and i s f ( T ) f ( 0 ) ~ ^F^(9) >0 a l s o . The second p a r t o f Prop. 8 f o l l o w s from (12) F ( T ) f l = 8F(T)/99 + f ( T ) - T ( 9 ) 9 = • f [ r V Q +WL• CdL (T) /d9) ] [ e r T - 1 ] -1 < 0 Prop. 9 has been i n c l u d e d i n t h i s paper m a i n l y f o r the sake o f completeness. I t s p r o o f i s t h e r e f o r e p r e s e n t e d i n Appendix 1 and we p r o c e e d here w i t h the main purpose o f t h i s s e c t i o n which i s the p r o o f o f Prop. 10. The p r o c e d u r e by which the f o r e s t manager maximizes the p r e s e n t v a l u e V i n (2) can be viewed as a two s t e p p r o c e d u r e . F i r s t , take T, F and F(T) to be g i v e n . Then W(L)e d t can be maximized w i t h the h e l p o f P o n t r y a g i n ' s maximum p r i n c i p l e . U s i n g the optimum L d e t e r m i n e d h e r e , V becomes a f u n c t i o n o f T, F q and F ( T ) . The f o r e s t manager can then determine the o p t i m a l v a l u e o f these v a r i a b l e s by o r d i n a r y c a l c u l u s t e c h n i q u e s . The f i r s t problem i s thus o - r t (13) MAX <L> S.T. F = f ( F , L ) and T, F , F(T) g i v e n . 35 -The H a m i l t o n i a n f o r t h i s p r o b l e m i s (14) H = [-W(L) + X • f ] e ' V t The K u h n - T u c k e r c o n d i t i o n s f o r m a x i m i z i n g H w i t h r e s p e c t to L a r e (15) [-WL +_ A f L ] • L = 0 WL + X £ L < 0 ; L > 0 The a d j o i n t e q u a t i o n i s X - rX = - H F e r T = - X f p o r (16) X = X [ r - f p ] H a v i n g d e s c r i b e d the optimum L we can now p r o c e e d t o m aximize V ( e v a l u a t e d on the optimum L) w i t h r e s p e c t to T, F and F ( T ) . We w i l l n e e d the p a r t i a l d e r i v a t i v e s o f VT - t - / W(L)e r d t w i t h r e s p e c t t o t h e s e v a r i a b l e s . These a r e H(T) = [-W(L(T)) + X ( T ) f ( F ( T ) , L ( T ) ) ] e " r T , X(0) and - X ( T ) e " r T r e s p e c t i v e l y . The f i r s t o r d e r c o n d i t i o n s f o r m a x i m i z i n g V w i t h r e s p e c t t o T, F Q and F ( T ) a r e t h e n (17) V T = [-rV-rR-W(L(T)) + X ( T ) • f ] [ e r T - 1 ] " 1 = 0 - 36 -(18) V p = [-Dp + X(0)][1 - e " r T ] _ 1 = 0 o o (19) V F ( T ) = [Rp - U T ) ] [ e r T - l ] " 1 = 0 C a l c u l a t i o n of the second order c o n d i t i o n requires some knowledge of the p a r t i a l d e r i v a t i v e s of X(0) and The f o l l o w i n g r e s u l t helps i n t h i s and i s also used to produce an important a l t e r n a t i v e form of V^ ,. A property of the Hamiltonian i s that 11 dH/dt = 3H/3t = rW(L)e I n t e g r a t i n g , i t f o l l o w s that W ( L ) e _ r ? d t = -[1 - e " r T ] r V + r R e " r T - r D u From the f i r s t e q u a l i t y , one gets by w r i t i n g out H(T) and H(0) (20) X ( T ) f ( F ( T ) , l ( T ) ) e ~ i ' T - X ( 0 ) f ( F Q , L ( 0 ) ) T W ( L ( T ) ) e " r T - W(L(0)) + r ( W(L)e" r tdt From the second e q u a l i t y , again s u b s t i t u t i n g f o r H(T) and H(0) - rT and c o l l e c t i n g terms i n v o l v i n g e one sees that (21) - 3 7 -V T = [-rV-rD-W(L(0)) + A(0)f(F Q,L(0))] n -rT-,-1 [1 - e ] We a r e now r e a d y t o l o o k a t the e f f e c t o f v a r i a t i o n s i n x. D i f f e r e n t i a t i n g the f i r s t o r d e r c o n d i t i o n s (17) - (19) (22) dT dF o .dF(T)-l = A V Tx V F x o L V F ( T ) x ' dx A i s the m a t r i x o f s e c o n d o r d e r p a r t i a l s o f V. A s i m p l i f i e d v e r s i o n o f A i s c a l c u l a t e d i n A p p e n d i x 2. I t i s t h e n a r o u t i n e c a l c u l a t i o n t o show t h a t (23) A" • V ^ f f O ) " ^ " . ^ ! ! ) " ^ " 1 -1 -1 -f(0) X -1 -1 f ( T ) Y :'-£("j3-)'"1X"1 f ( T ) h r X. 0 X and Y a r e d e f i n e d i n the a p p e n d i x . The main t h i n g t o n o t e h e r e i s t h a t t h e s e c o n d o r d e r c o n d i t i o n s f o r an i n t e r i o r maximum r e d u c e t o V T T < 0, X < 0 and Y < 0. Now c o m b i n i n g (22) and (23) one o b t a i n s (24) dF 1^-1 o / d x = f(0) A.x [ v T x - fC0)V ] (25) d F ( T ) / d x = - f f T ) " ^ " 1 [ V T x + f ( T ) V p ( T ) x ] - 38 -(26) dT/dx = -V'* V T x - £ ( 0 ) ~ ^ d F / d x ) + f ( T ) _ : L ( d F ( T ) / d x ) The c o n t i n u a t i o n o f t h i s c a l c u l a t i o n r e q u i r e s e x p r e s s i o n s f o r V T x Vp x and v p r x ) x ' F i r s t > f r o m t h e e n v e l o p e p r o p e r t y d e v e l o p e d i n A p p e n d i x 3, i t f o l l o w s t h a t when <L> i s c h o s e n f T - r t t o m aximize — / W(L)e d t (27) ^ / T W ( L ) e " r t d t = ^ T W x ( L ) e " r t d t Thus (28) V x = [ R x e " r T - D x -f t y L ) e " r t d t ] [ 1 - e " r T ] " 1 Now d i f f e r e n t i a t e t h e f i r s t o r d e r c o n d i t i o n (17) w i t h r e s p e c t to x. U s i n g (28) and t h e K u h n - T u c k e r c o n d i t i o n (15) (29) V T x = [ - r ( R x - D x ) ( l - e " r T ) " 1 /T W ( L ) e _ r t d t o W x ( L ( T ) ) + ( 9 A ( T ) / 9 x ) - f ( T ) ] [ e 1 " 1 - ! ] " 1 A n o t h e r f o r m o f t h i s e q u a t i o n i s g i v e n by d i f f e r e n t i a t i n g (21) 39 (30) V T x = [ - r ( R x - D x ) ( e r T - l ) - 1 + r ( l - e " r T ) _ 1 f W f L ) e ~ r t d t o W x ( L ( 0 ) ) + (3A (0). /9x) £(0)] r " r T n " l [e - 1] The o t h e r 2 d e r i v a t i v e s r e q u i r e d a r e o b t a i n e d f r o m . (18) and (19) (31) V p x = [-DF + ( 3 X ( 0 ) / 9 x ) ] [ l - e " r T ] _ 1 o o (32) V F ( T ) x = [ R F x ' O A ( T ) / 3 x ) ] [ e r T - i f 1 These e x p r e s s i o n s may now be f e d i n t o t h e c a l c u l a t i o n s (24) and (25) g i v i n g (33) V T x - £ ( 0 ) V F x = [-r(R -D ) C e r T - l ) ' " " 1 + D F f ( 0 ) o o + r ( l - e " r T ) _ 1 / * T W ( L ) e " r t d t o x - W x ( L ( 0 ) ) ] [1 - e _ r T ] _ 1 rT^ ~ 1 V T x + £ ^ V F ( T ) x = f - r ( R x - D x ) ( l - e ^ ) + R F x f ™ - 40 -+ r ( l - e " r T ) _ 1 f W ( L ) e " r t d t Jo x W X ( L ( T ) ) ] [ e r T - i f 1 I t w i l l now. be shown t h a t the assumptions made i n Prop. 10 enable one to s i g n V,p and the e x p r e s s i o n s i n (33) and (34) L o o k i n g a t ( 2 9 ) , i t f o l l o w s from assumption ( i ) t h a t the f i r s t two terms are n o n - n e g a t i v e . The assumption t h a t L(T) means t h a t W ( L ( T ) ) = W (0) = 0. Thus (35) V T x > 0 because i n Appendix 4 i t i s shown t h a t 3A(T)/3x > 0. I t i s a l s o c l e a r from assumption ( i ) and L(0) = 0 t h a t (36) V T x - f ( 0 ) V F x > a n d o (37) V T x + f ( T ) V F ( T ) x > 0 because R F x - f ( T ) - r ( R x - D x ) ( l - e " r T ) _ 1 = R p x . f ( T ) - ( R X - D X ) R F f ( T ) ( R - D - ^ T W ( L ) e " r t from f i r s t o r d e r c o n d i t i o n s (17) and (19) - 41 -> [ R p x - R x - R F - R _ 1 ] f ( T ) by assumption ( i ) > 0 since 8E D C/9x > 0. These determinations, of signs e s s e n t i a l l y conclude the proof of Prop. 10. Using (35) - (37) and the second order conditions V T T < 0, e t c . , one gets from (24) - (26) the f o l l o w i n g signs the f i n a l matter of t h i s s e c t i o n i s to show that the assumption L(0) = L(T) = 0 cannot be completely dropped from Prop. 10 (although i t might be replaced by some other c o n d i t i o n ) . To do t h i s , suppose the growth f u n c t i o n has the f o l l o w i n g separable form (38) dF Q/dx < 0, dF(T)/dx > 0 dT/dx > 0 (39) f(F,L) = f ( F ) g ( L ) Then, when L > 0, from (15) (40) -WL + Af(F)g'(L) = 0 and from (16) (41) A = A [r - f (F)g(L)] D i f f e r e n t i a t e (40) to get - 42 -(42) [W L L - A f ( F ) g " (L)]L = Af(F)g'(L) + A f ' ( F ) g ' ( L ) f ( F ) g ( L ) Then s u b s t i t u t i n g (41) (43) [W L L - A£(F)g" (L)]L = rA f ( F ) g » ( L ) and hence (44) L > 0 I f s i l v i c u l t u r e i s p r o f i t a b l e at any stage, then L(T) > 0. Moreover, from the mean value theorem f o r i n t e g r a l s , (45) J W x ( L ) e " r t d t = W x ( L ( s ) ) j e " r t d t = r - 1W ( L ( s ) ) [ l - e" r T] f o r some s, 0 < s <T Thus, from (44) and since WxL >0 (46) r ( l - e " r T ) _ 1 / W ( L ) e " r t d t - W (L(0)) W x(L(s)) - W x(L(0)) > 0 (47) r d - e " 1 1 ) " 1 / 1 W ( L ) e " r t d t - W v(L(T)) 43 -= W x ( L ( s ) ) - W x ( L ( T ) ) < 0 I t f o l l o w s t h e n t h a t t h e c o m b i n a t i o n ( 4 8 ) V T x - f ( 0 ) V F x > 0 , . V T x + £ ( T ) V p < 0 o i s p o s s i b l e . I t i s c e r t a i n l y t r u e i f c h a n g e s i n x d o n o t a f f e c t R o r D . I t f o l l o w s t h e n t h a t d F Q / d x < 0 a s i n P r o p . 10 b u t d F ( T ) / d x < 0 w h i c h i s t h e o p p o s i t e t o P r o p . 10 ( s e e ( 2 4 ) a n d ( 2 5 ) ) . M o r e o v e r , i t i s s t i l l t r u e t h a t V > 0 b u t n e v e r t h e l e s s d T / d x i s t h e s u m o f t w o p o s i t i v e t e r m s a n d o n e n e g a t i v e t e r m s o t h a t d T / d x < 0 i s a p o s s i b i l i t y . - 4 4 -C H A P T E R I I T H E F O R E S T R Y M A X I M U M P R I N C I P L E 1 . I n t r o d u c t i o n T h e p r o b l e m o f w h e n t o o p t i m a l l y l o g a f o r e s t h a s b e e n o f r e c u r r e n t i n t e r e s t t o e c o n o m i s t s s i n c e t h e n i n e t e e n t h c e n t u r y . I n 1 8 4 9 , M a r t i n F a u s t m a n n o b t a i n e d a s o l u t i o n t o t h i s p r o b l e m , w h i c h t o o k i n t o a c c o u n t t h e v a l u e o f t h e l a n d f o r g r o w i n g f u r t h e r c r o p s o f t r e e s . A f t e r m u c h d e b a t e , r e c o r d e d i n S a m u e l s o n ( 1 9 7 6 ) , t h i s s o l u t i o n , c a l l e d t h e F a u s t m a n n f o r m u l a , i s g e n e r a l l y a c k n o w l e d g e d t o b e t h e c o r r e c t s o l u t i o n t o d a y . I t m a y b e n o t e d , h o w e v e r , t h a t t h e F a u s t m a n n f o r m u l a r e q u i r e s a n u m b e r o f s i m p l i f y i n g a s s u m p t i o n s . T h e s e i n c l u d e t h e a s s u m p t i o n t h a t i n i t i a l l y t h e f o r e s t l a n d i s b a r e a n d t h e n i s c o m p l e t e l y p l a n t e d w i t h u n i a g e d s e e d l i n g s a t t h i s i n i t i a l m o m e n t " ' ' . F u r t h e r m o r e , u n i t h a r v e s t i n g c o s t s a r e a s s u m e d n o t t o v a r y w i t h t h e r a t e o f h a r v e s t i n g - w h i c h r e s u l t s i n t h e w h o l e f o r e s t s i t e b e i n g l o g g e d i n s t a n t a n e o u s l y a t r e g u l a r i n t e r v a l s . T h e n a t u r e o f t h e F a u s t m a n n s o l u t i o n -i t s r e g u l a r i t y - i n d i c a t e s t h a t t h e F a u s t m a n n s o l u t i o n i s p r o b a b l y a " s t e a d y s t a t e " s o l u t i o n t o a m o r e c o m p l e x f o r e s t r y m a n a g e m e n t p r o b l e m t h a n t h e F a u s t m a n n p r o b l e m . T h i s p a p e r w i l l s h o w t h a t t h i s i s i n d e e d t h e c a s e . T h e p r o b l e m d e a l t w i t h i n t h i s p a p e r w i l l b e c a l l e d t h e g e n e r a l i z e d f o r e s t r y m a n a g e m e n t p r o b l e m ( G F M P ) . I t i s o b t a i n e d b y w e a k e n i n g t h o s e a s s u m p t i o n s , m e n t i o n e d a b o v e , w h i c h l e a d t o t h e F a u s t m a n n m a n g e m e n t r u l e . F i r s t t h e n e t - 45 -value of the logs cut from a hectare of land i s allowed to vary with the age of the trees (as i s normal) and w i t h the rate of h a r v e s t i n g . This should be congenial to economists as i t allows f o r a U-shaped average harvesting cost curve. Such curves are often used when d i s c u s s i n g optimal e x t r a c t i o n 2 patterns f o r a non-renewable resource . Secondly, an a r b i t r a r y age d i s t r i b u t i o n of the f o r e s t i s allowed f o r at the i n i t i a l time^. The age d i s t r i b u t i o n of the f o r e s t becomes the important datum as f a r as the manager of the f o r e s t i s concerned. With the g e n e r a l i t y allowed f o r above, optimal logging need not take place instantaneously f o r the whole stand. Assuming that r e p l a n t i n g takes place at the same time as l o g g i n g , logging and the passage of time are operations which change the age d i s t r i b u t i o n of the trees i n the stand. The manager's problem w i l l be taken to be as usual to maximize the present net values of logs harvested from the f o r e s t . This may be viewed as a v a r i a n t of the optimal c o n t r o l problem. The manager c o n t r o l s the choice of the harvesting rate at time t . The s t a t e of the system i s given by the age d i s t r i b u t i o n of the stand at time t 4 . Conditions w i l l be derived i n t h i s paper which are neces-sary and s u f f i c i e n t to describe the optimal s o l u t i o n of the GFMP. These conditions are s i m i l a r to what i s known i n the l i t e r a t u r e as the maximum p r i n c i p l e f o r processes w i t h a delay^. The delay i n the case of f o r e s t r y i s the time between p l a n t i n g and h a r v e s t i n g . - 46 -The maximum p r i n c i p l e d e r i v e d below w i l l be c a l l e d t h e F o r e s t r y Maximum P r i n c i p l e . As u s u a l , i t l e a d s t o a d y n a m i c a l s y s t e m . However, t h i s c o n s i s t s o f f u n c t i o n a l r a t h e r t h a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n s ^ . N e v e r t h e l e s s , t h e u s u a l q u e s t i o n s a p p l y . F i r s t , a r e t h e r e s t e a d y s t a t e s ? A s t e a d y s t a t e s h o u l d be c o n s i d e r e d i n t h e c o n t e x t t h a t t h e s t a t e i s an age d i s t r i b u t i o n . Thus a s t e a d y s t a t e i s an i n i t i a l age d i s t r i b u t i o n f o r ' w h i c h t h e o p t i m a l h a r v e s t i n g p o l i c y i s t o 7 r e c o n s t i t u t e t h i s d i s t r i b u t i o n a t r e g u l a r i n t e r v a l s ( e . g . t r e e s a r e c u t when t h e y r e a c h t h e r o t a t i o n a g e ) . I t w i l l be shown t h a t t h e r e a r e s u c h s t e a d y s t a t e s and t h a t t h e r o t a t i o n p e r i o d i s alw a y s g i v e n by a Faustmann f o r m u l a . F u r t h e r m o r e , t h e c a s e where a v e r a g e h a r v e s t i n g c o s t s a r e v a r i a b l e i s p a r t i c u l a r l y i n t e r e s t i n g b e c a u s e t h i s p l a c e s r e s t r i c t i o n s on t h e i n i t i a l age d i s t r i b u t i o n s i n a s t e a d y s t a t e . T hese must be n o r m a l i n t h e f o r e s t e r ' s s e n s e t h a t t h e non-empty age g c o h o r t s c o n t a i n i d e n t i c a l numbers o f h e c t a r e s . The s e c o n d q u e s t i o n c o n c e r n s t h e a s y m p t o t i c p r o p e r t i e s o f t h e o p t i m a l h a r v e s t i n g p o l i c y . I n p a r t i c u l a r , do o p t i m a l h a r v e s t i n g p o l i c i e s a l w a y s c o n v e r g e , as t i m e p a s s e s , t o w a r d s a s t e a d y s t a t e ( e . g . i s t h e s y s t e m d e f i n e d by t h e f o r e s t r y maximum p r i n c i p l e g l o b a l l y a s y m p t o t i c a l l y s t a b l e ) ? T h i s i s an i m p o r t a n t q u e s t i o n b e c a u s e g i v e n t h e remarks above, i t seems t o c o n f i r m t h e f o r e s t e r ' s i n t u i t i o n t h a t i n t h e l o n g r u n n o r m a l age d i s t r i b u t i o n s a r e t h e p r e f e r a b l e age d i s t r i b u t i o n s 9 f o r f o r e s t s . The s t a b i l i t y q u e s t i o n t o g e t h e r w i t h t h e q u e s t i o n o f - 47 -w h e t h e r an o p t i m a l s o l u t i o n e x i s t s r e m a i n open q u e s t i o n s . T h i s p a p e r w i l l show o n l y t h a t i f t h e o p t i m a l h a r v e s t p o l i c y c o n v e r g e s a s y m p t o t i c a l l y t h e n i t c o n v e r g e s t o a s t e a d y s t a t e . However, i t a p p e a r s e x t r e m e l y l i k e l y t h a t g l o b a l a s y m p t o t i c s t a b i l i t y h o l d s and t h e r e w i l l be some d i s c u s s i o n o f t h i s q u e s t i o n b elow. The p a p e r i s o r g a n i z e d as f o l l o w s . S e c t i o n 2 s e t s o u t t h e GFMP p r o b l e m and e s t a b l i s h e s n o t a t i o n . The i m p o r t a n t p r i n c i p l e t h a t an o p t i m a l h a r v e s t i n g p o l i c y a l w a y s l o g s t h e o l d e s t t r e e s f i r s t i s e s t a b l i s h e d h e r e . S e c t i o n 3 s t a t e s and p r o v e s t h e f o r e s t r y maximum p r i n c i p l e and d e r i v e s some r e s u l t s a b o u t t h e H a m i l t o n i a n . S u f f i c i e n c y i s e s t a b l i s h e d i n S e c t i o n 4. The p r o o f s i n v o l v e p e r t u r b a t i o n t e c h n i q u e s so s h o u l d s t r i c t l y be v i e w e d as h e u r i s t i c . I t i s e s t a b l i s h e d h e r e t h a t t h e f o r e s t r y maximum p r i n c i p l e g e n e r a l i z e s Faustmann's f o r m u l a . The n e x t . t w o s e c t i o n s e s t a b l i s h t h e n a t u r e o f s t e a d y s t a t e s and d e a l w i t h t h e q u e s t i o n o f g l o b a l a s y m p t o t i c s t a b i l i t y . The f i n a l s e c t i o n g i v e s an economic i n t e r p r e t a t i o n o f t h e FMP and sums up. T h e r e i s a l s o an a p p e n d i x w h i c h r e l a t e s t h e r e s u l t s o b t a i n e d h e r e t o t h o s e o b t a i n e d i n Heaps and Neher (1979) . - 48 -2. The G e n e r a l i z e d F o r e s t r y Management P r o b l e m I t i s d e s i r e d t o d e v i s e a l o g g i n g p o l i c y f o r a f o r e s t o f t o t a l a r e a A w h i c h i s homogeneous w i t h r e s p e c t t o i t s b i o l o g i c a l and e c o n o m i c c h a r a c t e r i s t i c s . S u c h a p o l i c y may be d e s c r i b e d by t h e h a r v e s t i n g r a t e h ( t ) ( i n h e c t a r e s p e r u n i t t i m e ) a t t i m e t a f t e r t h e s t a r t o f t h e p l a n n i n g p e r i o d . The e c o n o m i c and b i o l o g i c a l c h a r a c t e r i s t i c s w i l l be assumed t o be d e s c r i b e d by a n e t a v e r a g e r e v e n u e f u n c t i o n p ( a ) and a t o t a l c o s t o f h a r v e s t i n g f u n c t i o n C ( h ) . Thus p ( a ) r e p r e s e n t s t h e n e t r e v e n u e o b t a i n a b l e p e r h e c t a r e f r o m l o g g i n g an a y e a r o l d f o r e s t and C( h ) r e p r e s e n t s t h e c o s t p e r u n i t t i m e o f l o g g i n g h h e c t a r e s " ^ . The p r o f i t s o b t a i n e d by t h e f o r e s t o p e r a t o r a t t i m e t a r e t h u s [ p ( a ( t ) ) h ( t ) - C ( h ( t ) ] e " r t when t h e y a r e d i s c o u n t e d t o t h e b e g i n n i n g o f t h e p l a n n i n g p e r i o d . The GFMP w i l l t h u s be t a k e n t o be t o m a x i m i z e [ p ( a ( t ) ) h ( t ) - C ( h ( t ) ) ] e " r t d t Some r e s t r i c t i o n s must be made on p ( a ) and C ( h ) i n o r d e r t h a t t h e p r o b l e m w i l l h ave a m e a n i n g f u l s o l u t i o n . The f o l l o w i n g r e s t r i c t i o n s seem t o be n e c e s s a r y t o t h e a u t h o r " ^ . (a) P ( 0 ) < 0 b u t p ( a ) > m f o r some a where m = m i n C ( h ) / h . h (b) p ( a ) > 0 f o r a l l a b u t l i m p ( a ) = 0 ( c ) p ( a ) - r p ( a ) < 0 f o r a l l a (d) C(0) > 0 and C'(h) > 0 f o r a l l h - 49 -(e) C(h)/h i s convex w i t h a minimum occuring at h = h. m m These . r e s t r i c t i o n s are j u s t i f i a b l e f o r the f o l l o w i n g reasons. R e s t r i c t i o n s (a) and (b) say that the value of a f o r e s t increases w i t h age, becomes p o s i t i v e but the rate of growth of t h i s value i s l i m i t e d . One might think of p(a) being the p r i c e per u n i t of logs times the volume per hectare of an a year o l d f o r e s t . In t h i s case, p(a) i s u s u a l l y assumed to have the l o g i s t i c shape of Fig. 3 below. $ P(a) a Figure 3 Value Growth Curve The consequence of (a) and (b) i s that the discounted value per hectare of an a year o l d f o r e s t p(a)e has a maximum value, and hence, a management p o l i c y which leaves p a r t of the 50 -forest uncut cannot be optimal. R e s t r i c t i o n (c) ensures that p(a)e has a unique l o c a l extremum which i s the global maxi-mum. Restrictions (d) and (e) are the economist's t r a d i t i o n a l 12 assumptions about average cost curves . Some preliminary knowledge of optimal logging programs is needed in order to present the GFMP i n a tractable form. This i s provided by the f i r s t result of this paper. Prop. 1: If an optimal logging p o l i c y involves cutting at time t, then i t is the oldest trees i n the forest which w i l l be cut. This proposition w i l l be proven at the end of this section. Its importance is that i t means that the data about the current age d i s t r i b u t i o n of the forest i s contained in the past harvesting p o l i c y . h h=0 v(0) v(t) 0 x-•1 t w(0) w(t) time Figure 4 A Logging Policy 51 T h i s may be i l l u s t r a t e d by means of F i g . 4 which g i v e s one example o f a h a r v e s t i n g p o l i c y . L e t a denote age a g a i n . In F i g . 4 , the age d i s t r i b u t i o n o f the f o r e s t which e x i s t s a t time 0 i s g i v e n by < h ( - a ) > , 0 < a < -v(0) . Logging does not b e g i n on t h i s f o r e s t u n t i l time x^. Then between x^ and w ( 0 ) , the whole f o r e s t i s l o g g e d o f f a t the r a t e i n d i c a t e d . S i n c e r e p l a n t i n g i s s i m u l t a n e o u s w i t h l o g g i n g , the age d i s t r i b u t i o n at time w(0) i s < h(w(0)-a) > , 0 < a < w(0) - x 1 . At w(0) l o g g i n g ceases a g a i n u n t i l X2. A f t e r X2, l o g g i n g i s conducted c o n t i n u o u s l y at the r a t e i n d i c a t e d . At x^ another c r o p p i n g o f the whole f o r e s t i s completed but i n s t e a d o f p a u s i n g , the l o g g e r s i m m e d i a t e l y s t a r t a g a i n on the f o r e s t c u t t i n g f i r s t the t r e e s p l a n t e d a t X2« At any time t , the age d i s t r i b u t i o n o f the f o r e s t w i l l be < h ( t - a ) >, 0 < a < t - v ( t ) , where v ( t ) i s the time a t which the o l d e s t t r e e s i n the f o r e s t a t time t were p l a n t e d . The l e n g t h of time between c u t t i n g and p l a n t i n g , t - v ( t ) , i s the r o t a t i o n p e r i o d . The c o n n e c t i o n between age d i s t r i b u t i o n and h a r v e s t i n g p o l i c y can now be used t o h e l p r e p r e s e n t the dynamics of h a r -v e s t i n g . L e t H ( t ) f o r t > 0 be the number of h e c t a r e s t h a t have been l o g g e d from time 0 t o time t . For t < 0, H ( t ) i s the number of h e c t a r e s c o v e r e d by t r e e s of age l e s s than - t at time 0. Then (2) H ( t ) = h ( t ) - 52 -I n t h e f o r e s t r y maximum p r i n c i p l e h ( t ) w i l l be t h e c o n t r o l and H ( t ) t h e s t a t e . N e x t , l e t w ( t ) be t h e d a t e a t w h i c h t h e y o u n g e s t t r e e s i n t h e f o r e s t . a t t i m e t w i l l be c u t . F o r e x a m p l e , i f t h e r e i s some l o g g i n g a t t i m e t , w ( t ) w i l l be t h e c u t t i n g d a t e o f t h e s e e d l i n g s p l a n t e d a t t i m e t . I n g e n e r a l , v ( t ) and w ( t ) have d i s c o n t i n u i t i e s when t h e p o l i c y i s t o c e a s e 13 l o g g i n g e n t i r e l y . However, when h ( t ) > 0 so w i s a s o r t o f i n v e r s e f u n c t i o n f o r v . Now, s i n c e t h e t r e e s a r e c u t by o r d e r o f a g e , v and w a r e o b t a i n a b l e f r o m a k n o w l e d g e o f t h e l o g g i n g p o l i c y . I n d e e d T hese l e a d t o t h e f o l l o w i n g i m p o r t a n t f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e r e l a t i o n s h i p s b e t w e e n h a r v e s t i n g r a t e and age o f t r e e s l o g g e d . R e l a t i o n s (4) o r (5) a c t as c o n s t r a i n t s on t h e f o r e s t m a n a g e r ' s d e c i s i o n m a k i n g , so t h e GFMP s h o u l d now be e x p r e s s e d as (3) v ( w ( t ) ) = t and w ( v ( t ) ) = t (4) H ( t ) - H d v ( t ) ) = A = H ( w ( t ) ) - H ( t ) (5) h ( v ) v = h ( t ) and h(w)w = h ( t ) MAX <h> V = [ p ( t - v ) h - C ( h ) ] e - r t d t - 53 ( 6 ) S.T. (a) H h (b) H ( t ) H(v) = A H ( t ) f o r t < 0 and v(0) are g i v e n (d) h e f t T h i s d i f f e r s from the o r d i n a r y c o n t r o l problem i n t h a t a d e l a y f u n c t i o n v ( t ) i s i n v o l v e d . N e v e r t h e l e s s , maximum p r i n c i p l e s have been found f o r p r o c e s s e s w i t h a d e l a y and i t w i l l be shown i n the nex t s e c t i o n t h a t t h e r e i s such a p r i n c i p l e v a l i d f o r the above problem. The c o n d i t i o n h e 9, r e p r e s e n t s any c o n s t r a i n t s one may w i s h to impose on the h a r v e s t i n g r a t e . I t may be viewed as b e i n g o f the form h < h < h where h may be 0 and h may be 0 0. I t w i l l be assumed t h a t h > h . to a v o i d degenerate c a s e s . — mm ° T h i s s e c t i o n c o n c l u d e s w i t h the p r o o f o f Prop. 1. Suppose at time t , t r e e s o f age a^ are c u t w h i l e t r e e s o f age a.^ ( a t time t ) are l e f t uncut u n t i l time t+dt, dt > 0. Take a p l o t P^ o f dh t r e e s c u t at time t , ( o f a^ y e a r o l d t r e e s ) and another i°2 o f dh t r e e s c u t a t time t+dt. An a l t e r n a t e h a r v e s t i n g p o l i c y i s t o f o l l o w the o p t i m a l s t r a t e g y i n ev e r y d e t a i l except t h a t at time t and a t every subsequent time when P^ was to be c u t , )?2 w i l l be c u t . C o n v e r s e l y , at time t+dt and e v e r y subsequent 14 time when was t o be c u t , P^ w i l l be c u t . The a l t e r n a t e p o l i c y has the same h a r v e s t i n g r a t e at ev e r y time as the o p t i m a l s t r a t e g y so d i s c o u n t e d h a r v e s t i n g c o s t s are the same f o r the two p o l i c i e s . Moreover, the o n l y times when t h e r e i s a change i n - 54 -the age of the trees being cut is at time t when dh hectares of a 2 year old trees are harvested instead of dh hectares of a^ year old trees. S i m i l a r l y at time t+dt, dh hectares of a-^  + dt year old trees are cut instead of dh hectares of a2 +dt year old trees. The change in present values of the alternate strategy i s thus (dropping the factor dh«e r t ) (7) A = p(a 2) + p ( a 1 + d t ) e " r d t - p f a ^ - p (a 2+dt) e " r d t Using the mean value theorem, there i s a 8, 0 < 6 < 1 , such that (8) A = {p(a 1 + 0dt) - r p f a ^ e d t ) - p(a 2+9dt) + r p(a 7+9dt)}e" r 9 d t-dt and as wellv^between a^+edt and a.2 + 9dt such that ( 9 ) A = [p(0 - r p ( 5 ) ] e " r 6 d 1 : . d t . ( a ] L - a 2 ) If the i n i t i a l strategy was optimal, then A < 0 and from r e s t r i c t i o n (c) i t follows that a^ > a 2 . - 55 -3. The F o r e s t r y Maximum P r i n c i p l e The f o r e s t r y maximum p r i n c i p l e w i l l be d e r i v e d by c o n s i d e r i n g the e f f e c t on p r e s e n t v a l u e of a p e r t u r b a t i o n o f a h a r v e s t i n g p o l i c y . A p e r t u r b a t i o n o f a l o g g i n g p l a n H ( t ) i s a f a m i l y o f h a r v e s t i n g p l a n s H ( t , e ) d e f i n e d f o r 0 < e < e Q , e Q > 0 such t h a t H ( t , 0 ) = H ( t ) and H i s d i f f e r e n t i a b l e i n e. t h The p r e s e n t v a l u e o f the e h a r v e s t i n g p l a n i s g i v e n by J- o o [ p ( t - v ( t , e ) ) h ( t , e ) - C ( h ( t , e ) ) ] e ~ r t d t O where (2) h ( t , e ) = H t ( t , e ) and (3) H ( t , e ) - H ( v ( t , e ) , e ) = A In the f o l l o w i n g c a l c u l a t i o n s , the d e r i v a t i v e V'(e) a t e = 0 w i l l be c a l c u l a t e d i n terms o f n ( t ) = H ( t , 0 ) . Note t h a t n ( t ) = h ( t , 0 ) . From (3) (4) H ( t , e ) - H e ( v ( t , £ ) , e ) = h ( v ( t , e ) , e ) • v £ ( t , e ) L e t t i n g e tend t o 0 15 (5) h (v) • v e ( t , 0 ) = n ( t ) - n(v) Now, l e t q ( t ) be any p i e c e w i s e d i f f e r e n t i a b l e m u l t i p l i e r . - 56 Then (6) V(e) = f [ p ( t - v ( t , e ) ) h ( t , e ) - C ( h ( t , e ) ) + q ( t ) ( H t ( t , e ) - h ( t , e ) ) ] e " r t d t - r t I n t e g r a t i n g the term q-H^'e by p a r t s and u s i n g H(0,e) = 0 (7) V(e) = f { [ p ( t - v ( t , e ) ) h ( t , £ ) - C ( h ( t , e ) ) q ( t ) h ( t , e ) J + [ r q - q ] H ( t , e ) } e " r t - r t + l i m q H ( t , e ) e t->°° The l a s t term can be e l i m i n a t e d by imposing the t r a n s v e r s a l i t y c o n d i t i o n (8) l i m q H ( t ) e " r t = 0 and r e s t r i c t i n g o u r s e l v e s t o p e r m u t a t i o n s such t h a t l i m H ( t , e ) = H ( t ) u n i f o r m l y . Now d i f f e r e n t i a t e (7) w i t h r e s p e c t t o (9) V (e) = / { - p ( t - v ( t , e ) ) v ( t , e ) h ( t , e ) + [ p ( t - v ( t , e ) ) - C ( h ( t , e ) ) - q ( t ) ] h £ ( t , e ) .+ [ r q - q ] H e ( t , e ) } e " r t d t 5 7 -L e t t i n g e t e n d to 0 (10) V'(0) = fQ { - p ( t - v ) h v £ + [ p ( t - v ) - C (h)-q]n + [rq-q]n}e r t d t Now, s i n c e h ( v ) v = h ( t ) and because o£ ( 5 ) (11) h ( t ) v £ = [ n ( t ) - n ( v ) ] v One term i n the i n t e g r a n d o f (10) i s , t h e r e f o r e , • * - r t + p ( t - v ) n ( v ) v e . A change of v a r i a b l e t = w(s) w i l l be ap-p l i e d t o t h i s over those i n t e r v a l s f o r which h ( t ) > 0 (which c o i n c i d e w i t h the i n t e r v a l s f o r which t h i s i n t e g r a n d i s non-z e r o ) . L e t h ( t ) = 1 i f h ( t ) > 0 and 0 o t h e r w i s e . Then (12) f p ( t - v ) n ( v ) v e ~ r t d t = f p ( w ( s ) - s ) h ( s ) n ( s ) e ~ r w ( s ) d s U s i n g v ( w ( s ) ) = s when h ( s ) > 0 and n ( s ) = 0 f o r s < 0. These c a l c u l a t i o n s then g i v e { [ - p ( t - v ) v + p ( w - t ) h ( t ) e " r ( w " t ) u + r q - q ] r , ( t ) + [p ( t - v ) - C • (h)-q] n ( t ) } e " r t d t 58 -T h e r e f o r e , t h e m u l t i p l i e r s h o u l d be c h o s e n t o s a t i s f y (14) q - r q = p (w-1) h ( t ) e " r - p ( t - v ) v i n w h i c h c a s e /oo [ p ( t - v ) - C (h) - q ] f 1 ( t ) e " r t d t u The i m p l i c a t i o n s o f (h,v) b e i n g o p t i m a l c a n now c o n v e n i e n t l y be e x p r e s s e d i n terms o f t h e c u r r e n t v a l u e H a m i l t o n i a n (16) J = p ( t - v ) h - C(h) - q h /"OO W i t h t h i s t e r m i n o l o g y V ( 0 ) = / (9J/9h) n ( t ) e~ r t < 0 f o r t h e o p t i m a l (h,v) and any a d m i s s i b l e n. A most i n t e r e s t i n g c a s e o f t h i s o c c u r s when n i s on i m p u l s e w h i c h changes t h e h a r v e s t r a t e a t t i m e t f r o m h ( t ) t o h'(t)+k b u t l e a v e s h unchanged e l s e w h e r e . I t i s shown t h e n i n A p p e n d i x 5 t h a t V ( 0 ) = - r t [J(h+k) - J ( h ) ] e . T h i s must be n o n p o s i t i v e f o r any a d m i s s i b l e h+k so t h e o p t i m a l h must maximize J ( h ) . The above r e s u l t s a r e summarized i n . . The F o r e s t r y Maximum P r i n c i p l e . The optimal- s o l u t i o n h t o t h e GFMP o f S e c t i o n 2 m a x i m i z e s t h e H a m i l t o n i a n J s u b j e c t t o t h e c o n s t r a i n t s h e f t . M o r e o v e r , t h e c o n d i t i o n H = h, H ( t ) - H ( v) = A and H ( t ) g i v e n f o r t <- 0 must be s a t i s f i e d as w e l l as t h e a d j o i n t e q u a t i o n . - 59 -(17) q - r q = p ( w - t ) h ( t ) e " r ^ - p ( t - v ) v and the t r a n s v e r s a l i t y c o n d i t i o n (18) l i m q H e " r t = 0 t-)-oo The f o r e s t r y maximum p r i n c i p l e d i f f e r s from the o r d i n a r y maximum p r i n c i p l e i n i t s a d j o i n t e q u a t i o n . O r d i n a r i l y ( g i v e n the way the H a m i l t o n i a n i s w r i t t e n here) (19) q - r q = 3J/3H(t) From H ( t ) - H(v) = A, i t f o l l o w s t h a t 3v/3H(t) = h ( v ) " 1 . Thus 3J/3H(t) = - p ( t - v ) h ( t ) h ( v ) _ 1 = - p ( t - v ) v - the second term on the RHS o f ( 1 7 ) . The o t h e r term p ( w - t ) h ( t ) e " r ( w - t ) a r i s e s because t h e dynamics o f f o r e s t r y i n c o r p o r a t e s a d e l a y between p l a n t i n g and c u t t i n g . P o n t r y a g i n e t . a l . (1964, 201-204) d evelop a maximum p r i n c i p l e f o r p r o c e s s e s w i t h a d e l a y . T h e i r s i t u a t i o n i s not as complex as the f o r e s t r y c a s e , however N e v e r t h e l e s s , P o n t r y a g i n ' s (1964, 204) a d j o i n t e q u a t i o n (34) can be i n t e r p r e t e d as (20) q - r q = 3J/3H(t) + 3J/3H(v) . e - r ( w - t ) t=w The second term i s t o be w r i t t e n i n terms o f l e a d s r a t h e r than l a g s . An ana l o g y w i t h our case i s now as f o l l o w s . The p a r t i a l 3J/3H(v) e q u a l s ( 3 J / 3 t ) • ( 3 t / 3 H ( v ) ) . From H ( t ) - H(v) = A, - 60 -9t / 9 H ( v ) = h ( t ) " 1 . T h i s i s when h ( t ) > 0. O t h e r w i s e , 9J/9H(v) = 0. Thus when h ( t ) > 0, 9J/9H(v) = p ( t - v ) h ( t ) h ( t ) _ 1 e - r ( w - t ) t=w = p ( t - v ) . I n terms o f l e a d s 9J/9H(v) = p ( w - t ) h ( t ) e r ^ w ^ w h i c h i s t h e f i r s t t e r m on t h e RHS o f ( 1 7 ) . Thus t h e f o r e s t r y maximum p r i n c i p l e i s v e r y s i m i l a r t o t h e c a s e d e v e l o p e d by P o n t r y a g i n e t . a l . F i n a l l y , t h i s s e c t i o n c o n c l u d e s w i t h t h e o b s e r v a t i o n t h a t h > h i m p l i e s t h a t h > h • where h . m i n i m i z e s — ^ _ —mm —mm (C(h) - C ( h ) ) / ( h - h ) . T h i s i s . t r u e i f i t c a n be shown t h a t t h e m a r g i n a l e x c e e d s t h e a v e r a g e , o r t h a t (21) (h-h)C* (h) > C(h) - C(h) Now, s i n c e h > h i m p l i e s = 0 by t h e maximum p r i n c i p l e , C (h) = p ( t - v ) - q S u b s t i t u t i n g t h i s i n (21) one s e e s t h a t (21) i s e q u i v a l e n t t o (22) J ( h ) > J ( h ) w h i c h i s t r u e , a g a i n by t h e f o r e s t r y maximum p r i n c i p l e . - 61 -4. S u f f i c i e n c y The argument t h a t the f o r e s t r y maximum p r i n c i p l e i s s u f f i c i e n t t o c h a r a c t e r i z e the o p t i m a l h a r v e s t p o l i c y under c o n c a v i t y assumptions i s based on a c a l c u l a t i o n o f the second v a r i a t i o n o f V f e ) . I t may be assumed t h a t H and h are 0. Then, d i f f e r e n t i a t i n g (3-9) w i t h r e s p e c t t o e and l e t t i n g £ tend to 0, (1) V" (0) = f [ p ( t - v ) v 2 h - p ( t - v ) v £ £ h - 2 p ( t - v ) v £ n - C" ( h ) n 2 ] e " r t d t Now, d o i n g the same to (3-4) g i v e s (2) h O ) v £ £ = - h ( v ) v £ - 2 v £ n ( v ) A l s o , from (3-4) i t f o l l o w s t h a t (3) h ( v ) v v £ + h ( v ) v £ = n ( t ) - n ( v ) v S u b s t i t u t i n g (2) and (3) i n t o (1) and some m a n i p u l a t i o n g i v e s /oo „ „ { [ p ( t - v ) h - p ( t - v ) h ( v ) v ] v ^ - C" (h)n o e - 2 p ( t - v ) h ( v ) v £ v £ } e " r t d t 62 -F i n a l l y , a p p l y i n g i n t e g r a t i o n by p a r t s t o t h e l a s t t e r m h e r e g i v e s (5) V " (0) =./ { [ p ( t - v ) - r p ( t - v ) ] h ( v ) v C " ( h ) n 2 } e " r t d t The f o l l o w i n g r e s u l t s c a n now be e s t a b l i s h e d . P r o p . 2: I f C(h) i s convex f o r h > h, t h e n t h e f o r e s t r y maximum p r i n c i p l e i s s u f f i c i e n t t o d e s c r i b e t h e o p t i m a l s o l u t i o n t o t h e GFMP. P r o o f : L e t be t h e optimum s o l u t i o n o f t h e GFMP and H 2 some o t h e r s o l u t i o n o f t h e f o r e s t r y maximum p r i n c i p l e . Then f o r a l l e, 0 < e < l , H ( t , e ) = ( l - e ) H 1 ( t ) + e H 2 ( t ) i s a f e a s i b l e h a r v e s t i n g p l a n b e c a u s e o f t h e c o n v e x i t y o f t h e c o n t r o l s e t . L e t t i n g V ( e ) be t h e n e t p r e s e n t v a l u e o f t h i s l o g g i n g p o l i c y , V ( e ) must have a r e l a t i v e minimum on [0,1] f o r some e > 0. Now H = H-.-H.. so H and h a r e 0. The c a l c u l a t i o n o f t h e e 2 1 ee ee s e c o n d v a r i a t i o n above t h e n a p p l i e s and by r e s t r i c t i o n (c) and th e c o n v e x i t y o f C(h) i t i s e s t a b l i s h e d t h a t V " (e) < 0. T h i s r u l e s o ut an i n t e r i o r e so e = 1. The d e r i v a t i o n o f t h e f o r e s t r y maximum p r i n c i p l e c o n t a i n s t h e f o l l o w i n g f o r m u l a ( 3 J / 3 h ) ( h 2 - h 1 ) e " r t d t where 3J/3h i s e v a l u a t e d a t ( h 2 , v 2 ) . S i n c e h 2 m a x i m i z e s 3J/3h - 63 -s u b j e c t t o h < h 2 < h, i t f o l l o w s t h a t (3J/3h) ( h ^ h ^ ) > 0 and V ' ( l ) >0. T h i s , t o g e t h e r w i t h . V" (1) < 0 r u l e s out 1 as a r e l a t i v e minimum. In summation, the f o r e s t r y maximum p r i n c i p l e has a unique s o l u t i o n and t h i s must be the optimum f o r the GFMP. The f o r e s t r y maximum p r i n c i p l e i s d i f f i c u l t t o use i n p r a c t i s e . In p a r t i c u l a r , i t does not appear to be s o l u b l e 17 n u m e r i c a l l y . The s u f f i c i e n c y theorem adds some power by a l l o w i n g us t o guess the s o l u t i o n t o some problems. In t h i s manner, the Faustmann s o l u t i o n can be v e r i f i e d f o r the case where average h a r v e s t i n g c o s t s a re c o n s t a n t , t h a t i s C(h) = c h. The Faustmann r o t a t i o n p e r i o d f o r t h i s s i t u a t i o n i s the s o l u t i o n x o f the Faustmann f o r m u l a . (6) p ( x ) = r [ p ( x ) - c ] [ l - e " r x ] _ 1 Suppose the i n i t i a l age d i s t r i b u t i o n i s such t h a t h < h ( t ) < h f o r t < 0. The Faustmann s o l u t i o n or r o t a t i o n i s d e f i n e d i n d u c t i v e l y f o r a l l t > 0 by (7) h ( t ) = h ( t - x ) In o t h e r words, the c u t t i n g r u l e i s to c u t the t r e e s as they r e a c h the Faustmann age x (and t h i s i s f e a s i b l e by the r e s t r i c t i o n on the i n i t i a l d i s t r i b u t i o n ) . With t h i s d e f i n i t i o n (8) v ( t ) = t - x and w ( t ) = t+x when h > 0 - 64 -(9) v ( t ) = s-x and w(t) = s+x when h = 0 where s i s the time f o l l o w i n g t a t which l o g g i n g b e g i n s a g a i n . Now l e t (10) q ( t ) = p ( t - v ) - c i f h > 0 and q ( t ) = [ p ( s - v ( t ) ) - c ] e r ( t _ S ' ) i f h = 0. I t w i l l be shown t h a t (h,v,w,q) s a t i s f i e s the f o r e s t r y maximum p r i n c i p l e . F i r s t = p ( t - v ) - c - q = 0 i f h > 0 by ( 1 0 ) . I f h = 0, J h = p(x+t- s ) - c - ( p ( x ) - c ) e r ( t _ s ) . I f p(x+t- s ) - c < 0, then J h < 0. Otherwise J h e " r ^ t _ s - ) < p(x+t-s) - c - p(x) + c < 0 s i n c e t < s. Thus, i n any o f these c a s e s , h maximizes the H a m i l t o n i a n J . S e c o n d l y , the a d j o i n t e q u a t i o n (3-17) must be checked. I f h ( t ) > 0 then from above h ( t ) = 1 and v = 1. Moreover, q-rq = 0 - r [ p ( x ) - c] = p ( x ) [ e - 1] by the Faustmann f o r m u l a ( 6 ) . But p ( x ) [ e r x - l ] i s a l s o p ( w - t ) e r'- w ^ - p ( t - v ) as r e q u i r e d . In the o t h e r case h ( t ) = 0 the a d j o i n t e q u a t i o n r e q u i r e s q-rq = 0 which f o l l o w s from the d e f i n i t i o n above. F i n a l l y , s i n c e q < p ( x ) the t r a n s v e r s a l i t y c o n d i t i o n (3-18) h o l d s . The common t e x t b o o k examples of the Faustmann f o r m u l a assume an i n i t i a l even-aged f o r e s t . In the f o r m u l a t i o n here t h i s i s r e p r e s e n t e d by h(-a) b e i n g an impulse a t the i n i t i a l age of the t r e e s a, and b e i n g zero e l s e w h e r e . The Faustmann s o l u -t i o n i s t o i n s t a n t a n e o u s l y c u t the whole f o r e s t , e v e r y time i t s age reaches - 6 6 -5. S t e a d y S t a t e s The s t a t e i n t h e GFMP s h o u l d be t a k e n t o be t h e age d i s t r i b u t i o n o f t h e f o r e s t . A s t e a d y s t a t e s h o u l d t h e n be an age d i s t r i b u t i o n w h i c h o p t i m a l l o g g i n g p o l i c i e s do n o t a l t e r t h r o u g h t i m e . Such a s t e a d y s t a t e w o u l d have t o be a n o r m a l f o r e s t by d e f i n i t i o n , and wou l d n o t i n c l u d e Faustmann r o t a t i o n s i n g e n e r a l . A l e s s r e s t r i c t i v e c o n c e p t o f s t e a d y s t a t e c an be o b t a i n e d by v i e w i n g t h e l o g g i n g p r o c e s s as a d i s c r e t e d y n a m i c a l s y s t e m . A f t e r e a c h h a r v e s t i n g c y c l e t h e i n i t i a l age d i s t r i b u -t i o n i s r e p l a c e d by a n o t h e r age d i s t r i b u t i o n . In s t e a d y s t a t e s , t h e two d i s t r i b u t i o n s s h o u l d be t h e same. S e e k i n g t h i s k i n d o f s t e a d y s t a t e amounts t o f i n d i n g o p t i m a l h a r v e s t i n g p o l i c i e s w h i c h a r e p e r i o d i c . T h a t i s , t h e r e s h o u l d be a r o t a t i o n p e r i o d x s u c h t h a t (1) h ( t + x ) = h ( t ) f o r a l l t > v ( 0 ) = -x. The Faustmann r o t a t i o n s o f t h e p r e v i o u s s e c t i o n a r e examples o f s u c h s t e a d y s t a t e s , i n t h e c a s e o f c o n s t a n t u n i t h a r v e s t i n g c o s t s . They a r e a l m o s t t h e o n l y s t e a d y s t a t e s i n t h i s c a s e . S i n c e by t h e f o r e s t r y maximum p r i n c i p l e , h must maximize t h e H a m i l t o n i a n J and (2) J = [ p ( x ) - q - c ] h t h e n J = 0 whenever h < h < h and, h e n c e , q = 0. T o g e t h e r w i t h t h e a d j o i n t e q u a t i o n (3-17) t h i s i m p l i e s t h a t x i s g i v e n by t h e - 67 -Faustmann f o r m u l a (4-6) . T h i s l e a d s t o P r o p . 3: I f C(h) = c h, t h e n t h e o n l y s t e a d y s t a t e s f o r the GFMP a r e Faustmann r o t a t i o n s o r h i d e n t i c a l l y e q u a l t o e i t h e r h o r K. The l a s t p a r t o f t h i s p r o p o s i t i o n f o l l o w s f r o m the u s e f u l Lemma. F o r a s t e a d y s t a t e h a r v e s t i n g p o l i c y w i t h r o t a t i o n p e r i o d x, q = ( p ( x ) / r ) [ l - e ] on any i n t e r v a l on w h i c h h > 0. Thus i n P r o p . 3, i t f o l l o w s t h a t i f h = h > 0 o r i f h = E, t h e n p ( x ) - q - c i s c o n s t a n t and, h e n c e , h=h o r h=E i d e n t i c a l l y . The p r o o f o f t h e lemma i n v o l v e s two c a s e s . I t f o l l o w s f r o m (3-17) t h a t on any i n t e r v a l on w h i c h l o g g i n g does n o t c e a s e (3) q = A e r t + ( p ( x ) / r ) [ l - e " ™ ] " 1 I n c a s e l o g g i n g n e v e r c e a s e s t h e n i t f o l l o w s f r o m t h e t r a n s v e r s -a l i t y c o n d i t i o n (3-18) t h a t A = 0 on s u c h an i n t e r v a l . On t h e o t h e r hand, suppose l o g g i n g b e g i n s a t t-^ and ends a t t ^ ( b e i n g n o n z e r o i n b e t w e e n ) . T h e r e a r e o t h e r t r a n s v e r s a l i t y c o n d i t i o n s J ( h , ( t ^ ) ) = J ( h . ( t 2 ) ) = 0 w h i c h a r e e s s e n t i a l l y t h e t r a n s v e r s a l i t y 18 c o n d i t i o n s f o r f r e e i n i t i a l o r t e r m i n a l t i m e p r o b l e m s . T h e s e i m p l y t h a t h = h m ^ n a t b o t h t ^ and t 2 and hence a g a i n A = 0 as q must e q u a l p ( x ) - C (h • ) a t b o t h t , and t 0 . n n r v J K mm 7 1 2 T h i s lemma i s a l s o t h e key i n d i s c u s s i n g t h e c a s e o f v a r i a b l e a v e r a g e c o s t s . Assume t h a t t h e m a r g i n a l c o s t c u r v e - 68 -C'(.h) i s s t r i c t l y c onvex w i t h minimum a t h*. T h e r e i s no u n i q u e Faustmann r o t a t i o n b u t r a t h e r a f a m i l y o f d i f f e r e n t Faustmann r o t a t i o n s f o r d i f f e r e n t l e v e l s o f h a r v e s t . In p a r t i c u l a r , g i v e n h > h* d e f i n e x ( h ) as t h e s o l u t i o n o f t h e Faustmann f o r m u l a w i t h c r e p l a c e d by C ' ( h ) . T h a t i s (4) p ( x ) = r [ p ( x ) - C ' ( h ) ] [ l - e " r x ] _ 1 D i f f e r e n t i a t i n g t h i s g i v e s (5) dx/dh = r C " (h) ( r p - p ) _ 1 ( l - e " r x ) " 1 > 0 by p r e v i o u s a s s u m p t i o n s . Thus as l o n g as t h e f o r e s t s i z e A i s s u f f i c i e n t l y l a r g e t h e r e w i l l be a u n i q u e h s u c h t h a t x ( h ) « h = A. T h i s x ( h ) and t h e l o g g i n g p o l i c y h ( t ) = h a r e t h e main c a s e o f a s t e a d y s t a t e . P r o p . 4: Suppose a s t e a d y s t a t e has h ( t ) > 0 f o r a l l t . Then h ( t ) i s a c o n s t a n t h and e i t h e r x = x ( h ) o r h i s i d e n t i c a l -_ 1 9 l y e i t h e r h. o r h . The p r o o f i s t h a t i f h < h ( t ) < h on some i n t e r v a l t h e n i t f o l l o w s t h a t = 0 on t h i s i n t e r v a l which, t o g e t h e r w i t h t h e lemma, i m p l i e s h i s c o n s t a n t on t h i s i n t e r v a l and x = x ( h ) . Then J i s t h e same e v e r y w h e r e so t h e h w h i c h m a x i m i z e s J i s a l s o t h e same. The t h i r d c a s e o f i n t e r e s t i s t h e p o s s i b i l i t y t h a t t h e r e 20 may be p e r i o d s o f no l o g g i n g i n a s t e a d y s t a t e p o l i c y . Here, P r o p . 5 : Assume 0 = h and h . < h. Then i f a s t e a d y 69 -s t a t e i s zero on some i n t e r v a l , h ( t ) = h . when h > 0 and v m m x = x f h . ). The argument that x = x f h . ) combines ( 3 ) with ^ mm^ & ^ mm^ v J the p r o o f o f the second h a l f o f the lemma above, and s i n c e t h i s p r o o f i m p l i e s as w e l l that h . maximizes J between t-, and t ~ , r v mm 1 2 the r e s t of the p r o p o s i t i o n f o l l o w s . - 70 -6 . A s y m p t o t i c Stab11 i t y The q u e s t i o n o f a s y m p t o t i c s t a b i l i t y i s the q u e s t i o n o f whether the o p t i m a l l o g g i n g p o l i c y must converge to a s t e a d y s t a t e p o l i c y . G l o b a l a s y m p t o t i c s t a b i l i t y r e f e r s t o whether t h i s convergence o c c u r s r e g a r d l e s s o f the i n i t i a l age d i s t r i b u t i o n o f the f o r e s t . T h i s i s , i n d e e d , the case when u n i t average h a r v e s t i n g c o s t s a re c o n s t a n t a t c. I t was shown i n S e c t i o n 4 t h a t once the f i r s t h a r v e s t i s f i n i s h e d , e.g. a f t e r w ( 0 ) , then h ( t ) = h ( t - x ) when x i s the Faustmann r o t a t i o n and so h a r v e s t i n g a f t e r w(0) i s done on a s t e a d y s t a t e b a s i s (see Prop. 3 ) . G l o b a l a s y m p t o t i c s t a b i l i t y a l s o appears t o be t r u e i n the v a r i a b l e average c o s t c a s e , a l t h o u g h a complete p r o o f o f t h i s has not been found. For 0 < t < w ( 0 ) , d e f i n e w n ( t ) and h n ( t ) i n d u c t i v e l y by (1) w x ( t ) = w ( t ) h 1 ( t ) = h ( t ) w n ( t ) = w ( w n _ 1 ( t ) ) h n ( t ) = h ( w n _ 1 ( t ) ) t h h n ( t ) r e p r e s e n t s the n round o f l o g g i n g the f o r e s t . These l o g g i n g p o l i c i e s must be u n i f o r m l y bounded i n o r d e r t o be o p t i m a l . I t must be t r u e t h e n t h a t each sequence { h n ( t ) } has a c l u s t e r p o i n t . There would thus be v e r y i r r e g u l a r changes i n the h a r v e s t p o l i c y from h a r v e s t t o h a r v e s t i n o r d e r t h a t the sequence o f f u n c t i o n s {h^} not converge to some f u n c t i o n k 7 1 -u n i f o r m l y . Now i t w i l l be shown t h a t such convergence i m p l i e s t h a t k i s a stea d y s t a t e . F i r s t , i t i s e a s i l y shown by i n d u c t i o n t h a t n n +-^ ( t ) w n ( t ) = h 1 ( t ) . Thus h n + 1 ( t ) w n = h n ( t ) w n _ 1 and ( 2 ) [h A l ( t ) - h ( t ) ] w = h ( t ) [w -w J Assuming k ( t ) > 0, i t f o l l o w s on t a k i n g l i m i t s t h a t {w n - w n _ ^ converges u n i f o r m l y t o 0 and, hence, w n - wn_^ converges t o a c o n s t a n t x. D e f i n e q. n(t) = q (w n_ ^  ( t ) ) . By the f o r e s t r y maximum p r i n c i p l e , {q ( t ) } converges t o p(x) - C ' ( k ( t ) ) i f h < k < h. Use o f the a d j o i n t e q u a t i o n and k(0) = k(w(0)) then e s t a b l i s h e s t h a t k i s c o n s t a n t and x = x ( k ) ^ . A f u r t h e r argument can be made which suggests t h a t i h n ) and {w - w ,} converge. n n - l ° L e t Z ( t ) = I [ p ( s - v ( s ) ) h ( s ) - C ( h C s ) ) ] e " r ( s _ t ) d s = e r t V ( t ) The p o l i c y ( h ( s ) : s > t } must be the s o l u t i o n o f the GFMP f o r the i n i t i a l age d i s t r i b u t i o n {h(s) : v ( t ) < s < t } . I t seems i n t u i t i v e t h a t a f t e r some r e g u l a r i t y has been a c h i e v e d i n the age d i s t r i b u t i o n o f the f o r e s t - t h e r e a f t e r , the o p t i m a l l o g g i n g p o l i c y w i l l change the age d i s t r i b u t i o n o f the f o r e s t towards s u c c e s s i v e l y more v a l u a b l e age d i s t r i b u t i o n s . Thus a t l e a s t f o r t s u f f i c i e n t l y l a r g e , Z ( t ) s h o u l d be i n c r e a s i n g . - 72 -Moreover, t h e r e must be some upper l i m i t t o the v a l u e o f i n i t i a l age d i s t r i b u t i o n s . Thus one can imagine t h a t l i m Z ( t ) e x i s t s t->«> and l i m Z ( t ) = 0. Now Z ( t ) = - [p ( t - v ( t ) ) h ( t ) - C ( h ( t ) ) ] t-^-oo + r Z ( t ) . I t would f o l l o w then t h a t (3 ) l i m [ p ( t - v ( t ) ) h ( t ) - C ( h ( t ) ) ] e x i s t s t->oo which suggests (but does not prove) t h a t b o t h l i m t - v ( t ) and l i m h ( t ) e x i s t . Another i n d i c a t i o n o f s t a b i l i t y i s o b t a i n e d t-*-°° i n the f o l l o w i n g c a l c u l a t i o n . U s i n g Appendix 6 r w ( t }- - r f s - t l ( 4 ) Z ( t ) = -qh + / p ( s - v ) h ( s ) e r L S t j d s T h i s can be r e w r i t t e n w i t h the h e l p o f the c o s t a t e e q u a t i o n from the f o r e s t r y maximum p r i n c i p l e . T h i s was (5 ) q - r q = p ( w - t ) e " r ^ - p ( t - v ) v w h i c h i s e q u i v a l e n t t o the i n t e g r a l e q u a t i o n p ( s - v ) v ( s ) e " r ( - s " t J d s S i n c e h ( v ( s ) ) v ( s ) = h ( s ) i t then f o l l o w s t h a t -w(t). ( 7 ) Z ( t ) = / p(s-v(s),) V ( s ) [ h ( v ( s ) ) - h ( t ) J e " r ( - s " t J d s 73 -T h i s s h o u l d converge to 0 which suggests t h a t h ( s ) - h ( t ) f o r v ( t ) < s < t a l s o converges to 0. In c o n c l u s i o n , t h e r e are s t r o n g h i n t s t h a t o p t i m a l l o g g i n g p o l i c i e s a re g l o b a l l y a s y m p t o t i c a l l y s t a b l e i n v a r i a b l e average c o s t f o r e s t r y . A complete p r o o f , however, remains to be found. 74 -7. C o n c l u s i o n T h i s p a p e r has d e v e l o p e d t h e dynamics o f f o r e s t r y management. I t has shown t h a t t h e age d i s t r i b u t i o n o f t h e f o r e s t m a t t e r s and t h a t t h e r e a r e economic b e n e f i t s o b t a i n a b l e by a l t e r i n g t h i s age s t r u c t u r e i n a c c o r d a n c e w i t h t h e h a r v e s t i n g r u l e g i v e n by t h e f o r e s t r y maximum p r i n c i p l e . W i t h i n t h i s d y n a m i c a l s y s t e m , t h e Faustmann r u l e g i v e s t h e s t e a d y s t a t e r o t a t i o n s and p r o b a b l y has t h e p r o p e r t y t h a t i n t h e l o n g r u n o p t i m a l i t y r e q u i r e s t h a t t h e f o r e s t age d i s t r i b u t i o n a p p r o a c h a u n i f o r m age d i s t r i b u t i o n w h i c h c a n be h a r v e s t e d a c c o r d i n g t o a Faustmann r o t a t i o n . T h i s i s an i m p o r t a n t p o i n t b e c a u s e i t s u g g e s t s t h a t i n a w o r l d w i t h an u n c e r t a i n b i o l o g i c a l and economic f u t u r e , t h e t a r g e t o f c o n v e r t i n g t h e age d i s t r i b u t i o n o f f o r e s t s t o n o r m a l d i s t r i b u t i o n s a f t e r s e v e r a l r o t a t i o n s may s e r v e as a p r a c t i c a l and a c c e p t a b l e p r o x y f o r o b t a i n i n g t h e maximum p r e s e n t v a l u e o f l o n g r u n b e n e f i t s f r o m f o r e s t management. The o p t i m a l i t y c o n d i t i o n s d e v e l o p e d i n t h i s p a p e r have t h e i n t e r p r e t a t i o n t h a t i s u s u a l i n n a t u r a l r e s o u r c e management p r o b l e m s . M a x i m i z i n g t h e H a m i l t o n i a n g i v e s (when h i s u n c o n s t r a i n e d ) p ( t - v ) - C'(h) - q = 0 Thus q r e p r e s e n t s t h e r e n t a . f i r m w o u l d be a t t i m e t t o manage t h e f o r e s t . M o r e o v e r , w i l l i n g t o pay o u t q r e p r e s e n t s , as ] - 75 -u s u a l , the m a r g i n a l u s e r c o s t o f d e l a y i n g a s m a l l p o r t i o n o f the h a r v e s t a t time t . Indeed, a c c o r d i n g t o the d e r i v a t i o n o f the f o r e s t r y maximum p r i n c i p l e the l o s s o f p r e s e n t v a l u e - r t from such a p o l i c y i s - [ p ( t - v ) - C ( h ) - q]e . The term - r t - [ p ( t - v ) - C ' ( h ) ] e i s the l o s s c o r r e s p o n d i n g t o the r e d u c t i o n - r t i n the h a r v e s t a t time t . The r e m a i n i n g l o s s qe must be a t t r i b u t e d t o the f a c t t h a t d e l a y i n g h a r v e s t i n g at time t on a p o r t i o n o f f o r e s t l a n d a l s o d e l a y s u s i n g t h a t l a n d f o r growing f u t u r e crops o f t r e e s . The d e r i v a t i o n o f the f o r e s t r y maximum p r i n c i p l e sheds l i g h t on the economic i n t e r p r e t a t i o n o f the H a m i l t o n i a n i t s e l f . - r t I t t u r n s out t h a t i f h = 0 , then J ( h ( t ) ) e i s the m a r g i n a l l o s s o f p r e s e n t v a l u e t h a t would be i n c u r r e d i f no l o g g i n g was conducted a t time t . - 76 -CHAPTER I I I ON THE OPTIMAL HARVESTING OF A REGIONAL FOREST 1. I n t r o d u c t i o n For many y e a r s , economists have debated w i t h f o r e s t r y e x p e r t s over the c o r r e c t method o f d e t e r m i n i n g o p t i m a l r o t a t i o n p e r i o d s f o r h a r v e s t i n g f o r e s t a r e a s . T h i s d i v e r g e n c e o f o p i n i o n s , w h i c h has been .summarized by*Samuelson (1976), a r i s e s because of a d i v e r g e n c e o f o p i n i o n s about the importance o f e x t e r n a l i t i e s i n p r o d u c t i o n from the f o r e s t . An example o f t h i s i s p r o v i d e d by the d e t e r m i n a t i o n o f an o p t i m a l f o r e s t management program f o r a r e g i o n c o n s i s t i n g o f a number o f f o r e s t a r e a s . An economist might argue t h a t the o p t i m a l r e -g i o n a l program i s t o choose f o r each a r e a s e p a r a t e l y the r o t a t i o n p e r i o d w h i c h maximizes d i s c o u n t e d r e t u r n s from the p e r p e t u a l s t r i n g o f f o r e s t s t o be grown i n the a r e a . The problem w i t h t h i s i s t h a t i t w i l l l i k e l y l e a d t o extreme f l u c t u a t i o n s i n the annual r e g i o n a l c u t . T h i s i s because l o g g i n g c o s t s and revenues v a r y from a r e a t o a r e a because o f d i f f e r e n c e s i n s o i l q u a l i t y , - g e o g r a p h i c a l c h a r a c t e r i s t i c s and d i s t a n c e s from r e g i o n a l wood-using c e n t e r s . The r e s u l t i s t h a t d i f f e r e n t areas have d i f f e r e n t o p t i m a l r o t a t i o n p e r i o d s (see Chapter I ) . Such an economist i s thus assuming t h a t such a f o r e s t r y management program w i t h an u n s t a b l e o u t p u t , w i l l not l e a d t o r e g i o n a l economic i n s t a b i l i t y or t h a t i f i t does, then such i n s t a b i l i t y i s not v e r y c o s t l y . (For example, f a c t o r s might c o s t l e s s l y m i g r a t e t o o t h e r employments d u r i n g low p e r i o d s i n the r e g i o n a l h a r v e s t . ) - 77 On the o t h e r hand, the f o r e s t r y e x p e r t might b e l i e v e t h a t f l u c t u a t i o n s i n r e g i o n a l l o g output would l e a d t o an e x t r e m e l y c o s t l y economic d e s t a b i l i z a t i o n o f the r e g i o n . Thus he would argue t h a t management programs w h i c h do n o t s a t i s f y a sus-t a i n e d y i e l d c o n s t r a i n t a re u n p r o f i t a b l e when d e s t a b i l i z a t i o n c o s t s a re accounted f o r and, hence, s h o u l d be i g n o r e d i n the d e c i s i o n making p r o c e s s . C l e a r l y , the q u e s t i o n o f the magnitude o f the c o s t s o f r e g i o n a l economic i n s t a b i l i t y u r g e n t l y r e q u i r e s f u r t h e r r e s e a r c h by economists ( f o r some comments see Pearse (1976, 230-231)). However, g i v e n the s t a t e o f the a r t , answers do not appear to be i n s i g h t and meanwhile s u s t a i n e d y i e l d i s the g e n e r a l l y a c c e p t e d p o l i c y . There i s thus the problem o f c h o o s i n g the o p t i m a l r e g i o n a l f o r e s t management program s u b j e c t t o the s u s t a i n e d y i e l d c o n s t r a i n t . I f the economist i s not ready to t a c k l e the c o s t o f i n s t a b i l i t y q u e s t i o n , he can s t i l l t r y and use h i s p r e s e n t c a p i t a l t h e o r e t i c t o o l s t o see i f he can c o n t r i b u t e t o the u n d e r s t a n d i n g o f the s o l u t i o n o f t h i s problem o f o p t i m a l management under s u s t a i n e d y i e l d . T h i s i s the purpose o f the p r e s e n t paper. Any f o r e s t management problem has a s t r u c t u r e f a m i l i a r to economists from growth t h e o r y . One b e g i n s w i t h an i n i t i a l age s t r u c t u r e o f t r e e s growing on the s i t e s i n q u e s t i o n . H a r v e s t i n g and r e p l a n t i n g ( o r n a t u r a l r e g e n e r a t i o n ) c o n v e r t t h i s t o another age s t r u c t u r e . Repeated c y c l e s o f h a r v e s t i n g and r e g e n e r a t i o n c o n t i n u a l l y change the age s t r u c t u r e o f the f o r e s t s . The problem i s thus s i m i l a r t o the growth problem. - 78 -One wants t o f i n d s t e a d y s t a t e s , t h a t i s e q u i l i b r i u m age s t r u c -t u r e s w h i c h t h e o p t i m a l h a r v e s t i n g p o l i c y w i l l p e r p e t u a t e . S e c o n d l y , one i s i n t e r e s t e d i n w h e t h e r o p t i m a l h a r v e s t i n g p o l i c i e s t r a n s f o r m any i n i t i a l age s t r u c t u r e t o t h e s t e a d y s t a t e and, i f s o , what f o r m t h e t r a n s i t i o n s h o u l d t a k e . An example o f t h i s k i n d o f a n a l y s i s may be f o u n d i n Heaps and Neher (1979) . D e t e r m i n a t i o n o f o p t i m a l r e g i o n a l f o r e s t management programs has b e en a t t e m p t e d by s i m u l a t i o n methods i n s p e c i f i c c a s e s . However, l i t t l e o f a g e n e r a l n a t u r e a p p e a r s t o be known a b o u t t h e s o l u t i o n t o s u c h p r o b l e m s . I t seems t o be commonly assumed t h a t t h e s t e a d y s t a t e f o r t h e management p r o b l e m s u b j e c t t o a s u s t a i n e d y i e l d c o n s t r a i n t i s t o c o n v e r t e a c h f o r e s t a r e a t o a " n o r m a l " forest"*" ( G r a y s o n and J o h n s t o n (1970, 7 1 ) ) . T h i s p a p e r w i l l show t h a t t h e s t e a d y s t a t e s f o r s u s t a i n e d y i e l d f o r e s t r y a r e i n d e e d n o r m a l o r q u a s i n o r m a l f o r e s t s . T h i s c o n c l u s i o n t h e n adds w e i g h t t o t h e f o r e s t e r ' s a s s e r t i o n s a b o u t t h e age d i s t r i b u t i o n s d e s i r a b l e i n t h e l o n g r u n . T hese a s s e r t i o n s w i l l be t r u e i f i t can be shown t h a t any o p t i m a l s o l u t i o n t o a s u s t a i n e d y i e l d p r o b l e m must c o n v e r g e t o a s t e a d y s t a t e , e.g. be a s y m p t o t i c a l l y s t a b l e . U n f o r t u n a t e l y , i t has n o t y e t b e e n p o s s i b l e t o p r o v e t h i s , a l t h o u g h t h e e v i d e n c e p o i n t s s t r o n g l y i n t h i s d i r e c t i o n ( s e e Heaps and Neher (1979) and C h a p t e r I I ) . The r e s u l t s o b t a i n e d i n t h i s p a p e r a r e o b t a i n e d f r o m an e x t e n s i o n o f t h e f o r e s t r y maximum p r i n c i p l e d e v e l o p e d i n 7 9 -Chapter I I . T h i s i s p r e s e n t e d i n s e c t i o n 2. S e c t i o n 3 l o o k s a t s u s t a i n e d y i e l d problems f o r a s i n g l e f o r e s t a r e a and s e c t i o n 4 o b t a i n s the s t e a d y s t a t e s f o r the m u l t i p l e a r e a problem. The f i n a l s e c t i o n sums up. - 80 -2. The R e g i o n a l F o r e s t Management Model The model u s e d h e r e i s t h e g e n e r a l i z e d f o r e s t r y manage-ment model d e s c r i b e d i n C h a p t e r I I . E a c h f o r e s t a r e a has i t s own r e v e n u e and c o s t f u n c t i o n s p ^ ( t ) and C^(h^) where h. i s t h e h a r v e s t r a t e i n t h e i f o r e s t a r e a . The r e g i o n a l f o r e s t manager's p r o b l e m i s t o c h o o s e t h e s e h a r v e s t r a t e s t o m a x i m i z e (1) V = E / [ p - C t - v ^ h . - C, ( h , ) ] e " r t d t s u b j e c t t o t h e s u s t a i n e d y i e l d c o n s t r a i n t E h. > h and t h e i o t h e r c o n s t r a i n t s w h i c h d e s c r i b e t h e d y n a m i c s o f f o r e s t manage-ment. The f o r e s t r y maximum p r i n c i p l e , d e v e l o p e d f o r t h e c a s e o f o n l y one f o r e s t a r e a , g e n e r a l i z e s d i r e c t l y t o t h i s s i t u a t i o n . Thus t h e r e a r e a d j o i n t v a r i a b l e s q - ( t ) s u c h t h a t t h e o p t i m a l h ^ t ) ( 2 ) MAX I J i = £ [ p i ( t - v i ) h i - C ( h i ) - q ^ ] s u b j e c t t o E > h i i M o r e o v e r , t h e a d j o i n t v a r i a b l e s s a t i s f y . ~ - - r ( w . - t ) % (3) q i - r q i = p i ( w i - t ) h i ( t ) e " P i ( t - v i ) v i (4) l i m q . e " r t = 0 t-x» as w e l l as t h e o t h e r c o n s t r a i n t s , t h e m a x i m i z a t i o n o f V was - 81 -s u b j e c t t o . A f u r t h e r i m p o r t a n t p o i n t i s t h a t the p r o o f o f the s u f f i c i e n c y o f the f o r e s t r y maximum p r i n c i p l e a l s o c a r r i e s over d i r e c t l y t o the m u l t i - h a r v e s t case p r o v i d e d a l l the C^(h^) are convex. T h i s i s because of the l i n e a r i t y o f the o b j e c t i v e and the c o n v e x i t y of the f e a s i b l e s e t c o r r e s p o n d i n g to > h. S u f f i c i e n c y e nables one t o guess a t the s o l u t i o n to the s u s t a i n e d y i e l d problem i n s i m p l e c a s e s . - 82 -3. The S i n g l e F o r e s t A r e a I t i s n e c e s s a r y to u n d e r s t a n d the s o l u t i o n to the sus-t a i n e d y i e l d problem i n a s i n g l e f o r e s t a r e a f i r s t , b e f o r e the m u l t i p l e a r e a problem can be t a c k l e d . Two cases w i l l a l s o be d i s t i n g u i s h e d . F i r s t the " c o n s t a n t average c o s t " case where C(h) = c*h. In t h i s c a s e , t h e r e i s a s i n g l e Faustmann r o t a t i o n p e r i o d g i v e n by the Faustmann f o r m u l a -1 (1) p(x) = r [ p ( x ) - c] [1 - e " r x ] The o t h e r case w i l l be r e f e r r e d to as the " v a r i a b l e average c o s t " case where C(h) i s s t r i c t l y convex. The Faustmann r o t a t i o n here depends on A, the s i z e o f the f o r e s t . I t i s d e f i n e d by the two e q u a t i o n s -1 - r x (2) p ( x A ) = r [ p ( x A ) - C ' ( h A ) ] [ l - e A ] h A ' *A = A . In e i t h e r case the s u s t a i n e d y i e l d c o n s t r a i n t i s h > h. I n t u i t i v e l y , the o p t i m a l l o g g i n g program under s u s t a i n e d y i e l d w i l l be the program " c l o s e s t " t o the o p t i m a l program when s u s t a i n e d y i e l d i s not imposed. In the c o n s t a n t average c o s t case the u n c o n s t r a i n e d optimum i s t o l o g each h e c t a r e when the t r e e s r e a c h the Faustmann age. T h i s becomes f e a s i b l e under s u s t a i n e d y i e l d a f t e r the f i r s t h a r v e s t o f the whole a r e a i s completed. The s u s t a i n e d y i e l d optimum w i l l then be t o " 83 " s e l e c t A - hx h e c t a r e s w h i c h w i l l be h a r v e s t e d when the t r e e s on t h a t s i t e r e a c h the Faustmann age. The r e m a i n i n g hx h e c t a r e s w i l l be c u t a t r a t e h, i n p e r p e t u i t y , the e l d e s t t r e e s b e i n g c u t f i r s t . . The s e l e c t i o n must be done i n the most advantageous way. In the example where the i n i t i a l s t a n d i s of u n i f o r m age y, the f i r s t h a r v e s t p a t t e r n i s i l l u s t r a t e d below i n F i g . 5 . h a r v e s t r a t e impulse o f a r e a A - hx h 0 x-y x time F i g u r e 5 The F i r s t H a r v e s t P a t t e r n Subsequent h a r v e s t r e p e a t t h i s p a t t e r n w i t h a r o t a t i o n p e r i o d o f x. T h i s may be f o r m a l l y proved from the f o r e s t r y maximum p r i n c i p l e as f o l l o w s . From the diagram v ( t ) = t+y f o r t < x and v(;t) = t - x f o r t > x. As w e l l w ( t ) = t+x f o r t > 0. D e f i n e now the a d j o i n t v a r i a b l e q by 84 -(3) r q = p ( x ) e " r x [ e r , ( : t + y ) - l ] f o r t < x r q = p ( x ) [ 1 - e ] f o r t > x A d i s c o n t i n u i t y i s a l l o w e d i n q at t=x because v ( t ) i s d i s c o n t i n u o u s at t h i s p o i n t . Now f o r t < x the H a m i l t o n i a n i s J = [p(t+y) - c - q]h and ) h = e r ^ t + ^ [ p ( t + y ) e - r ( : t + ^ - p ( x ) e - r x ] • - r s By assumption p ( s ) e i s d e c r e a s i n g i n s, so i t f o l l o w s t h a t has a maximum a t t = x-y. But J ^ ( x - y ) = 0 by the Faustmann f o r m u l a . Thus f o r t f x-y, < 0 and h = h maximizes J . For t = x-y, = 0 and the optimum c o n t r o l i s an i m p u l s e . As w e l l , f o r t < x, q - r q = p ( x ) e ~ r x = p ( w - t ) e " r ^ w - t ^ - p ( t - v ) v as r e q u i r e d . Thus the maximum p r i n c i p l e i s s a t i s f i e d f o r t < x and i t i s e a s i l y checked t h a t i t a l s o h o l d s f o r t > x. The r e s u l t then f o l l o w s from the s u f f i c i e n c y of the maximum p r i n c i p l e . The above d i s c u s s i o n assumes hx > A. I f hx < A then the s u s t a i n e d y i e l d optimum w i l l be t o c u t a t r a t e h i n p e r p e t u i t y , the t r e e s b e i n g o f age T = A/h when c u t . I t w i l l s i m p l i f y f u t u r e d i s c u s s i o n i f some t e r m i n o l o g y i s i n t r o d u c e d a t t h i s s tage t o d e s c r i b e the ty p e s o f age - 85 -d i s t r i b u t i o n s g e n e r a t e d by s u s t a i n e d y i e l d f o r e s t r y . The type o f age d i s t r i b u t i o n i l l u s t r a t e d above i s an example o f what w i l l be termed a q u a s i n o r m a l age d i s t r i b u t i o n . A q u a s i -normal (m,T) age d i s t r i b u t i o n c o n t a i n s o n l y t r e e s i n the age c l a s s e s 0 t o T and c o n t a i n s a t l e a s t m h e c t a r e s o f t r e e s i n each age c o h o r t . I f the d i s t r i b u t i o n c o n t a i n s e x a c t l y m h e c t a r e s i n each c o h o r t , i t w i l l be termed a normal (m,T) age d i s t r i b u t i o n . The r e s u l t above can now be s t a t e d as: A. I f hx < A then o p t i m a l h a r v e s t i n g under s u s t a i n e d y i e l d r e q u i r e s immediate c o n v e r s i o n o f the age d i s t r i b u t i o n o f the f o r e s t t o a q u a s i n o r m a l (h,x) age d i s t r i b u t i o n . B. I f hx > A a normal (h,A/h) age d i s t r i b u t i o n i s r e q u i r e d . The second s u s t a i n e d y i e l d case i s the v a r i a b l e average c o s t c a s e . Here, i t w i l l be a s s e r t e d t h a t i n the u n c o n s t r a i n e d case the o p t i m a l l o g g i n g p o l i c y , h ( t ) , converges a s y m p t o t i c a l l y 2 to h^ and the r o t a t i o n p e r i o d to x^. T h i s i s w i t h i n r e a s o n independent o f the i n i t i a l age d i s t r i b u t i o n o f the f o r e s t . The f o l l o w i n g r e s u l t s would then seem t o be t r u e . C. I f h < h^, then o p t i m a l h a r v e s t i n g under s u s t a i n e d y i e l d r e q u i r e s e v e n t u a l c o n v e r s i o n o f the age d i s t r i b u t i o n o f the f o r e s t to a normal ( h ^ , x^) age d i s t r i b u t i o n . D. I f h > h^ a normal (h, A/h) f o r e s t i s the d e s i r e d a s y m p t o t i c d i s t r i b u t i o n . - 86 4. M u l t i p l e F o r e s t Areas Assume now t h a t the r e g i o n a l f o r e s t c o n t a i n s N d i f f e r e n t f o r e s t a r e a s . The f o r e s t manager's problem i s now to maximize the sum o f d i s c o u n t e d n et revenues from l o g g i n g these a r e a s , t h a t i s t o maximize s u b j e c t t o the s u s t a i n e d y i e l d c o n s t r a i n t X h- > h. i = l 1 One can attempt to guess the s o l u t i o n t o t h i s problem and v e r i f y i t v i a the f o r e s t r y maximum p r i n c i p l e . However, s i m p l e s o l u t i o n s such as i n the one a r e a case do not appear t o be v a l i d h e r e . In the absence o f complete s o l u t i o n s , t h i s s e c t i o n w i l l f o c u s on the d e t e r m i n a t i o n o f stea d y s t a t e s f o r the s u s t a i n e d y i e l d problem. Steady s t a t e s a re i m p o r t a n t because i t seems l i k e l y t h a t g i v e n any r e a s o n a b l e s e t o f i n i t i a l age d i s t r i b u t i o n s , the o p t i m a l r e g i o n a l h a r v e s t p o l i c y 3 s h o u l d converge t o a s t e a d y s t a t e h a r v e s t i n g p o l i c y . A s t e a d y s t a t e i s a s e t o f i n i t i a l age d i s t r i b u t i o n s such t h a t the o p t i m a l r e g i o n a l management p l a n g i v e n t h e s e i n i t i a l d i s t r i b u t i o n s i s to h a r v e s t each a r e a i n such a way as to p e r i o d i c a l l y r e c r e a t e the i n i t i a l age d i s t r i b u t i o n s . T h i s means t h a t f o r each a r e a t h e r e s h o u l d be a r o t a t i o n p e r i o d T^ such t h a t the o p t i m a l p o l i c y s a t i s f i e s N ( 1 ) h ^ t ) = h ^ t - T i) f o r a l l t > 0 and a l l i . - 87 -T h e f o l l o w i n g w i l l s e e m m o r e r e a s o n a b l e i f we b e g i n w i t h a d i s c r e t e m o d e l i n w h i c h t i m e i s m e a s u r e d i n d i s c r e t e j u m p s . W i t h a n a p p r o p r i a t e c h o i c e o f t h e u n i t j u m p , i t c a n e a s i l y b e i m a g i n e d t h e n t h a t t h e r o t a t i o n p e r i o d s ( m e a s u r e d a s a n u m b e r o f j u m p s ) a r e r e l a t i v e l y p r i m e . F o r e x a m p l e , i f t h e r o t a t i o n p e r i o d s w e r e 5 0 , 60 a n d 70 y e a r s , t h e u n i t j u m p s w o u l d b e 10 y e a r p e r i o d s . L e t t ^ b e t h e t i m e i n [ 0 , T ^ ] a t w h i c h h ^ a c h i e v e s t h e m i n i m u m w h i c h w i l l b e d e n o t e d h ^ . I t f o l l o w s f r o m t h e C h i n e s e r e m a i n d e r t h e o r e m i n n u m b e r t h e o r y ( A p o s t o l ( 1 9 7 6 , 1 1 7 - 1 1 8 ) ) t h a t t h e r e a r e p o s i t i v e i n t e g e r s n ^ s u c h t h a t t = + i s t h e s a m e f o r a l l i . T h e n b y p e r i o d i c i t y £ / h . ( t ) = £ h . ( t . ) = E h . i i i a n d t h e f o l l o w i n g i m p o r t a n t c o n c l u s i o n i s o b t a i n e d . A p o s s i b l e s t e a d y s t a t e f o r r e g i o n a l f o r e s t m a n a g e m e n t s a t i s f i e s t h e s u s t a i n e d y i e l d c o n s t r a i n t £ > h f o r a l l t i f a n d o n l y i f i i t s a t i s f i e s ( 2 ) £ h± > h i T h i s p r o p o s i t i o n d e p e n d s o n t h e d e g r e e o f n o n m e s h i n g o f r o t a t i o n p e r i o d s b e i n g s u c h t h a t a n y t w o r o t a t i o n p e r i o d s a r e r e l a t i v e l y p r i m e . H o w e v e r , i n o r d e r t o m a k e t h e a p p l i c a t i o n o f m o d e r n c o n t r o l t h e o r y e a s i e r o u r m o d e l w a s s e t u p i n c o n t i n u o u s t i m e . I t i s s h o w n i n A p p e n d i x 8 t h a t t h e r e i s a c o n t i n u o u s a n a l o g u e o f t h e a b o v e p r o p o s i t i o n . I n t h i s c a s e n o n m e s h i n g o f - 88 " r o t a t i o n p e r i o d s must be taken to mean t h a t the r e c i p r o c a l s o f the r o t a t i o n p e r i o d s are l i n e a r l y independent over the r a t i o n a l number. Now l e t h. be the s o l u t i o n t o the s i n g l e a r e a problem / - r t o f m a x i m i z i n g I [ p ^ ( t - v ^ ) h ^ - C ^ ( h . ) ] e dt s u b j e c t t o the c o n s t r a i n t h^ > h-. These s a t i s f y the o v e r a l l s u s t a i n e d y i e l d c o n s t r a i n t X) ^ l l * Hence, they must be o p t i m a l f o r the s u s t a i n e d y i e l d problem. Consequently i n the m u l t i a r e a s u s t a i n e d y i e l d problem, a s t e a d y s t a t e must c o n s i s t o f a c o l l e c t i o n o f q u a s i n o r m a l or normal age d i s t r i b u t i o n s as were d e s c r i b e d i n s e c t i o n 3. The n e x t q u e s t i o n i s whether the s t e a d y s t a t e r o t a t i o n p e r i o d s must be Faustmann r o t a t i o n p e r i o d s . The answer, as might be e x p e c t e d , i s a f f i r m a t i v e i f Faustmann r o t a t i o n s are p o s s i b l e . L e t x^ be the Faustmann r o t a t i o n age f o r the i f o r e s t a r e a d e f i n e d by e i t h e r ( 3 - 1 ) or ( 3 - 2 ) . Then Prop. I f h < £ A i x i - 1 ' t h e n T i = x i £ o r a 1 1 i-The p r o o f o f t h i s i s i n two s t a g e s . F i r s t i t i s shown t h a t T. = x. f o r one i i m p l i e s T. = x. f o r a l l j . T h i s i s i i J J because T^ = x^ f o r a s t e a d y s t a t e r o t a t i o n i m p l i e s from the f o r e s t r y maximum p r i n c i p l e " r x i = P i ( x i ) [1 - e ] and, hence, t h a t the H a m i l t o n i a n i s maximized, e.g. 9J^/3h^ = 0. Now . the f o r e s t r y maximum p r i n c i p l e s t a t e s as w e l l t h a t the o p t i m a l r e g i o n a l f o r e s t management program - 89 maximizes XZ J - s u b j e c t t o X h- > h. Thus i f h. > 0, 3 3 j 3 3 as i t must be a t some p o i n t o f t i m e , then 8 / 3 h j = 0. T h i s , t o g e t h e r w i t h the f a c t t h a t h^ i s a s t e a d y s t a t e f o r a s i n g l e a r e a problem i m p l i e s t h a t T^  = x.. . The p r o p o s i t i o n w i l l now be proven i f i t can be shown t h a t T^ f x^ f o r a l l i i s impos-s i b l e . Note t h a t t h i s i m p l i e s T^ < x^ f o r a l l i . As w e l l , i f t h i s i s the c a s e , i t must be t r u e t h a t = h f o r a l l t . These a s s e r t i o n s f o l l o w s i n c e the m a x i m i z a t i o n o f X) J -s u b j e c t t o X, h. > h i m p l i e s 3J./3h- < 0 i f T. f x.. Now j j - — 1 i i i i X) H-(t) = h t . L e t [ t/T^ ] be the g r e a t e s t i n t e g e r l e s s j than or e q u a l t o t/T^ . By the p e r i o d i c i t y o f the h a r v e s t s H iCt) > [ t / T i ] A i . F u r t h e r m o r e , i f < x± f o r a l l i , then f o r t s u f f i c i e n t l y l a r g e t / x ^ < [ t / T ^ ] f o r a l l i . Combining the s e f a c t s g i v e s ( Z V i ' V « Z t t / T i ] A i <: E H ^ t ) = h t C a n c e l l i n g the t , t h i s c o n t r a d i c t s the h y p o t h e s i s o f the theorem. These r e s u l t s cannot be made any m o r e ^ s p e c i f i c , They say the f o l l o w i n g . In the c o n s t a n t average c o s t c a s e , take any c o l l e c t i o n o f q u a s i n o r m a l (h^,x^) f o r e s t s such t h a t 12—i - l l - Such a c o l l e c t i o n i s a s t e a d y s t a t e f o r the s u s t a i n e d y i e l d h a r v e s t i n g problem. In the v a r i a b l e average c o s t c a s e , a Faustmann h a r v e s t r a t e i s d e f i n e d by ( 3 - 2 ) as w e l l as the Faustmann r o t a t i o n p e r i o d . A c o l l e c t i o n o f normal - 90 (h^,x^) f o r e s t s i s t h e u n i q u e s t e a d y s t a t e f o r t h e s u s t a i n e d y i e l d h a r v e s t i n g p r o b l e m p r o v i d e d > h. I t i s a l s o p o s s i b l e t o a n a l y z e t h e c a s e where t h e Faustmann r o t a t i o n s a r e i n a d e q u a t e t o meet the s u s t a i n e d y i e l d . ^ - 1 T h a t i s , where h > Z^A^x^ . As s e e n i n t h e p r o o f a b ove, t h i s i m p l i e s T^ < x^ f o r a l l i and h = £ h ^ f o r a l l t . Now, s i n c e i t was shown t h a t £ h . > h t h i s means h. = h. f o r a l l i . ^ — 1 _ — l — i M o r e o v e r , h^ = A^T^ ^. Now t h e f o r e s t r y maximum p r i n c i p l e i m p l i e s t h a t - r T . = P i O j H l - e x ] and, h e n c e , 3 J i / 9 h i = p C T ^ - C ! ( A i T i X ) - r ' ^ C T ^ H l - e x ] F i n a l l y , m a x i m i z a t i o n o f s u b j e c t t o £ h ^ > h i m p l i e s ( 3 ) dJ±/dhi = 3 J j / 3 h j f o r a l l i and j L A i T i " 1 = h T h i s p r o v i d e s N e q u a t i o n s w h i c h can be s o l v e d f o r t h e T-. 91 -5. C o n c l u s i o n s I t was easy enough to f i n d the s o l u t i o n to the s u s t a i n e d y i e l d problem when o n l y a s i n g l e f o r e s t a r e a was i n v o l v e d . T h i s o b s e r v a t i o n does n o t , however, c a r r y over t o the case of two or more f o r e s t s as a l i t t l e work w i t h the f o r e s t r y maximum p r i n c i p l e w i l l show. Thus i t seems t h a t the b e s t t h a t can be done i n these cases i s t o l o o k f o r l o n g run r e s u l t s . The q u e s t i o n o f a s y m p t o t i c s t a b i l i t y i s thus a v e r y i m p o r t a n t u n s o l v e d problem i n f o r e s t r y e c o n o m i c s ; F i n a l l y , the r e s u l t s o f t h i s paper suggest t h a t i n the l o n g run i t i s o p t i m a l to do some l o g g i n g i n every a r e a i n e v e r y p e r i o d . There a r e , however, e x t e r n a l i t i e s between d i f f e r e n t areas w h i c h may make t h i s s o l u t i o n i m p r a c t i c a l . Is i t more c o s t l y t o m a i n t a i n c o n t i n u o u s l y the whole network o f l o g g i n g roads or to a l l o w roads to f a l l i n t o a s t a t e o f d i s r e -p a i r f o r a number of y e a r s and then to r e c o n s t r u c t them ? A nother problem c o u l d o c c u r i f the areas were too s m a l l to keep an e f f i c i e n t s i z e d l o g g i n g o p e r a t i o n busy f o r the whole season. Then crews and equipment would have t o be s h i f t e d from s i t e t o s i t e i n the s t r a t e g y s u g g ested h e r e . These e x t r a expenses might r u l e out the n o r m a l i z a t i o n o f each f o r e s t a r e a as b e i n g the o p t i m a l l o n g run s t r a t e g y . - 92 -APPENDIX 1 SUFFICIENT CONDITIONS FOR REPLANTING OR SILVICULTURE TO BE PROFITABLE E x t r a r e g e n e r a t i o n e f f o r t i s p r o f i t a b l e i f V* '(0) > 0 o where V* i s the s o l u t i o n t o MAX V = [R e " r T - D(F ) ] [ l - e ~ r T ] _ 1 {T} ° S.T. F = f ( F , 0 ) F(0) = F Q From ( 3 - 3 ) Vp = V F = R F ' f ( T ) f ( F ) - V r T - Dp 0 0 o so the c o n d i t i o n i s R F ( F ( T ) ) f ( F ( T ) ) f ( 0 ) - V r T - Dp (0) > 0 o where T i s the o p t i m a l r o t a t i o n p e r i o d when F Q = 0_. C o n s i d e r now the case o f s i l v i c u l t u r e . Suppose R F ( F ( T ) ) f ( F ( T ) ) f L ( F ( t ) , 0 ) e " r T - f (F ( t ) ) WL (0) e ' r t > 0 f o r s-^  < t < S2" L e t L ( t ) be a non-zero s i l v i c u l t u r a l p o l i c y w hich i s zero i f t < or t > s^, C o n s i d e r the problem MAX V = [R e " r T - D(0) - J W ( 0 L ) e " r t d t ] [ l - e " r T ] " 1 {T} ° 93 s u c h t h a t F = f ( F , 6 L ) F ( 0 ) = 0 I t w i l l f o l l o w t h a t s i l v i c u l t u r e i s p r o f i t a b l e i f V f i ( 0 ) > p . V 0 - [ R p e •F(T,e) . r T l f | I i - / L W ^ d t l f l - e - V 1 Now T '0 = / f ( F , 9 L ( t ( F , e ) ) ) x d F so 0 = f ( F ( T , 9 ) , 9 L ( T ) ) _ 1 9 F ^ > 9 ) . 89 rF (T,9) _ 2 . - J [ L * f L * £ 6 L t 0 £ L £ J d F L e t t i n g 9 t e n d t o 0 and c h a n g i n g v a r i a b l e s a g a i n -T £ ( T ) - 1 I F I T ^ ! = j L f L f - l d t Thus V e ( 0 ) = [f^ [ R F f ( T ) e ' r T f L ( t ) f " 1 ( t ) • W L e " r t ] L d t - r T -1 [1-e ] > 0 by t h e c h o i c e o f L, - 94 -APPENDIX 2 SECOND ORDER CONDITIONS FOR THE SILVICULTURE PROBLEM The s e c o n d o r d e r c o n d i t i o n s i n v o l v e d e r i v a t i v e s o f t h e shadow p r i c e s A ( 0 ) and A ( T ) . To s i m p l i f y t h e r e s u l t i n g e x p r e s s i o n s some p r e l i m i n a r y r e s u l t s a b o u t t h e s e d e r i v a t i v e s must be o b t a i n e d . S e c t i o n 3 o f C h a p t e r I I c o n t a i n s two forms o f t h e d e r i v a t i v e Vj, ((17) and ( 2 1 ) ) . T h e s e were (where - r T -1 t h e f a c t o r [1-e ] has b e en d r o p p e d f r o m t h e e q u a t i o n s below) (1) V T = - r V - rR - W ( L ( T ) ) + A(T) • f ( F ( T ) , L ( T ) ) (2) V T = - r V - rD - W ( L ( 0 ) ) + A ( 0 ) • f ( F Q , L ( 0 ) ) D i f f e r e n t i a t e t h e s e e x p r e s s i o n s a g a i n w i t h r e s p e c t t o e a c h o f t h e v a r i a b l e s T, F q and F ( T ) . E q u a t i n g t h e r e s u l t s g i v e s ( 3 ) ( 3 A ( T ) / 3 T ) f ( F ( T ) , L ( T ) ) e " r T = ( 3 A ( 0 ) / 3 T ) f ( F Q , L ( 0 ) ) (4) ( 3 A ( T ) / 3 F o ) f ( F ( T ) , L ( T ) ) e " r T = ( 3 A ( 0 ) / 3 F Q ) f ( F Q L ( 0 ) 0 + A ( 0 ) [ f F ( F o , L ( 0 ) ) - r ] (5) ( 3 A ( T ) / 3 F ( T ) ) f ( F ( T ) , L ( T ) ) e " r T = ( 3 A ( 0 ) / 3 F Q ) f ( F Q , L ( 0 ) ) + A ( T ) e " r T [ r - f F ( F ( T ) , L ( T ) ) ] ( H e r e , t h e f i r s t o r d e r c o n d i t i o n s ( 1 7 ) - ( 1 9 ) and t h e K u hn-Tucker c o n d i t i o n s (15) o f S e c t i o n 3 have been used.) - 9 5 Three more r e l a t i o n s a r e needed. These are o b t a i n e d by a p p l y i n g Young's theorem t o the second o r d e r c r o s s p a r t i a l s T o f - f W ( L ) e " r t d t . (6) ( 3 A ( T ) / 3 F Q ) f ( F ( T ) , L ( T ) ) e " r T = (3A(0)/3T) (7) ( 3 A ( T ) / 3 F ( T ) ) f ( F ( T ) , L ( T ) ) = -(3A(T)/3T) + [r - f F ( F ( T ) , L ( T ) ) ] A ( T ) (8) ( 3 A ( T ) / 3 F o ) e " r T = - ( 3 A ( 0 ) / 3 F (T) ) The f i r s t o r d e r c o n d i t i o n s ((18) and (19) o f S e c t i o n 3) f o r F q and F ( T ) were (9) V F = [-Dp + A ( 0 ) ] = 0 o o (10) V p ( T ) = [Rp - A ( T ) ] e " r T = 0 D i f f e r e n t i a t i n g (1) t o g e t h e r w i t h t h e s e g i v e s the second o r d e r p a r t i a l s (11) V T T = ( 3 A ( T ) / 3 T ) f ( T ) e " r T (12) Vp T = (3A(0)/3T) = f ( 0 ) _ 1 V T T by (3) and ( 1 1 ) . o (13) V F ( T ) T = - ( 3 A ( T ) / 3 T ) e " r T = - f ( T ) _ 1 V T T b y ( 1 1 ) . (14) V F ( T ) F = - ( 3 A ( T ) / 3 F o ) e " r T which by (6) = - f ( T ) _ 1 ( 3 A ( 0 ) / 3 T ) = - f ( 0 ) _ 1 f ( T ) " 1 V T T by ( 1 2 ) . (15) V F F = -Dp F + (3A(0)/3F ) which by (4) 0 0 o o - 96 = f ( 0 ) _ 1 [ - D r ; f . ( 0 ) + O X ( T ) / 3 F o ) £ ( T ) e " r T o o - A ( 0 . ) ( f F ( 0 ) - r ) ] = £ ( 0 ) _ 1 [ - D F F £ ( 0 ) - Dp ( f F ( 0 ) - r ) ] + f ( 0 ) ~ 2 V T T 0 0 o b y ( 9 ) a n d ( 1 4 ) . ( 1 6 ) V p C T ) F ( T ) = ( R p p - ( 3 A ( T ) / 3 F ( T ) ) ) e ~ r T w h i c h b y ( 7 ) = £ ( T ) " 1 [ R F F • £ ( T ) + ( 3 A ( T ) / 3 T ) - X ( T ) ( r - f F ( T ) ) ] e " r T = £ ( T ) _ 1 [ R F F • f ( T ) + R F ( £ p ( T ) - r ) ] e " r T + £ ( T ) " 2 V T T b y ( 1 0 ) a n d ( 1 1 ) . T h e m a t r i x o f s e c o n d o r d e r p a r t i a l s o f V i s t h u s ( 1 7 ) ; A V T T f ( 0 ) _ 1 V T T - f ( T ) _ 1 V T T f ( 0 ) _ 1 V T T f ( 0 ) " 2 V T T + X - f ( 0 ) _ 1 f ( T ) _ 1 V T T L - f ( T ) _ 1 V T T - f ( 0 ) - 1 f ( T ) " 1 V T T f ( T ) " 2 V T T + Y w h e r e X = - f ( 0 ) _ 1 [ D p p f ( 0 ) + Dp ( f p ( 0 ) - r ) ] ^ a n d o o o Y = f ( T ) " 1 [ R p F f ( T ) + R F ( f p ( T ) - r ) ] T h i s m u s t b e n e g a t i v e d e f i n i t e f o r a n i n t e r i o r m a x i m u m a n d t h i s i s e a s i l y s e e n t o i m p l y ( 1 8 ) V T T < 0 X < 0 Y < 0 - 9 7 " APPENDIX 3 ENVELOPE THEOREM FOR OPTIMAL CONTROL PROBLEMS Co n s i d e r the problem f1 MAX I = J J ( x , u , t ; a ) <u> t o S.T. x = f ( x , u , t ; a ) x o = x (^ to^ ) a n d x l = x C t 1 ) g i v e n a a parameter. Then l e t I * (a) be the s o l u t i o n o f t h i s problem. ch I*(a) = J [ J + Af - Ax]dt so d l * ( a ) / d a = I 1 [H + H (dx/da) + H (du/da) t o + ( d A / d a ) ( f - F) - A ( d x / d a ) ] d t = J 1 [Hn - A(dx/da) - A ( d x / d a ) ] d t t o = f 1 H dt - A(dx/da) O - H , J a dt ch - 98 ~ APPENDIX 4 THE SIGN OF 3 A ( T ) / 3 x (1) A ( T ) e " r T = 3( f W ( L ) e " r t d t ) / 3 F ( T ) so r l (2) ( 3 A ( T ) / 3 x ) e ~ r T = 3 ( 1 W ( L ) e ~ r t d t ) / 3 F ( T ) . Now Jo x /T rT W ( L ) e " r t d t = / W ( L ( F ) ) f ( F , L ( F ) ) " V r t ( F ) d F so u Jo T (4) ( 3 A ( T ) / 3 x ) e ~ r T = j {{^xL ~ W ^ ) £ " 2 (3L ( F ) / 3 F (T) ) - r W v f _ 1 ( 3 t ( F ) / 3 F ( T ) ) } e " r t d F .A. S i n c e W > 0, 3 X ( T ) / 3 x > 0 w i l l be shown by s h o w i n g t h a t (5) fW - W £ T > 0 when L ( F ) > 0 (6) 3 L ( F ) / 3 F ( T ) > 0 (7) 3 t ( F ) / 3 F ( T ) < 0 C o n c e r n i n g (5) t h e o p t i m a l i t y c o n d i t i o n -W^  + X£^ = 0 when L > 0 means we need t h e s i g n o f A f W T - W WT . I t a l s o Xi_i X Li f o l l o w s f r o m t h e o p t i m i z a t i o n c o n d i t i o n s t h a t H ( t ) = -W + A f > 0 f o r a l l t ( s i n c e f r o m f i r s t o r d e r c o n d i t i o n 3-21, H(0) > 0 and s i n c e H ( t ) i s i n c r e a s i n g ) . T hus, Af > W and AfW T -WW, > WW - W WT XJL X 1_J X L X Li 2 - 1 = W L ( 3 e W L / 3 x ) > 0 by a s s u m p t i o n ( i i ) . P r o o f o f t h e o t h e r two a s s e r t i o n s r e q u i r e s an a n a l y s i s o f t h e d y n a m i c a l s y s t e m A = A ( r - f p ) , F = f ( F , L ) . The t r a j e c t o r i e s o f t h i s s y s t e m s l o p e upwards as i l l u s t r a t e d i n F i g . 6 below. F(T) F(T)+AF(T) F i g u r e 6 The T r a j e c t o r i e s The o p t i m i z a t i o n c o n d i t i o n [-W^  + A£^]«L=0 d e f i n e s L = L ( A , F ) . When L > 0, ("WLL + A f L L ) L A + f L = 0. I t has been assumed t h a t f* L > 0 and -WLL + A f L L < 0 so L x < 0. I t f o l l o w s t h a t a t e v e r y f o r e s t s i z e F, t r a j e c t o r y A^ uses l e s s s i l v i c u l t u r e than t r a j e c t o r y . C o n s e q u e n t l y i t t a k e s a l o n g e r time to grow a f o r e s t t o s i z e F(T) a l o n g A 2 than a l o n g A^. Indeed, the problem o f f i n d i n g the o p t i m a l t r a j e c t o r y i s j u s t a q u e s t i o n o f s e l e c t i n g the t r a j e c t o r y which t a k e s time T to go from F Q to F ( T ) . Now i n c r e a s e the t e r m i n a l s t o c k to F(T) + A F ( T ) . Supposing A 2 was o p t i m a l f o r F ( T ) , i t cannot be o p t i m a l f o r F(T) + AF(T) because i t t a k e s l o n g e r than time T t o grow t h i s 100 s t o c k a l o n g t h i s p a t h . A p a t h w i t h a s h o r t e r growing time i s needed and t h e s e are the t r a j e c t o r i e s l y i n g above k^? Thus the o p t i m a l p a t h f o r F(T) + AF(T) w i l l be some p a t h l i k e k±. Moreover, i t f o l l o w s t h e n t h a t 9 L(F)/9F(T) = L A ( 9 A ( F ) / 3 F ( T ) ) > 0. F i n a l l y s i n c e on A.^  a t e v e r y s t o c k s i z e F, a l a r g e r s i l v i c u l t u r a l e f f o r t i s made i t t a k e s a s h o r t e r time to grow a s t o c k s i z e o f F on A^ than on k^ whence 9 t ( F ) / 9 F ( T ) < 0. - 101 -APPENDIX .5 AN EXAMPLE OF THE FIRST VARIATION Let (h,v,H) denote a f e a s i b l e h a r v e s t i n g p o l i c y and l e t k be an increment o f h a r v e s t i n g e f f o r t such t h a t h ( s ) + k i s a f e a s i b l e h a r v e s t i n g p o l i c y f o r t < s < t +£ 0- Now d e f i n e a p e r t u r b a t i o n o f H f o r 0 < e < e Q by H(s) f o r s < t H(s,e) =J H(s) + ( s - t j k f o r t < s < t+e ' H(s) + ek f o r t+e < s. Then h( s , e ) = h(s) except where t < s < t+e when h(s,e) = h(s)+k. Thus n i s j u s t an i m p u l s e o f amount k a p p l i e d t o h a t time t . The p l a n t i n g date v ( s , e ) i s unchanged f o r s < t and s > w(t)+e. In between these d a t e s , one can see t h a t ( s - t ) k f o r t < s < t+e H ( v ( s , e ) ) - H ( v ( s ) ) = { ek t+e < s < w(9) t'+e - v ( s , e j w ( 9 J < s < w(t)+e Here 9 i s d e f i n e d by H ( 9 ) = H ( t ) - ek. L e t t i n g e t e n d to 0 one sees t h a t h ( v ) v £ = k f o r t < s < w(t) and 0 e l s e w h e r e . Now w r i t e the p r e s e n t v a l u e o f the p e r t u r b e d h a r v e s t i n g p o l i c y as Jr t rt+e rw(0J /*w(t)+e r< + J + J + J + J r , o J t J t + z ^wfe) * /w(t)+e - 102 D i f f e r e n t i a t i o n o f these s u b i n t e g r a l s i n v o l v e s d i f f e r e n t i a t i n g them w i t h r e s p e c t to the e n d p o i n t s t+e, w ( 0 ) , w(t)+e. These w i l l c a n c e l out except a t t+e where h(s,e) i s reduced by k. Here d i f f e r e n t i a t i o n and l e t t i n g e tend t o 0 c o n t r i b u t e s { [ p ( t - v ) (h+k) - C(h+k)J - [ p ( t - v ) h - C ( h ) j } e ~ r t to V (0) . The r e s t o f V ( 0 ) i s o b t a i n e d by d i f f e r e n t i a t i n g under the i n t e g r a l y w ( t ) . _ r s s i g n and t h i s adds o n l y the term - / p ( s - v ) v h ( s ) e ds. w ( t ) . /t By above t h i s e q u a l s -k / p ( s - v ) v e ds. Now t h i s term - r t i s i n f a c t -kqe f o r i t may be seen t h a t r t /' w C 1 : ) - r - r s qe A u = J p ( s - v ) v e XJds i s one way o f w r i t i n g the a d j o i n t e q u a t i o n ( 3 - 1 4 ) . - r t - r t - r t F i n a l l y , w r i t e -kqe as -(h+k)qe + hqe and one sees t h a t V (0) = [Jfh+k) - J(h) J e " r t . - 103 " APPENDIX 6 SIMPLIFICATION OF PRESENT VALUE L e t V ( t ) = / [ p ( s - v ) h ( s ) - C ( h ( s ) ) ] e ~ r s d s t where (h,v) i s t h e o p t i m a l l o g g i n g p r ogram. Then i n t e g r a t i n g by p a r t s g i v e s VCt) = | [ p ( t - v ) h C ( h ) ] e _ r t CO + 7 J [ P ( s - v ) ( l - v ) h + ( p C s - v ) - C ' ( h ) ) h ] e " r s d s U s i n g t h e f o r e s t r y maximum p r i n c i p l e , t h i s i n t e g r a l i s f U P ( S - V ) + q - r q - p (w-s) e " r ( w " s ) ] h + q h } e " r S d s f» CO - r t I * - r s = qhe + / p ( s - v ) h e J t ' X(T) P ( s - v ) h ( s ) e " r s d s The l a s t i n t e g r a l i s o b t a i n e d f r o m t h e t e r m p ( w - s ) h ( s ) e r w d s by t h e change o f v a r i a b l e u = w(s) and the use o f d h ( u ) d u = h ( s ) d s . The p r e s e n t v a l u e t h u s s i m p l i f i e s t o - r t f w ( t ) . _ r s r V ( t ) = J ( t ) e +J p ( s - v ( s ) ) h e r b d s - 404 -APPENDIX 7 RELATIONSHIP OF THE FORESTRY MAXIMUM PRINCIPLE TO "THE ECONOMICS OF FORESTRY WHEN THE RATE OF HARVEST IS CONSTRAINED" The problem s t u d i e d i n t h i s paper (GFMP) i s the same as the on-going f o r e s t problem s t u d i e d i n Heaps and Neher (1979). The o n l y d i f f e r e n c e i s a change o f n o t a t i o n from .'\f) t o v and 0 t o w. The n e c e s s a r y c o n d i t i o n o b t a i n e d i n Heaps and Neher (1979) was (1) p ( t - v ) - e"r(w-t)p(w-tO = r p ( t - v ) + C" (h)h - rC' ( h ) T h i s i s o b t a i n a b l e from the f o r e s t r y , maximum p r i n c i p l e when h i s u n c o n s t r a i n e d . Then, d i f f e r e n t i a t i n g p ( t - v ) - C'(h) - q = 0 g i v e s (2) p ( t - v ) ( l - v ) - C M (h)h - q = 0 S u b s t i t u t i n g t h e s e two e q u a t i o n s i n the a d j o i n t e q u a t i o n f o r • q and q g i v e s ( 1 ) . The f o r e s t r y maximum p r i n c i p l e f o r m u l a t i o n i s , however, more p o w e r f u l than the f o r m u l a t i o n i n (1) as i t a l l o w s f o r c o n s t r a i n t s t o be imposed on the h a r v e s t r a t e . Moreover, i t enables one to get the s u f f i c i e n c y r e s u l t and to c h a r a c t e r i z e s t e a d y s t a t e s i n cases where the h a r v e s t becomes c o n t i n u o u s . - 1 0 5 -A P P E N D I X 8 C O N T I N U O U S N O N M E S H I N G O F R O T A T I O N P E R I O D S C o n s i d e r t h e s e q u e n c e o f v e c t o r s g i v e n b y { ( ( k T ] L + t 1 ) T 2 1 , ( k T 1 + t 1 ) T 3 1 , ( k T ^ t ^ T ^ ) : P r o v i d e d T^"*", T 2 \ • • • » a r e l i n e a r l y i n d e p e n d e n t o v e r t h e r a t i o n a l s i t f o l l o w s f r o m E x a m p l e 6 . 1 o f K u i p e r s a n d N i e d e r r e i t e r ( 1 9 7 4 , 4 8 ) t h a t t h i s s e q u e n c e i s u n i f o r m l y d i s t r i b u t e d m o d 1 i n ( n - 1 ) - d i m e n s i o n a l E u c l i d e a n s p a c e . T h i s r e q u i r e s t h a t t h e r e e x i s t s i n t e g e r s k ^ , k 2 , k n s u c h t h a t f o r i > 2 ( k l T l + t ^ " 1 ! . - 1 - k-i s a r b i t r a r i l y c l o s e ( a n d o n a p r e d e t e r m i n e d s i d e o f ) t o t .T . ' . ' ^ o r s u c h t h a t f o r i > 2 1 1 k-^T-^ + t ^ i s a r b i t r a r i l y c l o s e t o ( a n d o n a p r e d e t e r m i n e d s i d e o f ) k T . + t . i 1 1 A s s u m i n g t h a t h ^ , i > 1 a r e e i t h e r l e f t o r r i g h t c o n t i n u o u s a t t . , i t t h e n f o l l o w s t h a t t h e k ' s c a n b e c h o s e n s o t h a t 1' h . ( k , T , + t , ) i s a r b i t r a r i l y c l o s e t o h . ( k . T . + t . ) = h . . i l l 1 1 1 1 l — x . T h u s ]C h • i s a r b i t r a r i l y c l o s e t o X) ^ i ^ l ^ l + - — 5 0 ^ n i 1 i t h e l i m i t X h ; > h . 1 — 1 — - 106 " .NOTES INTRODUCTION: 1. The term used by Faustmann was Bodenerwartungswert. 2. See P o n t r y a g i n e t . a l . (1964, 204). 3. The q u e s t i o n o f whether the s o l u t i o n s to o p t i m a l m u l t i s e c t o r growth models are g l o b a l l y a s y m p t o t i c a l l y s t a b l e has a t t r a c t e d a t t e n t i o n i n the l i t e r a t u r e r e c e n t l y . See B r o c k and Scheinkman (1976). 4. See Grayson and J o h n s t o n (1970, 71-77). 5. See Goldman (1968) f o r a d i s c u s s i o n o f t h i s type of p l a n n i n g i n the c o n t e x t o f o p t i m a l growth models. 107 -CHAPTER I 1. See, f o r example,: Anderson (1976). 2. A n o n d e c r e a s i n g o p t i m a l s i l v i c u l t u r a l e f f o r t i s t r u e f o r c e r t a i n c l a s s e s o f f o r e s t growth f u n c t i o n s . 3. K i l k k i and V a i s a n e n (1969). For a b i b l i o g r a p h y o f o t h e r examples, see Cawrse (1979, 6-9). 4. See the c a l c u l a t i o n s i n the U n i t e d S t a t e s BLM's Timber P r o d u c t i o n Cost Manual. 5. That i s dF(T)/dT i s the time d e r i v a t i v e o f F ( T ) . - rT 6. The statement i s e q u i v a l e n t t o [1-e 1 < rT which i s t r u e s i n c e by mean v a l u e theorem -e" rT+i = r e " r ? T < rT f o r some £ > 0. 7. See n o t e 6. 8. See Smith and Tse (1977) and the U n i t e d S t a t e s BLM's Timber P r o d u c t i o n Cost Manual. 9. T h i s i s s i m i l a r t o the form used by Ledyard and Moses (1976, 151, f n . j ) . . Naslund (1977, 3) uses a s e p a r a b l e form, e s s e n t i a l l y F = f ( F ) + h ( L ) . 10. D e f i n e s ( A F Q ) by A F 0 = F ( s ( A F Q ) , F Q ) . D i v i d i n g by A F 0 and t a k i n g l i m i t s 1 = f ( F D ) • (ds (AF Qj/d.A'F 0) • at--.F£- = "<0_." Now F ( t , F 0 + A F 0 ) = F ( t + s ( A F 0 ) ,F G) so 9F(t)/8.F- = f ( F ( t ) ) s ' ( 0 ) = f ( F ( t ) ) f ( F Q ) " 1 . 11. The c o s t a t e t o xe r t so d(\e r t ) / % t = 0 has been used. - 108 CHAPTER I I : 1. See C l a r k (1976, 258) and H e l l s t e n (1980, 3-4). 2. See C l a r k (1976, 131-2). 3. Some r e g u l a r i t y c o n d i t i o n s s h o u l d be p l a c e d on a d m i s s i b l e age d i s t r i b u t i o n s . They s h o u l d be p i e c e w i s e c o n t i n u o u s and p o s s i b l y t h e r e s h o u l d be some l i m i t on the o l d e s t age a l l o w e d . 4. Thus l o g g i n g may be viewed as a dyn a m i c a l system i n the i n f i n i t e d i m e n s i o n a l space o f a d m i s s i b l e age d i s t r i b u -t i o n s . 5. See P o n t r y a g i n (1964, 204) or Gabasov and K i r i l l o v a (1976, 208). 6. See Hale (1977) . 7. A l t e r n a t i v e l y one can view t h i s as l o o k i n g f o r p e r i o d i c o p t i m a l h a r v e s t i n g p o l i c i e s . 8. These r e s u l t s are more g e n e r a l than the r e s u l t s o b t a i n e d i n Heaps and Neher (1979) because i t i s no l o n g e r n e c e s s a r y t o assume t h a t the o p t i m a l h a r v e s t s a re s e p a r a t e d from one another by p e r i o d s d u r i n g which t h e r e i s no l o g g i n g . 9. See Grayson and J o h n s t o n (1970, 71). 10. I t would be p r e f e r a b l e t o use a no n s e p a r a b l e c o s t f u n c t i o n o f the form C(h,a) as t h e r e are c o n s i d e r a b l e economies o f s c a l e i n v o l v e d i n l o g g i n g o l d e r t r e e s (See Chapter I ) . T h i s d i d n o t prove to be m a t h e m a t i c a l l y t r a c t a b l e , however. 11. I t has not been e s t a b l i s h e d , however, t h a t these c o n d i t i o n s are s u f f i c i e n t t o guarantee the e x i s t e n c e o f a s o l u t i o n to the GFMP. 12. I t s h o u l d be no t e d t h a t c o n s t a n t average c o s t s are a l l o w e d f o r h e r e . 13. For example, suppose the f o r e s t i s log g e d i n s t a n t a n e o u s l y every T y e a r s , the f i r s t time b e i n g at t=0. Then v ( t ) = nT and w(t) = (n+l)T f o r nT < t < ( n + l j f . That i s v ( t ) and w(t) are s t e p f u n c t i o n s . - 109 -14. More p r o p e r l y P, and P 2 s h o u l d be c u t o v e r a s m a l l i n t e r v a l o f t i m e . 15. Here n ( s ) = 0 f o r s ^ 0. 16. The c o s t a t e v a r i a b l e q may n o t be c o n t i n u o u s b u t i t c a n be t a k e n t o be c o n t i n u o u s on any i n t e r v a l on w h i c h v and w a r e c o n t i n u o u s . 17. See Heaps and Neher (1979, 310) f o r a d i s c u s s i o n o f t h i s p o i n t . 18. T h e s e t r a n s v e r s a l i t y c o n d i t i o n s c a n be d e d u c e d f r o m . t h e t r a n s v e r s a l i t y p r o p e r t y i n Heaps and Neher (1979, 3 1 6 ) . A l t e r n a t i v e l y J ( h ( t ^ ) ) = 0 f o l l o w s f r o m c o n t i n u i t y o f J ( h ) a t t w h i l e J ( h ( t 9 ) ) = 0 c a n be shown d i r e c t l y by 1 , £ c a l c u l a t i n g V (0) f o r t h e p e r t u r b a t i o n . H(s,e) = 0 f o r s < t ^ and s > t 2 + e H(s,e) = H ( t x + ( t 2 - t 1 ) ( t 2 + e - t 1 ) " 1 ( s - t 1 ) ) f o r t ^ s ^ + e The c a l c u l a t i o n i s s i m i l a r t o t h e example i n A p p e n d i x 5. T h i s i g n o r e s some s p e c i a l c a s e s . T h i s i s t h e s i t u a t i o n s t u d i e d i n Heaps and Neher ( 1 9 7 9 ) . A t l e a s t , i f t h e i n i t i a l age d i s t r i b u t i o n s a t i s f i e s some r e g u l a r i t y p r o p e r t i e s . I f q ( t ) = l i m q f t ) , t h e n q - r q = - p ( x ) [ l - e ] t a k i n g n>°° r t l i m i t s i n t h e a d j o i n t e q u a t i o n . Hence, r q = Ae + p ( x ) [ l - e " r x ] and p ( x ) - C ' ( k ( t ) ) = q ( t ) . Then A = 0 s i n c e k(0) = k ( w ( 0 ) ) whence k ( t ) i s c o n s t a n t and e l i m i n a t i o n o f q shows t h a t x = x ( k ) . 19. 20. 21. 22 . no -CHAPTER I I I : 1. These terms are e x p l a i n e d at the end o f S e c t i o n 3. 2. T h i s i s the q u e s t i o n o f g l o b a l a s y m p t o t i c s t a b i l i t y d i s c u s s e d i n Chapter I I . 3. See n ote 2. - I l l BIBLIOGRAPHY 1. Anderson, C.H. (1976) T e r r a c e - H a z e l t o n R e g i o n a l F o r e s t Resources Study, Logging Q p e r a b i l i t y Component, p r e p a r e d f o r E n v i r o n m e n t a l and Land Use S e c r e t a r i a t o f the Govt, o f B r i t i s h C olumbia, V i c t o r i a , B.C. 2. A p o s t a l , T.M. (1976) I n t r o d u c t i o n t o A n a l y t i c Number Theory (New York: S p r i n g e r - V e r l a g ) . 3. B r o c k , W.A. and J.A. Scheinkman (1976) " G l o b a l A y m p t o t i c S t a b i l i t y o f O p t i m a l C o n t r o l Systems w i t h A p p l i c a t i o n s t o the Theory o f Economic Growth". J o u r n a l o f  Economic Theory 12, 164-190. 4. Cawrse, D.C. (1979) Dynamic System M o d e l i n g i n Timber Management, A S e l e c t e d A n n o t a t e d B i b l i o g r a p h y , Dept. o f F o r e s t and Wood S c i e n c e s , C o l l e g e o f F o r e s t r y and N a t u r a l R e s o u r c e s , C o l o r a d o S t a t e U n i v e r s i t y , F o r t C o l l i n s , C o l . 5. C l a r k , C.W. (1976) M a t h e m a t i c a l Bioeconomics (New York: W i l e y ) . 6. Dobie, J . (1966) P r o d u c t Y i e l d and V a l u e s , F i n a n c i a l R o t a t i o n s and B i o l o g i c a l R e l a t i o n s h i p s o f Good  S i t e Douglas F i r , MSF T h e s i s , F a c u l t y o f F o r e s t r y , U n i v e r s i t y o f B r i t i s h C olumbia, Vancouver, B.C. 7. Faustmann, M. (1849) "On t h e D e t e r m i n a t i o n o f the V a l u e w h i c h F o r e s t Land and Immature Stands pose f o r F o r e s t r y " i n Gane, M. (ed.)(196#) M a r t i n Faustmann  and the E v o l u t i o n o f D i s c o u n t e d Cash FLow, O x f o r d I n s t i t u t e Paper 42, O x f o r d . 8. Gabasov, R. and F. K i r i l l o v a (1976). The Q u a l i t a t i v e Theory o f O p t i m a l P r o c e s s e s (New York: M a r c e l D e k k e r ) . 9. Goldman, S.M. (1968) " O p t i m a l Growth and C o n t i n u a l P l a n n i n g R e v i s i o n " . Review o f Economic S t u d i e s 35, 145-154. 10. Grayson, A . J . and D.R. J o h n s t o n (1970) "The Economics o f Y i e l d P l a n n i n g " . I n t e r n a t i o n a l Review o f F o r e s t r y  R e search 3, 69-122. 11. Gregory, G.R. (1972) F o r e s t Resource Economics, (New York: Ronald P r e s s ) . 12. H a l e , J . (1977) Theory o f F u n c t i o n a l D i f f e r e n t i a l E q u a t i o n s . (New York: S p r i n g e r - V e r l a g ) . 13. H a l e y , D. and J.H.G. Smith (1964) " A l l o w a b l e Cuts can be I n c r e a s e d S a f e l y by use o f F i n a n c i a l R o t a t i o n s " . B r i t i s h Columbia Lumberman. - 112 14. Heaps, T. and P.A. Neher (1979) "The Economics o f F o r e s t r y when the Rate o f H a r v e s t i s C o n s t r a i n e d " . J o u r n a l  o f E n v i r o n m e n t a l Economics and Management 6, 297-319. 15. H e l l s t e n , M. (1980) Are Trees any D i f f e r e n t from F i s h . ; Dept. o f Economics Research Paper 80-3, U n i v e r s i t y o f A l b e r t a , Edmonton, A l b . 1.6. Howe, C.W. (1979) N a t u r a l Resource Economics (New York: W i l e y ) . 17. K i l k k i , P. and U. V a i s a n e n (1969) " D e t e r m i n a t i o n o f the O p t i m a l P o l i c y f o r F o r e s t Stands by means o f Dynamic Programming". A c t a F o r e s t a l i a F e n n i c a 102, 100-112. 18. K u i p e r s , L. and H. N i e d e r r e i t e r (1974) U n i f o r m D i s t r i b u t i o n o f Sequences. (New York: WTley). 19. L e d y a r d , J . and L.N. Moses (1976) "Dynamics and Land Use: The Case o f F o r e s t r y " i n G r i e s o n , R.E. ( e d . ) , P u b l i c and Urban Economics ( L e x i n g t o n , Mass.: L e x i n g t o n ) . 20. N a s l u n d , B. (1977) "The P r i n c i p l e o f S u s t a i n e d Y i e l d and O p t i m a l F o r e s t Management". The S c a n d i n a v i a n J o u r n a l  o f Economics, 79, 1-7. 21. P e a r s e , P. (1976) Report o f the R o y a l Commission on F o r e s t r y P o l i c y ( V i c t o r i a , B.C.). 22. P o n t r y a g i n , L.S. and o t h e r s (1964) The M a t h e m a t i c a l Theory o f O p t i m a l P r o c e s s e s (New York: O x f o r d U n i v e r s i t y P r e s s ) . 23. Samuelson, P.A. (1976) "Economics o f F o r e s t r y i n an E v o l v i n g - Societyt"-.. * Economic I n q u i r y , XIV, 466-492. 24. S m i t h , D.G. and P.P. Tse (1977) Logging T r u c k s : Comparison o f P r o d u c t i v i t y and C o s t s , Tech. Report No. TR-18, F o r e s t E n g i n e e r i n g R esearch I n s t i t u t e o f Canada, Vancouver, B.C. 25. U n i t e d S t a t e s Bureau o f Land Management, Timber P r o d u c t i o n Costs Manual. 26. von B e r t a l a n f f y , L. (1938) "A Q u a n t i t a t i v e Theory o f O r g a n i c Growth". Human B i o l o g y , 10, 181-213. 27. Wackerman, A.E. (1949) H a r v e s t i n g Timber Crops (New York: M c G r a w - H i l l ) . 

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