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A commercial fisheries production function : the Skeena River sockey salmon gillnet fishery Roberts, Richard Frederick Anthony 1971

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A COMMERCIAL FISHERIES PRODUCTION FUNCTION: THE SKEENA RIVER SOCKEYE SALMON GILLNET FISHERY by R i c h a r d R o b e r t s B.A. , U n i v e r s i t y o f V i c t o r i a , 1965. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Department o f Economics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1971 In p r e s e n t i n g t h i s t h e s i s in 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 at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree tha 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 tudy . 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 tha t c o p y i n g o r 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 of Economics The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date A p r i l 6, 1971 ABSTRACT The p r o d u c t i o n f u n c t i o n s commonly s u g g e s t e d i n t h e f i s h e r y b i o - e c o n o m i c l i t e r a t u r e a r e reviewed, and m o d i f i c a t i o n s a r e s u g g e s t e d t o t a k e i n t o account d i r e c t and i n d i r e c t gear e x t e r n a l i t i e s and gear s a t u r a t i o n e f f e c t s i n a salmon g i l l n e t f i s h e r y . T h i s s u g g e s t s a d a i l y p r o d u c t i o n f u n c t i o n o f t h e Cobb-Douglas v a r i e t y , t e s t e d on Skeena R i v e r sockeye salmon d a t a . U s i n g t h e d a i l y p r o d u c t i o n f u n c -t i o n , a s e a s o n a l model i s dev e l o p e d , which r e f l e c t s t h e i n t r a - s e a s o n a l p e a k i n g n a t u r e o f t h e run, and t h e m u l t i p l e day s t o c k removal e f f e c t . The p r o p e r t i e s o f t h i s f u n c t i o n a r e a n a l y s e d , and t o demonstrate t h e s i g n i f i c a n c e o f t h e f u n c t i o n , i t s shape i s used t o i n d i c a t e t h e d i r e c t i o n o f b i a s i n t h e C r u t c h f i e l d and P o n t e c o r v o e s t i m a t e s o f d i s -s i p a t e d r e n t i n B r i s t o l Bay and Puget Sound. I l l ACKNOWLEDGMENTS The a u t h o r g r a t e f u l l y acknowledges t h e encouragement and support of Mr. B l a k e Campbell, C h i e f F i s h e r i e s Economist f o r t h e F i s h e r i e s S e r v i c e , P a c i f i c Region, o f t h e Department of F i s h e r i e s and F o r e s t r y , f o r making t h i s study p o s s i b l e . The a u t h o r i s i n d e b t e d t o Dr. C l i v e Southey o f t h e Department o f Economics o f t h e U n i v e r s i t y of B r i t i s h Columbia, f o r a d v i c e and g u i d -ance i n t h e p r e p a r a t i o n o f t h i s r e p o r t , and a l s o t o Dr. P a u l B r a d l e y , Dr. P e t e r P e a r s e , and Dr. R u s s e l l U h l e r f o r h e l p f u l comments and c r i t -i c i s m s . Many thanks f o r s t i m u l a t i n g d i s c u s s i o n s on t h e paper a r e due t o Dr. M i c h a e l Shepard, Mr. F r e d W i t h l e r , and Mr. J a c k McDonald o f t h e F i s h e r i e s R e search Board of Canada. The a u t h o r wishes t o g i v e s p e c i a l thanks t o Mr. P h i l Murray o f th e F i s h e r i e s S e r v i c e f o r h i s p a t i e n t and c a r e f u l h e l p i n p r o v i d i n g much of t h e data o n l y t o be f o u n d i n t h e musty f i l e s o f t h e Skeena R i v e r Management Committee, o f which Mr. Murray i s D i r e c t o r o f I n v e s -t i g a t i o n s . The computer programming h e l p p r o v i d e d by M i s s S h e i l a K i n n e a r and John Thomson i s g r a t e f u l l y acknowledged. The m e t i c u l o u s t y p i n g and p r o o f - r e a d i n g work done by Mi s s Mary H o l b r o o k d e s e r v e s s p e c i a l mention and thanks. The l o y a l and c h e e r f u l support of my w i f e B a r b a r a was e s p e c i a l l y a p p r e c i a t e d . i v TABLE OF CONTENTS Page T I T L E PAGE i ABSTRACT ' i i ACKNOWLEDGMENTS , i i i TABLE OF CONTENTS .... i v LIST OF FIGURES..... . v i LIST OF TABLES v i i LIST OF APPENDIX TABLES v i i i SUMMARY OF SYMBOLS USED i x CHAPTER I. . . 1 INTRODUCTION AND THEORETICAL BACKGROUND 1 1. I n t r o d u c t i o n . . . . 1 2. T h e o r e t i c a l Background 3 CHAPTER I I 1 3 THEORETICAL ASPECTS OF GAUNTLET FISHERY PRODUCTION FUNCTION 1 3 CHAPTER I I I . 19 ESTIMATION OF THE DAILY PRODUCTION FUNCTION 19 1. The F i s h e r y . 19 2. The Data 24 3. E c o n o m e t r i c Problems 28 4. D i s c u s s i o n o f R e g r e s s i o n R e s u l t s 31 CHAPTER IV 36 THE SEASONAL PRODUCTION FUNCTION 36 1. C o n s t r u c t i o n of S e a s o n a l F u n c t i o n 36 2. D i s c u s s i o n o f t h e S e a s o n a l P r o d u c t i o n F u n c t i o n 3 ^ 3. E s t i m a t i o n o f D i s s i p a t e d Rents . 46 V Page 4. C o n c l u s i o n 52 BIBLIOGRAPHY „ . .... 54 APPENDIX 56 v i LIST OF FIGURES F i g u r e Page 1 B r i t i s h Columbia S t a t i s t i c a l A r e a s 3, 4, and 5 20 2 Skeena R i v e r Sockeye T i m i n g E x p r e s s e d as C u m u l a t i v e P e r c e n t a g e of T o t a l S t o c k by End of Week. 23 3 S e a s o n a l C a t c h - E f f o r t R e l a t i o n s h i p U s i n g E q u a t i o n 10... 42 4 I s o - C a t c h Curves at D i f f e r e n t L e v e l s o f T o t a l Boat Days and Numbers of Boats 45 5 I s o - C a t c h Curves at D i f f e r e n t L e v e l s of Boat Days Per Boat and Numbers of Boats '45" 6 T o t a l S e a s o n a l C o s t s and Revenues as a F u n c t i o n o f F l e e t S i z e , f o r a G i v e n A l l o w a b l e C a t c h 47" 7 The C r u t c h f i e l d - P o n t e c o r v o P r o d u c t i o n F u n c t i o n and E s t i m a t e d Rents 49 8 Average S e a s o n a l C o s t s and Revenues P e r Boat as a F u n c t i o n o f F l e e t S i z e 5 i V l l LIST OF TABLES T a b l e Page I Skeena R i v e r (Area 4) Sockeye Salmon C a t c h , Escapement, and S t o c k S i z e , 1951-1969 21 I I D a i l y P r o d u c t i o n F u n c t i o n E s t i m a t e s 32 I I I D a i l y S t o c k S i z e and C a t c h o f Two H y p o t h e t i c a l G i l l n e t F i s h e r i e s , Showing M u l t i p l e Day S t o c k Removal E f f e c t •••• 3 9 IV D a i l y S t o c k S i z e and C a t c h of Three H y p o t h e t i c a l G i l l n e t F i s h e r i e s , Showing E f f e c t o f I n t r a -S e a s o n a l P e a k i n g 41 LIST OF APPENDIX TABLES T a b l e Page A-I D a i l y C a t c h , Numbers of B o a t s , S t o c k S i z e , and Dummy V a r i a b l e s Used i n R e g r e s s i o n E q u a t i o n s 57 A - I I D a i l y C a t c h , Average D a i l y C a t c h Per Boat, and M a r g i n a l D a i l y Boat C a t c h , at D i f f e r e n t D a i l y S t o c k S i z e and F l e e t S i z e L e v e l s , O b t a i n e d by U s i n g E q u a t i o n 4 59 A - I I I D a i l y C a t c h , Average D a i l y C a t c h Per Boat, and M a r g i n a l D a i l y Boat C a t c h , at D i f f e r e n t D a i l y S t o c k S i z e and F l e e t S i z e L e v e l s , O b t a i n e d by U s i n g E q u a t i o n 5 A-IV D a i l y C a t c h , Average D a i l y C a t c h Per Boat, and M a r g i n a l D a i l y Boat C a t c h , a t D i f f e r e n t D a i l y S t o c k S i z e and F l e e t S i z e L e v e l s , O b t a i n e d by U s i n g E q u a t i o n 9. A-V D a i l y C a t c h , Average D a i l y C a t c h P e r Boat, and M a r g i n a l D a i l y Boat C a t c h , at D i f f e r e n t D a i l y S t o c k S i z e and F l e e t S i z e L e v e l s , O b t a i n e d by U s i n g E q u a t i o n 10 A-VI S e a s o n a l C a t c h at D i f f e r e n t L e v e l s of S e a s o n a l S t o c k S i z e , F l e e t S i z e , and F i s h i n g Days P e r Week, D e r i v e d from t h e E q u a t i o n 10 D a i l y P r o d u c t i o n F u n c t i o n E s t i m a t e . . . SUMMARY OF SYMBOLS USED F i s h P o p u l a t i o n V a r i a b l e s N - the number of f i s h i n t h e f i s h i n g a r e a . N Q - t h e number of f i s h i n t h e f i s h i n g a r e a a t t h e s t a r t of t h e season. - t h e number of f i s h i n t h e f i s h i n g a r e a a t t h e s t a r t o f day t . N - average (over time) number o f f i s h i n t h e f i s h -i n g a r e a . S - t h e number of f i s h m i g r a t i n g t h r o u g h t h e f i s h -i n g area i n a g i v e n y e a r (the t o t a l r u n o f f i s h ) . S^ . - t h e number of f i s h e n t e r i n g t h e f i s h i n g a r e a d u r i n g day t . E - t h e number of f i s h e s c a p i n g c a p t u r e and e m i g r a t -i n g out of t h e f i s h i n g a r e a (escapement). E^ - t h e escapement i n numbers d u r i n g day t . C - t h e number of f i s h caught, on a d a i l y o r s e a s o n a l b a s i s , as a p p l i c a b l e . C.J. - t h e number o f f i s h caught d u r i n g day t . C* - " e q u i l i b r i u m c a t c h . " O t h e r V a r i a b l e s t h e number o f b o a t s f i s h i n g . the number o f bo a t s downstream f r o m boat i on day t . B i B t the number of days a f i s h t a k e s t o pass t h r o u g h t h e f i s h i n g a r e a . l e n g t h o f season i n days; time spent f i s h i n g , t h e number of f i s h days p e r week a l l o w e d , average c a t c h a b i l i t y c o e f f i c i e n t . = 1-p • c a t c h a b i l i t y c o e f f i c i e n t o f i ^ n boat on day t . dummy v a r i a b l e t a k i n g on t h e v a l u e o f 1 i f s t o c k s i z e b u i l d i n g t o a peak, 0 o t h e r w i s e , dummy v a r i a b l e t a k i n g on t h e v a l u e o f 1 i f f i r s t day o f t h e f i s h e r y , 0 o t h e r w i s e . c o n s t a n t i n Cobb-Douglas e q u a t i o n , exponent o f i n Cobb-Douglas e q u a t i o n , exponent of i n Cobb-Douglas e q u a t i o n , e s t i m a t e d parameters. CHAPTER I INTRODUCTION AND THEORETICAL BACKGROUND I n t r o d u c t i o n The e a r l y a r t i c l e s on f i s h e r i e s economics by Gordon (1954) and S c o t t (1955) and by C r u t c h f i e l d and Z e l l n e r (1963) d i s c u s s e d the d i s s i p a -t i o n of economic r e n t under c o n d i t i o n s ' o f f r e e e n t r y . The models d e v e l o p e d by t h e s e a u t h o r s were most r e a d i l y a p p l i c a b l e t o demersal f i s h e r i e s , and showed t h a t under f r e e e n t r y r e g i m e s , e f f o r t would i n c r e a s e beyond t h a t r e q u i r e d f o r maximum s u s t a i n a b l e p h y s i c a l y i e l d t o t h e p o i n t where r e t u r n s t o c a p i t a l and l a b o u r j u s t e q u a l l e d o p p o r t u n i t y c o s t s . The l e v e l of e f f o r t which would r e t u r n a maximum r e n t was below t h a t f o r maximum s u s t a i n a b l e y i e l d , and t o a c h i e v e t h i s optimum l e v e l , a- system o f t a x e s on l a n d i n g s , o r the a p p r o p r i a t i o n of t h e common p r o p e r t y r e s o u r c e i n t o t h e hands of a " s o l e owner," would be r e q u i r e d . The e a r l i e r b i o - e c o n o m i c models were r e f i n e d by Smith (1969). W h i l e t h e models by Gordon, S c o t t , and C r u t c h f i e l d and Z e l l n e r d i s c u s s e d e q u i l i b r i u m o r s t e a d y s t a t e s o l u t i o n s , Smith a n a l y s e d t h e dynamics of t h e model. Smith shows b a s i c a l l y t h a t e q u i l i b r a t i n g f o r c e s e x i s t i n t h e model, but s t a b i l i t y depends on t h e r a t e s o f change o f e f f o r t and o f t h e s i z e o f t h e e x p l o i t e d f i s h s t o c k . • ' Important t o Smith's a n a l y s i s , and l a t e r work by Southey (1969, u n p u b l i s h e d ) was t h e e x p l i c i t i n t r o d u c t i o n i n t o t h e model of a s h o r t - r u n p r o d u c t i o n f u n c t i o n , r e l a t i n g i n s t a n t a n e o u s c a t c h t o i n s t a n t a n e o u s f i s h -i n g e f f o r t and s t o c k s i z e . T h i s f u n c t i o n i s the c r u c i a l t i e - i n between b i o l o g i c a l f a c t o r s and economic v a r i a b l e s . The purpose of t h i s paper i s t o d e v e l o p a s h o r t - r u n p r o d u c t i o n f u n c t i o n r e l a t i n g as much as p o s s i b l e t o t h e s p e c i f i c s of an a c t u a l f i s h e r y , namely, the Skeena R i v e r sockeye salmon g i l l n e t f i s h e r y . A s e a s o n a l p r o d u c t i o n f u n c t i o n f o r t h i s f i s h e r y w i l l be c o n s t r u c t e d on t h e b a s i s o f e s t i m a t e d d a i l y r e l a t i o n s h i p s between c a t c h , e f f o r t and s t o c k s i z e . T h i s p r o d u c t i o n f u n c t i o n w i l l r e l a t e t o t h e c u r r e n t f i s h e r i e s management r e g u l a t i o n s o b t a i n i n g f o r t h e Skeena R i v e r and o t h e r such f i s h e r i e s on t h e P a c i f i c C o a s t . Some a s p e c t s of the Skeena R i v e r f i s h e r y , t y p i c a l o f o t h e r anad-romous f i s h e r i e s , a r e such t h a t the methods of a n a l y s i s d e v e l o p e d f o r demersal f i s h e r i e s a r e not a p p l i c a b l e . The unique a s p e c t s of such a f i s h e r y a r e t h e i n t r a - s e a s o n a l p e a k i n g n a t u r e of the run, and t h e day-t o - d a y management of f i s h i n g open and c l o s e d t i m e s . The development of a s e a s o n a l p r o d u c t i o n i s of i n c r e a s i n g impor-t a n c e . In B r i t i s h Columbia now, and i n t h e U n i t e d S t a t e s i n t h e f o r e s e e -a b l e f u t u r e , f i s h i n g r e g u l a t o r y a g e n c i e s a p p l y i n g e n t r y c o n t r o l measures i n an e f f o r t t o improve r e t u r n s t o f i s h e r m e n and r e a l i z e an "economic r e n t " from t h e i n d u s t r y , s h o u l d f u l l y u n d e r s t a n d t h e s e a s o n a l p r o d u c t i o n p r o c e s s . A p r o d u c t i o n f u n c t i o n , a l o n g w i t h many o t h e r s , c o u l d be used, say, i n a s i m u l a t i o n model t o determine the o p t i m a l number and d i s t r i b u -t i o n of u n i t s of f i s h i n g g e a r r e q u i r e d t o h a r v e s t r e t u r n i n g runs of salmon throughout t h e season. In a r e c e n t book, C r u t c h f i e l d and P o n t e c o r v o (1970) e s t i m a t e t h e magnitude o f r e n t s d i s s i p a t e d t h r o u g h f r e e e n t r y regimes i n B r i s t o l Bay, A l a s k a , and Puget Sound, Washington. These a n a l y s e s a r e c a r r i e d out w i t h o u t s p e c i f i c r e f e r e n c e t o t h e s e a s o n a l p r o d u c t i o n p r o c e s s , o r con-c r e t e knowledge of the shape of the s e a s o n a l p r o d u c t i o n f u n c t i o n . The shape of the p r o d u c t i o n f u n c t i o n d e v e l o p e d f o r t h e Skeena R i v e r w i l l be u s ed t o i n d i c a t e t h e d i r e c t i o n o f b i a s i n t h e C r u t c h f i e l d - P o n t e c o r v o e s t i m a t e s . T h i s paper w i l l f i r s t of a l l a n a l y s e t h e d a i l y p r o d u c t i o n f u n c t i o n from a t h e o r e t i c a l p o i n t of view; next an attempt w i l l be made t o e s t i -mate t h i s f u n c t i o n u s i n g Skeena R i v e r d a t a ; a s e a s o n a l p r o d u c t i o n f u n c -t i o n w i l l be g e n e r a t e d u s i n g t h e d a i l y f u n c t i o n ; t h e g e n e r a l p r o p e r t i e s of t h i s s e a s o n a l f u n c t i o n w i l l t h e n be reviewed; f i n a l l y , u s i n g t h e s e g e n e r a l p r o p e r t i e s , t h e d i r e c t i o n o f b i a s i n t h e C r u t c h f i e l d - P o n t e c o r v o e s t i m a t e s w i l l be i n d i c a t e d . T h e o r e t i c a l Background The purpose of t h i s s e c t i o n i s t o r e v i e w t h e most i m p o r t a n t " p r o duc-t i o n f u n c t i o n s " d e v e l o p e d t o date. We compare and c o n t r a s t f o u r major " p r o d u c t i o n f u n c t i o n s , " namely, the s t a n d a r d B e v e r t o n and H o l t (1957) f o r m u l a t i o n , t h e Smith model, and r e c e n t c o n t r i b u t i o n s by B r a d l e y (1970) and by Southey. We w i l l a l s o r e v i e w t h e S c h a e f e r model (1954), mainly f o r h i s t o r i c a l i n t e r e s t , and f o r i t s i n f l u e n c e on t h e development of " e q u i l i b r i u m " p r o d u c t i o n f u n c t i o n s . These p r o d u c t i o n f u n c t i o n s a r e a p p l i c a b l e t o non-anadromous f i s h e r i e s , and a r e e i t h e r i n s t a n t a n e o u s and/or i n t r a - s e a s o n a l , o r i n t e r -s e a s o n a l . T h i s c o n t r a s t s w i t h t h e model d e v e l o p e d l a t e r i n t h i s paper where t h e p r o d u c t i o n f u n c t i o n i s d e v e l o p e d on a d a i l y ( i n s t a n t a n e o u s ) b a s i s , and extended t o a s e a s o n a l model by t a k i n g i n t o account weekly r e g u l a t o r y measures. We s h a l l d i s c u s s t h e B e v e r t o n and H o l t ( h e r e i n a f t e r B-H) model f i r s t , s i n c e i t i s t h e s t a r t i n g p o i n t , and p r i n c i p a l b i o l o g i c a l r e f -e r e n c e , f o r most b i o - e c o n o m i c models. The B-H models a r e d e s c r i b e d as dynamic p o o l models ( S c h a e f e r , 1968), s i n c e B-H t a k e i n t o account the dynamic i n t e r a c t i o n s between r e c r u i t m e n t , growth, n a t u r a l m o r t a l -i t y and f i s h i n g m o r t a l i t y , by a n a l y s i n g t h e fundamental elements o f t h e f i s h l i f e c y c l e . The B-H approach i s o f i n t e r e s t i n t h i s s t u d y s i n c e i t g i v e s i n s i g h t i n t o the a c t u a l f i s h i n g p r o c e s s . We s h a l l use t h e f o l l o w i n g symbols: N = the s i z e of t h e s t o c k of f i s h under e x p l o i t a t i o n measured i n numbers o f f i s h . B = t h e number of f i s h i n g u n i t s (assumed homogeneous). F = i n s t a n t a n e o u s r a t e of f i s h i n g m o r t a l i t y , a = annual o r s e a s o n a l r a t e of f i s h i n g m o r t a l i t y . p = p r o p o r t i o n of t o t a l s t o c k t a k e n by one u n i t of f i s h i n g g e a r ( c a t c h a b i l i t y c o e f f i c i e n t ) . . . q = 1 - p C = c a t c h . t s u b s c r i p t r e f e r s t o time t i s u b s c r i p t r e f e r s t o t h e i t h v e s s e l F o r s i m p l i c i t y , we w i l l c o n s i d e r o n l y f i s h e r i e s where t h e r e i s no r e c r u i t m e n t , and no n a t u r a l m o r t a l i t y ( R l c k e r , 1958, Type 1A f i s h e r y ) . We a l s o d e f i n e , f o l l o w i n g and m o d i f y i n g B-H ( B e v e r t o n and 5 H o l t , 1957, p. 8 9 ) , t h e f o l l o w i n g c l a s s e s of f i s h e r y : C l a s s 1 - C o m p e t i t i v e . The c a t c h o f a v e s s e l i s i n f l u e n c e d by how much has been caught p r e v i o u s l y by t h e f l e e t as a whole, i n c l u d i n g t h e v e s s e l i n q u e s t i o n . C l a s s 2 - I n t e r a c t i v e . The. c a t c h e s of some v e s s e l s a r e i n f l u e n c e d by t h e c a t c h o f o t h e r v e s s e l s , but t h e c o n v e r s e i s not t r u e . C l a s s 3 - N o n - C o m p e t i t i v e . The c a t c h o f one v e s s e l does not i n f l u e n c e t h e c a t c h of a n o t h e r v e s s e l . An example o f a c o m p e t i t i v e f i s h e r y might be t h e P a c i f i c H a l i b u t f i s h e r y . An i n t e r a c t i v e f i s h e r y example would be a t y p i c a l e s t u a r i a l P a c i f i c salmon g i l l n e t f i s h e r y , and a n o n - c o m p e t i t i v e f i s h e r y would be an i d e a l i z e d v e r s i o n o f t h e salmon s e i n e f i s h e r y a t t h e e n t r a n c e t o t h e S t r a i t o f Juan de Fuca on t h e " b l u e l i n e . " We w i l l c o n c e n t r a t e f o r t h e moment on t h e c o m p e t i t i v e f i s h e r y , say, a t r a w l f i s h e r y . F o l l o w i n g B e v e r t o n and H o l t (1957, pp. 90-91), we assume: (1) N i s u n i f o r m l y d i s t r i b u t e d o v e r t h e f i s h i n g a r e a ; (2) t h e v e s s e l can be r e g a r d e d as c a t c h i n g d u r i n g each h a u l a c e r t a i n p r o p o r t i o n of t h e f i s h i n i t s p a t h ; (3) t h e f i s h r e d i s t r i b u t e them-s e l v e s between h a u l s . Then, i n a s i n g l e v e s s e l f i s h e r y , t h e f i r s t h a u l w i l l y i e l d a 1 c e r t a i n p r o p o r t i o n p of the t o t a l s t o c k at the b e g i n n i n g o f t h a t h a u l , N. The f i r s t h a u l w i l l y i e l d pN, l e a v i n g qN. A f t e r t h e f i s h r e d i s t r i b -2 u t e t h e m s e l v e s , t h e second h a u l w i l l y i e l d pqN, l e a v i n g q N. On t h e r —1 r r t h h a u l t h e number caught w i l l be pq N, and q N w i l l remain. I f each h a u l i s of t h e same l e n g t h and e f f i c i e n c y , p and q a r e assumed c o n s t a n t . L e t t i n g q = e ^ Then r a i s i n g q t o t h e r t h power f o r r h a u l s r - b r g i v e s q = e We l e t r be assumed c o n s t a n t , so b r = u and q = e~u/r - u / r , so q = e = 1-p o r - u / r = In (1-p) I - 1 and when p i s v e r y s m a l l , (I - 1) can be a p p r o x i m a t e d by + u / r = (p I f t h e time o f t h e h a u l i s s h o r t r e l a t i v e t o t h e ti m e i t t a k e s t o do r h a u l s , t h e n t h e r a t e o f change of N w i t h r e s p e c t t o time i s g i v e n by dN dt I F " r -pN = H i '• N I f t h e r e a r e B v e s s e l s f i s h i n g s i m u l t a n e o u s l y , and t h e f i s h do not r e d i s t r i b u t e t h e m s e l v e s d u r i n g h a u l s , t h e n d u r i n g any g i v e n h a u l , 1 Another d e f i n i t i o n o f p i s t h e p r o b a b i l i t y o f a g i v e n f i s h en-c o u n t e r i n g a g i v e n u n i t of f i s h i n g g e a r . t h e c a t c h i s BpN so . N but w i t h a f l e e t of g i v e n s i z e B, and g i v e n e f f i c i e n c y , Bu can be assumed c o n s t a n t , r L e t t i n g Bu = F j t h e n r F i s known as t h e i n s t a n t a n e o u s r a t e of f i s h i n g m o r t a l i t y . B-H a l s o show t h a t f o r a s t r i c t l y i n t e r a c t i v e f i s h e r y (say a g a u n t l e t f i s h e r y w i t h u n i t s o f gear s t r u n g a l o n g a r i v e r bank) = -B l o g (1-p) N = -BpN 1 - 3 and f o r a n o n - c o m p e t i t i v e f i s h e r y = l o g (1-Bp) and i n t h i s case C = BpN Summarizing t h e n , f o r c o m p e t i t i v e and i n t e r a c t i v e f i s h e r i e s , F i s l i n e a r l y r e l a t e d t o B, and l o g a r i t h m i c a l l y r e l a t e d t o p. T h i s l a t t e r r e l a t i o n s h i p i s o f t e n a p p r o x i m a t e d by a l i n e a r r e l a t i o n s h i p . We th e n have t h e f o l l o w i n g p r o d u c t i o n f u n c t i o n s ( y i e l d e q u a t i o n s ) : 1. C o m p e t i t i v e f i s h e r y ) s o l v i n g d i f f e r e n t i a l e q u a t i o n s ) 2. I n t e r a c t i v e f i s h e r y ) I - 2 o r I - 3. -pBt c t = N o "No 3. N o n - c o m p e t i t i v e f i s h e r y C t = B P N t I t w i l l a l s o be n o t i c e d t h a t c a t c h p e r u n i t e f f o r t , C/B, i s not always p r o p o r t i o n e d t o N q (or an i n d e x o f abundance) except i n the case of a n o n - c o m p e t i t i v e f i s h e r i e s : E m ploying t h e l o g a r i t h m a p p r o x i m a t i o n u s ed i n e q u a t i o n (I - 1) — F — F C l = N o ~ N o e = ( 1 - e J N °1 T - F o r = 1 -e 1 0 1 1 - — = e V m ( 1 ^ > - - F = --pa N o c l and o n l y when — , o r a, t h e annua l e x p l o i t a t i o n r a t e , i s v e r y No-s m a l l C _ do we have — - PN B ° C T h i s use o f —— as an i n d e x o f i n i t i a l abundance u s e s t h e equa-B t i o n 1 a p p r o x i m a t i o n t w i c e , and a i s not l i k e l y t o be s m a l l i n a mature °1 f i s h e r y . T h e r e f o r e , o n l y when the removal r a t e i s s m a l l i s — a n B a c c u r a t e i n d e x o f i n i t i a l abundance, o r i n i t i a l s t o c k s i z e . Thus i n t e s t f i s h e r i e s , and i n n o n - c o m p e t i t i v e f i s h e r i e s do we have a s i t u a t i o n C where — i s p r o p o r t i o n e d t o i n i t i a l s t o c k s i z e . B-H show, however, B C-, t h a t _ i s p r o p o r t i o n a l t o average (over t i m e ) s t o c k s i z e , (1957, pp. B 39-41). The next p r o d u c t i o n f u n c t i o n we s h a l l examine, t h a t o f S c h a e f e r (1954), i s o f i n t e r e s t h i s t o r i c a l l y and because i t t y p i f i e s many o t h e r s . 9 The h i s t o r i c a l i n t e r e s t a r i s e s because t h e S c h a e f e r paper, contempora-n e o u s l y w i t h t h e Gordon-Scott a r t i c l e s , f o reshadowed t h e dynamic models of Smith. Smith's work i s perhaps more e l e g a n t , but t h e e s s e n -t i a l f e a t u r e s of t h e Smith model a r e i n h e r e n t i n t h e S c h a e f e r approach. The S c h a e f e r approach i s i m p o r t a n t a l s o as a comparison t o B-H. The l a t t e r term t h e methodolog of S c h a e f e r and o t h e r s " n o n - a n a l y t i c a l " , (Beverton and H o l t , 1957, pp. 329-330), s i n c e t h e s e p a r a t e p r o c e s s e s u n d e r l y i n g the b e h a v i o u r of a p o p u l a t i o n a r e not d i s t i n g u i s h e d and r e p r e s e n t e d m e t h e m a t i c a l l y . R a t h e r , a "macro" a p p r o a c h i s used, based on a l o g i s t i c growth model. R e c e n t l y , however, improvements have been made i n t h e b a s i c model t o t a k e i n t o a c c o u n t t h e " e n g i n e e r i n g " a s p e c t s o f t h e f i s h i n g p r o c e s s , and t h e s e improvements have been a p p l i e d t o p e l a g i c and demersal f i s h e r i e s ( C a r l s o n , 1969; B e l l , e t a l , 1970). W h i l e the B-H p r o d u c t i o n f u n c t i o n examines t h e f i s h i n g p r o c e s s w i t h i n t h e f i s h i n g season ( i . e . , an i n t r a - s e a s o n a l p r o d u c t i o n model), S c h a e f e r examines an i n t e r - s e a s o n a l p r o d u c t i o n p r o c e s s . He p o s t u l a t e s a v e r age c a t c h p e r u n i t e f f o r t e q u a l t o t h e c a t c h a b i l i t y c o e f f i c i e n t , p, t i m e s t h e average s t o c k s i z e . By i n t e r p o l a t i o n , between s u c c e s s i v e y e a r s , he i s a b l e t o compute i n i t i a l a n n u a l s t o c k s i z e s , and by assum-i n g p o p u l a t i o n s i n e q u i l i b r i u m ( f i s h i n g m o r t a l i t y r a t e = r e c r u i t m e n t p l u s growth minus n a t u r a l m o r t a l i t y r a t e s ) he i s a b l e t o e s t i m a t e e q u i l i b r i u m c a t c h c u r v e s . The p o i n t h e r e i s t h a t t h e S c h a e f e r model does not examine t h e f i s h i n g p r o c e s s w i t h i n t h e season, where ti m e spent f i s h i n g by the f l e e t i s an i m p o r t a n t economic v a r i a b l e . We w i l l see l a t e r t h a t , i n d i r e c t l y a t l e a s t , t h i s i s one o f t h e i m p o r t a n t f e a t u r e s o f t h e Smith and B r a d l e y a p p r o a c h e s . C r u t c h f i e l d and Z e l l n e r ( h e r e i n a f t e r C-Z) and Z e l l n e r (1962) e x p l i c i t l y i n t r o d u c e d a p r o d u c t i o n f u n c t i o n , r e l a t i n g c a t c h t o s t o c k s i z e and e f f o r t , a " g e n e r a l i z e d i n p u t . " C-Z p o s t u l a t e a Cobb-Douglas t y p e p r o d u c t i o n f u n c t i o n w i t h c o n s t a n t r e t u r n s t o s c a l e and d i m i n i s h -i n g m a r g i n a l p r o d u c t s , and compare t h i s t o t h e S c h a e f e r f u n c t i o n ( i n c r e a s i n g r e t u r n s t o s c a l e ) . C-Z do not e x p l a i n why t h e i r p r o d u c-t i o n f u n c t i o n s h o u l d conform t o t h e law of d i m i n i s h i n g r e t u r n s , and do not m a t h e m a t i c a l l y a n a l y s e t h e i m p l i c a t i o n s of such a p r o d u c t i o n f u n c t i o n . The C-Z h y p o t h e s i s concerning d i m i n i s h i n g r e t u r n s , i s , how-ev e r , i m p o r t a n t . The C-Z p r o d u c t i o n f u n c t i o n i s i m p l i c i t l y , a t l e a s t , an i n t r a - s e a s o n a l r e l a t i o n s h i p . The S c h a e f e r a p p r o a c h i s i n t e r -s e a s o n a l . The C-Z p r o d u c t i o n f u n c t i o n , on an i n t r a - s e a s o n a l b a s i s , a l l o w s f o r a n a l y s i s of n o n - s t a t i o n a r y s o l u t i o n s . The S c h a e f e r f u n c -t i o n : Cj. = p B t , (N^ = average s t o c k s i z e ) does n o t . W i t h t h i s p r o d u c t i o n f u n c t i o n and a l o g i s t i c growth f u n c t i o n , S c h a e f e r (1954, p. 34) shows, however, t h a t : * _JL = a -b B + Bt where a and b a r e parameters r e l a t e d t o t h e V e r h u l s t - P e a r l l o g i s t i c c u r v e and where C^ a r e e q u i l i b r i u m c a t c h e s . T h e r e a r e two problems w i t h t h i s a p p r o a c h — f i r s t , e q u i l i b r i u m c a t c h e s a r e almost n e v e r o b s e r v e d , so o b s e r v e d c a t c h and e f f o r t data must be m a n i p u l a t e d , and 2 s e c o n d l y t h e " p r o d u c t i o n f u n c t i o n " i s not at a l l r e a l i s t i c . 2 See B e l l , et a l (1970) f o r a d i s c u s s i o n o f t h i s a pproach, and s u g g e s t i o n s f o r i m p r o v i n g and e s t i m a t i n g the model. S c h a e f e r t h e n d e r i v e d a b e h a v i o u r a l model f r o m which i t i s pos-s i b l e t o a r r i v e a t an e x p r e s s i o n f o r dN, but t h e i n t r a - s e a s o n a l dB t e c h n o l o g i c a l r e l a t i o n s h i p , as p r o p o s e d by C-Z, i s n o t e x p l o r e d i n depth by S c h a e f e r . The C-Z model i s e x p l i c i t l y i n terms o f t e c h n o l o g i c a l p r o d u c t i o n ; t h e S c h a e f e r model i s i n terms o f " p o p u l a t i o n dynamics." Smith e s s e n t i a l l y f o r m a l i z e s and makes dynamic p r e v i o u s models. We w i l l c o n c e r n o u r s e l v e s h e r e w i t h h i s p r o d u c t i o n f u n c t i o n . The Smith f o r m u l a t i o n i s as f o l l o w s : the r a t e o f change i n biomass i s a f u n c t i o n o f t h e s t o c k s i z e , mesh s i z e , t h e number o f f i s h i n g v e s s e l s , and t h e " c a t c h r a t e " of each v e s s e l . Under o u r assumptions ( R i c k e r Type 1A f i s h e r y ) , w i t h no mesh s i z e c o n s i d e r a t i o n s : * I = f <B . x ) , dt where x i s the c a t c h o f f i s h per v e s s e l p e r u n i t t i m e . (Smith, 1969, p. 183). In a d d i t i o n , x i s s u b j e c t t o t h e f o l l o w i n g : (1) Resource S t o c k E x t e r n a l i t i e s - t h e c o s t o f a f i s h i n g v e s s e l ' s c a t c h d e c r e a s e s as t h e p o p u l a -t i o n i n c r e a s e s . (2) Crowding E x t e r n a l i t i e s - c o n c e n t r a t e d f i s h p o p u l a t i o n s cause v e s s e l c o n g e s t i o n and thu s i n c r e a s e v e s s e l o p e r a t i n g c o s t s f o r any g i v e n c a t c h . N o t i c e t h a t t h e Smith t e r m i n o l o g y i s i n terms o f c o s t s p e r f i s h , n ot c a t c h p e r u n i t o f e f f o r t . In t h e Smith model, f i s h i n g e f f o r t i s measured by o p e r a t i n g c o s t s , and hence, i n t r a - s e a s o n a l l y , t i m e spent f i s h i n g i s an i m p l i c i t v a r i a b l e . The i m p o r t a n t a s p e c t o f t h e Smith model i s t h e p o s s i b i l i t y (not t r e a t e d i n h i s examples) o f a non-p r o p o r t i o n a l r e l a t i o n s h i p between c a t c h and e f f o r t . The Smith model b r i n g s t o g e t h e r t h e p o p u l a t i o n dynamics and th e t e c h n o l o g i c a l e x t e r n a l i t i e s o f t h e f i s h i n g p r o c e s s . In a d d i t i o n , Smith b r i n g s f o r w a r d a n o t h e r e x t e r n a l i t y — t h e g e a r s e l e c t i v i t y e x t e r n a l i t y , where mesh s i z e o r o t h e r such gear s e l e c t i v i t y f a c t o r i n f l u e n c e s t h e growth o f t h e f i s h p o p u l a t i o n , i n t h e sense t h a t growth, r e c r u i t m e n t and n a t u r a l m o r t a l i t y might be d e n s i t y dependent. Southey (1969, u n p u b l i s h e d , pp. 12-17) a l s o has a p r o d u c t i o n f u n c t i o n i n h i s b i o - e c o n o m i c model. The f u n c t i o n h o l d s between seasons as w e l l as w i t h i n each season. Southey c l a i m s t h i s f u n c t i o n i s " w e l l behaved," i . e . , t h e law of v a r i a b l e p r o p o r t i o n s o b t a i n s ; t h e r e i s a phase of c o n s t a n t r e t u r n s t o s c a l e , and e v e n t u a l l y d i m i n i s h i n g m a r g i n a l r e t u r n s t o e f f o r t . Southey i n p o s t u l a t i n g t h i s shape o f the p r o d u c t i o n f u n c t i o n r e f e r s t o p o s s i b l e e x t e r n a l economies of f i s h f i n d i n g and d i s e c o n o m i e s of o v e r c r o w d i n g . Southey uses d i f f e r e n c e e q u a t i o n s , i n s t e a d o f d i f f e r e n t i a l e q u a t i o n s ; t h i s i s more p r a c t i c a l t h a n Smith, s i n c e i t a l l o w s him t o a p p l y t h e model on a s e a s o n a l b a s i s . Bradley. (1970, pp. 34-37) e x p l i c i t l y i n t r o d u c e d f i s h i n g time as an independent v a r i a b l e i n the i n t r a - s e a s o n a l p r o d u c t i o n f u n c t i o n : i . e . C = C (B, T, N q; p) where T i s time spent f i s h i n g B r a d l e y t h e n shows how t h e p r o d u c t i o n f u n c t i o n d i f f e r s i n d i f -f e r e n t t y p e s of f i s h e r i e s — c o m p e t i t i v e ( B r a d l e y Type A) and non-c o m p e t i t i v e f i s h e r i e s (Type B ) . (See t h e B-H d i s c u s s i o n a b o v e ) . B r a d l e y does not t a k e i n t o a c c o u n t , i n t h e a p p l i c a t i o n o f h i s p r o d u c t i o n f u n c t i o n t o a salmon f i s h e r y , t h e c o m p l i c a t i o n of t h e i n t r a -s e a s o n a l p e a k i n g of t h e s t o c k of f i s h under e x p l o i t a t i o n , and t h e day-t o - d a y management s t r a t e g i e s c u r r e n t l y used. As we s h a l l see below, t h e s e f a c t o r s a r e v e r y i m p o r t a n t i n t h e s e a s o n a l p r o d u c t i o n p r o c e s s . CHAPTER I I THEORETICAL ASPECTS OF GAUNTLET FISHERY PRODUCTION FUNCTION In t h i s c h a p t e r we d e v e l o p a t e s t a b l e h y p o t h e s i s c o n c e r n i n g t h e r e l a t i o n s h i p between d a i l y c a t c h , e f f o r t , and s t o c k s i z e f o r an anad-romous f i s h e r y , d e p a r t i n g from t h e " p r o d u c t i o n f u n c t i o n s " d i s c u s s e d u n i t o f time (a day) and by a l l o w i n g p,the c a t c h a b i l i t y c o e f f i c i e n t o f B-H,to be a v a r i a b l e . We assume a " g a u n t l e t " t y p e f i s h e r y , w i t h f i s h moving i n and t h r o u g h the f i s h i n g a r e a , and out o f t h e a r e a , a t a c o n s t a n t speed. T h e r e i s no n a t u r a l m o r t a l i t y . F i s h i n g e f f o r t , B, w i l l be d e f i n e d as t h e number of f i s h i n g b o a t s f i s h i n g p e r day. We d e f i n e t h e f i s h i n g power, > of each v e s s e l as t h e p r o p o r -t i o n o f t h e t o t a l s t o c k p r e s e n t h a r v e s t e d by one u n i t o f f i s h i n g e f f o r t a t time t . In o t h e r words, i f v e s s e l i c a t c h e s C a t time t , i n t h e p r e v i o u s c h a p t e r by c o n c e r n i n g o u r s e l v e s w i t h a d i s c r e t e s m a l l i t t h e n ^ p t C = i t II - 1 With one boat f i s h i n g : dN-= - i * t N t I I - 2 And w i t h B b o a t s f i s h i n g s i m u l t a n e o u s l y : I I - 3 Where i s t h e average f i s h i n g power of a l l v e s s e l s . Now equatiozi 1 g i v e s t h e d e f i n i t i o n o f p i t C 1 * i . e . p = _ — I I - 1 i t N t E q u a t i o n 1 i s an i d e n t i t y — t h e problem now i s t o o b t a i n a d e t e r m i n i s t i c r e l a t i o n s h i p between p and o t h e r r e l e v a n t v a r i a b l e s . i t We p o s t u l a t e t h e r e l a t i o n s h i p : p = p ( B N ) I I - 4 i t i t t Where R i s the number of v e s s e l s between the mouth o f the i t g a u n t l e t and v e s s e l i , B i s assumed t o remain c o n s t a n t t h r o u g h o u t i t t h e day. We f u r t h e r p o s t u l a t e n e g a t i v e m a r g i n a l r a t e s of change of p i t w i t h r e s p e c t t o .B^ . i t 3 i P t y < 0 I I - 5 M t and t o s t o c k s i z e , N I I B N t In o t h e r words, the f i s h i n g power of each v e s s e l a t time t i s r e l a t e d t o t h e number of v e s s e l s f i s h i n g , and t h e s i z e of t h e s t o c k . Now combining e q u a t i o n s I I - 2 and I I - 4, we have: B d t ' B • We now c o n c e i v e of an average f i s h i n g power: p = p (N t , B t ) ' I I - 8 15 T h i s y i e l d s : ( o f ) pB = " ( P <W Bt > Nt T h i s d i f f e r s from the s t a n d a r d B e v e r t o n and H o l t t y p e f o r m u l a -t i o n : dN \ dt / = - (p B ) N pB s i n c e i n our e q u a t i o n I I - 9 t h e f i s h i n g power of each v e s s e l v a r i e s as N arid B v a r i e s . I f o u r time i n t e r v a l i s s h o r t enough, say a day, then e q u a t i o n I I - can be a p p r o x i m a t e d by: ^ - / ^ N = -C = -p (B,N) . B.N I I - 10 dt E q u a t i o n I I - 10 t h e n i s our p r o d u c t i o n f u n c t i o n . C e n t r a l t o o u r f o r m u l a t i o n i s t h e b e h a v i o u r of p 03,N), o r the average f i s h i n g power, and the m a r g i n a l p r o p e r t i e s of p h y p o t h e s i z e d i n e q u a t i o n s I I - 5 and I I - 6. We p o s t u l a t e t h a t t h e average f i s h i n g power d e c l i n e s as B and/or N i n c r e a s e due t o the f o l l o w i n g e f f e c t s : (a) D i r e c t Gear E x t e r n a l i t y . T h i s i s t h e d i r e c t e f f e c t of one u n i t o f gear on the f i s h i n g power of a n o t h e r u n i t i n an i n t e r a c t i v e o r c o m p e t i t i v e f i s h e r y . In such a f i s h e r y (as t h e Skeena R i v e r g i l l n e t f i s h e r y ) t h e f a c t t h a t one u n i t o f g e a r downstream makes a g i v e n c a t c h r e d u c e s the p r o p o r t i o n o f t h e t o t a l s t o c k a v a i l a b l e f o r c a p t u r e by u n i t s upstream. Thus t h e average p r o p o r t i o n of t h e t o t a l s t o c k taken by one u n i t o f g e a r d e c r e a s e s as the number of u n i t s of gear i n c r e a s e s . T h i s i s s i m p l y due t o t h e " i n t e r a c t i v e " n a t u r e o f t h e f i s h e r y . I n d i r e c t Gear E x t e r n a l i t i e s . These are i n d i r e c t e f f e c t s s e p a r a t e from the a c t u a l f i s h i n g p r o c e s s . Some of t h e s e e f f e c t s can be d e s c r i b e d as f o l l o w s : (1) Space L i m i t a t i o n . On t h e Skeena, f o r i n s t a n c e , t h e r e a r e o n l y so many good " d r i f t s , " where f i s h i n g i s more p r o d u c -t i v e than e l s e w h e r e i n t h e a r e a . Once t h e s e d r i f t s a r e o c c u p i e d , a d d i t i o n a l b o a t s must f i s h i n l e s s p r o d u c t i v e s p o t s , r e d u c i n g average f i s h i n g power. (2) Overcrowding. The net of one v e s s e l may a c t u a l l y b l a n k e t a n o t h e r v e s s e l ; downstream o r seaward v e s s e l s may d r i v e t h e f i s h t o o deep f o r upstream v e s s e l s t o c a t c h ; c o n c e n t r a t i o n s o f v e s s e l s and n e t s make movement o f v e s s e l s f r o m one l o c a t i o n t o a n o t h e r s l o w e r , t h u s r e d u c -i n g e f f e c t i v e f i s h i n g t i m e . Gear S a t u r a t i o n E f f e c t s . The s t a n d a r d g i l l n e t used on t h e Skeena R i v e r has l i m i t e d c a t c h i n g c a p a b i l -i t i e s . I t s i n k s w i t h two hundred o r more f i s h . The p r e s e n c e of f i s h i n t h e net may s c a r e o f f o t h e r f i s h . Thus t h e h i g h e r t h e s t o c k s i z e , t h e s m a l l e r t h e p r o p o r t i o n of t h e t o t a l s t o c k h a r v e s t e d by one u n i t o f g e a r . Thus, d i r e c t and i n d i r e c t g e a r e x t e r n a l i t i e s and gear s a t u r a t i o n e f f e c t s cause t h e average f i s h i n g power of t h e gear t o d i m i n i s h as t h e amount of gear and/or the s i z e o f t h e f i s h s t o c k i n c r e a s e s . One p o s -s i b l e r e l a t i o n s h i p f o r p ( N , B ) might t h e n be: V b 2 p = a B N 1 1 - 1 1 and i f b and b ^ 0, t h e n ^ p and c) p ^ 0, thus d e m o n s t r a t i n g t h e above e x t e r -~ B B a N n a l i t i e s , Then from e q u a t i o n 1 1 - 1 0 C . = p ( B , N ) . B . N . b, + 1 b + 1 1 2 a B N The q u e s t i o n t h e n a r i s e s as t o what s o r t o f r e t u r n s t o s c a l e we s h o u l d e x p e c t , as i n d i c a t e d by t h e sum o f t h e exponents of B and N, and what s o r t o f m a r g i n a l p r o d u c t s t o B and N . The l a t t e r q u e s t i o n can be d e a l t w i t h f i r s t . S i n c e we assume above t h a t b^ C 0 and b^ 0, t h e n ( b x + 1) < 1 and ( b 2 + 1) < 1 ; hence we would expect d i m i n i s h i n g m a r g i n a l p r o d u c t s t h r o u g h o u t . C o n c e r n i n g t h e r e t u r n s t o s c a l e f o r t h e p r o d u c t i o n f u n c t i o n , t h e S c h a e f e r b i o l o g i c a l y i e l d e f f o r t r e l a t i o n s h i p ( i . e . , C - p B . N . ) shows i n c r e a s i n g r e t u r n s t o s c a l e . S i n c e p i s n u m e r i c a l l y v e r y s m a l l , i t i s l i k e l y t h a t r e l a t i v e changes i n p and B and N w i l l not be as g r e a t as 18 changes i n (B.N), hence we might expect, o v e r some range at l e a s t , i n c r e a s i n g r e t u r n s t o s c a l e . The a u t h o r can t h i n k of no a p r i o r i r e a s o n , at l e a s t , f o r t h e p r o d u c t i o n f u n c t i o n t o show c o n s t a n t r e t u r n s t o s c a l e , o r d e c r e a s i n g r e t u r n s t o s c a l e except at v e r y h i g h s t o c k s i z e and f i s h i n g e f f o r t l e v e l s . CHAPTER I I I ESTIMATION OF THE DAILY PRODUCTION FUNCTION The F i s h e r y The f i s h e r y chosen f o r t h i s s t u d y i s t h e Skeena R i v e r sockeye salmon g i l l n e t f i s h e r y . T h i s f i s h e r y i s a t t h e same t i m e b o t h v e r y complex and f a i r l y s i m p l e . I t i s complex t o t h e e x t e n t t h a t a l l f i s h e r i e s a r e complex: t h e f i s h i n g a r e a ( S t a t i s t i c a l Area 4) i s q u i t e l a r g e (See F i g u r e 1 ) , and c a t c h and e f f o r t i s d i s t r i b u t e d w i d e l y and n o n - u n i f o r m l y o v e r i t . The f i s h e r y i s s u b j e c t t o day-to-day r e g u l a -t i o n , and s u b j e c t a l s o t o t h e v a g a r i e s o f wind, weather, t i d e , and sea-bed topography. The f i s h e r y i s s i m p l i f i e d t o t h e e x t e n t t h a t i t i s b e t t e r documented t h a n most, i f not a l l , o t h e r E a s t e r n P a c i f i c salmon f i s h e r i e s , t h e area i s f i s h e d by o n l y one t y p e o f g e a r , and n e a r l y a l l t h e sockeye caught are d e s t i n e d f o r t h e B a b i n e Lake spawn-i n g systems. The v e s s e l s f i s h i n g t h e a r e a a r e by no means homogeneous, but t h e gear, d r i f t g i l l n e t , 200 fathoms long, t and 60 meashes deep, of r e l a t i v e l y u n i f o r m mesh s i z e and s e l e c t i v i t y , can be c o n s i d e r e d homogeneous. O t h e r s p e c i e s of salmon a r e h a r v e s t e d by Skeena R i v e r g i l l n e t t e r s , but the f i s h e r m e n a r e a p p a r e n t l y s e e k i n g sockeye, and c a t c h e s of o t h e r salmon s p e c i e s a r e c o n s i d e r e d i n c i d e n t a l . S i n c e t h e gear i s mainly s e l e c t i v e towards sockeye, t h i s g e a r would y i e l d a d i f f e r e n t p r o d u c t i o n f u n c t i o n f o r o t h e r s p e c i e s , such as p i n k and coho salmon. D i f f e r e n t s p e c i e s of salmon p r o b a b l y i n t e r a c t w i t h sockeye, but t h i s i n t e r a c t i o n would be d i f f i c u l t t o o b s e r v e and q u a n t i f y . 21 TABLE I SKEENA RIVER (AREA 4) SOCKEYE SALMON CATCH, ESCAPEMENT, AND STOCK SIZE, 1951-1969 Landed Landed E s c a p e - T o t a l Y e a r Weight V a l u e P i e c e s ment S t o c k Cwt. $'000 000's 000's (C+ ) 1951 43 433 1 086 691 140 831 1952 77 507 1 938 1 294 335 1 629 1953 42 471 934 659 685 1 344 1954 39 781 876 571 494 1 065 1955 9 481 228 157 71 228 1956 8 793 233 149 ,355 504 1957 16 128 455 280 430 710 1958 38 595 1 085 602 810 1 412 1959 11 997 374 196 781 977 1960 10 191 360 186 263 449 1961 52 510 1 748 895 922 1 817 1962 30 473 1 020 484 548 1 032 1963 7 967 271 142 588 730 1964 49 419 1 781 765 828* 1 593 1965 16 256 603 294 580 874 1966 37 121 1 377 593 389 982 1967 62 315 2 340 1 044 603 1 647 1968 53 930 2 039 780 552 1 332 1969 32 117 1 245 , 536 659 1 186 Average 543 527 1 070 * No f e n c e . E s t i m a t e o n l y . S o u r c e s : B r i t i s h Columbia C a t c h S t a t i s t i c s (Annual) B.C. l - A ' s (U n p u b l i s h e d Economics Branch R e c o r d s ) . Records of t h e S.R.M.C. Sockeye c a t c h and escapement data f o r A r e a 4 a r e shown i n T a b l e I . The v a l u e of sockeye l a n d i n g s t o f i s h e r m e n has r a n g e d f r o m $228,000 i n 1955 and $233,000 i n 1956 (the c y c l e y e a r s f o l l o w i n g t h e d i s a s t r o u s 1951 Skeena R i v e r s l i d e ) t o $2.3 m i l l i o n i n 1967. Over t h e p e r i o d 1951 t o 1969 the average c a t c h has been 543,000 f i s h , w i t h an average e s c a p e -ment of 527,000 f i s h , f o r an average t o t a l r u n o f 1.07 m i l l i o n f i s h . The s l i d e i n 1951, however, caused a d e c l i n e i n t h e runs a f t e r 1955, from which the f i s h e r y d i d not b e g i n t o r e c o v e r f o r some ti m e . I t i s c o n s i d e r e d t h a t now t h e average run has r e t u r n e d t o a more normal l e v e l o f 1.3 t o 1.6 m i l l i o n f i s h p roduced under n a t u r a l c o n d i t i o n s . The y e a r s chosen f o r a n a l y s i s i n c l u d e a good y e a r (1967), an average y e a r (1968), and a poor y e a r (1969). I t i s not p o s s i b l e t o o b s e r v e , when a f i s h e r y i s i n p r o g r e s s , t h e e x a c t t e m p o r a l d i s t r i b u t i o n o f f i s h p a s s i n g t h r o u g h t h e f i s h e r y : a f i s h might be caught on a g i v e n day, o r , i f n o t caught, be " o b s e r v e d " (through t e s t f i s h i n g ) p a s s i n g t h e upstream boundary s e v e r a l days l a t e r . The t i m i n g r e p r e s e n t a t i o n o f Skeena R i v e r s o c k e y e , as shown i n F i g u r e 2, r e p r e s e n t s c u m u l a t i v e c a t c h e s and escapements, and does not t h e r e f o r e show the t i m i n g o f t h e f i s h e n t e r i n g t h e a r e a , n o r the t i m i n g o f t h e f i s h l e a v i n g t h e a r e a . S i n c e i t t a k e s a f i s h anywhere f r o m t h r e e t o seven days, w i t h an average of about f o u r o r f i v e days, t o pass t h r o u g h the a r e a , F i g u r e 2 p r o b a b l y u n d e r e s t i m a t e s t h e number of f i s h e n t e r i n g t h e f i s h i n g area d u r i n g t h e peak o r m i d d l e o f t h e r u n . F o r t h e purposes of t h i s r e p o r t , and l a t e r a n a l y s i s , i t has been assumed t h a t t h e numbers of f i s h e n t e r i n g t h e area each day a r e n o r m a l l y d i s t r i b u t e d , w i t h t h e mean about t h e end of the f o u r t h week i n J u l y and w i t h 90 p e r c e n t o f t h e f i s h e n t e r i n g t h e area from t h e b e g i n n i n g o f t h e f i r s t week i n J u l y Per Cent o f T o t a l Run - 1956-69 Week 2 3 4 1 2 3 4 5 1 2 3 4 Month June * J u l y • August F i g u r e 2. Skeena R i v e r sockeye t i m i n g e x p r e s s e d as c u m u l a t i v e p e r c e n t a g e of t o t a l s t o c k by end of week. t o t h e end of t h e second week i n August. T h i s r e p r e s e n t s , i n e f f e c t , a s i x week season. A c c o r d i n g t o t h e Skeena R i v e r Management Committee, 90 p e r c e n t o r more of the Skeena sockeye s t o c k s a r e produced i n t h e Babine Lake system. Those which a r e not a p p a r e n t l y r e t u r n e a r l y t o t h e Skeena R i v e r . Management p r e d i c t i o n s a r e made o n l y f o r t h e r e t u r n t o t h e Babine system, so f o r t h e purposes of t h i s r e p o r t , t h e f i s h p o p u l a -t i o n s b e i n g d e a l t w i t h w i l l be assumed t o be t h o s e produced i n t h e B a b i n e Lake system, and w i l l be a v a i l a b l e f o r c a p t u r e i n i t s e n t i r e t y d u r i n g t h e s i x week p e r i o d d e s c r i b e d above. The Data The f i s h e r y i s under t h e j u r i s d i c t i o n of t h e Skeena R i v e r Manage-ment Committee (S.R.M.C.) of the Department of F i s h e r i e s and F o r e s t r y . T h i s Committee, under i t s D i r e c t o r o f I n v e s t i g a t i o n s , a l o n g w i t h i t s management r e s p o n s i b i l i t i e s , c o l l e c t s and c o m p i l e s c a t c h , escapement, and e f f o r t s t a t i s t i c s f o r t h e Skeena R i v e r Management a r e a . Much o f the data f o r t h i s study were made a v a i l a b l e t h r o u g h t h e good o f f i c e s o f t h e above Committee, and were used o r c a l c u l a t e d as f o l l o w s : C^ - d a i l y c a t c h - i n numbers of sockeye, o b t a i n e d by a d j u s t i n g d a i l y f i e l d e s t i m a t e s made by t h e S.R.M.C. t o conform t o t h e weekly t o t a l s o f t h e c a t c h o f sockeye i n S t a t i s t i c a l Area 4 p u b l i s h e d i n B r i t i s h Columbia C a t c h S t a t i s t i c s , 1967, 1968, and 1969. 25 - d a i l y numbers of g i l l n e t v e s s e l s f i s h -i n g i n Area 4, o b t a i n e d by t h e S.R.M.C. thro u g h a e r i a l f l i g h t s . - d a i l y escapement i n numbers of f i s h a t the f i s h i n g boundai-y, o b t a i n e d ' b y t h e S.R.M.C. t h r o u g h t e s t f i s h i n g and a d j u s t e d by Ba b i n e Lake f e n c e c o u n t s , and advanced one day t o t a k e i n t o account t h e m i g r a t i o n time between t h e boundary and t h e t e s t f i s h i n g s i t e . D - m i g r a t i o n speed, i n days; assumed t o be e i t h e r f o u r o r f i v e days. - t h e number of f i s h i n t h e area at t h e b e g i n n i n g o f each day. C^ and E^ were used t o make e s t i m a t e s o f as f o l l o w s : N t = C t + E t + E t + 1 + ••• + E t + D - l + D - l I I I - 1 D ' t + 1 + D-2 C. _ + ... D - ( D - l ) C — t+2 g •' t+D-1 S e t t i n g t h e data out i n a t a b l e , and assuming t h a t i t t a k e s f i v e days (D=5) f o r a f i s h t o pass t h r o u g h t h e a r e a , Nj. e s t i m a t e s a r e shown as f o l l o w s : DAY CATCH ESCAPEMENT STOCK z t t t C E C + 5L E + 4/5 C + 3/5 C 1 1 1 Y" t 2 3 6 C E C + / E + 4/5 C 2 2 2 c — t 3 27 °3 E3 C3 + ^ \ 3 4 E 4 5 E 5 6 E 6 7 E 7 T h i s r e p r e s e n t s an example o f a t h r e e - d a y f i s h i n g week, w i t h t h e f i s h e r y o r d i n a r i l y o p e n i n g a t 6:00 P.M. on one day (day t ) and b e i n g d e f i n e d as t h e 24-hour p e r i o d e n d i n g at 6:00 P.M. t h e f o l l o w i n g day. F o r t h e l a s t day of t h e week, t h e computation f o r N^ i s f a i r l y s i m p l e . Ng, i n t h i s example, e q u a l s t h e c a t c h on t h a t day, C^» p l u s t h e e s c a p e -ment on t h e day, and each s u c c e e d i n g day u n t i l f i v e days' escapement i s summed, f i v e days r e p r e s e n t i n g t h e l e n g t h o f time i t t a k e s a f i s h t o pass t h r o u g h t h e a r e a . On days 1 and 2, however, t h e computation f o r N and N must t a k e i n t o account t h e f a c t t h a t some o f t h e f i s h 1 « i n t h e a r e a on, say, day 2, d i d not escape, but were i n f a c t caught on day 3. The breakdown o f c a t c h i n t o e q u i - p r o p o r t i o n a l segments i s p r o b a b l y not t o o s e r i o u s a l i b e r t y , s i n c e the m i g r a t i o n t i m e i s s h o r t r e l a t i v e . t o the l e n g t h o f the season. The l e n g t h o f time i t t a k e s f o r a f i s h t o pass f r o m t h e e n t r a n c e o f t h e f i s h e r y t o t h e f i s h i n g boundary c o u l d be c r i t i c a l . T a g g i n g 3 data would i n d i c a t e e i t h e r f o u r o r f i v e days m i g r a t i o n r a t e s . Two s e r i e s o f were used, one employing the f o u r day assumption, t h e o t h e r f i v e days. Data were c o m p i l e d f o r t h e y e a r s 1967, 1968, and 1969, y i e l d i n g 61 o b s e r v a t i o n s of f i s h i n g days when was g r e a t e r than 10,000 f i s h . The data f o r a l l t h r e e y e a r s were lumped t o g e t h e r on t h e assumption t h a t t h e r e had been no s i g n i f i c a n t t e c h n o l o g i c a l change o v e r t h e p e r i o d , and so t h e t e c h n o l o g i c a l r e l a t i o n s h i p s b e i n g e s t i m a t e d remained c o n s t a n t w i t h i n as w e l l as between seaso n s . Dummy V a r i a b l e s D l = 1 i f s t o c k b u i l d i n g t o a peak = 0 o t h e r w i s e D2 = 1 i f f i r s t day o f a f i s h e r y = O o t h e r w i s e D l was used t o t a k e i n t o account e s t i m a t i n g e r r o r s i n due t o t h e e q u i - p r o p o r t i o n a l i t y breakdown o f C^. N w o u l d t e n d t o be o v e r -e s t i m a t e d w i t h the s t o c k s i z e b u i l d i n g , and u n d e r e s t i m a t e d w i t h t h e s t o c k s i z e f a l l i n g o f f . D2 was used t o t a k e i n t o account t h e phenomenon of b o a t s c o n c e n t r a t -i n g n e a r the f i s h i n g boundary on t h e f i r s t day, and t h e n d i s p e r s i n g on f o l l o w i n g days. 3 Aro and McDonald (M.S., 1968, p. 48) g i v e data on t h e d i s t r i b u -t i o n of t h e number of t a g s r e c o v e r e d a t t h e boundary area r e l a t e d t o " t h e number of days o u t , " o b t a i n e d by t a g g i n g a t Dundas I s l a n d . The modal v a l u e i s f o u r days and the median v a l u e f i v e days, and t h e average j u s t , o v e r f i v e days. F i s h e n t e r i n g t h e area s o u t h o f Dundas I s l a n d would t a k e l e s s time t o pass t h r o u g h t h e a r e a . 28 3. E c o n o m e t r i c Problems The e c o n o m e t r i c models t e s t e d were of t h e g e n e r a l form: c< N C t B u t and where C i s t h e c a t c h on day t B^ . i s t h e number of b o a t s f i s h i n g on day t Nj. i s t h e s t o c k s i z e on day t u . i s a random v a r i a b l e w i t h = c o n s t a n t i f t = s mean l o g (u.) = 0, Cov ( l o g u l o g u ) t r s = 0 i f t / s • T h e r e appear, on c l o s e e x a m i n a t i o n , t o be two major e c o n o m e t r i c problems. The prime d i f f i c u l t i e s a r i s e out o f t h e d e f i n i t i o n and e s t i m a t i o n of N.. On any g i v e n day, t h e number of f i s h i n t h e a r e a , N ^ t i s c o m p r i s e d of two d i f f e r e n t s e g m e n t s — t h o s e which a r e caught, and t h o s e which a r e not caught, say, E^_• Thus, = C t + E t " W n e n w e s u b s t i t u t e t h i s i d e n t i t y i n t o o u r model, o b t a i n i n g : i t a ppears we have a s i t u a t i o n where t h e r e a r e s t r u c t u r a l i n t e r d e p e n d -e n c i e s between t h e dependent v a r i a b l e and t h e i n d e p e n d e n t v a r i a b l e N^. T h i s appears analogous t o t h e s i m p l e income d e t e r m i n a t i o n model a n a l y s e d by J o h n s t o n (1963, pp. 231-234). I n t h e J o h n s t o n example, t h e r e i s a s i m p l e consumption f u n c t i o n r e l a t i n g consumption, as a dependent (endogenous) v a r i a b l e t o n a t i o n a l income. N a t i o n a l income i s i t s e l f endogenous, and e q u a l s consumption p l u s e x o g e n o u s l y g i v e n i n v e s t m e n t . T h e r e f o r e , i n t h i s income d e t e r m i n a t i o n model, v a r i a t i o n i n consumption f e e d s back t o v a r i a t i o n i n n a t i o n a l income, and, as shown by J o h n s t o n , t h e a p p l i c a t i o n o f the o r d i n a r y l e a s t s q u a r e s (OLS) e s t i m a t o r y i e l d s u. t 29 b i a s e d e s t i m a t e s o f t h e s t r u c t u r a l p a r a m e t e r s . T h i s problem, how-e v e r , does not a r i s e i n t h e f i s h e r i e s p r o d u c t i o n model s p e c i f i e d above. N i s composed of C and E , but i s not d e t e r m i n e d by C . t t t y t i s d etermined by n a t u r e , and p r e v i o u s days' c a t c h e s . V a r i a t i o n s i n , t h e n , l e a d t o e q u a l and o p p o s i t e v a r i a t i o n s i n E^_, s i n c e i s p r e d e t e r m i n e d . Thus, i f i n p e r i o d t , because of weather c o n d i -^ v t i o n s , t h e d i s t u r b a n c e term u^ _ = u / 0 t h e n t h e number o f f i s h not caught, E , w i l l be u* g r e a t e r than E would have been had t t u^ = 0. T h e r e f o r e , u^ _ w i l l be u n c o r r e l a t e d w i t h t h e e x p l a n a t o r y v a r i a b l e = (C^. + E^. ), and OLS can be l e g i t i m a t e l y a p p l i e d . Problems do not a r i s e , t h e n , i n t h e n a t u r e of N , but d i f -t f i c u l t i e s may be e n c o u n t e r e d i n t h e e s t i m a t i o n o f . In t h i s paper, has been e s t i m a t e d by t h e f o l l o w i n g e x p r e s s i o n : N t = C t + E t + E t + 1 + ••• + E t + D - l 1 1 1 " 1 + D - l + D-2 + . . . + D - ( D - l ) D t+1 D t+2 D t+D-1 U s i n g t h i s e x p r e s s i o n t h e r e appears t o be a s e r i o u s problem of a u t o -c o r r e l a t i o n , s i n c e subsequent v a l u e s of C ( i . e . C ... C ) t t+1 t+D-1 a r e used. S i n c e t h e s e a r e c o r r e l a t e d w i t h contemporaneous v a l u e s of u^ ( i . e . u t + 1 ••• Ut+D 1^' t h e n i * m i g n t appear t h a t cov (u^ _ u^)^0, t s. However, by the same argument as i n t h e above d i s c u s s i o n about t h e s t r u c t u r a l independence of C and N., v a r i a t i o n s i n t h e C 's t ^ t due t o contemporaneous u^_' s w i l l be o f f s e t by v a r i a t i o n s i n t h e E '.s, " and we would not expect an a u t o c o r r e l a t i o n problem from t h i s s o u r c e at l e a s t . However, f o r e x p r e s s i o n I I I - 1 t o be c o r r e c t r e q u i r e s : (a) no e r r o r s i n o b s e r v a t i o n of (b) no e r r o r s i n t h e e s t i m a t i o n . . o f t h e E + ' s and (c) no e r r r s i n e s t i m a t i o n of D and (d) no f i sh 30 m o r t a l i t y o t h e r t h a n o b s e r v e d f i s h i n g m o r t a l i t y ( c a t c h ) . The problem i s t h a t t h e r u l e t h a t v a r i a t i o n s i n l e a d s t o e q u a l and o p p o s i t e v a r i a t i o n s i n i s not i n d e p e n d e n t l y v e r i f i a b l e , s i n c e , o f c o u r s e , we don't know what E would have been i f t h e r e had t been no C^. J o h n s t o n (1960, pp. 162-165), however, shows t h a t i f t h e r e e x i s t measurement problems, a l t h o u g h s t r u c t u r a l parameters might be b i a s e d , as l o n g as t h e method of measurement p e r s i s t e n t l y y i e l d s t h e . same e r r o r s , t h e n the r e g r e s s i o n c o e f f i c i e n t s can be used f o r p r e d i c -t i o n p u r p o ses. We may say, t h e n , t h a t t h e f o l l o w i n g h o l d : (a) E v e r y t h i n g e l s e b e i n g measured c o r r e c t l y , i f t h e component o f i s s u b j e c t t o measurement e r r o r , the r e g r e s s i o n c o -e f f i c i e n t s w i l l be b i a s e d , and t h e r e s u l t -ant e q u a t i o n cannot be used f o r p r e d i c -t i o n purposes, due t o t h e " f e e d b a c k " a s p e c t s w i t h C a p p e a r i n g on b o t h s i d e s o f t h e r e g r e s s i o n e q u a t i o n . (b) E v e r y t h i n g e l s e b e i n g measured c o r r e c t l y , i f t h e E and D measurements a r e s u b j e c t t o measurement e r r o r , t h e r e g r e s s i o n c o -e f f i c i e n t s w i l l be b i a s e d , but t h e r e s u l t -ant e q u a t i o n can be used f o r p r e d i c t i o n p u r p o s es. (c) I f t h e r e a r e no e r r o r s i n measurement o f , t h e E ^ ' s , o r the magnitude o f D, but t h e r e i s some unobserved m o r t a l i t y , t h e n the r e g r e s s i o n c o e f f i c i e n t s w i l l be b i a s e d , 31 but i f t h e uno b s e r v e d m o r t a l i t y i s un-c o r r e c t e d w i t h C , t h e n t h e r e s u l t a n t t e q u a t i o n can be u s e d f o r p r e d i c t i o n p u r p o s e s . In a c t u a l f a c t , i s p r o b a b l y n o t s u b j e c t t o v e r y much measure-ment e r r o r . We know n o t h i n g about u n o b s e r v e d m o r t a l i t y a l t h o u g h i t may be s i g n i f i c a n t , and the E^_'s a r e l i k e l y t o be s u b j e c t t o some e r r o r . In view o f t h e above, i t would p r o b a b l y be b e s t t o e r r on t h e h i g h s i d e as f a r as t h e D assumption i s c o n c e r n e d i n o r d e r t o negate p a r t i a l l y t h e unobserved m o r t a l i t y problem. The c o e f f i c i e n t s w i l l not be r e a l l y u n b i a s e d e s t i m a t e s o f t h e s t r u c t u r a l p a r a m e t e r s , but t h e e s t i m a t e s can be used f o r p r e d i c t i o n p u r p o s e s , y i e l d i n g r e s u l t s i n d i c a t i v e , at l e a s t , o f t h e s e a s o n a l p r o d u c t i o n r e l a t i o n s h i p between c a t c h , e f f o r t , and s t o c k s i z e . 4. D i s c u s s i o n o f R e g r e s s i o n R e s u l t s R e g r e s s i o n e s t i m a t e s were o b t a i n e d f o r each o f t h e y e a r s 1967, 1968, and 1969, and f o r a l l y e a r s p o o l e d t o g e t h e r , u s i n g b o t h t h e f o u r day and f i v e day m i g r a t i o n speed a s s u m p t i o n . These e s t i m a t e s a r e 2 t a b u l a t e d i n T a b l e I I . In a l l c a s e s , t h e R v a l u e s a r e i n excess o f .80, and t h e r e g r e s s i o n e q u a t i o n s a r e s i g n i f i c a n t at t h e 95% l e v e l . In a l l c a s e s , t h e N exponent i s s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o at t h e 95% l e v e l . The B exponent i s s i g n i f i c a n t l y d i f f e r e n t from z e r o i n e q u a t i o n s 6 and 9 a t t h e 95% l e v e l , and i n e q u a t i o n s 4 and-10 a t t h e 90% l e v e l . However, t h e f a c t t h a t exponents a r e s i g n i f i c a n t l y d i f f e r e n t from z e r o i s r e a l l y i r r e l e v a n t . Even i f t h e exponents were z e r o , i . e . TABLE I I DAILY PRODUCTION FUNCTION ESTIMATES lua -.on " Y e a r D n C o n s t a n t N B 5 DI 5 D2 2 R D u r b i n -Watson d F V* s 1 1967 4 26 1.401 .7047*# (.0902) .3345# (.2562) -.23S5* (.0865) -.0851 .88 1.33 (3) 38.1 1.04 2 1968 4 20 0.983 .7705*# (.0986) .2179## (.3791) .3630* (.1398) -.0458 (.1412) .88 2.40 (1) 27.2 . 99 3'' '• 1969 4 15 0.605 .9415* (.1526) .0673 (.4684) -.3174* (.1478) .0321 (.1035) .96 2.55 (1) 55.3 1.01 4 1967-69 4 61 0.541 .7788*# (.0652) .3335** (.1948) .84 1.20 (2) 154.8 • 1.11 5 1967-69 4 61 0.779 .7981*# (.0689) .2462# (.2155) -.0795 (.0757) .0168 (.0717) .85 1.25 (2) 76. 5 1.04 '6 1967 5 26 0.331 ,686*# (.0926) .5805* (.2481) -.2774* (.0820) -.1008 (.0897) .87 1.38 (3) 34.7 1.23 |7 1968 5 20 0.449 .8212* (.1038) .2320## (.3742) .3299* (.1397) -.0561 (.1396) .88 2.46 (1) 27.9 1.05 8 1969 5 15 0.^51 .8988* (.1519) .2659 (.4579) -.3920* (.1602) .0826 (.1071) .95 2.49 (1) 51.7 1.16 9 1967-69 5 61 0.194 .7538*# (.0695) ,5220*# (.1996) .82 1.18 (2) 132.1 1.28 P 1967-69 5 61 0.330 ,7904*# (.0740) .3785#** (.2219) -.1255 (.0814) .0273 (.0761) .83 1.21 (2) 67.1 1.17 * s i g n i f i c a n t l y d i f f e r e n t from z e r o at 957o l e v e l . ** s i g n i f i c a n t l y d i f f e r e n t from z e r o at 907o l e v e l . # s i g n i f i c a n t l y d i f f e r e n t from u n i t y a t 95% l e v e l . ## s i g n i f i c a n t l y d i f f e r e n t from u n i t y at 90% l e v e l . 4 F i g u r e s i n p a r e n t h e s i s below c o e f f i c i e n t s r e p r e s e n t s t a n d a r d e r r o r s . 5 In t h e r e g r e s s i o n e q u a t i o n s as e s t i m a t e d , t h e dummy v a r i a b l e s D l and D2 e n t e r t h e e q u a t i o n i n a m u l t i p l i c a -D l D2 . t i v e form, t a k i n g t h e v a l u e s 1 and e and 1 and e (1) A c c e p t h y p o t h e s i s t h a t no p o s i t i v e a u t o c o r r e l a -t i o n e x i s t s , at 95% l e v e l . (2) R e j e c t h y p o t h e s i s t h a t no p o s i t i v e a u t o c o r r e l a -t i o n e x i s t s , at 95% l e v e l . (3) T e s t f o r a u t o c o r r e l a t i o n i n c o n c l u s i v e . 10 * X tT ^ 0 0 ^ C = C < N . . B = <X N B ;= • C< t h e n the average f i s h i n g power, p = p (B,N) C y i e l d s p = B.N B.N and P and c> p ^ Q ^ N £ B t h u s showing the e x i s t e n c e of d i r e c t and i n d i r e c t g e a r e x t e r n a l i t i e s and g e a r s a t u r a t i o n e f f e c t s . The i m p o r t a n t s t a t i s t i c a l r e s u l t s a r e exponents f o r B and N s i g n i f i c a n t l y d i f f e r e n t from u n i t y . F o r i f t h e exponents e q u a l one, t h e n C = oC B.N, so p - <X , and ^ p and h p _ 0 y i e l d i n g no g e a r e x t e r n a l i t i e s o r s a t u r a t i o n e f f e c t s . The N e x p o n e n t s a r e s i g n i f i c a n t l y d i f f e r e n t from u n i t y i n a l l e q u a t i o n s except 3, 7, and 8, at t h e 95% l e v e l . The B exponents a r e s i g n i f i c a n t l y d i f f e r e n t from u n i t y i n e q u a t i o n s 1, 4, 5, 9, and 10 a t t h e 95% l e v e l , and i n e q u a t i o n s 2 and 7 a t t h e 90% l e v e l . The magnitudes of t h e B and N c o e f f i c i e n t s i n d i c a t e t h e s t r e n g t h o f the g e a r e x t e r n a l i t i e s and s a t u r a t i o n e f f e c t s . The s m a l l e r t h e e xponents, the l a r g e r t h e s e e f f e c t s . The magnitude of t h e N exponent g i v e s an i n d i c a t i o n of the g e a r s a t u r a t i o n e f f e c t . The c o e f f i c i e n t s a r e m o s t l y between .7 and .9, and a r e f o r t h e most p a r t s t a t i s t i c a l l y d i f f e r e n t from u n i t y . These c o e f f i c i e n t s , t h e n , show a weak, but s i g n i f i c a n t gear s a t u r a t i o n e f f e c t . The o n l y y e a r where g e a r s a t u r a -t i o n e f f e c t s a r e n o n - s i g n i f i c a n t was i n 1969, t h e p o o r e s t y e a r of t h e t h r e e chosen, w i t h a run of o n l y 1,186,000 sockeye (Ta b l e I ) . The B exponent shows the s i z e of t h e d i r e c t and i n d i r e c t g e a r e x t e r n a l -i t i e s . Seven out of t h e t e n e q u a t i o n s y i e l d exponents s i g n i f i c a n t l y d i f f e r e n t from u n i t y at e i t h e r t h e 90% o r 95% l e v e l , w i t h t h e c o -e f f i c i e n t s r a n g i n g between .2 and .6. These lower exponents i n d i c a t e g e a r e x t e r n a l i t i e s of some magnitude, and demonstrate t h e " p r o d u c t i v e i n t e r d e p e n d e n c e " between competing u n i t s o f f i s h i n g g e a r . I t i s n o t e d t h a t , except f o r e q u a t i o n s 2 and 7, t h e N expo-n e n t s a r e lower i n the f i v e day m i g r a t i o n r a t e e q u a t i o n s t han i n t h e c o r r e s p o n d i n g f o u r day e q u a t i o n s , w h i l e t h e B exponents a r e i n e v e r y case h i g h e r . T h i s we would expect, s i n c e N u s i n g t h e f i v e day assump-t i o n i s i n e v e r y o b s e r v a t i o n l a r g e r t h a n t h e N d e r i v e d from t h e f o u r day assumption. U s i n g t h e h i g h e r N s e r i e s , t h e apparent m a r g i n a l r a t e of s u b s t i t u t i o n of N f o r B would be lower, and so we would e x p e c t t h e apparent m a r g i n a l p r o d u c t of N t o be l e s s and t h e a p p a r e n t m a r g i n a l p r o d u c t of B t o be g r e a t e r , and thus we would expect l a r g e r B expo-n e n t s u s i n g t h e f i v e day assumption. A A T a b l e I I a l s o shows ( ) , t h e sum o f t h e B and N expo-n e n t s . I t w i l l be n o t e d t h a t t h e s e sums a r e u n i f o r m l y h i g h e r f o r t h e e q u a t i o n s u s i n g t h e f i v e day m i g r a t i o n r a t e assumption than f o r t h o s e u s i n g t h e f o u r day assumption. The f o u r day m i g r a t i o n r a t e assumption y i e l d s r e s u l t s which a r e v e r y c l o s e t o c o n s t a n t r e t u r n s t o s c a l e , w h i l e the f i v e day assumption g i v e s a p r o d u c t i o n f u n c t i o n which i s i n t e r e s t i n g i n terms of i n c r e a s -i n g r e t u r n s t o s c a l e . As e x p l a i n e d e a r l i e r (page 27) two dummy v a r i a b l e s were used. DI was used t o t a k e i n t o account t h e b i a s i n c a l c u l a t i n g due t o t h e e q u i - p r o p o r t i o n a l breakdown i n the C^'s o n f o l l o w i n g days ( e q u a t i o n I I I - 1 ) . Thus, w i t h t h e s t o c k b u i l d i n g t o a peak and D l = 1, we would expect N t o be o v e r s t a t e d , and t h i s c o r r e c t e d by D l <^ 0, y D l , S \ o r 0 \ e ^ 1, where D l i s t h e e s t i m a t e d c o e f f i c i e n t o f D l . The c o e f f i c i e n t o f D l i n t h e r e g r e s s i o n e s t i m a t e s i s s i g n i f i c a n t i n a l l s i n g l e g e a r e q u a t i o n s , w i t h t h e e x p e c t e d s i g n i n ev e r y y e a r e x c e p t 1968. E x a m i n a t i o n o f S.R.M.C. f i l e s r e v e a l s t h a t i n 1968 t h e r u n came i n sooner, and pa s s e d t h r o u g h f a s t e r t h a n u s u a l . I n a l l l i k e l i h o o d , D v a l u e s ( m i g r a t i o n speed r a t e s ) o f f o u r and f i v e days o v e r s t a t e d the t r u e v a l u e o f D i n t h e e a r l y p a r t o f t h e r u n i n 1968, so t h i s l e d t o an u n d e r e s t i m a t e o f N , and t h i s was compensated by a p o s i t i v e e s t i m a t e d c o e f f i c i e n t f o r D l . The second dummy v a r i a b l e , D2, was i n t r o d u c e d t o t a k e i n t o account t h e d i f f e r e n t s p a t i a l d i s t r i b u t i o n o f b o a t s f i s h i n g on t h e f i r s t day of t h e f i s h i n g week compared w i t h t h e d i s t r i b u t i o n on subsequent days. The c o e f f i c i e n t s of D2 were a l l n ot s i g n i f i c a n t l y d i f f e r e n t from z e r o , so t h i s phenomenon does n o t seem t o o i m p o r t a n t . T a b l e I I a l s o shows t h e v a l u e s o f t h e Durbin-Watson d t e s t s t a t i s t i c . F o r t h e s i n g l e y e a r e q u a t i o n s , we r e j e c t t h e h y p o t h e s i s of p o s i t i v e a u t o c o r r e l a t i o n a t t h e 95% l e v e l f o r e v e r y y e a r b u t 1967,where t h e t e s t i s u n c o n c l u s i v e . F o r t h e e q u a t i o n s w i t h t h e data p o o l e d f o r 1967-69, s i n c e t h e o b s e r v a t i o n s f o r each y e a r would p r o b -a b l y l i e s y s t e m a t i c a l l y above, o r s y s t e m a t i c a l l y below t h e e s t i m a t e d r e g r e s s i o n p l a n e , t h e Durbin-Watson t e s t r e v e a l s t h e p r e s e n c e o f p o s i t i v e s e r i a l c o r r e l a t i o n . S i n c e a u t o c o r r e l a t i o n l e a d s t o se v e r e u n d e r e s t i m a t e s of t h e s t a n d a r d e r r o r s o f t h e r e g r e s s i o n c o e f f i c i e n t s ( J o h n s t o n , 1963, p. 179), some doubt must be p l a c e d on t h e t e s t s o f s i g n i f i c a n c e f o r t h e B and N exponents i n e q u a t i o n s 4, 5, 9, and 10. CHAPTER IV THE SEASONAL PRODUCTION FUNCTION C o n s t r u c t i o n o f S e a s o n a l F u n c t i o n The b a s i s f o r t h e s e a s o n a l p r o d u c t i o n f u n c t i o n i s t h e d a i l y p r o d u c t i o n f u n c t i o n and assumptions made c o n c e r n i n g t h e te m p o r a l d i s t r i b u t i o n of f i s h e n t e r i n g t h e f i s h e r y . Due t o t h e ma t h e m a t i c a l c o m p l e x i t i e s of d e v e l o p i n g the s e a s o n a l f u n c t i o n from t h e d a i l y f u n c t i o n , i t was n e c e s s a r y t o c o n s t r u c t a computer program t o show s e a s o n a l c a t c h e s at v a r i o u s l e v e l s of f i s h i n g e f f o r t and s e a s o n a l s t o c k s i z e . In e s t i m a t i n g t h e d a i l y f u n c t i o n , e s t i m a t e s o f d a i l y s t o c k s i z e were o b t a i n e d by a d d i n g d a i l y c a t c h t o d a i l y escapement, and depending on t h e m i g r a t i o n r a t e assumption used, f o l l o w i n g days' escapements, p l u s some f u n c t i o n of f o l l o w i n g days' c a t c h e s . T h i s was t o t a k e i n t o account t h e f a c t t h a t some f i s h i n t h e area on a p a r t i c u l a r day may not have been caught on t h a t day, but might have been caught on some f o l l o w i n g day. In d e t e r m i n i n g t h e s t o c k s i z e i n t h e area on any g i v e n day f o r use i n t h e s e a s o n a l program, account must be t a k e n o f p r e v i o u s days' c a t c h e s , u s i n g t h e f o l l o w i n g p r o c e d u r e : F o r any g i v e n s e a s o n a l s t o c k , S, 90 p e r c e n t o f t h e run of f i s h would e n t e r the f i s h i n g area i n a s i x week (42 day) p e r i o d . (See F i g u r e 2 ) . The r e m a i n i n g 10 p e r c e n t i s assumed t o r e p r e s e n t sockeye not d e s t i n e d f o r t h e Babine Lake system, not under i n t e n s i v e management by t h e S. R. M. C., 37 and not s u b j e c t t o e x p l o i t a t i o n by t h e f i s h e r y . These assumptions a r e made t o a v o i d t h e d i f f i c u l t i e s i n h e r e n t i n c o n s i d e r i n g t h e e x p l o i t a t i o n of t h e e a r l y comers and " t a i l - e n d C h a r l i e s , " and a r e not a g r o s s o v e r s i m p l i f i c a t i o n o f r e a l i t y . I f 90 p e r c e n t of t h e run, S, e n t e r t h e f i s h e r y i n s i x weeks, w i t h t h e peak O c c u r r i n g a t t h e end of t h e t h i r d w e e k — b e g i n n i n g of t h e f o u r t h week--and i f t h i s t e m p o r a l d i s t r i b u t i o n i s n o r m a l l y d i s t r i b u t e d , then t h r e e weeks o r 21 days i s e q u i v a l e n t t o 1.64 s t a n d a r d d e v i a t i o n s . Then the s t a n d a r d d e v i a t i o n o f t h e r u n i s 21 days, o r 1.64 12.8 days. U s i n g t h i s , and t h e o r d i n a t e s of t h e normal d i s t r i b u t i o n , i t i s p o s s i b l e t o compute the number o f f i s h , S^, e n t e r i n g t h e area on each day. I f we assume t h a t i t t a k e s a f i s h f o u r days t o pass t h r o u g h t h e a r e a , t h e n t h e number of f i s h w hich would be i n t h e area on any g i v e n day t , p r o v i d i n g t h e r e was no f i s h e r y i n p r e v i o u s days, would be = S t + S t_^ + ... + S t_g.. S i m i l a r l y , i f t h e m i g r a t i o n speed was f i v e days, t h e n N = S + S, 1 + ... + S . . I f t h e r e were a f i s h e r y on a p r e v i o u s day, t h e n some f r a c t i o n of t h e f i s h which would have been i n t h e a r e a h a d t h e r e been no f i s h e r y , would have been removed. Then N t= S t + S t _ 1 + ... + S t _ 3 ~§ Ct_x " | C t _ 2 ~ \ C t _ 3 H I - 2 under t h e f o u r day m i g r a t i o n speed assumption, and 4 3 2 —1 N t = S t + S t - l + ... + S t _ 4 ~^ Ct_t C t_2 ~ F C t _ 3 ^ C t _ 4 in - 3 under the f i v e day m i g r a t i o n speed assumption. In e s t i m a t i n g s e a s o n a l c a t c h e s a t v a r i o u s s e a s o n a l s t o c k s i z e l e v e l s , t h e above method was u sed t o compute d a i l y s t o c k s i z e s . In a d d i t i o n , i t was assumed t h a t t h e same number o f b o a t s would f i s h e v e r y 38 day t h e f i s h e r y was open, and t h a t t h e f i s h e r y would be open t h e same number of days p e r week. R e g r e s s i o n e s t i m a t e s 4 and 5 ( f o u r day assumption) and 9 and 10 ( f i v e day assumption) (Table I I ) were used on d a i l y s t o c k s i z e s computed u s i n g t h e above method on s e a s o n a l runs r a n g i n g from 1.0 m i l l i o n f i s h t o 3.0 m i l l i o n f i s h , i n i n c r e m e n t s of 200,000 f i s h , w i t h f l e e t s i z e s r a n g i n g f r o m 100 b o a t s t o 800 b o a t s , , w i t h i n c r e m e n t s of 100 b o a t s , f i s h i n g one t o seven days per week. Due t o t h e m a t h e m a t i c a l i n t r a c t a b i l i t y o f t h i s approach, computer " s i m u l a t i o n " was used, so t h a t f o r each c o m b i n a t i o n a b o v e , d a i l y c a t c h e s were summed, y i e l d i n g t o t a l c a t c h , t o t a l escapement, average c a t c h p e r boat d u r i n g t h e s i x week season, and average c a t c h per boat day. R e s u l t s o b t a i n e d by u s i n g e q u a t i o n 10 i n T a b l e I I , showing t o t a l c a t c h e s a g a i n s t t o t a l e f f o r t i n boat days, a t d i f f e r e n t s t o c k s i z e and f l e e t s i z e l e v e l s , a r e shown i n F i g u r e 3 and Appendix T a b l e A-VI. 2. D i s c u s s i o n of t h e S e a s o n a l P r o d u c t i o n F u n c t i o n The d a i l y p r o d u c t i o n f u n c t i o n e s t i m a t e d e a r l i e r had two " i n p u t " v a r i a b l e s - B and N. The s e a s o n a l p r o d u c t i o n r e l a t i o n s h i p c o n s t r u c t e d on t h e b a s i s of t h e d a i l y f u n c t i o n , however, has one more v a r i a b l e : t h e number of days p e r week the f i s h e r y i s open. F o r the s e a s o n a l r e l a t i o n s h i p , t h e n , t h e r e i s a f i x e d c o s t i n p u t , B, a v a r i a b l e c o s t element, f i s h i n g days p e r week (F) and t h e r e s o u r c e v a r i a b l e , S. The s e a s o n a l p r o d u c t i o n r e l a t i o n s h i p has i n h e r e n t i n i t what might be c a l l e d " t h e m u l t i p l e day s t o c k r e d u c t i o n e f f e c t " due t o t h e e f f e c t o f f i s h i n g on one day h a v i n g an e f f e c t on the s t o c k s i z e i n t h e area on f o l l o w i n g days. T h i s e f f e c t i s s t r o n g e r t h e g r e a t e r t h e number of f i s h -i n g days per week. T h i s can be thought of as an i n t e r t e m p o r a l e x t e n -39 s i o n of t h e d i r e c t g e a r e x t e r n a l i t y . T h i s e f f e c t i s b e s t i l l u s t r a t e d w i t h a n u m e r i c a l example. Assume t h e i m m i g r a t i o n of t h e f i s h i n t o t h e area and escapement out of t h e area i s such t h a t i f t h e r e were no f i s h e r y on p r e c e d i n g days, t h e n t h e r e would be 100,000 f i s h i n t h e area at t h e s t a r t of each day, w i t h D, o r the m i g r a t i o n speed, b e i n g f o u r days. In f i s h e r y A below, t h e r e a r e 400 b o a t s f i s h i n g t h r e e days per week and i n f i s h e r y B, 300 b o a t s f i s h f o u r days per week, so f o r both f i s h e r i e s , t h e t o t a l number of boat days i s 1,200. We assume a c o n s t a n t p = .001, so t h a t = .001 B.N^. N.J. i s computed a c c o r d i n g t o t h e f o r m u l a e x p r e s s e d i n e q u a t i o n I I I - 2 on page 37,. TABLE I I I DAILY STOCK SIZE AND CATCH OF TWO HYPOTHETICAL GILLNET FISHERIES, SHOWING MULTIPLE DAY STOCK REMOVAL EFFECT p = .001 D = 4 FISHERY A FISHERY B B=400 B=300 N. N + c. N. c. t t t t t (No F i s h e r y ) Day 1 100 000 100 000 40 000 100 000 30 000 Day 2 100 000 70 000 28 000 77 500 23 250 Day 3 100 000 59 000 23 600 67 198 20 159 Day 4 100 000 65 756 19 729 TOTAL 91 600 93 138 40 I t i s no t e d t h a t , because of t h e m u l t i p l e day s t o c k r e d u c t i o n e f f e c t on N^, g i v e n t h e f l e e t s i z e , c a t c h e s d e c r e a s e on s u c c e s s i v e days. A l s o , s i n c e t h i s e f f e c t i s a te m p o r a l e x t e n s i o n o f t h e d i r e c t g e a r e x t e r n a l i t y , fewer b o a t s can c a t c h more f i s h a t t h e same number of boat days as a l a r g e r f l e e t f i s h i n g fewer days p e r week. T h i s e f f e c t w i l l be a c c e n t u a t e d by t h e r e d u c t i o n o f p a t h i g h e r f l e e t s i z e l e v e l s due t o t h e g e a r e x t e r n a l i t i e s , but t h i s might be p a r t i a l l y o f f s e t by i n c r e a s e s i n p as N i s drawn down t h r o u g h t h e week, t h u s l e s s e n i n g t h e gear s a t u r a t i o n e f f e c t . The s e a s o n a l f u n c t i o n a l s o t a k e s i n t o account t h e i n t r a - s e a s o n a l p e a k i n g e f f e c t o f t h e r u n . The g r e a t e r t h e p e a k i n g n a t u r e o f t h e run, t h e l e s s t h e c a t c h w i l l be at g i v e n l e v e l s o f S, B and F. T h i s i s due t o gear s a t u r a t i o n e f f e c t s b e i n g g r e a t e r at t h e "peak," and e f f o r t b e i n g "wasted" at t h e b e g i n n i n g and t h e end of t h e r u n when t h e numbers of f i s h i n t h e area a r e r e l a t i v e l y s m a l l . T h i s e f f e c t can be demonstrated as f o l l o w s : u s i n g t h e data p r e s e n t e d i n T a b l e A-V f o r t h e d a i l y p r o d u c t i o n f u n c t i o n u s i n g equa-t i o n 10, assume a t h r e e day season, and a one day m i g r a t i o n speed. F i s h e r y A i s one i n which t h e r e i s no i n t r a - s e a s o n a l p e a k i n g , and F i s h e r y B i s one i n which t h i s phenomenon i s p r e s e n t , and F i s h e r y C i s one i n which t h e p e a k i n g e f f e c t i s more pronounced. 41 TABLE IV DAILY STOCK SIZE AND CATCH OF THREE HYPOTHETICAL GILLNET FISHERIES, SHOWING EFFECT OF INTRA-SEASONAL PEAKING B = 300 D = 1 F i s h e r y A F i s h e r y B F i s h e r y C N t C t N t C t N t Day 1 80 000 21 743 60 000 17 323 20 000 7 273 Day 2 80 000 21 743 120 000 29 952 200 000 44 842 Day 3 80 000 21 743 60 000 17 323 20 000 7 273 TOTAL 240 000 65 229 240 000 64 598 240 000 60 388 I t can be seen t h a t t h e g r e a t e r the p e a k i n g phenomenon, the l e s s t h e s e a s o n a l c a t c h at a g i v e n l e v e l of s e a s o n a l s t o c k s i z e and f l e e t s i z e . T h i s i s i m p o r t a n t i n c o n s i d e r i n g t h e i m p l i c a t i o n s f o r spawning j ch a n n e l development, and i t s impact on t h e pe a k i n g o f t h e r u n . ^ The s e a s o n a l p r o d u c t i o n r e l a t i o n s h i p d e r i v e d from e q u a t i o n 10 ( f i v e day m i g r a t i o n r a t e assumption, w i t h dummy v a r i a b l e s ) i s shown i n F i g u r e 3 , and i n Appendix T a b l e A-VI. T h i s w i l l be t h e s e a s o n a l r e l a t i o n s h i p u s ed i n t h i s and subsequent a n a l y s i s f o r i l l u s t r a t i v e pur-poses o n l y . There i s no r e a s o n t o b e l i e v e i t i s any " b e t t e r " t h a n s e a s o n a l r e l a t i o n s h i p s d e v e l o p e d from any o f t h e o t h e r e q u a t i o n s . The s e a s o n a l p r o d u c t i o n f u n c t i o n has t h e f o l l o w i n g p r o p e r t i e s : (a) At any g i v e n l e v e l of S and B, t o t a l . c a t c h i n c r e a s e s but a t a d e c r e a s i n g r a t e , w i t h i n c r e a s e s i n F o r t o t a l boat days. 2 < ° S, B T h i s i s due t o t h e m u l t i p l e day s t o c k Or h_C > 0 > ^ 2 a F S.,B * F i g u r e 3. S e a s o n a l c a t c h - e f f o r t r e l a t i o n s h i p u s i n g e q u a t i o n 10. 43 r e d u c t i o n e f f e c t , (b) At any g i v e n l e v e l o f S, t h e r a t e of i n c r e a s e of t o t a l c a t c h w i t h r e s p e c t t o t o t a l boat days d e c r e a s e s w i t h i n c r e a s e s i n B. Or 2 b (F.B) d B < o T h i s i s due t o d i r e c t and i n d i r e c t g e a r e x t e r n a l i t i e s . (c) At any g i v e n l e v e l o f S and t o t a l boat days, t o t a l c a t c h d e c r e a s e s and a t an i n c r e a s i n g r a t e , w i t h i n c r e a s e s i n B. Or b B d B S, (F.B) T h i s too i s due t o t h e g e a r e x t e r -n a l i t i e s . (d) At any g i v e n l e v e l o f B and F o r t o t a l boat days, t o t a l c a t c h i n c r e a s e s a t a d e c r e a s i n g r a t e w i t h i n c r e a s e s i n S. Or t) S B,F £ <0 S,(F.B) B,F T h i s shows t h e g e a r s a t u r a t i o n e f f e c t , (e) At any g i v e n l e v e l o f B, t h e r a t e o f i n c r e a s e of c a t c h w i t h F and t o t a l boat 44 days i n c r e a s e s w i t h i n c r e a s e s i n S. \ 2 Or _0 C )> 0 2>B £ S F T h i s i s due t o m u l t i p l e day s t o c k r e d u c t i o n e f f e c t o f f s e t t i n g t h e gear, s a t u r a t i o n e f f e c t . F o r t h e a n a l y s i s t h a t f o l l o w s , t h e shape o f i s o - c a t c h c u r v e s at d i f f e r i n g l e v e l s o f F (and t o t a l boat days) and B i s of i n t e r e s t . Due t o p r o p e r t i e s (a) and (b) above, i s o - c a t c h c u r v e s f o r g i v e n S a g a i n s t B and t o t a l boat days a r e shown i n F i g u r e 4. From F i g u r e 4 i t can be seen t h a t t h e g r e a t e r the number of b o a t s , t h e g r e a t e r t h e number of boat days r e q u i r e d , but s i n c e the number of boat days r e q u i r e d i n c r e a s e s a t a d e c r e a s i n g r a t e w i t h t h e number of b o a t s , t h e number of F r e q u i r e d d e c r e a s e s l e s s t h a n p r o p o r t i o n a t e l y w i t h t h e number of b o a t s . The h i g h e r t h e s t o c k s i z e , t h e lower t h e i number of t o t a l days f i s h i n g , and F r e q u i r e d . I s o - c a t c h c u r v e s f o r g i v e n S a g a i n s t f i s h i n g days per boat and B g e n e r a t e d from F i g u r e 4 a r e shown i n F i g u r e 5. I t i s apparent t h a t due t o t h e p r o p e r t i e s which determine, t h e shape o f t h e s e a s o n a l p r o d u c t i o n f u n c t i o n , t h e number of boat days p e r boat r e q u i r e d t o h a r v e s t a g i v e n c a t c h a t a g i v e n s t o c k s i z e d e c l i n e s , but at a d i m i n i s h i n g r a t e , as f l e e t s i z e i n c r e a s e s . T h i s i s c r i t i c a l . Due t o d i r e c t and i n d i r e c t g e a r e x t e r n a l i t i e s , and due t o t h e i n t r a - s e a s o n a l " p e a k i n g " of t h e r u n and t h e m u l t i p l e -day s t o c k removal e f f e c t , o n l y a l e s s t h a n p r o p o r t i o n a l i n c r e a s e o f the number of f i s h i n g days p e r week, and p e r season, must be made 45 T o t a l Boat Days u Number of Bo a t s F i g u r e 4. I s o - C a t c h c u r v e s a t d i f f e r e n t l e v e l s o f t o t a l boat days and number of b o a t s . Boat Days P e r Boat 0 Number of Boats F i g u r e 5. I s o - C a t c h c u r v e s at d i f f e r e n t l e v e l s of boat days p e r boat and numbers of b o a t s . 46 when f l e e t s i z e i s reduced i f h a r v e s t l e v e l s a r e t o be m a i n t a i n e d . 3. E s t i m a t i o n of D i s s i p a t e d Rents The purpose of t h i s s e c t i o n i s t o r e v i e w t h e methods used by C r u t c h f i e l d and P o n t e c o r v o ( h e r e i n a f t e r C-P) i n e s t i m a t i n g t h e magnitude of the r e n t d i s s i p a t e d due t o f r e e e n t r y , and t o show, by u s i n g t h e p r o d u c t i o n f u n c t i o n d e v e l o p e d e a r l i e r , how t h e i r e s t i m a t e s might be b i a s e d . We w i l l c o n s i d e r a f i s h e r y i n i s o l a t i o n , and assume no m o b i l i t y between f i s h e r i e s . The model w i l l not d e a l w i t h t e c h n i c a l change i n g i l l n e t g e ar, and w i l l p r e c l u d e t h e p o s s i b i l i t y o f o t h e r g e a r h a r v e s t -i n g t h e f i s h . F i s h e r i e s management p o l i c y and t e c h n i q u e s w i l l be t h a t employed at p r e s e n t by t h e Department of F i s h e r i e s and F o r e s t r y , and o t h e r such r e g u l a t o r y a g e n c i e s : a maximum s u s t a i n e d y i e l d p o l i c y a c h i e v e d by p r e d e t e r m i n i n g t h e a l l o w a b l e c a t c h l e v e l s , and o p e n i n g the f i s h e r y t o e x p l o i t a t i o n f o r a g i v e n number of days per week.. Whether t h i s i s an " o p t i m a l " p o l i c y i s a l t o g e t h e r a n o t h e r q u e s t i o n . The purpose h e r e i s o n l y t o demonstrate a t e c h n i q u e , and t o i n d i c a t e t h e b i a s i n the C-P e s t i m a t e s . The i m p o r t a n t p o i n t w i l l be t h e r o l e p l a y e d by t h e s e a s o n a l p r o d u c t i o n f u n c t i o n , and t h e b i a s which can r e s u l t from not knowing i t s shape. Our model i s shown i n F i g u r e 6. I t i s assumed, f o l l o w i n g C-P, t h a t a v e s s e l ' s c o s t s can be viewed as t h e sum of (a) l a b o u r c o s t s , (b) f i x e d c o s t s , and (c) v a r i a b l e c o s t s . L a b o u r c o s t s can be viewed as c o n s t a n t t hroughout the season, s i n c e i t can be assumed t h a t f i s h -i n g even one day p e r week p r e c l u d e s employment e l s e w h e r e . F i x e d c o s t s i n c l u d e i n s u r a n c e , d e p r e c i a t i o n , wharfage, e t c . and a normal r e t u r n t o 47 A 0 B* Boats F i g u r e 6. T o t a l s e a s o n a l c o s t s and revenues as a f u n c t i o n of f l e e t s i z e , f o r a g i v e n a l l o w a b l e c a t c h . c a p i t a l . Average l a b o u r c o s t s and f i x e d c o s t s , t h e n independent on f l e e t s i z e , and a r e shown i n t h e diagram as FC. V a r i a b l e c o s t s , shown as c u r v e VC, a r e c o s t s o f f u e l and o i l , f o o d , and i n t h e case o f t h e g i l l n e t f i s h e r y , r e p a i r and maintenance o f t h e n e t . These c o s t s a r e dependent on F, which, as shown i n the p r e v i o u s s e c t i o n , depend on t h e f l e e t s i z e , t h e a l l o w a b l e c a t c h l e v e l , and t h e s t o c k s i z e . Assuming a f i s h p o p u l a t i o n i n s t a b l e e q u i l i b r i u m , t h e a l l o w a b l e c a t c h l e v e l i s g i v e n , and t h e VC c u r v e i s g e n e r a t e d f r o m t h e a p p r o p r i a t e i s o - c a t c h curve r e l a t i n g t o t a l boat days p e r season t o f l e e t s i z e , as shown i n F i g u r e 4 o f t h e p r e v i o u s s e c t i o n . The sum of VC and FC y i e l d s the TC c u r v e . I t must be r e a l i z e d i n t h e above t h a t t h e c o s t c u r v e s , r e f l e c t -i n g t h e shape o f t h e i s o - c a t c h c u r v e s o f t h e p r e v i o u s s e c t i o n , a r e d e r i v e d from t h e s e a s o n a l p r o d u c t i o n f u n c t i o n which i s c o n s t r a i n e d by F ( a l l o w a b l e f i s h i n g days per week). The model assumes, t h e n , t h e r e a c t i o n of the r e g u l a t o r y a u t h o r i t y t o a g i v e n s e a s o n a l s t o c k s i z e S, and f l e e t s i z e B, i n s e t t i n g F f o r t h e s e a s o n . F o r g i v e n S, t h e r e g u l a t o r y a u t h o r i t y w i l l s e t a d i f f e r e n t F when f a c e d w i t h d i f f e r e n t f l e e t s i z e s B, t h u s s e t t i n g t h e shape of t h e VC c u r v e i n t h e above model. One of t h e more i n t e r e s t i n g i m p l i c a t i o n s t o be d e r i v e d from t h e g e n e r a l shape of t h e p r o d u c t i o n f u n c t i o n i s t h a t t h e r e i s no un-c o n s t r a i n e d optimum f l e e t s i z e as l o n g as a v e r a g e c o s t s p e r boat day do not i n c r e a s e s i g n i f i c a n t l y r e l a t i v e t o t h e a v e r a g e number of days f i s h i n g p e r b o a t . In F i g u r e 6 a g a i n , s i n c e VC assumes c o n s t a n t average c o s t s p e r boat day r e l a t i v e t o a v e r age number of days f i s h i n g , and s i n c e t o t a l c o s t s i n c r e a s e at a d e c r e a s i n g r a t e w i t h f l e e t s i z e , m a r g i n a l c o s t s w i l l always be p o s i t i v e , w h i l e m a r g i n a l r e v e n u e s w i l l be z e r o p a s t t h a t l e v e l o f B where the a l l o w a b l e c a t c h i s t a k e n . The "optimum" f l e e t s i z e w i l l be t h e c o n s t r a i n e d maximum where a g i v e n number of b o a t s , say B*, a r e a l l o w e d t o f i s h t h e maximum seven days a week th r o u g h o u t the season. To t h e e x t e n t t h a t a v e r a g e v e s s e l c o s t s per day i n c r e a s e w i t h r e s p e c t t o t h e average number of days f i s h i n g per b o a t , and t h i s i s r e f l e c t e d i n a p o r t i o n of t h e VC c u r v e h a v i n g p o s i t i v e second o r d e r d e r i v a t i v e s ( i . e . , t h e dashed p o r t i o n of t h e VC and TC c u r v e ) , t h e n t h e r e w i l l e x i s t a " w e l l o r d e r e d " optimum where r e n t s a r e maximized. C r u t c h f i e l d and P o n t e c o r v o e s t i m a t e t h e magnitude o f r e n t s d i s -s i p a t e d due t o f r e e e n t r y w i t h o u t e x p l i c i t r e f e r e n c e t o any p r o d u c -t i o n f u n c t i o n . 0 . B 2 B 1 Boats F i g u r e 7. The C r u t c h f i e l d - P o n t e c o r v o p r o d u c t i o n f u n c t i o n and e s t i m a t e d rents'. In t h e B r i s t o l Bay case, C-P i m p l i c i t l y assume a p r o d u c t i o n f u n c t i o n w i t h c o n s t a n t average r e t u r n s t o b o a t s up t o a c e r t a i n p o i n t , f o l l o w e d by z e r o m a r g i n a l p r o d u c t s . T h i s i s diagrammed above i n F i g u r e 7. ' The C-P argument runs as f o l l o w s : a s i m i l a r and r e c o r d v a l u e of average p r o d u c t s was o b s e r v e d d u r i n g t h e war y e a r s a t B' and C'; t h i s a v e r age p r o d u c t i s termed the " o p t i m a l e f f i c i e n c y . " F o r a g i v e n y e a r , OA i s t h e a l l o w a b l e c a t c h , and t h e p r o d u c t i o n f u n c t i o n i s OA'A". I f average v e s s e l c o s t s a r e c o n s t a n t , t h e t o t a l c o s t c u r v e i s OD. I f t h e r e a r e B^ b o a t s f i s h i n g , t h e n r e n t i s c o m p l e t e l y d i s -s i p a t e d . A c c o r d i n g t o C-P, t h e f l e e t s i z e c o u l d be r e d u c e d t o B^ (B„ B.. b e i n g t h e amount of " u n n e c e s s a r y gear") t o a c h i e v e t h e • s i B B 1 2 " o p t i m a l e f f i c i e n c y " , and . OA would be t h e p o t e n t i a l r e n t . 50 The t r o u b l e w i t h t h i s approach i s t h a t t h e r e i s no s i n g l e s i m p l e p r o d u c t i o n f u n c t i o n f o r any g i v e n a l l o w a b l e c a t c h l e v e l . T h e r e would be a whole s e r i e s o f s e a s o n a l p r o d u c t i o n f u n c t i o n s f o r each S, F, and a l l o w a b l e c a t c h l e v e l . By a l l o w i n g more f i s h i n g days p e r week, th e s e a s o n a l p r o d u c t i o n f u n c t i o n would s h i f t upwards, say f r o m OP t o OP'. The i n d u s t r y t o t a l c o s t c u r v e would a l s o s h i f t upwards from OD t o OD'. F o r r e n t t o be maximized at B , w i t h a v e r a g e p r o d u c t OA 1, . 2 r e q u i r e s t h a t B be a c o r n e r s o l u t i o n when b o a t s a r e f i s h i n g t h e most days p e r week ( i . e . , seven d a y s ) . T h i s w i l l a g a i n be a c o n s t r a i n e d s o l u t i o n , l i m i t e d by t h e number of days per week, and t h e a l l o w a b l e c a t c h . I f B 2 i s t h e c o n s t r a i n e d o p t i m a l , r e n t w i l l be o v e r e s t i m a t e d by C-P t o t h e e x t e n t t h a t OD' l i e s above OD. T h i s b i a s i n t h e e s t i m a t e s of B r i s t o l Bay p o t e n t i a l r e n t s i s due t o not c o n s i d e r i n g t h e e f f e c t on o p e r a t i n g c o s t s o f a l l o w i n g i n c r e a s i n g f i s h i n g time f o r the s m a l l e r f l e e t s i z e s . The C-P t e c h n i q u e used i n B r i s t o l Bay might be a p p r o p r i a t e t h e r e , where t h e run l a s t s a s h o r t p e r i o d of t i m e , and o p e r a t i n g c o s t s a r e not s i g n i -f i c a n t r e l a t i v e t o c a p i t a l and l a b o u r c o s t s . The t e c h n i q u e , however, s t i l l depends on a r b i t r a r i l y c h o o s i n g a c o n s t a n t a v e rage p r o d u c t which r e f l e c t s " o p t i m a l e f f i c i e n c y " , and t h e assumption t h a t t h i s " o p t i m a l e f f i c i e n c y " i s independent of t h e s i z e o f t h e t o t a l r u n . These assumptions are d e n i e d i n t h e s e a s o n a l p r o d u c t i o n f u n c t i o n d e v e l o p e d i n t h i s s t u d y . In e s t i m a t i n g d i s s i p a t e d r e n t s i n Puget Sound, C-P use data r e p o r t e d e a r l i e r i n a study of salmon gear l i m i t a t i o n i n N o r t h e r n Washington waters (Royce, Bevan, et a l , 1963). The t e c h n i q u e s can be seen by r e f e r e n c e t o F i g u r e 8, which shows i n p a r t t h e p e r boat c o s t s and.revenues shown f o r t h e t o t a l f l e e t i n F i g u r e 6. Average 51 i O B* F i g u r e 8. Average s e a s o n a l c o s t s and revenues per boat as a f u n c t i o n of f l e e t s i z e . r e v e n u e s , AR, i s a r e c t a n g u l a r h y p e r b o l a , average f i x e d c o s t s , AFC, a r e c o n s t a n t , and average v a r i a b l e c o s t s , AVC, a r e d e r i v e d from t h e a p p r o p r i a t e i s o - c a t c h c u r v e shown i n F i g u r e 4. ATC i s t h e average t o t a l c o s t c u r v e . I f t h e r e a r e B b o a t s f i s h i n g now, and average v a r i a b l e c o s t s a r e BC, then a r e d u c t i o n i n t h e f l e e t s i z e w i l l i n c r e a s e v a r i a b l e c o s t s i n p r o p o r t i o n . C-P w r i t e (p. 161): " T o t a l r u n n i n g c o s t s f o r t h e f l e e t would remain e s s e n t i a l l y t h e same, s i n c e r u n n i n g time f o r a l a r g e r number of v e s s e l s w i l l s i m p l y be r e a l l o c a t e d t o a s m a l l e r number f i s h i n g a g r e a t e r number o f days." T h i s assumption g i v e s an average v a r i a b l e c o s t c u r v e i n t h e shape of a r e c t a n g u l a r h y p e r b o l a , C C , and a t o t a l c o s t curve DD'. Though C-P do not e s t i m a t e maximum p o t e n t i a l r e n t s , but merely show r e s u l t s which c o u l d be a c h i e v e d w i t h a 50% r e d u c t i o n i n gear, t h e i r v a r i a b l e c o s t assumption c o u l d s e r i o u s l y u n d e r e s t i m a t e p o t e n -t i a l r e n t s . U s i n g t h e i r approach w i t h v a r i a b l e c o s t s , p o t e n t i a l r e n t at f l e e t s i z e B* would be HEFJ, as compared w i t h AEFG. T h i s approach, th e n , u n d e r e s t i m a t e s p o t e n t i a l r e n t s . C o n c l u s i o n Knowledge o f t h e shape of the s e a s o n a l p r o d u c t i o n f u n c t i o n i s i m p o r t a n t i n e s t i m a t i n g t h e magnitude of d i s s i p a t e d r e n t s . Such a p r o d u c t i o n f u n c t i o n c o u l d a l s o be used i n b e n e f i t / c o s t a n a l y s i s , f o r e s t i m a t i n g a s s o c i a t e d c o s t s o f h a r v e s t i n g f i s h produced by spawn-i n g c h a n n e l s and h a t c h e r i e s . The p r o d u c t i o n f u n c t i o n f o r one f i s h e r y c o u l d be used w i t h s i m i l a r f u n c t i o n s f o r o t h e r f i s h e r i e s i n a s i m u l a t i o n model d e v e l o p e d t o determine t h e o p t i m a l f l e e t s i z e and d i s t r i b u t i o n r e q u i r e d t o h a r v e s t r e t u r n i n g runs of salmon. However, f o r b e t t e r e s t i m a t e s of t h i s f u n c t i o n , i t i s n e c e s s a r y t o o b t a i n b e t t e r data on d a i l y s t o c k s i z e and f i s h m i g r a t i o n t i m i n g and r a t e . T h i s paper, though, has attempted t o demonstrate a t e c h n i q u e of d e v e l o p i n g . s u c h a f u n c t i o n , and t h e i m p o r t a n c e o f c e r t a i n phenomena i n i t s c o n s t r u c t i o n . On a d a i l y b a s i s , d i r e c t and i n d i r e c t g e a r e x t e r n a l i t i e s and gear s a t u r a t i o n e f f e c t s must be t a k e n i n t o account i n i n f l u e n c i n g the average f i s h i n g power. Two o t h e r f a c t o r s a r e o p e r a t i v e throughout t h e s e a s o n : t h e p e a k i n g n a t u r e of t h e r u n , and t h e m u l t i p l e day s t o c k removal e f f e c t . 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FAO: Rome, 1962: 495-510. 5 6 APPENDIX 57 TABLE A-I DAILY CATCH, NUMBERS OF BOATS, STOCK SIZE, AND DUMMY VARIABLES USED IN REGRESSION EQUATIONS S t o c k S i z e F o u r Day F i v e Day Number Assump- Assump-C a t c h of t i o n t i o n Date ( P i e c e s ) B o a t s ( P i e c e s ) D l D2 June 26/67 15 100 310 43 200 58 700 1 1 June 27/67 15 100 300 41 100 74 800 1 0 J u l y 3/67 15 400 560 61 400 75 600 1 1 J u l y 4/67 18 500 500 62 600 76 200 1 0 J u l y 5/67 17 500 450 57 400 66 000 1 0 J u l y 10/67 24 700 532 110 700 165 600 1 1 J u l y 11/67 31 300 532 150 400 170 500 1 0 J u l y 12/67 38 900 532 145 200 178 100 1 0 J u l y 16/67 52 900 662 204 400 233 000 1 1 J u l y 17/67 70 600 662 184 900 222 800 1 0 J u l y 18/67 52 900 662 149 500 192 500 1 0 J u l y 19/67 32 300 550 144 900 192 200 1 0 J u l y 20/67 38 200 550 168 900 204 900 1 0 J u l y 21/67 47 000 550 172 300 208 300 1 0 J u l y 24/67 70 800 656 222 000 260 900 1 1 J u l y 25/67 57 900 600 197 300 235 200 0 0 J u l y 26/67 51 500 600 188 300 216 500 0 0 J u l y 27/67 51 500 550 165 800 185 900 0 0 J u l y 28/67 45 000 500 137 100 163 300 0 0 J u l y 29/67 45 000 500 119 900 148 800 0 0 J u l y 31/67 39 000 600 146 500 177 700 0 1 Aug. 1/67 58 600 600 157 300 179 400 0 0 Aug. 2/67 54 700 550 131 900 144 800 0 0 Aug. 3/67 43 000 500 93 500 106 300 0 0 Aug. 7/67 29 400 636 61 800 68 700 0 1 Aug. 8/67 20 400 600 36 500 44 000 0 0 June 24/68 13 500 363 31 800 43 000 1 1 June 25/68 15 800 363 30 900 50 100 1 0 J u l y 1/68 39 300 406 88 200 105 400 1 1 J u l y 2/68 25 300 406 72 800 90 800 1 0 J u l y 3/68 29 000 406 70 800 81 900 1 0 J u l y 8/68 28 000 609 64 700 72 000 1 1 J u l y 9/68 30 400 500 47 100 53 100 1 0 J u l y 15/68 75 800 545 334 500 374 400 1 1 J u l y 16/68 91 600 550 322 900 351 700 1 0 J u l y 18/68 91 600 550 245 700 267 200 1 1 J u l y 20/68 56 800 740 134 800 178 800 1 1 J u l y 22/68 48 200 620 179 600 212 200 1 1 J u l y 23/68 51 800 517 174 200 191 100 1 0 58 TABLE A - I — C o n t i n u e d S t o c k S i z e F o u r Day F i v e Day Number Assump- Assump-Date C a t c h ( P i e c e s ) of B o a t s t i o n t i o n ( P i e c e s ) D l DS J u l y 24/68 39 300 517 123 200 146 500 0 0 J u l y 27/68 39 300 670 100 300 130 900 0 1 J u l y 28/68 25 600 600 96 200 116 200 0 0 J u l y 29/68 17 400 . 400 97 700 108 500 0 0 J u l y 30/68 39 700 400 93 400 104 300 0 0 Aug. 5/68 9 400 560 38 300 42 400 0 1 Aug. 6/68 8 300 500 31 200 36 900 0 0 June 23/69 8 200 294 23 900 30 100 1 1 June 24/69 7 600 300 21 300 27 700 1 0 June 30/69 8 500 317 29 400 40 200 1 1 J u l y 1/69 15 900 320 33 500 47 200 1 0 J u l y 7/69 34 100 423 97 800 112 000 1 1 J u l y 8/69 26 800 423 78 500 92 700 1 0 J u l y 14/69 39 500 550 120 400 145 700 1 1 J u l y 15/69 39 500 550 110 100 136 700 1 0 J u l y 21/69 62 400 660 168 800 196 600 1 1 J u l y 22/69 52 000 660 145 500 166 200 0 0 J u l y 23/69 58 900 660 116 800 141 000 0 0 J u l y 28/69 88 600 554 129 600 141 800 0 1 Aug. 7/69 47 200 525 109 900 117 500 0 1 Aug. 8/69 30 200 525 70 900 75 200 0 0 Aug. 13/69 11 300 465 19 400 19 500 0 1 59 TABLE A - I I DAILY CATCH, AVERAGE DAILY CATCH PER BOAT, AND MARGINAL DAILY BOAT CATCH, AT DIFFERENT DAILY STOCK SIZE AND FLEET SIZE LEVELS, OBTAINED BY USING EQUATION 4 D a i l y S t o c k S i z e No. of Boats 300 400 500 600 700 D a i l y C a t c h 20 000 8 148 8 970 9 664 10 271 10 813 40 000 13 982 15 392 16 583 17 624 18 555 60 000 19 175 21 109 22 742 24 170 25 447 80 000 23 992 26 411 28 455 30 242 31 840 100 000 28 547 31 426 33 857 35 983 37 884 120 000 32 903 36 222 39 024 41 475 43 666 140 000 37 101 40 843 44 003 46 766 49 237 160 000 41 169 45 321 48 827 51 893 54 635 180 000 45 125 49 676 53 519 56 880 59 885 200 000 48 984 53 925 58 097 61 745 65 007 D a i l y No. o f S t o c k Boats 300 400 500 600 700 S i z e Average C a t c h Per Boat 20 000 27 22 19 17 15 40 000 46 38 33 29 26 60 000 63 52 45 40 36 80 000 79 66 56 50 45 100 000 95 78 67 59 54 120 000 109 90 78 69 62 140 000 123 102 88 77 70 160 000 137 113 97 86 78 180 000 150 124 107 94 85 200 000 163 134 116 102 92 D a i l y No. o f S t o c k B o a t s 300 400 500 600 700 S i z e " " M a r g i n a l D a i l y Boat C a t c h 20 000 9 7 6 5 5 40 000 15 12 11 9 8 60 000 21 17 15 13 12 80 000 26 22 19 16 15 100 000 31 26 22 20 18 120 000 36 30 26 23 20 140 000 41 34 29 26 23 160 000 45 37 32 28 26 180 ooo" 50 41 35 31 28 200 000 54 45 38 34 31 TABLE A - I I I DAILY CATCH, AVERAGE DAILY CATCH PER BOAT ', AND MARGINAL DAILY BOAT CATCH, AT DIFFERENT DAILY STOCK SIZE AND FLEET SIZE LEVELS, OBTAINED : BY ' USING EQUATION 5 •I D a i l y S t o c k No. o f S i z e Boats 300 400 500 600 700 D a i l y C a t c h 20 000 8 056 8 646 9 134 9 553 9 923 40 000 14 006 15 034 15 882 16 610 17 252 60 000 19 357 20 777 21 949 22 956 23 843 80 000 24 353 26 139 27 614 28 880 29 997 100 000 29 099 31 233 32 996 34 509 35 843 120 000 33 657 36 125 38 163 39 914 41 456 140 000 38 062 40 853 43 159 45 138 46 883 160 000 42 342 45 447 48 012 50 214 52 155 180 000 46 515 49 926 52 743 55 162 57 294 200 000 50 595 54 305 57 369 60 001 62 320 D a i l y S t o c k No. of S i z e Boats 300 400 500 600 700 Average C a t c h P er Boat 20 000 27 22 18 16 14 40 000 47 38 32 28 25 60 000 65 52 44 38 34 80 000 81 65 55 48 43 100 000 97 78 66 58 51 120 000 112 90 76 67 59 140 000 127 102 86 75 67 160 000 141 114 96 84 75 180 000 , 155 125 105 92 82 200 000 169 136 115 100 89 D a i l y S t o c k No. o f S i z e B o a t s 300 400 500 600 700 M a r g i n a l D a i l y Boat C a t c h 20 000 7 5 4 4 3 40 000 11 9 8 7 6 60 000 16 13 11 9 8 80 000 20 16 14 12 11 100 000 24 19 16 14 13 120 000 28 22 19 " 16 15 140 000 31 25 21 19 16 160 000 35 28 24 21 18 180 000 38 31 26 23 20 200 000 41 33 28 25 22 61 TABLE A-IV DAILY CATCH, AVERAGE DAILY CATCH PER BOAT, AND MARGINAL DAILY BOAT CATCH, AT DIFFERENT DAILY STOCK SIZE AND FLEET SIZE LEVELS, OBTAINED BY USING EQUATION 9 D a i l y S t o c k No. of S i z e Boats 300 400 500 600 700 D a i l y C a t c h 20 000 6 666 7 746 8 702 9 571 10 373 40 000 11 241 13 063 14 676 16 142 17 494 60 000 . 15 261 17 734 19 925 21 914 23 750 80 000 18 958 22 030 24 751 27 222 29 503 100 000 22 432 26 066 29 286 32 210 34 909 120 000 25 737 29 907 33 602 36 957 40 054 140 000 28 909 33 594 37 744 41 512 44 991 160 000 31 972 37 152 41 742 45 910 49 756 180 000 34 941 40 602 45 618 50 173 54 377 200 000 37 830 43 960 49 390 54 322 58 873 D a i l y S t o c k No. o f S i z e Boats 300 400 500 600 700 Average C a t c h Per Boat 20 000 22 19 17 16 15 40 000 37 33 29 27 25 60 000 ' 51 44 40 37 34 80 000 63 55 50 45 42 100 000 75 65 59 54 50 120 000 86 75 67 62 57 140 000 96 84 75 69 64 160 000 107 93 83 77 71 180 000 116 102 91 84 78 200 000 126 110 99 91 84 D a i l y S t o c k No. of S i z e Boats 300 400 500 600 700 M a r g i n a l D a i l y Boat C a t c h 20 000 12 10 9 8 8 40 000 20 17 15 14 13 60 000 27 23 21 19 18 80 000 33 29 26 24 22 100 000 39 34 31 28 26 120 000 45 39 35 32 30 140 000 50 44 39 36 34 160 000 56 48 44 40 37 180 000 61 53 48 44 41 200 000 66 57 52 47 44 62 TABLE A-V DAILY CATCH, AVERAGE DAILY CATCH PER BOAT, AND MARGINAL DAILY BOAT CATCH, AT DIFFERENT DAILY STOCK SIZE AND FLEET SIZE LEVELS, OBTAINED BY USING EQUATION 10 D a i l y S t o c k No. o f S i z e Boats 300 400 500 600 700 D a i l y C a t c h 20 000 7 273 8 110 8 826 9 458 10 027 40 000 12 575 14 023 15 261 16 353 17 337 60 000 17 323 19 318 21 023 22 527 23 882 80 000 21 743 24 247 26 387 28 275 29 976 100 000 25 934 28 922 31 474 33 726 35 755 120 000 29 952 33 402 36 350 38 951 41 294 140 000 33 831 37 728 41 058 43 995 46 642 160 000 37 595 41 926 45 626 48 890 51 831 180 000 41 261 46 014 50 075 53 657 56 886 200 000 44 842 50 008 54 421 58 315 o 61 823 D a i l y S t o c k No. o f S i z e Boats 300 400 500 600 700 Average C a t c h Per Boat 20 000 24 20 18 16 14 40 000 42 35 31 27 25 60 000 58 48 42 38 34 80 000 72 61 53 47 43 100 000 86 72 63 56 51 120 000 100 84 73 65 59 140 000 113 94 82 73 67 160 000 125 105 91 81 74 180 000 138 115 100 89 81 200 000 149 125 109 97 88 D a i l y S t o c k No. o f S i z e Boats 300 400 500 600 700 Ma r g i n a l D a i l y Boat C a t c h 20 000 9 8 7 6 5 40 000 16 13 12 10 9 60 000 22 18 16 14 13 80 000 27 23 20 18 16 100 000 33 27 24 21 19 120 000 38 32 28 25 22 140 000 43 36 31 28 25 160 000 47 40 35 31 28 180 000 52 44 38 34 31 200 000 57 47 41 37 33 TABLE A-VI SEASONAL CATCH AT DIFFERENT LEVELS OF SEASONAL STOCK SIZE, FLEET SIZE, AND FISHING DAYS PER WEEK, DERIVED FROM THE EQUATION 10 DAILY PRODUCTION FUNCTION ESTIMATE PCPULAT10N• NO. OF ?OATS DAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT DAYS AVG CATCH / BOAT DAY ESCAPEMENT . 1,000,000 100 1 104,116 1,041 600 ' 174 895,884 1,000,000 100 2 195,698 1,957 1,200 163 304,302 1,030,000 n o 3 281,116 2,811 1,800 156 718,834 1,000,000 100 4 361,6*9 3,617 2,400 151 638,311 1,000,000 100 5 433,386 4, 384 3,000 . 146 561,614 1,000,000 100 6 511,378 5,114 3,600 142 488,622 I,000,000 100 7 576,243 5,762 4,200 137 423,757 1,000,000 2^0 1 . 135, 400 677 1,200 113 864,600 1,000,000 200 2 • 250,160 1,251 ' 2 ,400 104 749,840 1,000,000 200 3 354,632 1,773 3,600 59 64-;. 363 I,000.000 200 4 451,650 2,258 4, 800 94 543,J10 1,000,000 . 200 5 543,329 2,717 6,000 91 456,67.. 1,000.000 200 6 629,354 3, 149 7,200 37 370,146 1,000,000 200 7 704,637 3,523 8,400 84 295,363 1,COO,000 300 1 157,892 526 1,800 88 842,103 1,000,000 300 2 2 8 5.053 960 3,600 80 711,947 1,000,000 300 3 404,471 1, 348 5,400 75 595,529 1,000,000 300 • 4 511,602 1, 7C5 7,200 71 483,398 1,000,000 300 5 612,409 2,041 9, 000 63 387,591 1,000,000 300 6 707,216 2,357 10,800 65 292.784 1,000,000 300 7 787,506 2,625 12,600 63 212,494 1,000,000 410 • 1 176,083 440 2,400 73 323,917 1.000,000 400 2 317,907 795 4,800 66 . 682,093 1,000,000 400 3 ' 442,955 1,107 7,200 62 557,045 1.000,000 4^0 4 557,264 . 1,393. 5,600 58 442,736 1,000,000 400 5 664,683 1 ,662 12,000 55 335,317 1,000,000 400 6 765,495 1,914 14,400 53 234,505 1.000,000 400 . 7 849,435 2, 124 16,800 51 150,565 1,000,000 500 1 191,622 383 3,000 64 803,378 1,000,000 500 2 342,349 686 6,000 57 657,151 1,000.000 500 3 474,563 949 9, 000 53 525,437 1,000,000 500 4 594,378 1, 189 12,000 50 405,622 1,000,000 500 5 706,961 1,414 15,000 . 47 293,039 1,000,000 500 6 312,487 1,625 18,000 . 45 187,513 I.000.000 500 7 899,062 1,798 21,000 43 100,938 1.000,000 600 1 205,332 342 3,600 57 794,668 1.000,000 600 2 364,422 607 7,200 51 635,573 I,COO,000 600 • 3 501,489 836 10,800 . 46 498,511 1,000,000 600 4 625,719 1,043 14,400 43 374,281 1,000,000 600 5 . 742,528 1,238 18,000 41 257,472 1.000,000 600 6 851,951 1,420 21,600 39 148,049 1,000,000 600 7 940,531 1,568 25,200 - 37 59,469 .1.000,000 ' 700 1 217,685 311 4,200 52 782, 3 15 1,000,000 700 2 383,507 548 8,400 •46. 616,493 I,COO,000 700 3 524,992 750 12,600 42 475,008 1 ,000,000 700 4 652.872 933 16,800 39 347,128 1,000,000 700 5 773,268 1, 105 21,000 37 226,732 1,000,000 700 6 886,005 1,266 25,200 35 113,995 1,000,000 700 7 976,167 1,395 29,400 33 23,833 JABLE A-VI (Continued) PGPULATION NO. OF BOATS DAYS / WEEK . TOTAL CATCH AVG CATCH / BOAT NO. BOAT OAYS AVG CATCH / BOAT DAY ESCAPEMENT 1,200,000 100 1 120,247 1,202 600 200 1,079,753 1,200,000 100 2 226,497 2,265 1,200 189 973,503 1,200,000 100 3 325,803 ' 3,259 1,800 181 874, 107 1,200,000 100 4 419,842 4, 198 2,400 175 . 780,158 1,200,000 . 100 5 509,393 5,094 3.000 170 690,607 1,200,000 100 6 594,716 5 ,947 3,600 165 605,284 1,200,000 100 7 670,806 6,708 4,200. 160 529,194 1,200,000 200 1 156,377 782 1,200 130 1,043,623 1,200,000 200 . 2 230,735 1 ,449 2,400 121 910,265 1,200,000 20.0 3 411,619 2,058 3,600 114 788,381 . 1 ,200,000 20.0 4 525,117 2,626 4,800 109 674,883 1,200,000 200 5 632,391 3,162 6,000 105 567,609 1,200,000 . 200 6 733,791 3,669 7,200 102 466,209 1,200,000 200 7 8 21,830 4, 109 8,400 98 373,170 1,200,000 300 1 182,354 608 1,800 101 1,017,646 1 ,200,000 . 300 2 333,798 1,113 3,600 93 866,202 I,200,000 300 3 469,888 1,566 5,400 87 730,112 1,200,000 300 4 595,423 1,985 7,200 83 - 604,577 1,200,000 • 30.0 5 713,627 2,379 9,000 79 486,373 1,200.000 300 6 824,892 2,750 10,800 76 375,103 1,200,000 300 7 919,637 3,065 12,600 73 230,363 1,200,000 400 1 203,365 508 2,400 85 996,635 1",200,000 400 2 368,562 921 4,800- 77 331,438 I,200,000 400 3 514,980 1.287 7,200 72 685,020 1,200,000 400 4 649,136 1,623 9,600 68 550,364 1,200,000 400 5 775,240 1,938 ~ 12,000 • 65 424,760 1,200,000 400 6 893,676 2,234 14,400 62 306,324 1,200,000 400 • 7 992,897 2,482 16,800 59 207,103 1,200,000 500 1 221,312 443 3,000 74 978,688 1,200,000 500 2 397,636 795 6,000 66 802,364 1,200,000 ' 500 3 552,083 1,104 . 9,000 61 647,917 1,200,000 ' 500 4 692,835 1,386 12,000 58 507,115 1,200,000 500 5 825,174 1,650 15,000 55 374,326 1,200,000 500 6 949,236 1,898 18,000 53 250,764 1,200,000 500 7 1,051,725 2, 103 21,000 50 . 143,275 1,200,000 600 1 237,145 395 3,600 66 962,855 1,200,000 . 600 2 422,804 705 7,200 59 777,196 1,200,000 600 ' 3 583,745 973 10,800 54 616,255 1,200,000 600 ' 4 729,897 1,216 . 14,400 51 •470, 103 1,200,000 600 5 867,262 . 1,445 18,000 48 332,738 1,200,000 600 6 995,972 1,660 21,600 46 204,028 1,200,000 600 7 1,100,958 1,835 25,200 • 44 99,042 1,200,000' 700 . 1 251,411 359 4,200 60 948,589 1,200,000 700 2 445,092 636 8, 40.0 53 754,908 1,200,000' 700 3 611,418 873 12,600 49 588,582 1,200,000. 700 4 762,021 1,089 16,800 45 437,979 1,200,000 700 5 903,673 1,291 21,000 43 296,327 1,200,000 700 6 1,036,350 1.481 25,200 41 163,650 1,200,000 700 7 1,143,313 1,633 29,400 39 56,687 1,200,000 • 800 1 264,466 331 4,800 55 935,534 1,200,000 300 2 465,163 581 9,600 48 734,837 1,200,000 800 3 636,041 795 14,400 44 563,959 1,200,000 800 4 790,405 988 19,200 41 409,595 1,200,000 800 5 935,789 1,170 24,000 . 39 264,211 1,200,000 800 . 6 1 ,071,926 1,340 28,800 37 123,074 1,200,0*00 800 7 - 1,180,501 1,476 33,600 35 19,499 TABLE A-VI (Continued) POPULATION NO. OF BOATS DAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT DAYS AVG CATCH / BOAT DAY ESCAPEMENT 1,400,000 100 1 135,823 1,358 600 226 1,264,177 1,400,000 n o 2 256,272 2,563 1,200 2 14 1.143,728 1,400,000 TOO 3 369,232 3,692 1,800 205 - 1,030,768 1,400,000 100 4 476,184 4,762 2,400 193 923,816 1,400,000 100 5 578,233 5,782 3,000 193 821,767 1,400,000 .100 6 675,570 6,756 3,600 138 724,430 1,400.000 100 7 762,618 7,62o 4,200 . 182 637,382 1,400,000 2 no 1 176,633 8 83 1,200 147 1,223,367 1,400,000 200 2 328,016 1,640 2,400 137 I ,071,984 1,400,000 200 3 466,822 2,334 3,600 130 933,178 1,400,000 200 4 596,324 2,982 4,500 124 803,676 l ,400,000 200 5 718,8 46 3, 594 6,000 120 ' 681,154 1,400,000 200 6 . 834.759 4,174 7,200 116 565,241 1,400,000 200 7 935,785 4,679 8,400 111 464,215 1,400,000 300 I 205,971 687 1,300 114 1,194,029 1,400,000 300 2 378,059 1,260 3,660 105 1,021,941 1,4 00,000 300 3 533.289 1 ,778 5,400 99 866,711 1,400,000 300 4 676,761 2,256 7,200 94 723,239 1,400,000 ' 300 5 81 1,953 • 2,707 9,000 90 583,047 1,400,000 3">0 6 939,318 3,131 10,800 87 460,682 1,400,000 300 7 1,048,241 3,494 12,600 83 351,759 l ,400,000 400 1 229,700 574 2,400 96 1,170,300 i , 4 0 0 , o o o 400 2 417,580 1 ,044 4,800 87 982,420 1,400,000 400 3 584,813 1,462 7,200 81 815.187 1,400,000 400 . 4 738,345 1,846 9,600 77 661,655 1,400,000 400 5 882,721 2,207 12,000 74 517,279 1,400,000 400 6 1,018,402 2,546 14,400 71 381,598 1,400,000 400 7 • 1,132,639 2,832 16,800 67 267,361 1,400,000 500 1 249,974 500 3,000 83 1,150,026 1,400,000 510 ' • 2 450,670 901 6,000 75 949,330 1,400,090 500 3 627,279 1,255 9,000 70 772,721 1,400,000 500 4 788,591 1,577 12,000 66 611,409 1,400,000 510 5 940,163 . 1,880 15,000 63 459,837 1,400,000 '500 6 1,082,382 2,165 18,000 60 317,618 1,400,000 500 . .7 1,200,523 2,401 21,000 57 199,477 1,400,000 600 1 267,857 446 3,600 74 1,132,143 1,400,000 600 2 479,332 799 7,200 67 920,663 1,400,000 610 3 663,560 . 1,106 10,800 61 736,440 1,400,000 6"0 4 831,162 1 ,3es 14,400 58 568,838 1,400,000 610 5 988,640 1,648 18,000 55 411,360 1,400,000 600 6 1,136,260 1,894 21,600 53 263,740 1,400,000 . 600 7 1,257,404 2,096 25,200 50 142,596 1,40u,000 710 1 283,971 406 4,200 68 1,116,029 I , 4 . .0,000 . 7.10 2 504,733 721 8,400 60 895,267 i,420 , 0 0 0 7^0 3 695,317 993 12,600 55 704,633 1,4.,,J00 10 4 868.158 1,240 16,800 52 531,842 1,400,000 700 5 1,030,645 1,472 21,000 49 369,355 1,400,000 710 6 1,182,866 1,690 25,200 47 217,134 1,400,000 710 7 1,306,401 1,866 29,400 44 93,599 1,400,000 810 1 298,714 373 4,800 62 1,101,286 1,400.000 , 810 2 527,619 660 9,600 55 872,381 1,400,000 810 3 723,599 904 14,400 50 676,401 1,400,000 800 4 900,895 1, 126 19,200 47 499,105 1,400,000 8^0 ' 5 1,067,724 1,335 24,000 . 44 332,276 1,400,000 800 6 1,223,962 . 1,530 . 28,800 42 176,038 l,4C0«000 800 7 1,349,447 1,687 33,600 40 50,553 TABLE A-VI (Conti nued) PC PULAT I ON NO. OF BOATS CAYS / WEEK TOTAL CATCH I,600,000 100 1 150,933 1,600,000 no 2 285,195 1,600,000 100 3 4 11,367 I,600,COO 100 4 531,002 1,600,000 100 5 645,263 1,600,000 100 6 754,344 1,600,000 • 100 7 852,120 1,600,000 290 I 196,284 1,600,000 200 2 365,212 1,600,000 200 3 520,524 1,600,000 . 200 4 665,670 1,600,000 2^ 0 5 303,107 1,600,000 200 6 933, 245 1,600,000 210 7 1,047,009 1,600,000 300 1 228,890 1 ,600,000 310 2 421,084 1,600,000 300 3 • • 595,004 1,600,000 300 4 756,029 1,600,000 300 5 907,862 1,600,000 300 6 1,051,003 1,600,000 300 7 1,173,874 1,600,000 400 1 255,260 1,600,000 400 2 465,246 1,600,000 400 3- 652,822 1,600,000 400 4 825,327 1,600,000 400 5 937,613 1,600,000 400 6 1,140,214' 1,600,000 - 400 7 ' 1,269,243 1,600,000 500- 1 277,785 1,600,000 50,0 2 502,235 1,600,000 500 3 700,523 1,600,000 500 4 881,944 1,600,000 500 5 1,052,436 1,600,000 500 6 1,212,484 1,600,000 500 .7 1,346,056 1,600,000 600 1 297,658 1,600,000 600 2 534,310 1,600,000 6"0 •' 3 741,335 1,600,000 600 4 929,977 1,600,000 600 5 1.107,211 1,600,000 600 6 1,273,409 1,600,000 600 ? 1,410,488 1,600,000 , 700 1 315,568 1,600,000 700 2 562,754 1,600,000 710 3 777,094 1,600,000 • 700 4 971,762 1,600,000 700 5 1,154,714 1,600,000 700 6 1,326,152 I,600,000 . 700 7 1,466,045 1,600,000 800 1 331,950 1,600,000 800 2 538,391 1,600,000 800 3 308,965 1,600,000 800 . 4 1,003,777 1 ,600,000 810 5 1,196,692 1,600,000 800 6 1,372,706 1,600,000 800 7 1.514,896 CATCH / BOAT NO. OOAT DAYS AVG CATCH / 30AT DAY ESCAPEMENT 1 , 509 600 252 1,449,067 2,852 1,200 23S 1,314,805 4,114 1,300 229 1,188,633 5, 310 2,400 221 1,063,993 6,453 3,000 215 .954, 737 7,54 3 , 3,600 210 345,656 8, 521 4,200 203 747,380 98 1 1,200 164 1 ,403, 716 1,826 2,400 152 1,234,788 2,603 '3,600 145 I ,079,476 3,323 4,800 139 934,330 4,016 6,000 134 796,893 4,666 7,200 . 130 666,755 5,235 3,400 i25 552,991 763 1 , 800 127 1,371, 110 1,404 . 3,600 1 17 I.173,916 1,983 5,400 1 10 1,004,996 2,520 7,200 105 843,971 3,026 9,000 101 692,138 3,503 10,800 97 543,997 3,913 12,600 93 426,126 638 2,400 106 1,344,740 1, 163 4,800 97 1,134,754 1,632 7,200 91 947,178 2,063 9,600 86 774,673 2,469 12,000 82 612,387 2,851 14,400 79 459,736 3,173 16,800 76 330,752 556 3,000 93 1,322,215 1,004 6,000 84 1,097,765 1,401 9.000 78 899,477 1,764 12,000 73 718,056 2,105 15,000 70 547,564 2,425 18,000 67 387,516 2,692 21,000 64 253,944 496 3,600 83 1,302,342 891 7,200 74 1,065,690 1,236 10,800 69 858,665 1 ,550 14,400 65 670,023 1,845 18,000 62 492,789 2, 122 21.600 59 326,591 2,351 25,200 56 189,512 451 4,200 75 1,284,432 804 8,400 67 1,037,246 1,110 ' 12,600 62 822,906 1,388 16,800 . 58 628,238 1,650 21.000 55 445,286 1,895 25,200 53 273,348 2,094 29,400 50 133,955 415 4, 800 69 1,268,050 735 9,600 61 1,011,609 1,011 14,400 56 791,035 1,261 19,200 . 53 591,223 1,496 24,000 50 403,308 1,716 28,800 48 227,294 1,894 33,600 4S 85,104 TABLE A-VI (Continued) POPULATION NO. CF BOATS DAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. GOAT DAYS AVG CATCH / BOAT DAY ESCAPEMENT 1,800,000 100 1 165,654 1,657 600 276 1,634,346 1,800,000 100 2 313,396 3,134 1,200 261 1,486,604 .1,800,000 100 3 452,482 4,525 1,800 251 1,347,518 1,800,000 100 4 584,528 5, 845 2,400 244 1,215,472 1,800,000 100 • . 5 . 710,747 7,107 3,000 237 1,089,253 1,800,000 100 6 831,339 8,313 3,600 231 968,661 1,800,000 - 100 7 939,653 9, 397^  4,200 224 860,347 1,800,000 200 1 215,424 ' • 1,077 1,200 180 1,584,576 1,800,000 200 2 401,488 2,007 2,400 167 1,398,512 1,800,000 200 3 572,954 2, 865 3,600 159 1,227,046 1,800,000 200 4- 733,424 3,667 4,800 153 1,066,576 1,800,COO 200 5 885,494 4,427 6,000 148 914,506 1,800,000 200 6 1,029,537 5, 148 7,200 143 770,413 1,800,000 200 7 l,l55i 890 5, 779 8,400 138 644,110 1,800,000 300 1 251,212 837 1,800 140 1,543,738 1,800,000 300 2 463,054 1,544 3,600 129 1,336,9^6 1,800,000 300 3 655,277 2,184 , 5,400 121 1,144,7.23 1,800,000 300 4 833,520 2,778 7,200 116 966,480 1,800,000 '300 5 1,001,684 3,339 9,000 111 798,316 1,800,000 300 6 1,160,339 3, 868 10,800 107 • 639,661 1,800,000 300 7 1,296,945 . 4,323 12,600 103 503,055 1,800,000 400 1 . 280,150 700 2,400 117 1,519,850 1,800,000 400 2 511,744 1,279 4,800 107 1,288,256 1,800,U00 400 3 719,257 1,798 7,200 100 1,080,743 1,800,000 400 4 910,392 2,276 9,600 95 339,608 1,800,000 400 5 1,090,285 2,726 12,000 91 709,715 1,800,000 . 400 6 1,259,530 3,149 14,400 87 540,470 1,800,000 400 . . 7 1 ,403,144 3,508 . 16,800 84 396,856 1,800,000 !>00 I 304,874 610 3,000 102 1,495,126 1,800,000 500 2 552,562 . 1,105 6,000 92 1,247,438 1,800,000 500 3 772,109 1,544 9,000 86 1,027,891 1,800,000 '500 4 973,274 1,947 12,000 81 826,726 1,800,000 500 5 1,162,372 2,325 15,000 77 637,628 1,800,000 500 6 1,339,967 2,680 18,000 74 460,033 1,800,000 500 7 1,488,772 2,978 21,000 7.1 311,228 1,800,000 600 i 326,684 544 3,600 91 1,473,316 1,800,000 600 2 5R7,967' 980 7,200 82 1,212,033 1,800,000 600 3 817,361 1,362 10,800 76 982,639 1,800,000 600 4 1,026,685 1,711 14,400 71 773,315 1,800,000 600 5 1,223,357 2,039 18,000 68 576,643 1,300,000 600 6 1,407,845 2,346 21,600 65 392,155 1,800,000 600 7 1,560,673 2,601 25,200 62 239,327 1,800,000 ' 700 1 346,341 495 4,200 82 1,45.3,659 1,800,000 . 700 2 619,384 833 8,400 74 1,180,616 1,800,000 700 3 857,045 1,224' . 12,600 68 942,955 .1,800,000 710 4 1,073,189 1,533 16,800 64 726,811 1,800,000 700 5 - 1,276,287 1,823 21,000 61 523,713 1,800,000 700 6 1,466,653 2,095 25,200 58 333,347 1,800,000 700 7 1,622,712 2,318 29,400 55 177,288 1,800,000 800 1 364,322 455 4,800 76 1,435,673 1,800,000 800 2 647,716 810 9,600 67 1,152,2 84 1,800,000 800 3 392,451 1,116 14,400 62 907,549 1,800,000 800. 4 1,114,425 1,393 19,200 . 58 685,575 1,800,000 800 5 1,323,096 1,654 24,000 • 55 476,904 1,800,000 eoo 6 1,518,591 1,898 28,800 53 281,409 1,800,000 800 ' 7 1,677,310 2,097 33,600 50 122,690 TABLE A-VI (Continued) POPULATION NO. CF BOATS DAYS / WEEK . TOTAL CATCH AVG CATCH / BOAT NO. BOAT DAYS AVG CATCH / BOAT OAY ESCAPEMENT 2,000,000 no 1 180,034 1,800 600 • 300 1,819,966 2,000,000 100 2 340,969 3,410 1,200 284 1,659,031 2,000,000 100 3 492,707 4,927 1,800 274 1,507,293 2,000,000 no 4 636,924 6,369 2,400 265 1,363,076 2,000,000 100 5 774,877 7,749 . 3,000 25B 1,225,123 2,000,000 100 6 906,772 9,068 3,600 252 1,093,228 2,000,000 too • 7 1,025,452 10,255 4,200 244 974,548 2,000,000 2C0 1 234,124 1,171 1,200 195 1.765,876 2,000,000 ; 2 0 0 2 436,964 2, 185 2,400 182 1,563,036 2,000,000 200 3 624,272 3,121 3,600 173 1,375,728 2,000,000 . 200 4 799,791 3,999 4,800 167 1,200,209 2,000,000 200 5 966,239 4,831 6,000 161 1,033,761 2,000,000 200 ' 6 1,124.066 5, 620 7,200 156 875,934 2,000,000 200 7 1,262,726 6,314 8,400 150 737,274 2,000,000 . 300 1 273,015 910 1,800 152 1,726,985 2,000,000 300 2 504,103 1,680 ' ... 3,600 140 1,495,897 2,000,000 300 . 3 714,290 2,381 5,400 132 1,285,710 2,000,000 300 4 ._ 909,460 3,032 7,200 126 1,090,540 2,000,000 300 5 1,093,698 3,646 9,000 122 906,302 2,000,000 300 6 1,267,613 4,225 10,800 117 732,387 2,000,000 300 7 •1,417,783 4,726 12,600 113 532,217 2,000,000 400 1 r . . 304,465 761 2,400 127 1,695,535 2,000,000 400 2 557,235 1 ,393 4,800 116 1,442,765 2,000,000 400 3 784,329 1,961 7,200 109 1,215,671 2,000,000 400 4 993,780 2,484 9,600 . - 104 1,006,220 2,000,000 400 5 1,191,004 . 2,978 • 12,000 99 808,996 2,000,000 400 • 6 1,376,649 3,442 14,400 96 623,351 2,000,000 400 7 1,534,674 3,837 16,800 . 91 465,326 2,000,000 500 1 331,338 663 3,000 110 1,668,662 2,000,000 500 2 601,799 1,204 6,000 100 1,398,201 2,000,000 500 3 842,237 1,684 9,000 94 1,157,763 2,000,000 500 4 1,062,842 2,126 12,000 89 937,158 2,000,000 500 5 1,270,267 2,541 15,000 85 . 729,733 2,000,000 500 6 1,465,154 2,930 18,000 81 534,846 2,000,000 . 500 '7 1,629,011 3,258 21,000 78 370,989 2,000,000 600 ' 1 355,044 592 3,600 99 1,644,956 2,000,000 600 2 640,474 1,067 7,200 89 1,359,526 2,000,000 600 3 891,858 1,486 10,800 83 1,108,142 2,000,000 600 4 1,121,550 1,869 14,400 78 878,450 2,000,000 600 5 1,337,372 2,229 18,000 • 74 662,628 2,000,000 600 6 1,539,899 2,566 21,600 71 460,101 2,000,000 • 600 7 1,708,301 2,847 25,200 68 291,699 2,000,000 700 1 376,402 538 4,200 90 1,623,598 2,000,000 • 700 2 674,804 964 • 8,400 80 1,325,196 2,000,000 700 3 935,407 1,336 12,600 74 1,064,593 2,000,000 700 4 1,172,712 1,675 16,800 70 827,288 2,000,000 700 5 1,395,669 1,994 21,000 66 604,331 2,000,000 700 6 1,604,707 2,292 25,200 64 395,293 2,000,000 700 7 1,776,766 2,538 29,400 60 223,234 2,000,000 800 1 395,945 495 4, 800 82 1,604,055 2.000,000 8"0 2 705,779 882 9,600 74 1,294,221 2,000,000 800 3 974,287 1,218 14,400 68 1,025,713 2,000,000 810- 4 1,218,106 1,523 19,200 63 781,894 2,000,000 800 5 1,447,254 1,809 24,000 ' 60 552,746 2,000,000 BOO 6 1,661,974 2,077 28,800 58 338,026 2,000,0Ti0, 800 7 1,837,056 2.296 33,600 55 162,944 TABLE A-VI (Continued) POPULATION NO. CF ROATS CAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT DAYS AVG CATCH / BOAT DAY ESCAPEMENT 2.200,000 100 1 194,112 1,941 600 324 2,005,888 2,200,000 100 2 367,983 3,630- 1,200 307 1,8 32,017 2,200,000 100 3 532,142 5,321 1, eoo 296 1,667,858 2,200,000 100 4 688,318 6,883 2,400 287 1,511,682 2,200,000 100 5 837,8 17 3,378 3,000 2 79 1,362,163 2,200,000 100 6 930,841 9,808 . 3,600 272 1 ,219,159 2,200,000 100 7 1,109,723 11,097 4,200 264 1,090,277 2,200,000 200 . 1 252,436 1,262 1,200 210 1,947,564 2,200,000 200 2 471,736 2, 359 2,400 197 1,726,264 2,200,000 2T0 3 674,609 3,373 3,600 187 I,525,391 2,200,000 . 200 4 864,935 4,325 4,800 180 1,335,065 2,200,000 200 5 1,045,535 5,228 6, 000 1 74 1,154,465 2,200,000 • 200 6 1,216,884 6,084 7,200 169 9S3, 116 2,200,000 200 7 1,367,743 6,839 8,400 163 832,257 2,200,COO 300 1 294,366 901 1,800 164 1,905,634 2 ,200', 000 300 2 544,345 1,814 3, 600 151 1,655,655 2,200,000 300 3 772,194 2,574 5,400 143 1.427,806 2,200,000 300 4 934,024 3,280 7,200 137 1,215,976 2,200,000 300 5 1,184,097 3,947 9,000 132 1,015,903 ,2,200,000 300 6 1,373,064 4,577 10,800 127 826,936 2,200,000 300 7 1,536,62R 5,122 12,600 122 663,372 2,20n ;CCO 400 .1 328,275 821 2,400 137 1,871,725 2,200,000 400 2 601,837 1,505 4,800 125 1,598,163 2,200,000 400 3 848,193 2,120 7,200 118 1,351,807 2 ,200,000 400 4 1,075,696 . 2,689 9,600 112 " 1,124,304 2,200,000 400 5 1,290,001 3,225 12,000 108 909,999 2,200,000 400 6 1,491,818 3,730 14,400 104 708,182 2,200,000 400 7 1,664,093 4, 160 16,800 99 535,907. 2,200,000 500 " 1 357,249 714 3,000 119 1.842,751 2,200,000 500 2 650,080 1,300 • 6,000 ; 108 1,549,920 2,200,000 . 500 3 911,075 1,822 9,000 101 1,288,925 2,200,000 500 4 1,150,841 2,302 12,000 96 1,049,159 2,200,000 500 5 1,376,342 2,753 15,000 92 823,658 2,200,000 500 6 1,588,303 3.177 18,000 88 ' 611,697 2,200,000 500 • 7 1,767,053 3,534 21,000 84 432,947 2,200,000 6^0 1 382,808 • 638 3.600 106 1,317,192 2,200,000 600 2 691,965 1,153 7,200 96 1,508,035 2,200,000 '. 600 3 965,001 . 1,608 10,800 89 1,234,999 2,200,000 600 . 4 • ' ' 1,214,771 2,025 14,400 84 985,229 2,200,000 600 5 1,449,499 2,416 18,000 81 750,501 2,200,000. 600 6 1,669,835 2,783 21,600 77 530,165 2,200,000 • 600 7 1,853,655 3,089 25,200 74 346,345 2,200,000 • .' 700 1 405,841 580 4,200 97 . 1,794,159 2,200,000 700 2 729,165 1,042 8,400 87 1,470,835 2,200,000 7"0 3 1,012,364 1,446 12,600 80 1,187,636 2,200,000 700 4 1,270,534 1,815 • 16,800 76 929,466 2,200,000 700 5 1,513,096 2,162 21,000 72 686,904 2,200,000 700 6 1,740,579 2,487 25,200 69 459,421 2,200,000 700 7 1,928,486 2,755 29,400 . 66 271,514 2,200,000 800 1 426,909 534 4,800. 89 1,773,091 2 ,200,000 eoo 2 762,732 953 9,600 79 1,437,268 2,200,000 800 3 • 1 ,054,663 1,318 14,400 73 1 , 145,337 2,200,000 eoo 4 . 1,320,040 .1,650 19,200 69 879,960 2,200,000 eoo 5 1,569,411 1,962 24,000 65 630,589 2,200,000 800 - 6 1 ,803,126 2,254 28,800 63 396,874 2,200,000 600 7 1,994,422 2,493 33,600 59 205,573 cn TABLE A-VI (Continued) POPULATION NO. CF "OATS OAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT OAYS AVG CATCH / BOAT DAY ESCAPEMENT .2,400,000' 100 1 207,926 • 2,079 600 347 2,192,074 2,400,000 100 2 394,505 3,945 1,200 329 2,005,495 2,400,000 . 100 3 570,879 5,709 1,800 317 1,829,121 2,400,000 100 4 738,827 7,388 2,400 308 1,661,173 2,400,000 100 5 899,683 8,997 . 3,000 300 1,500,317 2,400,000 100 6 1,053,667 10,537 3,600 293 1,346,333 2,400,000 . 100 7 1,192,625 11,926 ' 4,200 284 1,207,375 2 ,400,000 200 I 270,398 1,352 1,200 225 2,129,602 2 ,400,000 200 2 505,876 2,529 2,400 211 -1,894, 124 2 ,400,000 200 3 724,065 3,620 3,600 201 1,675,935 2,400,000 200 4 928,976 4,645 4, 800 194 1,471,024 2,400,000 ' - 200 5 1,123,531 5,618 6,000 187 1,276,469 2 ,400,000 200 6 1,308,225 6,541 7,200 ' 182 1,091,775 2,400,000 ' 200 7 1,471,124 7,356 8,400 175 928,876 2,400,000 300 " 1 315,314 1,051 1,800 175 2,084,686 2,400,000 300 -. 2 583,867 1,946 3,600 162 1,816,133 2,400,000 300 3 829,109 2,764 5,400 154 . 1,570,891 2,400,000 300 4 1,057,358 3,525 7,200 147 1,342,642 2 ,400,000 300 5 1,273,052 4,244 9,000 141 1,126,948 2,400,000, 300 6 1,476,876 4,923 10,800 137 923,124 2,400,000 300 7 1,653,684 5,512 12,600 131 746,316 2 ,400,000 400 1 351,637 879 2,400 147 2,048,363 2,400,000 400 2 645,645 1,614 • 4,800 135 1,754,355 2,400,GOO 400 3 910,976 2,277 • 7,200 127 1,489,024 2,400,000 400 4 1,156,278 2,891 9,600 120 1,243,722 2,400,000 400 5 1,387,444 3,469 12,000 • -116 1,012,556 2,400,000 400 ' 6 1,605,242 4,013 14,400 111 794,758 2,400,000 400 7 1,791,609 4,479 16,800 107 608,391 2,400,000 . 500 . I • 3B2.670 765 3,000 128 2,017,330 2,400,000 500 2 697,507 1,395 6,000 116 1,702,493 2,400,000. 500 3 • • 978,765 1,958 9,000 109 1,421,235 .2,400,000 . 5 0 0 4 1,237,435 2,475 12,000 103 1,162,565 2,400,000 . 500 • 5 1,480,786 2,962 15,000 99 919,214 2,400,000 500 6 1,709,613 3,419 • 18,000 95 690,387 2,400,000 500 .7 1,903,118 3,806 21,000 91 496,882 2,400,000 600 1 410,051 683 3,600 114 1,989,949 2,400,000 . • 600 2 742,552 1,238 7,200 103 1,657,448 2,400,000 600 3 1,036,935 1,728 10,800 96 1.363,065 2,400,000 600 4 1,306,527 .2.178 14,400 91 1,093,473 2,400,000 600 5 1,559,931 2,600 18,000 87 840,069 2 ,400,000 600 6 1,797,876 2,996 21,600 • 83 602,124 2,400,000 600 7 1,996,966 3,328 25,200 79 403,034 2,400*000 700 1 434,718 621 4,200 104 1,965,282 2,400,000 700 2 782,567 1,118 8,400 93 1,617,433 2,400,000 700 3 1,088,047 1,554 12,600 86 1,311,953 2,400,000 700 4 1,366,833 1,953 16,800 81 1,033,167 2,400,000 700 5 1,628,769 2,327 21,000 78 771,231 2,400,000 700 6 1,874,492 2,678 25,200 . 74 525,508 2,400,000 700 7 2,078,110 2,969 29,400 71 321,690 2,400,000 800 1 457,285 572 4,800 95 1,942,715 2,400,000 eoo 2 818,692 1,023 9,600 85 1,581,303 2,400,000 800 3 1,133,734 1,417 14,400 79 1,266,266 2,400,000 800 . 4 1,420,412 1,776 19,200 74 979,588 2,400,000 810 5 1,689,765 2, 112 24,000 70 710,235 2,400,000 810 6 1,942,270 2,428 28,800 67 457,730 2,400,000 800 7 2,149,648 2,687 33,600 64 250,352 TABLE A-VI '(Continued) POPULATION NO. OF 30ATS DAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT DAYS AVG CATCH / BOAT DAY ESCAPEMENT 2,600,000 100 1 221,499 2,215 600 369 2,378,501 2,600,000 100 2 420,583 4,206 1,200 350 2,179,417 2,600,000 no 3 608,9S5 6,090 1,800 338 • 1,991,015 2,600,000 no 4 788,525 7,385 2,400 329 1,811,475 2,600,000 no 5 960.5 90 9,606 3,000 . 320 1,639,410 2.600,000 100 6 1,125.389 11,254 3,600 313 1,474,611 2,600,000 100 7 1,274,293 12,74'3 4,200 303 1,325,707 2,600,000 200 1 238,045 1,440 1,200 • 240 2,311,955 2,600,000 200 2 539,448 2,697 2,400 225 2,060,552 2,600,000 210 3 772,731 3,364 3,600 215 1,827,269 2,600,000 200 4 992,028 4,960 4,800 207 1,607,972 2,600,000 200 5 1,200,345 6,002 6,000 . 200 1,399,655 2,600,000 200 6 1,398,218 6,991 7,200 194 1,201,782 2,600,000 200 7 1,573,033 7, 865 8,400 187 1,026,967 2,600,000 300 1 335,895 1 ,120 1,800 187 2,264,105 2,600,000 300 2 622,733 2,076 3,600 173 1,977,267 2,600,000 310 3 885,121 2,950 5,400 164 1 ,714,879 2,600,000 300 4 1,129.575 3,765 7,200 157 1,470,425 2,600,000 ' 300 5 1,360,698 4,536 . 9,000 151 1,239,302 2,600,000 310 • 6 1,579,206 5,264 10,800 146 1,020,794 2,600,000 \ 310 7 1,769,116 5,897 12,600 140 830,884 2,600,000 400 . 1 374,594 936 2 ,400 156 2,225,406 2,600,000 400 2 688,738 I, 722 4,800 143 1,911,262 2,600,000 400 3 972,783 2,432 7,200 135 1,627,217 2,600,000 . 410 4 1,235,661 3,089 9,600 129 1,364,339 2,600,000 400 5 . 1,483,484 3,709 12.00U 124 1,116,516 2,600,000 410 6 1,717,079 ' 4, 293 14,400 . 1 1 9 382,921 2,600,000 400 7 1,917,406 4,794 16,800 114 682,594 2.600,000 500 1 407,652 315 3,000 136 2,192,348 2,600,000 500 . 2 744,162 1,488 6,000 124 1,855,338 2,600,000 500 3 1,045,411 2,091 9,000 116 1,554,589 2.600,000. 500 4 1,322.751 2,646 12,000 110 1,277,249 2,600,000 500 5 1,583,748 3,167 15,000 106 1,016,252 2,600,000 510 6 1,829,258 3,659 13,000 102 770,742 2,600,000 500 . 7. 2,037,377 4, 075 21,000 97 . 562,623 2,600,000 610 1 436,813 728 3,600 121 2,163,187 2,600,000 610 2 792,314 1,321 7,200 110 1,807,686 2,600,000 600 3 1,107,766 1 ,346 10,800 103 1,492,234 2,600.000 600 4 1,396,954 2,328 14,400 97 1,203,046 2,600,000 610 5 1,668,818 2,781 18,000 93 931,182 2,600,000 600 6 1,924,180 3,207 21,600 89 675,320 2,600,000 600 7 2,138,416 3,564 25,200 35 461,534 2,600,000 700 ' 1 463,097 662 4,200 110 2,136,903 2,600,000. 7"0 2 835,110 1,193 8,400 99 ' 1,764,890 2,600,000 710 3 1,162,594 1,661 12,600 92 1,437,406 2,600,000 710 4 1,461,758 2,038 16,800 87 1,138,242 2,600,000 700 5 1,742,858 2,490 21,000 83 857,142 2,600,000 700 6 2,006,625 2,867 25,200 80 593,375 2,600,000 710 7 2,225,324 3, 180 29,400 76 374,176 2,600,000 . 810 1 487,136 609 4, 800 101 2,112,364 2,600,000 800 2 873,756 1,092 9,600 91 1,726,244 2,600,000 810 3 1,211,622 1,515 14,400 84 1,388,378 2,600,000 810 4 1,519,357 1, 399 19,200 79 1,080,643 2,600,000 810 5 1,808,486 2,261 24,000 . 75 791,514 2,600,000 810 6 2,079,588 2,599 28,800 72 520,412 2,600,0X10 800 7 2,302,909 2,879 33,600 69 297,091 TABLE A-VI (Continued) POPULATION. NO. OF BOATS • DAYS / WEEK TOTAL CATCH AVG CATCH / BOAT !I0. BOAT 04YS AVG CATCH / BOAT DAY ESCAPEMENT 2,800,000 100 I 234,353 2,349 600 39 I 2,565,147 2 ,800,000 1<>1 2 446,252 4,463 1,200 372 2, 353, 74?. 2,100,000 100 3 646,510 6,465 1 ,800 359 2,153,490 2,800,000 100 4 837,495 8,375 2,400 349 1,962,505 2,800,000 . 100 5 1,020,614 10,206 3,000 340 1,779,386 2,800,000 1"0 6 1,196,osn 11,961 , 3,600 332 1,603,912 2,800,000 ino 7 1,354,824 13,548 4,200 323 1,445,176 2,800,000 200 . 1 305,416 1.527 1,200 255 2,494,584 2,800,000 200 2 572,510 • 2,863 2,400 239 2,227,490 2,800,000 200 . 3 820,680 4, 103 3,600 228 1,979,320 2,800,000 2"0 4 1,054,175 5,271 4, 800 220 1,745,825 2,800,000 200 • 5 1,276,097 6,380 6,000 213 1,523,903 2,800,000 200 . 6 1,486,989 7,435 7,200 207 1,313,011 2,800,000 200 7 1,673,587 8,368 3,400 199 1,126,413 2,800,000 300 1 356,149 1,187 1,800 198 2,443,851 2,800,000 3"0 2 661,011 2,203 3,600 184 2,138,939 2,800,000 300 3 ' 940,318 3, 134 5,400 174 1,859,682 2,800,000 300 4 1,200,780 4,003 7,200 . 167 1,599,220 2,800,000 3"0 5 1,447,143 4,824 9,000 161 1,352,852 2,800,000 300 6 1,680,171 5,601 10,800 156 1,119,829 2,800,000 300 7 1,883,065 6,277 12,600 149 9 16,935 2,800,000 400 1 397, 175 993 2.400 165 2,402,825 2,800,000 400 2 731,173 1, 828 •4,800 152 2,068,327 2,800,000 400 3 1,033,694 2,584 7,200 144 1,766,306 2.BOO,000 400 4 1,313,942 3,285 9,600 137 1,486,058 2,800,000 400 •• 5 1,578,237 3,946 12,000 132 1,221,763 2,800,000 400 6 1,827,462 4,569 14,400 127 972,538 2,800,000 400 7 2,041,622 5,104 : 16,800 122 753, 3 78 2,800,000 '500 1 432,229 864 3,000 144 2,367,771 2,800,000 500 2 790,113 1,580 6,000 132 2,009,887 2,800,000 500 " 3 1,111,105 2,222 9,000 123 1,688,895 2,800,000 500 4 1,406,913 2,814 12,000 . 117 1,393,087 2,800,000 500 5 1,685,360 • 3,371 15,000 112 1,114,640 2,300,000 ' 500 '. 6 1.947,332 3,895 18,000 108 852,613 2,800,000 500 7 2,169,988 4,340 21,000 103 630,012 2,800,000 600 1 463,155 772 3,600 . 1 2 9 2,336,845 2,800,000 600 2 841,339 1,402 7,200 117 1,958,661 2,800,000 600 3 1,177,605 1.963 10,800 109 . 1,622,395 2,800,000 600 4 . 1,486,164 2,477 14,400 103 1,313,336 2,800,000 610 . 5 1.776,296 2,960 18,000 99 1,023,704 2,800,000 600 6 2,048,906 3,415 21,600 95 751,094 2,800,000 '. 600 7 2,278,157 3,797 25,200 90 521,843 2,800,000 7^0 1 491,018 701 4,200 117 2,308.,982 2,300,000 700 2 886,871 1,267 8,400 106 1,913,129 2,800,000 7no 3 1,236,098 1,766 12,600 98 1,563,902 2,800,000 700 4 1,555,415 2,222 16,800 93 1,244.585 2,800,000 700 5 1,855,487 2,651 21,000 88 944,513 2,800,000 700 . 6 2,137,126 3,053 25,200 85 662,874 2,800,000 700 7 2,371,782 3,388 29,400 81 428,218 2,800,000 . 800 1 516,510 646 4,800 108 2,263,490 2,800,000 800 2 928,004 1, 160 9,600 97 1,871,996 2,800,000 800 3 1,288,431 1,611 14,400 89 1,511,569 2,800,000 8^0 • 4 1,617,007 2,021 19,200 84 1, 182,993 , 2,800,000 800 5 1,925,718 2,407 24,000 80 874,282 2,800,000 . 800 6 2,215,237 2,769 28,800 77 584,763 2,800,000 7 2,454,390 3,068 33,600 73 345,610 TABLE A-VI (Continued) PCPULATION NO. CF ^OATS OAYS / WEEK TOTAL CATCH AVG CATCH / BOAT NO. BOAT OAYS AVG CATCH / BOAT DAY ESCAPEMENT .3,091).000 100 1 24*,009 2,430 600 413 2,751,991 3,000,000 - 110 2 471,552 4,716 1,200 393 2,523,448 3.000,000 100 3 683,510 6,835 . 1 • 800 380 2,316,490 3,000,000 ' 100 4 885,785 8,353 2.400 369 2,114,212 3,000,000 100 5 1,079,835 10,743 3.000 360 1,920,165 3 , 0 0 0 , 0 0 0 100 6 1 ,265,864 12,659 3,600 352 1,734,136 3,000,000 1 0 0 7 1 , 4 3 4 , 3 2 4 14, 343' 4,200 342 1,565,676 3,000,000 200 ,1 322,524 1,613 1,200 269 2,677,476 3,000,000 200 2 605,096 3..025 2,400 252 2,394,904 3,000,000 210 3 867,966 4,340 3,600. 241 2, 132,034 3,000,000 200 4 1,115,490 5,577 4,800 232 1,884,510 3,000,000 2 0 0 5 1,350.858 6,754 6,000 225 1,649,142 3,000,000• 2 0 0 . 6 1,574,633 7,673 7,200 219 1,425,367 3,000,000 210 7 1, •'72,900 3,865 8,400 211 1,227,100 3,000,000 3 1 0 1 376,093 1,254 1,300 209 2,623,902 3,000,000 300 . 2 698,744 2,329 3.600 194 2,301,256 3,000,000 3 1 0 3 994,766 3,316 ' 5,400 1R4 2,005,234 3,COO,000 300 4 1,271,055 4,237 7 , 2 0 0 177 1,728,945 3,000,000 300 5 1,532,503 5,108 9,000 170 1,467,497 3,000,000 300 6 1,779,897 5,933 10,800 165 1,220,103 3,000,000 5-'iU .7 1,995,654 6, 652 12,600 158 1,004,346 3.OHO,COO 410 1 419,427 1,049 2, 400 175 , 2.580,573 3,000,000 400 2 773,016 1,933 4,800 161 2,226,984 3,000,000 400 3 1,093,793 2,734 7,200 152 1,906,207 3,000,000 410 4 1,391,219 3, 478 9,600 145 1,603,781 3,000,000 4-10 5 1,671,813 4,180 12,000 139 1,328,187 3,000,000 400 6 1,936,511 4,34 1 14,400 134 1,063,489 3,COO,000 410 7 2,164,383 5,411 16,800 129 835,612 3,000,000 500 1 456,442 913 3,000 152 2,543,558 3,000,000 500 2 935,423 1,671 6,000 139 2,164,577 1.000.00.0 500 3 1,175,930 2, 352 9,000 131 1 ,824,070 .3,000,000 500 ' 4 1,490,001 2,930 12,000 124 1,509,999 • 3,000,000 500 5 1,785,726 3,571 15,000 119 1,214,274 3,000.000 500 6 2,064,104 4, 128 13,000 115 935,390 3,030,000 500 7 • 2,301,085 4,602 21,000 110 693,915 3,000,000 600 1 489,099. 815 3,600 136 2,510,901 3,COO,000 610 2 , 889,673 1,483 7,200 124 2,110,322 3,000,000 610 3 1,246,523 2,078 10,300 115 1,753.477 3,000,000 610 4 1,574,262 2,624 14,400 109 1,425,738 3,000,000 . 600 5 1,882,435 3, 137 18,000 105 1, 117,515 3,000,000 610 6 . 2,172,176 3,620 21,600 101 827,824 3,000,000 610 7. 2,416,334 4,02 7 .25,200 96 583,666 3,000,000 710 1 51",523 74 1. 4,200 123 2,481,477 3,000,000 . 710 2 937,913 1,340 8,400 112 2,062,087 3,000,000 7 1 0 3 1,308,641 1, 869 12,600 104 1,691,359 5,000,000 710 4 1,647,919 2,354 16,300 98 • 1,352,081 3.000,000 7 1 0 5 1 ,966.781 2,810 21,000' 94 1,033,219 3,000,000 7 1 0 6 2,266,125 3,237 .25, 200 90 733,375 3,000,000 700 7 2,516,134 3,594 29,400 86 483,866 3,000,000 8"0 1 545,440 6?2 4 , eoo 114 2,454,560 3,000,000 8"0 2 981,498 1.227 9,600 102 2,018,502 3,000,000 8-10 3 1,364,240 1,705 14,400 95 1,635,760 3,000,000 BIO 4 1,713,461 2, 142 19,200 89 1,286,539 3,000,000 810 5 2,041,566 2,552 24,000 85 953,434 .3,000.000 810 6 2,349,343 2,937 28,800 82 650,657 3,000,000 810 7 2,604,221 3,255 33,600 78 395,779 

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