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Optimal management of the Fraser River sockeye salmon Gardner, Peter Nigel 1980

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OPTIMAL MANAGEMENT OF THE FRASER RIVER SOCKEYE SALMON By Peter N i g e l Gardner B.A., U n i v e r s i t y of V i c t o r i a , 1968 M.A., Queen's U n i v e r s i t y , 1970 THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Economics We accept t h i s t h e s i s as conforming to the required, standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1980 (c^ Peter N i g e l Gardner, 1980 In p resent ing t h i s t he s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 re fe rence and study. I f u r t h e r agree that permiss ion f o r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n permi s s ion . Department n f ^ C O M & W I G ' S The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 D E - 6 B P 75-51 1 E OPTIMAL MANAGEMENT OF THE •PHASER RIVER SOCKEYE SALMON by Peter N. Gardner ABSTRACT The question to which t h i s study i s addressed i s : can the Eraser River sockeye salmon fi s h e r y be managed i n such a way as to maximize i t s present worth? A review of the exist i n g b i o l o g i c a l and economics l i t e r a t u r e would suggest that such optimal management i s indeed possible. Putting numbers into the th e o r e t i c a l equations and solving for an optimal solution has been based on a three part approach. F i r s t , a Ricker form of recruitment function was used to model the basic interseasonal r e l a t i o n s h i p between spawning escapement and subsequent future recruitment. Second, nonlinear production functions were used to model the harvesting process in a highly c y c l i c a l fishery spread out over a f a i r l y extensive f i s h i n g gauntlet. And t h i r d , i t i s assumed that the manager i s faced with two in t e r - r e l a t e d problems which must be solved simultaneously: he must decide the optimal escapement which has future revenue consequences in terms of size of catch and future cost consequences i n terms of size of the subsequent recruitment (the larger the recruitment, the lower the harvesting costs), and he must decide the least cost s p a t i a l combination of harvesting gear to take the specified catch. The major finding of t h i s study i s that i t i s possible to I i i manage the F r a s e r R i v e r sockeye salmon f i s h e r y i n an optimal manner and to do so would i n c r e a s e i t s present worth s u b s t a n t i a l l y . The use of c y c l e dummy v a r i a b l e s to allow f o r the marked f o u r year c y c l e s i n both r e c r u i t m e n t and h a r v e s t i n g p a t t e r n s plays a major r o l e i n improving parameter e s t i m a t i o n . Nonlinear programming technigues can be developed to allow the simultaneous determination of the o p t i m a l i n t e r t e m p o r a l spawning escapement and the l e a s t c o s t s p a t i a l a l l o c a t i o n of e f f o r t to harvest the optimal c a t c h . The o r i g i n a l c o n t r i b u t i o n o f t h i s d i s s e r t a t i o n l i e s i n i t s use of d e t e r m i n i s t i c models t o e m p i r i c a l l y s o l v e the problem of optimal management of a f i s h e r y . iv TABLE OF CONTENTS ABSTR ACT ...... ... i i LIST OP TABLES v i LIST OF FIGORES v i i i LIST OF FLOWCHARTS ................................. ....-. . i x LIST OF MAPS X ACKNOWLEDGEMENTS . . . . . x i CHAPTER ONE: FISHERIES ECONOMICS AND THE FRASER RIVER SOCKEYE SALMON I INTBODOCTION ................................... 1 (1) The model .2 (2) Making the model operational ...............4 (3) Alternative systems of management ..........10 (4) Survey of the thesis .......................13 I I MANAGEMENT OF THE FfiASEE RIVER SOCKEYE ......... 14 III THE BIOLOGY OF THE FRASER RIVER SOCKEYE ........22 IV ECONOMIC THEORY OF FISHERIES MANAGEMENT ........26 (1) The harvesting model 27 (2) Optimal f i s h e r i e s management ............... 31* (3) Management model ........................... 48 V SUMMARY AND CONCLUSIONS ........................ 53 APPENDIX Derivation of Clark-Munro equilibrium equation ..57 CHAPTER THO: THE RECRUITMENT FUNCTION I STOCK MANAGEMENT 59 II THE RECRUITMENT MODEL ... 65 III SOCKEYE RECRUITMENT FONCTION ...................72 APPENDIX Compilation of catch and escapement data ........80 Recruitment function estimation ................. 85 V CHAPTER THREE: HARVESTING: THE PRODUCTION AND COST FUNCTIONS X IISITfiODCJC'EXON • • • * • • « • * « » • » • * • # • * » • > • • • * • • » S3 I I PRODUCTION FUNCTION ESTIMATION (1) Production f u n c t i o n s p e c i f i c a t i o n ..........98 (2) T e c h n i c a l progress . . . . . . . . . . . . . . . . , v . - 1 0 4 (3) Biomass a v a i l a b i l i t y and c a t c h a b i l i t y . . . . . . 1 0 8 (4) The h a r v e s t i n g p r o d u c t i o n f u n c t i o n s . . . . . . . . 1 1 1 I I I THE HARVESTING COST FUNCTIONS (1) P r o d u c t i o n f u n c t i o n c o n v e r s i o n ............. 115 (2) Least c o s t programming ...••...•.•.••....•*.120 (3) Net p r o f i t comparisons ••i»v-*:Vv«»>,>,»''»'*;>-'":*''*%'»'•«>••'• T27 APPENDIX Com p i l a t i o n of cat c h and e f f o r t data ............ 136 T e c h n i c a l - change .• • « t;*;W'V(/«.»«.>'4*>«vvvf'*>;*'*»•/*•:..'./"..,.>T39 C a l c u l a t i o n of c o s t s and p r i c e s ................. 140 Gear And V e s s e l Value Indexes ,........,.,..,,...151 CHAPTER FOUR: THE OPTIMAL MANAGEMENT SOLUTION X INTfiOD0CTXON -• «-* «< •<• #V*-* •-••» ••• * • •-• *•••*-•/••:•••'•>'"*••-«.*• •,* !l 56 I I CASE ONE: HARVESTING SOCKEYE ONLY . . , , . . . . . . , , . . 1 6 4 I I I CASE TWO: ALTERING DISCOUNT RATES 170 IV CASE THREE: VARYING HARVEST COSTS .............. 173 V CASE FOUR: HARVESTING OTHER SALMON .............176 VI MAXIMIZING PRESENT WORTH 180 VII SUMMARY AND CONCLUSIONS 182 3 IB JLX OG fi A I? HIT • »/»•>: v • *.».: • ** *:-«. • »••- • * * • »•»-:#•. *-• • *••••-••- #•:». •:•-••>>' 1 3 9 v i LIST OF TABLES Table page (3-1) Unconstrained Average H i s t o r i c a l Cycle Catch Of Sockeye Only In 000 fs Of Pounds, Proportions Of Gear Type Area Catches And Resulting Costs And P r o f i t s In OOO's Of 1951 Dollars .. , 1 28 (3-2) Constrained Average H i s t o r i c a l Cycle Catch Of Sockeye Only In 000's Of Pounds, Proportions Of Gear Type Area Catches And Resulting Costs And P r o f i t s In OOO's Of 1951 Dollars .130 (3-3) Unconstrained Average H i s t o r i c a l Cycle Catch Of Sockeye And Other Salmon In OOO's Pounds, Proportions Of Gear Type Area Catches of Sockeye And Resulting P r o f i t s In OOO's Of 1951 Dollars .,131 (3-4) Constrained Average H i s t o r i c a l Cycle Catch Of Sockeye And Other Salmon In OOO's Of Pounds, Proportions Of Gear Type Area Catches Of Sockeye And Resulting P r o f i t s In OOO's Of 1951 Dollars ...... 132 (4-1) Constrained Optimal Catch Of Sockeye Only, H i s t o r i c And Optimal Cycle Year Escapement, Recruitment And Catch In OOO's Of Pounds; Proportion Of Gear Type Area Catches; And Resulting Net P r o f i t In OOO's Of 1951 Dollars Using A Discount Rate Of Three And A Half Percent . . . . . . . . . . . i . . . . . . . . . . . . 165 (4-2) Unconstrained Optimal Catch Of Sockeye Only, Optimal Cycle Year Escapement, Recruitment And Catch In OOO's Of Pounds; Proportion Of Gear Type Area Catches; And Resulting Net P r o f i t In OOO's Of 1951 Dollars Using A Discount Rate Of Three And A (4-3) Constrained Optimal Catch Of Sockeye Only, Optimal Escapement, Recruitment And Catch In OOO's Of Pounds, And Resulting Net P r o f i t In 000's Of 1951 D O.X 1 c U T 3 • • ; • ' * * ' • : # / * * > , « > ' • ' * mm m •* • • ' m - m • ••,*•">.•>"*'•"*.- •/* •••••#171 (4-4) Constrained Optimal Catch Of Sockeye Only, Optimal Cycle Year Escapement, Recruitment And Catch In OOO's Of Pounds And Resulting Net P r o f i t In OOO's Of 1951 Dollars Using A Discount Rate Of Three And A Half Percent And Assuming Opportunity Costs Are 15% Higher For A l l Gear Types .......,.,..,..173 (4-5) Unconstrained Optimal Catch Of Sockeye Only, Optimal Cycle Year Escapement, Recruitment And Catch In OOO's Of Pounds; Proportion Of Gear Type Area Catches; And Resulting Net P r o f i t In 000's Of 1951 Dollars Using A Discount Rate Of Three And A Half Percent And Assuming Opportunity Costs Are v i i Lowered By 15% For O.S. gear Only ...............175 (4-6) Unconstrained Optimal Catch Of Sockeye And Average H i s t o r i c a l C y c l e Catch Of Other Salmon In OOO's Of Pounds; Gear Type Area P r o p o r t i o n s ; And R e s u l t i n g Net P r o f i t In 000«s Of 1951 D o l l a r s Osing A Disccunt Rate Of Three And A H a l f Percent 177 (1-7) C o n s t r a i n e d Optimal Catch Of Sockeye And Average H i s t o r i c a l Cycle Catch Of Other Salmon In 000»s Of Pounds; Gear Type Area P r o p o r t i o n s ; And R e s u l t i n g Net P r o f i t In 000«s Of 1951 D o l l a r s Using A Discount Rate Of Three And A H a l f Percent 179 (4-8) Compounded Net P r o f i t s Or Losses In 000»s Of 1975 D o l l a r s During The Pe r i o d 1947 To 1975, Assuming A l l Management Adjustments C a r r i e d out In The F i r s t Four Years Of The C y c l e And Compounding With A S o c i a l Bate Of Return Of Three And A H a l f Percent ............... ............ ........... ... 181 v i i i LIST OF FIGURES F i g u r e page (2-1) Simultaneous S o l u t i o n of H a r v e s t i n g And Recruitment ,..65 LIST OF FLOWCHABTS Flowchart page (1-1) Optimal Management Program ..........................51 (3-1) Least Cost Program to Harvest Sockeye Assuming Average H i s t o r i c a l Cycle Recruitment, Catch and Escapement ............ ...... . . 1 22 (3-2) Least Cost Program to Harvest Sockeye And Other Salmon Assuming Average H i s t o r i c a l Cycle Recruitment, Catch And Escapement ...............124 (4-1) Program to Maximize Present Worth Of Sockeye Fishery Assuming Sockeye Harvesting Only ................163 X LIST OF MAPS Sap - - , . v ' page (1-1) D i s t r i b u t i o n o f Sockeye Spawning Grounds i n t h e F r a s e r R i v e r Watershed . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 (1-2) Department o f t h e E n v i r o n m e n t , F i s h e r i e s S e r v i c e , S t a t i s t i c a l Map, Showing A r e a s o f C a t c h f o r B r i t i s h C o l u m b i a Waters (Southern H a l f ) . . . , . . . . ,28 x i ACKNOWLEDGEMENTS The i n s p i r a t i o n f o r t h i s t h e s i s came from Dr. A.D. S c o t t and Dr. G.S. Munro both of whom have played an important and s u p p o r t i v e r o l e over the s e v e r a l years i n v o l v e d i n w r i t i n g t h i s d i s s e r t a t i o n . Dr. R.S. Uhler has played an i n s t r u m e n t a l r o l e i n the e s t i m a t i o n and programming i n v o l v e d i n the p r o j e c t . D i s c u s s i o n s with Dr. C.W. C l a r k of the Mathematics department and Dr. C. Walters of the I n s t i t u t e of Animal Resource Ecology have helped t o c l a r i f y the mathematical and b i o l o g i c a l concepts i n v o l v e d i n bio-economic management., O f f i c i a l s i n the Departments of F i s h e r i e s i n Vancouver, B.C. and Olympia, Washington have been very h e l p f u l both i n terms of knowledge of the sockeye s p e c i e s and the p e c u l i a r i t i e s of h a r v e s t i n g and i n a l l o w i n g access to unpublished data. The U n i v e r s i t y of Calgary has very generously donated s e v e r a l thousand d o l l a r s t o the p r o j e c t . And the Department of Economics a t the U n i v e r s i t y o f Calgary has been very understanding i n terms of the l e n g t h o f time r e q u i r e d to complete the t h e s i s . , There are many i n d i v i d u a l s who have helped i n so many ways th a t t h e i r combined c o n t r i b u t i o n has been s u b s t a n t i a l . A l l e r r o r s i n e x p l a n a t i o n and computation, however, are s o l e l y my r e s p o n s i b i l i t y . In p a r t i c u l a r , I wish to acknowledge my wife and c h i l d r e n f o r t h e i r patience and understanding d u r i n g these d i f f i c u l t t h e s i s years. The best thank you I can give them w i l l be my c l o s e a t t e n t i o n from now on.. My pray e r s have been answered: the task i s completed. 1 CHAPTER ONE: FISHERIES ECONOHICS AND THE FJASER SUM SOCKEYE SALMON I INTRODUCTION T h i s t h e s i s i s concerned with i l l u s t r a t i n g how best to manage the E r a s e r River sockeye salmon f i s h e r y . The study does not attempt t o c o n t r i b u t e to the theory o f the optimal management of f i s h e r y s t o c k s . As Peterson and F i s h e r (1977, p.691) have pointed c u t , most of the t h e o r e t i c a l models n cannot be used t o manage a c t u a l n a t u r a l r e s o u r c e s , because t h e i r f u n c t i o n a l forms are too simple and t h e i r e m p i r i c a l content too low. With most f i s h e r i e s , f o r i n s t a n c e , we do not know what the maximum s u s t a i n a b l e y i e l d i s with any degree of accuracy, l e t alone the parameters of the growth and pr o d u c t i o n f u n c t i o n s . With expanded c o - o p e r a t i o n between n a t u r a l s c i e n t i s t s and economists, the e m p i r i c a l base might be developed t o the po i n t where the dynamic o p t i m i z i n g models could o f f e r p r a c t i c a l guidance t o managers of f i s h e r i e s . " The purpose of t h i s t h e s i s i s t o expand the " e m p i r i c a l base" f o r the E r a s e r R i v e r sockeye system and to show how these data can be used i n a dynamic o p t i m i z i n g model to guide the managers of t h i s salmon stock. The b a s i s f o r t h i s dynamic o p t i m i z i n g i s the t h e o r e t i c a l model developed by C l a r k and Munro (1975). Optimal management has been the concern of resource economists ever s i n c e H o t e l l i n g ' s f i r s t attempts t o grapple with the problem i n 1931 ( H o t e l l i n g , 1931). And i t i s only r e c e n t l y t h a t economists have sol v e d the problem on a t h e o r e t i c a l l e v e l . Recent works i n economic l i t e r a t u r e have reduced the t h e o r e t i c a l c o m p l e x i t i e s to the point where i t i s p o s s i b l e to s t a r t i n j e c t i n g e m p i r i c a l m a t e r i a l to handle the p r a c t i c a l problem of 2 optimal management. However, the task of transforming theory in t o empirically useful models i s not simple and requires rather large amounts of information. For t h i s reason the Eraser Eiver sockeye salmon fishery was chosen for study as i t has had a comprehensive data base for a long period of time. JLil 1M Model The need for management of the Eraser Eiver sockeye arises from the i n t e r n a t i o n a l interception problem, the problem of s p a t i a l misallocation of harvesting e f f o r t on the f i s h i n g grounds and the problem of economic rent d i s s i p a t i o n which res u l t s from the open access nature of the resource. The Fraser River sockeye fishery i s a gauntlet type where the f i s h migrate throught the S t r a i t of Juan de Fuca and the S t r a i t of Georgia and into the mouth cf the Fraser River. ,• During migration they pass through both U.S. and Canadian waters and are vulnerable to the f i s h i n g gear of fishermen from both countries. Since 1937 there has been an egual catch agreement between both countries regardless of harvesting costs. One of the problems addressed by t h i s study i s the measurement of the s o c i a l costs imposed by such an agreement. Because the sockeye are not equally dispersed throughout the gauntlet but swim through the gauntlet i n one d i r e c t i o n , gear harvesting e a r l i e r i n the gauntlet can impose costs on gear harvesting l a t e r in the gauntlet by reducing the migration of sockeye. I f the f i s h are easier to catch i n the early part of the gauntlet, there i s no problem. , However, i f harvesting i s more e f f i c i e n t i n l a t e r parts of the gauntlet, there i s a s o c i a l 3 cost involved in thi s s p a t i a l misallocation of harvesting e f f o r t . This study attempts to measure those s o c i a l costs and to indicate how they could he reduced through s p a t i a l r e l o c a t i o n of harvesting e f f o r t . The common property problems associated with open access resources can best be summarized as an intertemporal misallocation of harvesting e f f o r t . Investment i n future migrations of sockeye are made by abstaining from current harvesting. However, i f a fisherman knows he cannot reap the f u l l reward from his investment because of the open access nature of the fish e r y , he w i l l not abstain from current f i s h i n g . The r e s u l t i s declining future migrations leading to smaller and smaller catches. Once again, t h i s study attempts to measure the s o c i a l costs which arise from such a common property problem and to indicate how they could be reduced by a l t e r i n g the intertemporal a l l o c a t i o n of harvesting e f f o r t . In order to manage the Fraser River sockeye f i s h e r y , a manager must work with a three part model. The f i r s t part i s a b i o l o g i c a l system i n which parent f i s h escaping from the nets of fishermen ascend the Fraser River system to spawn and die. The second part of the management model i s an economic system in which fishermen have twelve weeks to harvest the f i s h migrating through the gauntlet. and the t h i r d part of the model i s a management system which allocates harvesting e f f o r t so as to maximize the present value of the fishery through optimal intertemporal a l l o c a t i o n cf that harvesting e f f o r t , taking into account the s o c i a l costs of int e r n a t i o n a l sharing of the catch and the costs of harvesting through least cost s p a t i a l 4 a l l o c a t i o n of that harvesting e f f o r t . J21 Making The Model Operational To make thi s model operational requires a three stage process of obtaining data, estimating parameters and programming to achieve the optimal values. There are nearly two hundred thousand observations on the Fraser River sockeye fishery scattered in various journals and books i n B . C . and Washington which means that data r e t r i e v a l i s a very involved task. Despite these data, many aspects of the management model had to be ignored because of the lack of observations. Furthermore, even where data did e x i s t , seme could not be used because of the need to make the model operational. That i s , there are certain aspects of the problem over which a manager has l i t t l e or no control and these must be excluded even though data ex i s t to aid in devising solutions. When sockeye escape from f i s h i n g gear they migrate up the Fraser Biver system u n t i l they reach t h e i r natal stream where they spawn, bury the f e r t i l i z e d eggs and die. The eggs hatch the following spring and the immature salmon swim to a lake where they spend a year preparing for the migration to ocean feeding grounds the following spring. They spend two winters i n those ocean areas and return to spawn as mature, four year old sockeye by migrating through the f i s h i n g grounds. , The f i s h e r y manager can increase that migration either by enhancing the migration routes and spawning beds or by abstaining from f i s h i n g to allow a greater escapement of parents. Data on enhancement methods fo r the Fraser River 5 sockeye do e x i s t but are r e s t r i c t e d i n a p p l i c a t i o n and d i f f i c u l t to work with. T h e r e f o r e , only escapement and c a t c h data w i l l be analysed. I f we can assume t h a t migration i s the sum of escapement and c a t c h , we can then attempt to e s t i m a t e the parameters o f the r e l a t i o n s h i p between escapement and subsequent re c r u i t m e n t . Data a l s o e x i s t on the escapement of the eleven major r a c e s of F r a s e r fiiver sockeye. However, as i t i s a very complex problem to model the race p r o p o r t i o n s i n the subsequent recru i t m e n t , escapement and r e c r u i t m e n t were analysed i n aggregate form only. (This i s not s t r i c t l y c o r r e c t : see S i c k e r , 1S73b.) Chapter Two of t h i s study p r o v i d e s the parameter e s t i m a t e s f o r t h i s b i o l o g i c a l r e l a t i o n s h i p and i s the b a s i s of the b i o l o g i c a l p a r t of the management model. T h e r e f o r e , the f i r s t r e s t r i c t i v e assumption i s that the manager c o n t r o l s the b i o l o g i c a l system only to the extent that he c o n t r o l s the escapement of spawning parents. Sockeye are harvested by f o u r d i f f e r e n t gear types o p e r a t i n g i n seven f i s h i n g zones i n the g a u n t l e t . Data were not a v a i l a b l e f o r the p r o c e s s i n g i n d u s t r y (canning and f r e e z i n g plants) but were a v a i l a b l e f o r c a t c h and d e l i v e r i e s by gear type i n each zone f o r the twelve week sockeye f i s h i n g season. Data a l s o e x i s t f o r the c a t c h of f o u r other s p e c i e s of salmon bes i d e s sockeye. I f we can assume t h a t d e l i v e r i e s can be transformed i n t o h a r v e s t i n g e f f o r t , i t i s then p o s s i b l e t o e s t i m a t e the parameters o f the r e l a t i o n s h i p between catch and e f f o r t and the m i g r a t i o n of sockeye c a l c u l a t e d e a r l i e r . Chapter Three p r o v i d e s the parameter estimates f o r those h a r v e s t i n g production 6 functions and they form the basis of the economic part of the management model. The second r e s t r i c t i v e assumption of the study i s that the manager controls harvesting only to the extent that certain gear types are permitted to harvest sockeye in ce r t a i n f i s h i n g zones for certain periods of time. No attempt i s made to account for the influence of the processing sector on the harvesting of sockeye, however, i t can be assumed that competition amongst processors i s adequate from an economic point of view. Despite increasing concentration over the l a s t twenty-five years, Japanese, O.S. and Canadian investment in processing in both Washington and B.C. has turned the salmon f i s h i n g industry into an i n t e r n a t i o n a l l y financed industry operating in international trade markets. Fortunately, there i s no need to include processors in the analysis i f we can assume that fishermen are f a i r l y mobile between processors. From discussions with processors, fishermen and o f f i c i a l s i n the departments of f i s h e r i e s for both Washington and Canada, mobility of fishermen does not appear to be a problem. Processors engage i n a l l types of non-price competition, p a r t i c u l a r l y subsidized c r e d i t costs f o r purchase of boats and gear, to a t t r a c t fishermen. The r e s u l t i s that most fishermen remain " l o y a l " during a single season but many switch l o y a l t i e s between seasons. Although b i o l o g i c a l data exist f o r the four species of salmon other than sockeye, the j o i n t production nature of the harvesting production functions was ignored i n parameter estimation because of the complexities of analysing a multi-species fishery. However, this j o i n t production problem was 7 i n c l u d e d i n the management programming by using a system of c a p a c i t y c o n s t r a i n t s to allow f o r the c a t c h o f salmon ot h e r than sockeye. The t r a n s n a t i o n a l i n t e r c e p t i o n problem can be s t u d i e d from the p o i n t of view o f d i f f e r i n g n a t i o n a l t a s t e s , s o c i a l r a t e s of d i s c o u n t and h a r v e s t i n g c o s t s (Munro, 1979). Only l i m i t e d data e x i s t f o r gear type u n i t h a r v e s t i n g c o s t s i n Washington but adequate data do e x i s t f o r B.C. T h e r e f o r e , d i f f e r e n c e s i n c o s t s of h a r v e s t i n g are r e s t r i c t e d to production f u n c t i o n e f f e c t s . That i s , i t was assumed that gear type u n i t h a r v e s t i n g c o s t s were the same f o r both c o u n t r i e s but t h a t gear e f f i c i e n c y was d i f f e r e n t as i n d i c a t e d by the parameter e s t i m a t i o n f o r the h a r v e s t i n g p r o d u c t i o n f u n c t i o n s , landed p r i c e s f o r sockeye were a v a i l a b l e f o r the two c o u n t r i e s and p r i c e d i f f e r e n t i a l s were permitted at one p o i n t (Chapter Four) to a f f e c t h a r v e s t i n g p r o p o r t i o n s . , Although the data probably do e x i s t to determine d i f f e r e n c e s i n consumer p r e f e r e n c e s and the s o c i a l r a t e s of d i s c o u n t , the e x c l u s i o n of the p r o c e s s i n g s e c t o r and the d i f f i c u l t y i n determining r e a c t i o n f u n c t i o n s i n a b a r g a i n i n g model precluded work i n these areas. T h e r e f o r e , the t h i r d r e s t r i c t i v e assumption of the study i s t h a t the manager i s allowed to a l l o c a t e country c a t c h p r o p o r t i o n s o n l y on the b a s i s of d i f f e r e n c e s i n h a r v e s t i n g c o s t s . Programming to achieve the o p t i m a l values was d i v i d e d i n f o two p a r t s . In the f i r s t p a r t , landed p r i c e s and u n i t h a r v e s t i n g c o s t s were obtained f o r each gear type o p e r a t i n g i n the seven major f i s h i n g zones; and, u s i n g h i s t o r i c a l v a l u e s f o r m i g r a t i o n . 8 c a t c h and escapement, a p r o f i t maximizing r o u t i n e was d e v i s e d t o permit r e l o c a t i o n of h a r v e s t i n g gear i n the g a u n t l e t to take advantage of d i f f e r e n c e s i n l e v e l s of m i g r a t i o n i n each f i s h i n g zone, d i f f e r e n c e s i n gear type e f f i c i e n c y and d i f f e r e n c e s i n e f f o r t c o s t s . Because h a r v e s t i n g production f u n c t i o n s are n o n l i n e a r , more than one gear type could be chosen t o h a r v e s t the sockeye c a t c h . In f a c t , two gear types o p e r a t i n g i n f o u r f i s h i n g zones proved to be f a r more p r o f i t a b l e than the r e s t . These were chosen f o r subsequent programming work and t h e i r c o n f i g u r a t i o n of h a r v e s t i n g p r o p o r t i o n s was used as a b a s i s f o r judging a c t u a l h i s t o r i c a l h a r v e s t i n g p r o f i t s . The p r o f i t d i f f e r e n t i a l between the h y p o t h e t i c a l c o n f i g u r a t i o n and the a c t u a l h i s t o r i c a l one i n d i c a t e d t h a t s o c i a l c o s t s were i n c u r r e d by an i n a p p r o p r i a t e s p a t i a l a l l o c a t i o n of e f f o r t . The r e s u l t s are recorded i n the l a s t p a r t of Chapter Three. , The second p a r t of the programming f o r o p t i m a l v a l u e s was designed t o check both the l e a s t c o s t s p a t i a l l o c a t i o n of gear and the i n t e r t e m p o r a l a l l o c a t i o n of h a r v e s t i n g e f f o r t . That i s , the program checks i n two dimensions, the s p a t i a l and the i n t e r t e m p o r a l , t o maximize p r o f i t s over time. The b a s i s of the i n t e r t e m p o r a l dimension i s the Clark-Hunro e q u i l i b r i u m equation which takes the r a t e of r e t u r n on the f i s h e r y , a l l o w i n g f o r the p h y s i c a l p r o d u c t i v i t y of the sockeye themselves, the p r o f i t s from c a t c h i n g sockeye and the c o s t r e d u c i n g a s p e c t s of l a r g e r sockeye m i g r a t i o n s , and s e t s t h i s r a t e of r e t u r n equal to the s o c i a l r a t e of r e t u r n . , The b a s i s of the s p a t i a l dimension i s the p r o f i t maximizing program used to choose the most p r o f i t a b l e s p a t i a l c o n f i g u r a t i o n 9 of h a r v e s t i n g e f f o r t . T h i s c o n f i g u r a t i o n i s very s e n s i t i v e to a l t e r a t i o n s i n the r e l a t i o n s h i p between c a t c h and m i g r a t i o n s , whereas the i n t e r t e m p o r a l a l l o c a t i o n of h a r v e s t i n g e f f o r t i s very s e n s i t i v e to the p r o f i t s from h a r v e s t i n g a s p e c i f i e d c a t c h . T h e r e f o r e , the program must i t e r a t e between these two dimensions u n t i l p r o f i t s are maximized f o r both s i m u l t a n e o u s l y . .Optimal e q u i l i b r i u m values are found f o r the c a t c h , m i g r a t i o n , escapement and gear type c a t c h p r o p o r t i o n s by checking v a r i o u s combinations of c a t c h and r e c r u i t m e n t a g a i n s t v a r i o u s h a r v e s t i n g c o n f i g u r a t i o n s t o f i n d the combination and c o n f i g u r a t i o n which maximizes the present value of the f i s h e r y . The r e s u l t s of the optimal value programming are given i n Chapter Four. In the i n t e r t e m p o r a l model there should be three c a p i t a l s t o c k s of i n t e r e s t : the stock of f i s h , the s t o c k of v e s s e l s and the stock of c a p i t a l equipment used i n p r o c e s s i n g the c a t c h . The l a c k of a v a i l a b l e data f o r the p r o c e s s i n g i n d u s t r y p r e c l u d e s any a n a l y s i s of p r o c e s s i n g c a p i t a l equipment. Data do e x i s t f o r the stock of v e s s e l s but modeling r e q u i r e s a knowledge of c a p i t a l m a l l e a b i l i t y (Clark, C l a r k e and Hunro, 1979) and the data do not e x i s t f o r e s t i m a t i n g t h i s m a l l e a b i l i t y parameter. I n s t e a d , i t w i l l be assumed t h a t v e s s e l c a p i t a l i s p e r f e c t l y m a l l e a b l e . E x t e n s i v e data do e x i s t , however, f o r the s t o c k of f i s h . For these reasons, i t w i l l be assumed t h a t the s t o c k of f i s h i s the only c a p i t a l stock i n the model. , 10 (3) A l t e r n a t i v e Systems Of Management To make the model o p e r a t i o n a l a number of r e s t r i c t i v e assumptions have been made. In p a r t i c u l a r , the study i s r e s t r i c t e d t o a p a r t i a l e q u i l i b r i u m approach. I t w i l l be assumed t h a t : o p p o r t u n i t y c o s t s are constant; the h a r v e s t r a t e i s equal t o the consumption r a t e (Clark and Munro, 1975, p.95); the s o c i a l r a t e o f d i s c o u n t r e f l e c t s p e r f e c t l y the marginal product of c a p i t a l p l u s percent c a p i t a l g a i n s and i s , t h e r e f o r e , equal to the s o c i a l time preference r a t e ; s y s t e m a t i c r i s k can be i g n o r e d ; and that output from the sockeye f i s h e r y i s too small t o i n f l u e n c e the o v e r a l l p r i c e of salmon. To avoid second-best problems which can a r i s e from monopoly power, management w i l l be vested i n an i n t e r n a t i o n a l agency and i t w i l l be assumed t h a t a r t i f i c i a l s c a r c i t i e s cannot take p l a c e . Even i f output c o n s t r a i n t s were permitted and p r i c i n g r u l e s devised to a t t a i n second-best (Allingham and A r c h i b a l d , 1973), i t i s not c l e a r that these would be v a l i d i n an i n t e r t e m p o r a l framework. That i s , there are other s e c t o r s of the economy which may be a t f i r s t b e s t s t a t i c optimum and yet not at f i r s t best i n t e r t e m p o r a l optimum (see the p u b l i c s e c t o r example i n Arrow and Kurz, 1970). The c o m p l e x i t i e s i n v o l v e d i n c a l c u l a t i n g time paths f o r the p r i c i n g r u l e s to reach some kind of second best i n t e r t e m p o r a l optimum would be f o r m i d a b l e . For these reasons, t h i s study w i l l assume that the r e s t of the economy i s at f i r s t best i n t e r t e m p o r a l optimum. Within the sockeye f i s h e r y , only an economic p r o f i t d e r i v e d from the i n h e r e n t nature of the problem w i l l be p e r m i t t e d . And the r e n t d e r i v e d therefrom w i l l be assumed d i s p e r s e d i n lump sum 11 form so as to c r e a t e no a l l o c a t i v e d i s t o r t i o n s . Most of these assumptions are made because the data do not e x i s t t o allow the model to be g e n e r a l i z e d or because i n c l u s i o n of some of these f a c t o r s would complicate an alr e a d y complex model. That i s , the model i n c l u d e s o n l y those f a c t o r s which can be q u a n t i f i e d and which are e s s e n t i a l to o p t i m a l management of the F r a s e r E i v e r sockeye f i s h e r y . The data r e t r i e v a l , parameter e s t i m a t i o n and o p t i m a l value programming even f o r these most e s s e n t i a l b i o l o g i c a l and economic f a c t o r s have been very i n v o l v e d . Comparison of the model used i n t h i s study with a l t e r n a t i v e models i n d i c a t e s which e s s e n t i a l p a r t s o f the model must be r e t a i n e d to give some i n d i c a t i o n of optimal management. For many years the F r a s e r River sockeye f i s h e r y has been managed a c c o r d i n g t o b i o l o g i c a l c r i t e r i a and c e r t a i n economic c r i t e r i a . Great p r o g r e s s has been made i n i n c r e a s i n g the sockeye-run p o t e n t i a l and i n preventing economic demise through o v e r f i s h i n g and d e s t r o y i n g the stock of sockeye. But very l i t t l e has been w r i t t e n about the economics or about fishermen and t h e i r r e t u r n s . H a r v e s t i n g e f f o r t has been r e g u l a t e d to f a c i l i t a t e c o n t r o l of sockeye escapement and to prevent uneconomic o v e r f i s h i n g ; and some attempt has been made by r e s p e c t i v e governments to stop the steady i n c r e a s e i n h a r v e s t i n g e f f o r t capacity.„The economic d i s t r e s s caused by t h i s i n c r e a s e i n h a r v e s t i n g c a p a c i t y has r e s u l t e d i n both B.C. and Washington imposing l i c e n s e l i m i t a t i o n schemes. A management model must, t h e r e f o r e , i n c l u d e economic f a c t o r s i f i t i s t o be expected t o le a d to optimal e x p l o i t a t i o n of the resource. 12 A management model with a b i o l o g i c a l element and e f f o r t l i m i t a t i o n c o u l d lead to s t a t i c p r o f i t maximization. .That i s , h a r v e s t i n g gear c o u l d be a l l o c a t e d t o c e r t a i n areas of the g a u n t l e t i n r e s t r i c t e d numbers so as to take the s p e c i f i e d c a t c h i n the l e a s t c o s t manner. T h i s would, i n f a c t , maximize p r o f i t s each season. The r e s u l t s would appear t o be s u p e r i o r to a management model based on b i o l o g y alone. However, the c a p i t a l t h e o r e t i c problem i n v o l v e d i n the d e c i s i o n to i n v e s t i n f u t u r e m i g r a t i o n s has s t i l l been i g n o r e d . In the e a r l y h i s t o r y o f f i s h e r y management there was a tendency to emphasize the h a r v e s t i n g on a s u s t a i n e d b a s i s of the l a r g e s t c a t c h t h a t the b i o l o g i c a l system c o u l d bear., T h i s i s u s u a l l y a t some intermediate s i z e of stock i n terms of m i g r a t i n g parents. I f too few a d u l t s escape, subsequent m i g r a t i o n s and catches w i l l be s m a l l e r . I t i s e q u a l l y true t h a t i f too many a d u l t s escape, subsequent m i g r a t i o n s and c a t c h e s w i l l a l s o be smaller, because of the d e s t r u c t i o n of spawning beds by l a r g e numbers of spawning parents. However, h a r v e s t i n g the l a r g e s t c a t c h i s u n l i k e l y t o be the most p r o f i t a b l e way o f h a r v e s t i n g . I t i s true t h a t p r o f i t s are i n f l u e n c e d by the t o t a l revenue d e r i v e d from s e l l i n g the c a t c h . But p r o f i t s are a l s o i n f l u e n c e d by the c o s t s of t a k i n g that c a t c h . H a r v e s t i n g production f u n c t i o n s i n d i c a t e that the l a r g e r the c a t c h , the more e f f o r t r e q u i r e d to land t h a t c a t c h unless there i s some o f f s e t by i n c r e a s e d migrations of sockeye which makes i t e a s i e r to c a t c h them. I t i s h i g h l y u n l i k e l y t h a t the l a r g e s t s u s t a i n a b l e c a t c h i s a s s o c i a t e d with the l a r g e s t s u s t a i n a b l e m i g r a t i o n . 13 S t a t i c p r o f i t maximization i s not good enough: th e r e must be i n t e r t e m p o r a l p r o f i t maximization i n order t o maximize the present value c f t h e f i s h e r y . The model developed i n t h i s study i n c l u d e s a l l these e s s e n t i a l f e a t u r e s : the b i o l o g i c a l system, the l e a s t c o s t h a r v e s t i n g system and the i n t e r t e m p o r a l view of h a r v e s t i n g i n which the rate of r e t u r n on the f i s h e r y i s equated with the s o c i a l r a t e o f r e t u r n . D e s p i t e the many c o n s i d e r a t i o n s which have been ignored, these t h r e e e s s e n t i a l p a r t s of the model can give some i n d i c a t i o n of the op t i m a l management of the F r a s e r B i v e r sockeye f i s h e r y . Once the data become a v a i l a b l e , a more gen e r a l model can i n c l u d e some o f these c o n s i d e r a t i o n s and w i l l probably give s u p e r i o r r e s u l t s . JJJl Survey Of The Th e s i s Part I I of t h i s chapter reviews the j u r i s d i c t i o n a l l i m i t s to the management of the F r a s e r B i v e r sockeye salmon f i s h e r y . I n c l u d e d i s a h i s t o r y of management p r a c t i s e s i n the p a s t . Part I I I o f t h i s chapter reviews the b i o l o g y of the F r a s e r B i v e r sockeye and d i s c u s s e s the c o m p i l a t i o n of b i o l o g i c a l data and the e s s e n t i a l elements of parameter e s t i m a t i o n f o r the b i o l o g i c a l r e l a t i o n s h i p s . Part IV surveys the economic theory of f i s h e r i e s management and d i s c u s s e s the main f e a t u r e s of the Clark-Munro e q u i l i b r i u m e q u a t i o n . Included i n t h i s part i s a d i s c u s s i o n of the h a r v e s t i n g model and the management model. In Chapter Two a b i o l o g i c a l model i s developed which r e l a t e s the escapement of sockeye from fishermen's nets to the r e t u r n m i g r a t i o n ( c a l l e d recruitment) f o u r years l a t e r . In Chapter Three a h a r v e s t i n g model i s developed which w i l l t e l l us 14 the c o s t s i n v o l v e d i n t a k i n g a s p e c i f i c c a t c h given a c e r t a i n run or recruitment. Chapter Four takes the recruitment model and the h a r v e s t i n g c o s t s model and combines them i n t o a management model. In i t , the manager decides both how t o h a r v e s t a given run of sockeye a t l e a s t c o s t , and how to o b t a i n a d i v i s i o n of t h a t run i n t o c a t c h and escapement so as to get a subseguent cat c h and escapement which w i l l maximize the present value of the f i s h e r y . With the data p r e s e n t l y a v a i l a b l e i t w i l l be p o s s i b l e to develop the recruitment model and the h a r v e s t i n g model f o r the F r a s e r sockeye. Using the p r i c e s and c o s t s which are a l s o a v a i l a b l e , i t w i l l then be p o s s i b l e to estimate the optimal stock l e v e l and the optimal amount o f e f f o r t needed to harvest from t h a t s t o c k . The r e s u l t s permit comparison of the a c t u a l management experience over the p e r i o d 1951 to 1975 with a h y p o t h e t i c a l s i t u a t i o n i n which a s o l e owner had been charged with the task of maximizing the present value of the f i s h e r y . The r e s u l t s can be used t o t e s t d i f f e r e n t managerial systems and t h i s has been done a t d i f f e r e n t p o i n t s i n the study. II MANAGEMENT OF THE FfiASJI BIVEfi SOCKEYE There are a number o f j u r i s d i c t i o n a l l i m i t s on the management of the F r a s e r B i v e r sockeye because of the t r a n s n a t i o n a l nature of the f i s h i n g g a u n t l e t . These l i m i t s place c e r t a i n c o n s t r a i n t s on the management model and i t i s u s e f u l to have some idea of the h i s t o r i c a l experience of sockeye management w i t h i n these l i m i t s . 15 H i s t o r i c a l l y the F r a s e r B i v e r sockeye salmon have been managed through a t r a n s n a t i o n a l commission o p e r a t i n g through the r e l e v a n t l e g a l a d m i n i s t r a t i o n i n each j u r i s d i c t i o n . And, although the commission i s charged only with the b i o l o g i c a l , not the economic, management of the f i s h e r y , many of i t s e f f o r t s have been d i r e c t e d toward p r e v e n t i n g the economic c a t a s t r o p h e which can r e s u l t from the common property problem. Despite the very s e r i o u s nature of t h i s problem, management has been c o n s t r a i n e d by the s t r u c t u r e of the f i s h i n g i n d u s t r y and the economic r a t i o n a l i z a t i o n programs i n s t i t u t e d r e c e n t l y . From 1900 to 1913 catches were very l a r g e but s t e a d i l y d e c l i n i n g as c a t c h i n g e f f i c i e n c y and p r o f i t s drove fishermen to e x p l o i t the resource more h e a v i l y . In 1913 there was a s e r i o u s s l i d e i n the F r a s e r B i v e r Canyon which, together with the heavy e x p l o i t a t i o n , reduced the F r a s e r B i v e r sockeye to a l e s s s i g n i f i c a n t f i s h e r y . Pressure from p r o c e s s o r s and fishermen culminated i n the s i g n i n g of a t r e a t y convention i n 1930 which i n c l u d e d p r o v i s i o n s f o r an equal d i v i s i o n o f the sockeye c a t c h and enhancement programs to r e s t o r e y i e l d s t o previous l e v e l s . However, t h i s t r e a t y was not r a t i f i e d u n t i l 1937 because of the p o l i t i c a l nature of the i n t e r c e p t i o n problem. P r i o r t o that date, the Canadian Department of F i s h e r i e s had j u r i s d i c t i o n over Canadian t e r r i t o r i a l waters and the S t a t e of Washington Department o f F i s h e r i e s had j u r i s d i c t i o n over U.S. waters. The Sockeye Saloon F i s h e r i e s Convention " a p p l i e s to the t e r r i t o r i a l waters and high seas westward of Canada and the United States from a l i n e between Bo.ni.lla P o i n t , Vancouver I s l a n d , and Tatoosh I s l a n d , Washington. I t i n c l u d e s a l l such waters between 48 and 49 degrees north l a t i t u d e , excepting Barkley Sound and N i t i n a t Lake. Eastward of t h i s l i n e , 16 i t includes the S t r a i t of Juan de Fuca, the S t r a i t of Georgia as far as Lasgueti Island — excepting Howe Sound and the waters east of Whidbey Island — and the Fraser Biver and i t s t r i b u t a r i e s . Regulations enacted by the International P a c i f i c Salmon Fisheries Commission are enforced within the t e r r i t o r i e s of each nation s o l e l y by the government of that nation. The Commission has no power to authorize any type of f i s h i n g gear contrary to laws of the State of Washington or the Dominion of Canada. The Convention required research f o r eight years before power to regulate the catch was given to the Commission. During t h i s time studies were made of the salmon runs and of the r i v e r , and obstructions to the migration of salmon were removed. In addition, the Commission co l l e c t e d d e t a i l e d and excellent s t a t i s t i c s of the t o t a l sockeye runs. Regulation of the Fraser Biver and Puget Sound sockeye f i s h e r i e s was undertaken for the f i r s t time by the Commission i n 1946. The r e s p o n s i b i l i t i e s of the Commission were enlarged to include pink salmon stocks of the Fraser River by an ammendment which was signed on 28, December, 1956. Es s e n t i a l l y the Commission has the r e s p o n s i b i l i t y to decide cn the required escapement and the permissable catch from each run of pink and sockeye salmon and to recommend regulations to the Government of Canada and to the State of Washington which w i l l allow an even d i v i s i o n of the catch between the fishermen of Canada and the United S t a t e s . ( B o y c e -Bevan, 1963, pp.6-8) The f i r s t attempt at regulation of f i s h i n g e f f o r t was made by the IPSFC in 1946. Their techniques were refined so that by 1951, mesh si z e , length of nets, area closures by gear type and weekly l i m i t s on days of f i s h i n g had settled into a consistent pattern. The IPSFC also worked on enhancement through removal of obstructions and the construction of f i s h ladders. After the removal of the Hell's Gate obstruction, recruitment to the fishery increased s i g n i f i c a n t l y . This led to increasing investment i n gear and vessels to enhance catching a b i l i t y during permitted times and in open areas. The ratio of c a p i t a l 17 investment t o c a t c h rose r a p i d l y (flctfynn, 1965, p. 152) l e a d i n g to i n c r e a s i n g l y r e s t r i c t i v e measures on the p a r t of the IPSFC: "the e f f e c t o f i n c r e a s i n g u n i t s of gear, g r e a t e r gear e f f i c i e n c y , and expanded f i s h i n g area i n convention waters has been l a r g e l y o f f s e t by reduced f i s h i n g time" (IPSFC, 1961, p.16). The r e s u l t of most of these r e g u l a t i o n s designed to prevent d i s a s t r o u s o v e r f i s h i n g has been to c r e a t e d i s t o r t i o n s i n the s t r u c t u r e of the sockeye f i s h i n g i n d u s t r y . The i n t r a - s e a s o n a l c l o s e d p e r i o d s are normally too short and i r r e g u l a r t o permit fishermen to look f o r a l t e r n a t i v e employment duri n g the f i s h i n g season. Increased gear e f f i c i e n c y and a v a i l a b i l i t y of gear has l e d t o i n c r e a s e d c o n g e s t i o n and gear i n t e r f e r e n c e . And the bunching of d e l i v e r i e s because o f the severe c u r t a i l m e n t of permissable f i s h i n g days (as few as two per week i n some areas) has l e d to more packers and c o l l e c t o r boats, more overtime hours i n shore-based p l a n t s , g r e a t e r investment i n p l a n t c a p a c i t y and f i s h storage f a c i l i t i e s ( S i n c l a i r , 1960, p.135). The reason f o r these d i s t o r t i o n s a r i s e s from the l e g a l c o n t r a i n t s on the IPSFC attempts to l i m i t e f f o r t . Support and reinforcement o f these IPSFC measures by the r e s p e c t i v e governments has been r e l u c t a n t and inadequate. , Turning to the Canadian a d m i n i s t r a t i o n o f salmon f i s h i n g , i t i s important to note t h a t the f e d e r a l government has j u r i s d i c t i o n over the salmon u n t i l they are a c t u a l l y landed. The l e g i s l a t i v e j u r i s d i c t i o n over seacoast and i n l a n d f i s h e r i e s i s vested i n the f e d e r a l government under S e c t i o n 92 of the BNA Act, but under the same s e c t i o n the p r o v i n c e s get property 18 r i g h t s over anything taken or caught i n those waters, as a r e s u l t of P r i v y C o u n c i l d e c i s i o n s i n 1898, 1913 and 1920, the f e d e r a l government handed over the a d m i n i s t r a t i o n of n o n - t i d a l s p o r t and commercial f i s h e r i e s t o the p r o v i n c i a l government i n B.C. but r e t a i n e d the s o l e r i g h t to enact any p r o v i n c i a l l e g i s l a t i o n which a f f e c t s them. ,The f e d e r a l government r e t a i n e d a l l r i g h t s t o marine and anadromous s p e c i e s . One of the major sources of c o s t to f i s h e r i e s i n B.C. i s the c o n f u s i o n over j u r i s d i c t i o n s and the pre-emptive r i g h t s of the f e d e r a l government i n f i s h e r i e s a f f a i r s through d i f f e r e n t p i e c e s of f e d e r a l l e g i s l a t i o n c o v e r i n g f i s h e r i e s , n a v i g a t i o n , and transboundary i n t e r e s t s . For p r o j e c t s a f f e c t i n g f i s h e r i e s there i s seldom a c o - o r d i n a t e d j o i n t impact study which means tha t p r i o r i t i e s are based on chronology r a t h e r than on e f f i c i e n c y . The f e d e r a l government d e a l s with f i s h i n g w h i le the p r o v i n c i a l government d e a l s with landed f i s h . ,• Despite these j u r i s d i c t i o n a l problems and because of p r o t e c t i o n and enhancement, the value of the salmon c a t c h has r i s e n , investment i n gear and v e s s e l s has i n c r e a s e d and t e c h n i c a l change has been r a p i d . And yet the incomes of salmon fishermen have not kept pace even though a s t r o n g union a c t s as a u n i t i n b a r g a i n i n g with p r o c e s s o r s to s e t p r i c e s (sea Campbell, 1969 and Canada, 1971). I t appears t h a t t o t a l c o s t s had been pushed to an e q u i l i b r i u m with t o t a l revenues. To c o u n t e r a c t t h i s , i n the B.C. f i s h e r y , the f e d e r a l government, using i t s powers under the F i s h e r i e s Act, i n s t i t u t e d a r a t i o n a l i z a t i o n p o l i c y i n 1968. For s i m i l a r reasons the S t a t e of Washington enacted a r a t i o n a l i z a t i o n p o l i c y i n 1974, 19 From a study of the h i s t o r i c a l record i t appears that the U.S. and Canada have had some type of c a t c h agreement s i n c e 1937. The U.S. has attempted to maintain a p r o p r i e t a r y i n t e r e s t i n F r a s e r S i v e r sockeye salmon by p a r t i c i p a t i n g i n a b s t e n t i o n through the IPSFC and by h e l p i n g t o f i n a n c e enhancement programs on the F r a s e r . T h i s has l e d to an egual s h a r i n g of the sockeye catch but n e i t h e r party has been i n t e r e s t e d i n maximizing the resource r o y a l t y p o t e n t i a l from the f i s h e r y . The i n s t i t u t i o n a l arrangements have probably worked w e l l i n terms of minimizing t r a n s a c t i o n , agreement and compliance c o s t s (and the economic c o s t s a s s o c i a t e d with d e s t r u c t i v e o v e r f i s h i n g ) . However, they appear to have f a i l e d i n terms of minimizing user and a b s t e n t i o n c o s t s . The management model developed i n t h i s study w i l l be c o n s t r a i n e d by a number of these c o n s i d e r a t i o n s . The major reason i s the l a c k of data which c o u l d have been used as a guide when r e l e a s i n g such c o n s t r a i n t s . I t w i l l be assumed that management w i l l remain i n the hands of the IPSFC because of the t r a n s n a t i o n a l nature o f the i n t e r c e p t i o n problem and because of the very capable management of the IPSFC gi v e n the h i g h l y r e s t r i c t i v e l e g a l c o n s t r a i n t s i n the e x i s t i n g convention. T h e r e f o r e , the terms of r e f e r e n c e o f the IPSFC w i l l be assumed to be broadened to i n c l u d e economic management as w e l l . The IPSFC could take the whole sockeye ca t c h with a few, w e l l placed t r a p s i n the mouth of the F r a s e r B i v e r ( C r u t c h f i e l d and Pontecorvo, 1969). However, onl y l i m i t e d data e x i s t f o r t r a p s . /For sockeye catches taken by t r a p s s i n c e 1951 data do e x i s t f o r the t r a p s i n Sooke Harbour but only f o r seven years. 20 I t i s also unlikely that either government would wish to reduce employment i n the f i s h i n g industry by introducing such a harvesting system. For these reasons, the IPSFC w i l l be constrained to using the usual gear types, although they w i l l be able to assign or exclude c e r t a i n gear types from certain f i s h i n g zones. The use of f i s h finding equipment and monofilament nets (which cannot be seen by sockeye) must also be excluded for the same reasons. There does not appear to be any data available for changes i n gear type productivity associated with these innovations but one can presume that productivity increases were considerable given the speed with which t h i s gear was banned by both governments. The employment-reducing e f f e c t s w i l l be assumed to be not p o l i t i c a l l y f e a s i b l e given the transnational background to the catching agreement., It w i l l be assumed that the IPSFC i s r e s t r i c t e d to managing i n the physical areas delineated above and to a twelve week period including July, August and September (two weeks e a r l i e r i n the f i r s t part of the gauntlet and two weeks l a t e r i n the l a s t p a r t ) . These s p a t i a l and temporal l i m i t s would encompass ninety-five percent of the sockeye catch. This harvesting of sockeye can have a serious impact on stocks of the other four species of salmon. This i s p a r t i c u l a r l y true for the run of pink salmon which occurs every other year only but takes place at the same time as the sockeye run. However, the IPSFC currently has j u r i s d i c t i o n over the pink salmon escapement; and this j u r i s d i c t i o n i s assumed to include harvesting. Area and time closures, the methods used f o r c o n t r o l l i n g 21 escapements i n the past, w i l l only apply to areas which are not l e a s t c o s t i n terms of h a r v e s t i n g . That i s , c e r t a i n gear types o p e r a t i n g i n c e r t a i n f i s h i n g zones w i l l be excluded d u r i n g the sockeye-run because they are not economic and i n t e r f e r e with m i g r a t i o n s of sockeye to other zones and more p r o d u c t i v e gear. The most e f f i c i e n t gear types w i l l be assigned to the most e f f i c i e n t areas f o r the whole season but i n r e s t r i c t e d numbers. Gear types w i l l no longer be r e s t r i c t e d to two or three days a week but w i l l be a b l e t o f i s h f o r seven days a week i f they wish. The r e l e a s e of t h i s major time c o n s t r a i n t w i l l a llow the present f l e e t to be able t o h a r v e s t the l a r g e r catches proposed by the o p t i m a l values d e r i v e d from the management programming. However, t h i s may e n t a i l severe l i m i t s on hours per day to prevent decimation of m i l l i n g s t o c k s and, t o a v o i d i n t e r - g e a r i n t e r f e r e n c e , purse s e i n e s w i l l continue to be r e s t r i c t e d to daytime o p e r a t i o n s and g i l l n e t s t o nightime o p e r a t i o n s , k second major change i s t h a t the IPSFC w i l l no l o n g e r manage by p e r m i t t i n g o t h e r s t o f i s h . I n s t e a d , i t w i l l be assumed t h a t the IPSFC i t s e l f h a r v e s t s the s p e c i f i e d c a t c h by h i r i n g v e s s e l s and a s s i g n i n g them t o the a p p r o p r i a t e a r e a s . ., To a v o i d over-investment i n boat c a p i t a l and the subsequent problems a s s o c i a t e d with the i r r e v e r s i b l e nature of such an investment, i t w i l l be assumed t h a t the IPSFC simply h i r e s u n i t s of e f f o r t {fishermen p l u s gear and boat) to take the s p e c i f i e d c a t c h i n the s p e c i f i e d areas. Only l e a s t c o s t gear and v e s s e l s w i l l be used and the r e s t w i l l be o f f e r e d lump sum compensation f o r t h e i r c a p i t a l l o s s . 22 F i n a l l y , given constant opportunity costs, i t w i l l be assumed that the most rapid approach to the equilibrium biomass l e v e l i s optimal. This w i l l , of course, only occur once i n the l i f e of the fishery. Given the past success of the IPSFC i n co n t r o l l i n g catch (despite severe constraints on i t s a b i l i t y to l i m i t e f f o r t ) , i t w i l l be assumed that t h i s task w i l l be made easier by the IPSFC harvesting for i t s e l f only. Therefore, i f a larger escapement i s optimal, the IPSFC w i l l simply reduce the catch. I f a smaller escapement i s optimal, the IPSFC w i l l simply increase the catch. The former e n t a i l s a s a c r i f i c e while the l a t t e r results in greater p r o f i t . But these would be once-and-f o r - a l l as subsequent catches would always be the same a f t e r the i n i t i a l year of adjustment. I l l THE BIOLOGY 0? THE FBASJR M I J ? B SOCKEYE The f i r s t part of the management model must provide a l i n k between escapement of spawning parents and the subsequent return migration four years l a t e r . This l i n k i s often referred to as the recruitment r e l a t i o n s h i p because of the "recruitment" of migrating sockeye to the fishery. As there are f i v e species of salmon available to the nets cf fishermen i n the fishery through which the Fraser River sockeye salmon migrate on t h e i r way to thei r spawning grounds, t h i s i s a multi-species fishery. The sockeye f i s h i n g season l a s t s for a twelve week period every summer during which time sockeye i s the dominant species both in terms of weight and i n terms of value per pound. For t h i s reason and because of the complexities involved in modeling a multi-species f i s h e r y which includes the other four species of salmon. 23 i t was d e c i d e d t h a t a r e c r u i t m e n t model would be developed f o r sockeye o n l y . Map (1-1) i l l u s t r a t e s the e x t e n t of the sockeye spawning i n the f o r t y - f o u r c r e e k s and s t reams of the F r a s e r R i v e r w a t e r s h e d . As soon as t h e spawn h a t c h , they migra te to e l e v e n f r e s h water l a k e systems connected with those c r e e k s and s t r e a m s . They spend two w i n t e r s i n those l a k e s and streams and then migrate to the ocean when the m e l t - w a t e r has s w o l l e n t h e F r a s e r R i v e r . They migrate t o the o c e a n i c f e e d i n g grounds which are r i c h i n z o o p l a n k t o n and spend two w i n t e r s i n t h a t e n v i r o n m e n t . The next s p r i n g , the f o u r year o l d s o c k e y e m i g r a t e back t o spawn i n t h e i r n a t a l s t r e a m s . The r e c r u i t m e n t model g i v e s us some i d e a of the r e t u r n on the " i n - k i n d " form of i n v e s t m e n t produced by a b s t a i n i n g from f i s h i n g i n any p e r i o d . The model i n d i c a t e s t h a t i f escapement i s e i t h e r too s m a l l or t o o l a r g e , subsequent r e c r u i t m e n t w i l l be s m a l l . That i s , the l a r g e s t r e c r u i t m e n t s a r i s e f rom i n t e r m e d i a t e escapements . The work i n t h i s s t u d y s u p p o r t s p r e v i o u s s t u d i e s which have found t h a t r e c r u i t m e n t i s very s e n s i t i v e to d i f f e r e n c e s i n average weight of s o c k e y e , the r a c e o f s o c k e y e , and t h e year c l a s s o f s o c k e y e . The c y c l e year c l a s s of sockeye i t s e l f appears to i n f l u e n c e average w e i g h t , f e r t i l i t y , s u r v i v a b i l i t y and growth i n a h i g h l y s i g n i f i c a n t manner. There has been e x t e n s i v e r e s e a r c h b o t h by b i o l o g i s t s working f o r the IPSFC and by o t h e r b i o l o g i s t s i n t o the n a t u r e of the sockeye salmon s t o c k . These s t u d i e s were of g r e a t a s s i s t a n c e i n c o m p i l i n g the data and f o r m u l a t i n g the b a s i c m o d e l . No attempt was made t o improve on t h e b i o l o g i c a l n a t u r e of the MAP (1-1)  DISTRIBUTION OF SOCKEYE SALMON SPAWNING GROUNDS 25 r e c r u i t m e n t r e l a t i o n s h i p . However, every attempt was made to ensure good parameter e s t i m a t e s as the r e c r u i t m e n t r e l a t i o n s h i p i s of fundamental importance i n the management model. , Because of the four year l a g between escapement and subsequent r e c r u i t m e n t , there are f o u r d i s t i n c t year c l a s s c y c l e s i n the annual m i g r a t i o n s . In e s t i m a t i n g the parameters f o r the sockeye rec r u i t m e n t f u n c t i o n , the major i n n o v a t i o n was the i n c l u s i o n of a four year c y c l e through the use of slope dummy v a r i a b l e s which led to a s i g n i f i c a n t improvement i n f i t . I t should be made c l e a r t h a t b i o l o g i s t s have worked with c y c l e s f o r some time but there do not appear to be any s t u d i e s which have i n c o r p o r a t e d c y c l e e f f e c t s through the use of c y c l e dummies. Di s a g g r e g a t i n g the escapement data a c c o r d i n g to the eleven freshwater l a k e s (to give race o f sockeye) l e d to a f u r t h e r s i g n i f i c a n t improvement i n f i t . The IPSFC uses time of e n t r y models to f a c i l i t a t e c o n t r o l of escapement by race and i t would be very f e a s i b l e t o use these parameter e s t i m a t e s f o r the management model. To i n c l u d e race escapement improves the parameter e s t i m a t e s f o r r e c r u i t m e n t i n aggregate form. However, to determine race p r o p o r t i o n of the subseguent recruitment and the i n f l u e n c e of gear type c a t c h on recruitment t o subsequent f i s h i n g zones would r e q u i r e i n t r a - s e a s o n a l a n a l y s i s and complicate an a l r e a d y complex model. For t h i s reason the r e c r u i t m e n t model developed i n t h i s study d i s t i n g u i s h e s between the recruitment/escapement r e l a t i o n s h i p on a year c l a s s b a s i s but not a c c o r d i n g to r a c e . 26 I I ECONOMIC THEORY OF FISHERIES MANAGEMENT The second part of the management model involves the harvesting of the sockeye as they migrate through the fi s h i n g gauntlet. What i s cf primary i n t e r e s t i n the harvesting model i s the amount of gear type e f f o r t required to land a s p e c i f i e d catch of sockeye i n d i f f e r e n t f i s h i n g zones i n the gauntlet.. As one would expect, the larger the migration of sockeye, the less e f f o r t required to take that catch. Taking into account the size of migration, the e f f i c i e n c y of various gear types and the costs associated with gear type e f f o r t , the manager must choose the least cost method of harvesting. Once t h i s s p a t i a l location problem has been solved, one must then consider the intertemporal nature of the fishery. , A survey of the l i t e r a t u r e indicates that, given certain assumptions, the c a p i t a l theoretic problem can be solved by setting the rate of return on the fishery equal to the s o c i a l rate qf return. This fishery rate of return must take into account the physical productivity of the stock of f i s h i t s e l f and the impact on harvesting p r o f i t s of d i f f e r e n t combinations of catch and recruitment. Once eguilibrium i s achieved, the present value of the fishery i s maximized. Therefore, the management model consists of a recruitment function which provides the l i n k between escapement and subsequent recruitment and a harvesting cost function which indicates the costs of harvesting a s p e c i f i e d catch given a certain l e v e l of recruitment. These components are combined into a management program which varies the s p a t i a l location of the harvesting gear and the intertemporal flow of e f f o r t , catch and 27 r e c r u i t m e n t u n t i l the f i s h e r y r a t e of r e t u r n equals the s o c i a l r a t e of r e t u r n , (1) The H a r v e s t i n g Model Given the recruitment or s i z e of run and the escapement r e q u i r e d to achieve that l e v e l of r e c r u i t m e n t , the manager must decide which method of h a r v e s t i n g the c a t c h i s l e a s t c o s t . The higher the c a t c h , the higher the c o s t s i n terms of h a r v e s t i n g e f f o r t . But the l a r g e r the r e c r u i t m e n t , the l o s e r the c o s t s , because of g r e a t e r a v a i l a b i l i t y of sockeye to the nets (see Bradley, 1970). That i s , the same gear may have the same t e c h n i c a l e f f i c i e n c y i n d i f f e r e n t f i s h i n g areas but i f the f i s h are more concentrated i n one area, i t i s e a s i e r to take the s p e c i f i e d c a t c h with a minimum of e f f o r t . But, even i f the c o n c e n t r a t i o n or " a v a i l a b i l i t y " of the f i s h i s g r e a t , t h e r e i s a second q u e s t i o n , whether they are " c a t c h a b l e " . A given r e c r u i t m e n t a v a i l a b i l i t y or c o n c e n t r a t i o n of f i s h swimming very deeply and out o f the reach of the fishermens' nets i s f a r l e s s " c a t c h a b l e " than i f swimming near the s u r f a c e . Becruitment a v a i l a b i l i t y i s dependent on the recruitment/escapement model which i s under the c o n t r o l of the manager. However, the model suggests t h a t c a t c h a b i l i t y i s dependent on year c l a s s c y c l e , over which the manager has no c o n t r o l . For t h i s reason, the h a r v e s t i n g model i s d i f f e r e n t f o r each of the f o u r year c l a s s c y c l e s . The m i g r a t i o n route o f the sockeye i s i l l u s t r a t e d i n Map (1-2). A f t e r spending time i n ocean f e e d i n g grounds, f o u r year o l d sockeye migrate toward the F r a s e r 'River by passing through READ CAREFULLY 1. PIN UP IN WHEELHOUSE. 2. WHEN DELIVERING YOUR CATCH, GIVE TALLY MAN THE MAP NUMBER,OR NUMBERS SHOWING THE AREA IN WHICH YOUR FISH WERE CAUGHT. 3. ACCURATE CATCH REPORTS WILL HELP PRESERVE YOUR FISHERIES. 4. FOR COMPLETE DETAILS, CONSULT BRITISH COLUMBIA FISHERIES REGULATIONS. STATISTICAL AREAS ARE DIVIDED BY RED LINES — SALMON FISHING WITH NETS OF ANY KIND IS NOT PERMITTED OUTSIDE OF -THAT IS SEAWARD. OF — THE HEAVY BLACK LINE. • FISHERIES SERVICES OFFICES DEPARTMENT OF THE ENVIRONMENT FISHERIES SERVICE STATISTICAL MAP SHOWING AREAS OF CATCH FOR BRITISH COLUMBIA WATERS (SOUTHERN HALF) 3 :ITISH.CgLUBBm_,r, U. S. A. SCALE IN MIU1 FOUtTH CQTION < AMU. n i l 29 the southern half of area 24 and through areas 23 and 21. They migrate along the north side of the S t r a i t of Juan de Fuca through areas 20 and the southern half of 19. They then enter U.S. waters and migrate through the San Juan Islands and past Point Roberts into the mouth of the Fraser River. Migrating sockeye in the Fraser River after Mission are only vulnerable to native Indians who f i s h for t h e i r own consumption according to treaty r i g h t s . Ninety percent of the migrating sockeye pass through the fi s h i n g gauntlet i n about s i x weeks with the peak of the run usually about the second week of August. The sockeye a r r i v e i n area 24 by the middle of June and have a l l passed Mission by the middle cf October. Sixteen gear type f i s h i n g areas can be distinguished i n t h i s f i s h i n g gauntlet. , T r o l l s , g i l l n e t s , reefnets and purse seines i n these areas account for ninety-five percent of the sockeye catch. However, g i l l n e t s and purse seines in only six of these areas account for ninety percent of the t o t a l sockeye catch. The harvesting model developed i n t h i s study i s based on these six gear type areas only and assumes that costs of harvesting for these areas are dependent on size of catch (direct relationship) and a v a i l a b i l i t y of recruitment (modified by cat c h a b i l i t y ) to that f i s h i n g area (inverse r e l a t i o n s h i p ) . C a t c h a b i l i t y by gear type i s assumed to vary on a year cycle basis f o r each gear type area. The analysis of the optimal s p a t i a l d i s t r i b u t i o n of e f f o r t i n a gauntlet type of f i s h e r y such as this one, i s f a i r l y straight forward. The salmon follow customary routes and are 30 c a t c h a b l e i n p r e d i c t a b l e ways by the s i x t e e n d i f f e r e n t gear type h a r v e s t i n g areas, common property causes a problem because fishermen w i l l tend to move out to e a r l i e r p o s i t i o n s i n the g a u n t l e t t o overcome the crowding and c o n g e s t i o n of gear i n l a t e r areas (see Bradley, 1970). T h i s can r e s u l t i n q u i t e severe r e d u c t i o n s i n re c r u i t m e n t a v a i l a b i l i t y i n other areas where gear may be more e f f i c i e n t . Furthermore, i n t e r - g e a r i n t e r f e r e n c e between p u r s e - s e i n e s and g i l l n e t s can have a s e r i o u s impact on a v a i l a b i l i t y to a l a t e r a r e a where gear e f f i c i e n c y may be higher. The IPSFC r e g u l a t i o n s c o n f i n i n g purse s e i n e s to daytime o p e r a t i o n s and g i l l n e t s t o nightime o p e r a t i o n s should e l i m i n a t e t h i s l a s t problem. The s p a t i a l a l l o c a t i o n problem can be handled, using non-l i n e a r mathematical programming, to permit many d i f f e r e n t h a r v e s t i n g c o n f i g u r a t i o n s to be t r i e d . Only i n t h i s way can the complex i n t e r a c t i o n s between gear e f f i c i e n c y , r e c r u i t m e n t a v a i l a b i l i t y and biomass c a t c h a b i l i t y be r e s o l v e d to g i v e the l e a s t c o s t h a r v e s t i n g areas. The p r o d u c t i o n f u n c t i o n f o r each gear type area i s s e n s i t i v e t o changes i n c a t c h a b i l i t y between seasons and a l s o to changes i n re c r u i t m e n t a v a i l a b i l i t y f o r any giv e n season..Thus re c r u i t m e n t can be permitted t o flow through to any s i n g l e area f o r each year of the fou r year c y c l e without being reduced by f i s h i n g i n areas e a r l i e r i n the g a u n t l e t . The gear type areas with the most e f f i c i e n t gear and the g r e a t e s t c a t c h a b i l i t y of th a t given recruitment w i l l h a r v e s t the s p e c i f i e d c a t c h at l e a s t c o s t . , 31 With more recruitment f l o w i n g through t o c e r t a i n zones and l a r g e r catches to be taken by c e r t a i n gear types o p e r a t i n g i n those zones, i t i s c l e a r that more e f f o r t w i l l be needed than the h i s t o r i c a l l e v e l s of e f f o r t . T h i s should not l e a d t o s e r i o u s c o m p l i c a t i o n s such as c o n g e s t i o n because the number of days of f i s h i n g per week i s no longer r e s t r i c t e d . That i s , most gear type areas have h i s t o r i c a l l y been reduced to an average of three f i s h i n g days per week, whereas the management model used i n t h i s study can allow f i s h i n g seven days per week. In f a c t , only f i v e days per week should be s u f f i c i e n t as the maximum i n c r e a s e i n any gear type e f f o r t appears to be about t h i r t y percent. Note th a t t h i s change i n number of days per week w i l l not a f f e c t c o s t s as u n i t opportunity c o s t s are assumed to be c o n s t a n t . However, because of the p o s s i b l e decimation o f a m i l l i n g stock, i f the numbers of days per week i s i n c r e a s e d , there may be a need to r e s t r i c t the hours of f i s h i n g per day. The f i r s t s tep i n developing the h a r v e s t i n g model i s t o estimate the parameters o f the p r o d u c t i o n f u n c t i o n s f o r the s i x t e e n gear type a r e a s . E f f o r t i s adjusted f o r t e c h n i c a l change over the p e r i o d under study. Gear type areas w i t h i n each f i s h i n g zone are a l l assumed to experience the same a v a i l a b i l i t y of sockeye. However, each subsequent zone i n the g a u n t l e t e x p e r i e n c e s a r e d u c t i o n i n the m i g r a t i o n a v a i l a b l e by the amount of f i s h i n g by gear type areas i n the previous zone. There are no data on these reduced a v a i l a b i l i t i e s ; however, we know tha a c t u a l m i g r a t i o n s from the work with the r e c r u i t m e n t f u n c t i o n and we know the gear type area catches and e f f o r t from published r e p o r t s . T h e r e f o r e , i t i s a simple matter to c a l c u l a t e reduced 32 m i g r a t i o n to subsequent zones. Because no data were a v a i l a b l e to a i d i n e s t i m a t i n g the c a t c h a b i l i t y parameter, the e s t i m a t i o n process i t s e l f was used to d e r i v e estimates by p u t t i n g r e c r u i t m e n t a v a i l a b i l i t y i n dummy c y c l e form. That i s , a c t u a l c a t c h e s were regressed on e f f o r t a d j u s t e d f o r t e c h n i c a l change and on rec r u i t m e n t a d j u s t e d f o r a v a i l a b i l i t y i n each zone i n f o u r year c y c l e dummy form. This meant t h a t the exponent on re c r u i t m e n t a v a i l a b i l i t y c o u l d be d i f f e r e n t f o r each o f the f o u r year c l a s s e s . A l l o w i n g f o r v a r y i n g c a t c h a b i l i t y i s a w e l l e s t a b l i s h e d p r a c t i s e i n b i o l o g i c a l s t u d i e s . However, there do not appear to be any s t u d i e s which i n c o r p o r a t e f l u c t u a t i o n s i n c a t c h a b i l i t y by us i n g c y c l e dummies. These l e d to a s i g n i f i c a n t improvement i n f i t . The second step i n de v e l o p i n g the h a r v e s t i n g model was to convert the pr o d u c t i o n f u n c t i o n s i n t o c o s t f u n c t i o n s given a v a i l a b l e c o s t data. (Ising the a c t u a l h i s t o r i c a l averages f o r c a t c h and recrui t m e n t , a n o n l i n e a r c o s t m i n i m i z a t i o n program was then employed t o achieve l e a s t c o s t h a r v e s t i n g . P r o f i t c a l c u l a t i o n s using these l e a s t c o s t h a r v e s t i n g s p a t i a l c o n f i g u r a t i o n s were then compared with c a l c u l a t e d p r o f i t s f o r the a c t u a l h i s t o r i c a l h a r v e s t i n g experience. The r e s u l t s i n d i c a t e d t h a t s i g n i f i c a n t g a i n s c o u l d be made by r e l o c a t i n g gear type e f f o r t i n the g a u n t l e t . As the v a r i o u s r e a c t i o n s to c a t c h , recruitment a v a i l a b i l i t y and biomass c a t c h a b i l i t y are d i f f e r e n t f o r each of the gear type areas f o r each of the f o u r c y c l e y e a r s , the op t i m a l h a r v e s t i n g c o n f i g u r a t i o n i n terms of c a t c h p r o p o r t i o n s amongst the gear type areas w i l l be d i f f e r e n t f o r each of the f o u r c y c l e years. 33 The parameter estimates f o r the p r o d u c t i o n f u n c t i o n s i n d i c a t e d i n c r e a s i n g r e t u r n s to s c a l e f o r the s i x major gear type areas f o r each of the f o u r c y c l e years, although the s c a l e does not change p r o p o r t i o n a t e l y f o r each i n p u t i n each c y c l e year, the change i n r e c r u i t m e n t a v a i l a b i l i t y from one c y c l e year to another i s so l a r g e that the c y c l e i n f l u e n c e d c a t c h a b i l i t y exponents (recruitment e l a s t i c i t i e s ) s i g n a l q u i t e l a r g e s h i f t s i n h a r v e s t i n g e f f o r t from one f i s h i n g zone to another and from one gear type t o another. The p r o f i t improvements r e s u l t i n g from these d i f f e r i n g h a r v e s t i n g c o n f i g u r a t i o n s a r i s e o n l y from i n t e r - s e a s o n a l changes i n c o n f i g u r a t i o n . ,,No attempt was made to a l t e r c o n f i g u r a t i o n s on an i n t r a - s e a s o n a l basis.,A f u l l i n t r a - s e a s o n a l h a r v e s t i n g model would i n v o l v e the a n a l y s i s of thousands of d a i l y r e c o r d s f o r a l l gear type areas i n the g a u n t l e t . And i t i s not c l e a r t h a t the r e s u l t i n g s p a t i a l , c o s t minimizing program would be s i g n i f i c a n t l y d i f f e r e n t from the one based on the assumption of one l e a s t c o s t s p a t i a l c o n f i g u r a t i o n f o r each of the f o u r c y c l e years. Loose (1977) g i v e s an a n a l y s i s of the s i g n i f i c a n t p r o f i t improvements t h a t can be r e a l i z e d through c o s t minimizing i n t r a -s e a s o n a l h a r v e s t i n g p a t t e r n s i n a g a u n t l e t type of f i s h e r y . However, i n t h i s study i t w i l l be assumed t h a t the l e a s t cost h a r v e s t i n g c o n f i g u r a t i o n f o r any p a r t i c u l a r year of the f o u r year c y c l e remains the same f o r the whole f i s h i n g season during t h a t c y c l e year but w i l l be d i f f e r e n t f o r each o f the f o u r c y c l e years. 34 (2) Optimal JLishgcigs Management The purpose of the f o l l o w i n g survey i s to h i g h l i g h t the t h e o r e t i c a l c o n s i d e r a t i o n s i n v o l v e d i n managing a f i s h e r i e s resource i n an optimal manner. The word optimal i m p l i e s maximizing the f e a s i b l e present value of the resource t a k i n g i n t o account the best r e l a t i o n s h i p between c a t c h , escapement and subsequent recruitment i n terms of the b i o l o g i c a l model and i n terms o f the c o s t s and revenues i n v o l v e d i n t a k i n g that c a t c h . The v a r i a b l e under c o n t r o l of the manager i s assumed to be the e f f o r t i n v o l v e d i n l a n d i n g the c a t c h ; and the manager must a l l o c a t e t h i s e f f o r t i n the l e a s t c o s t s p a t i a l c o n f i g u r a t i o n and i n an o p t i m a l manner over time. The f i n a l i n f l u e n c e s on the o p t i m i z i n g process are the types of e f f o r t c o n t r o l used and the management regime i t s e l f . : The most important development i n f i s h e r i e s economics has been the r e c e n t work on the c a p i t a l t h e o r e t i c problem encountered when working with s t o c k s of f i s h . R e s o l u t i o n of t h i s problem, however, has depended on the c o r r e c t a n a l y s i s of the economic e x p l o i t a t i o n of a f i s h e r y . Although Gordon's (1954) a n a l y s i s was s t a t i c r a t h e r than i n t e r t e m p o r a l , i t must be s i n g l e d out as one of the most important e a r l y works because of the i s o l a t i o n and c l a r i f i c a t i o n of four i s s u e s which are r e l e v a n t t c t h i s t h e s i s . Gordon's f i r s t p o i n t was t h a t because f i s h e r i e s are e x p l o i t e d f o r economic reasons and not b i o l o g i c a l , the concept of net economic y i e l d must have precedence over the b i o l o g i c a l concept of maximum s u s t a i n a b l e y i e l d . Gordon's second point was that the e q u i l i b r i u m s o l u t i o n i s 35 a composite of b i o l o g i c a l and economic f a c t o r s . T h i s "bionomic" s o l u t i o n i n v o l v e s some type of b i o l o g i c a l f u n c t i o n to determine p o p u l a t i o n s i z e (the recruitment f u n c t i o n ) , the i n t r o d u c t i o n of a h a r v e s t i n g production f u n c t i o n with i n p u t s of the f i s h stock and f i s h i n g e f f o r t , and the c o n v e r s i o n of the l a t t e r i n t o a c o s t f u n c t i o n . The comparison of the p r i c e of f i s h and the c o s t of l a n d i n g them determines the f e a s i b i l i t y o f f i s h i n g . The i n t e r a c t i o n of e f f o r t (and i t s cost) with the p o p u l a t i o n dynamics of h a r v e s t i n g (and i t s subsequent revenue) si m u l t a n e o u s l y determines both the b i o l o g i c a l and the economic e g u i l i b r i u m . Gordon's t h i r d p o i n t was t h a t the c a p i t a l component (stock of f i s h ) of the production process and not the law of d i m i n i s h i n g r e t u r n s i s r e s p o n s i b l e f o r the d i m i n i s h i n g r a t e of c a t c h . That i s , the e f f e c t o f h a r v e s t i n g i s t o reduce the f i s h p o p u l a t i o n thereby reducing the a v a i l a b i l i t y of the stock. Gordon's f o u r t h point was t h a t most f i s h e r i e s are common pr o p e r t y . And t h i s open access t o a common f i s h stock reduces and d i s s i p a t e s the economic p r o f i t or rent. I f t h i s open access e x t e r n a l i t y can be i n t e r n a l i z e d somehow, the maximized net economic y i e l d which i s d e r i v e d from e x p l o i t a t i o n o f the f i s h e r y r e s u l t s i n an economic p r o f i t . T h i s economic r e n t a r i s e s from the bounty of nature and not from an a r t i f i c i a l s c a r c i t y . , Scott (1955a) was the f i r s t to p o i n t out t h a t Gordon had missed the e s s e n t i a l i n t e r t e m p o r a l nature o f the investment problem. A fundamental a n a l y s i s of c a p i t a l t h e o r e t i c problems i n an i n t e r t e m p o r a l s e t t i n g r e v e a l s two p a r t s to the problem of investment: an " i n v e n t o r y " h a n d l i n g aspect and a " r e - s t o c k i n g " 36 aspect. I g n o r i n g the c o n g e s t i o n and crowding problems t h a t a r i s e with open a c c e s s , S c o t t (1955a) recognized that i n the s h o r t ran a monopoly regime and a c o m p e t i t i v e regime would handle the i n v e n t o r y i n much the same way. Both would "mine" to the p o i n t where s h o r t run marginal c o s t e q u a l s p r i c e . Only i n the long run would the behaviour of the monopolist be s i g n i f i c a n t l y d i f f e r e n t ; i t would pay the monopolist to i n v e s t " i n - k i n d " by a b s t a i n i n g from f i s h i n g . The reason i s the user c o s t encountered by using some of your c a p i t a l today r a t h e r than l e a v i n g i t f o r tomorrow. That i s , by l a n d i n g an e x t r a f i s h today you not only reduce the st o c k a v a i l a b l e f o r tomorrow but you a l s o reduce the p o t e n t i a l growth of that f u t u r e s t o c k . A f i s h e r y manager i s concerned with maximizing p r o f i t over time and not j u s t a t a p o i n t i n time. T h e r e f o r e , the management o b j e c t i v e becomes the maximization of t o t a l discounted net revenue d e r i v e d from e x p l o i t i n g the r e s o u r c e . T h i s dynamic o p t i m i z a t i o n , or c o n t r o l , problem r e q u i r e s the manager to take e x p l i c i t account of time, the c a p i t a l stock (or s t a t e v a r i a b l e ) and the e f f o r t (or c o n t r o l ) v a r i a b l e . P r o f i t over time i s determined by revenue which c o n s i s t s of p r i c e per pound (assumed constant) times the c a t c h and a l s o by c o s t s which are d e r i v e d from the per u n i t c o s t s (assumed constant) times the amount o f e f f o r t r e q u i r e d to land the c a t c h . The c a t c h i t s e l f i s determined by the b i o l o g i c a l recruitment f u n c t i o n and i t s i n t r i n s i c p r o d u c t i v i t y i n terms of escapement, recruitment and c a t c h . And f i n a l l y , recruitment i t s e l f a f f e c t s the c o s t s of h a r v e s t i n g by reducing the amount of e f f o r t needed to take the 37 catch. .Therefore, revenue can be raised by increasing catch but with the penalty of reduced recruitment to sustain that catch. Or costs can be reduced by increasing recruitment but with the penalty of reduced catch to allow for that larger recruitment. If certain assumptions are made, the s o l u t i o n to a dynamic problem of t h i s sort can be s i m p l i f i e d . By assuming prices and costs are constant over time, we can eliminate the e x p l i c i t role of time. I f we put the analysis i n discounted present value terms, time i s s t i l l i m p l i c i t but the model i s autonomous (arrow and Kurz, 1970, pp. 50-51). This s t a t i o n a r i t y assumption together with a further assumption of concavity i n a l l functions and a l i n e a r harvesting function permits the derivation of an i n t e r i o r solution. The control problem then becomes one of maximizing an objective functional of the form: (1-1) PV = J e x p (-st) (p-c (x (t)) )h (t)dt where PV = present value s = s o c i a l rate of discount p = constant price of the resource c = costs which are l i n e a r i n harvesting x(t) = biomass at time t (state variable) h(t) = harvest rate (control variable) This i s the present value of net economic p r o f i t over a l l time periods (starting from the i n i t i a l period): that i s , the present value of current p r o f i t s plus the imputed value, at the shadow price, of the current rate of investment i n restocking (see Neher, 1974b, p.40). Osing the maximum p r i n c i p l e t h i s can be seen more c l e a r l y by forming the Bamiltonian: (1-2) H = exp(-st) {(p-c(x))h(t) • k(t) (F(x)-h(t))} where k (t) = costate variable (shadow price) P(x) = dx/dt, the natural rate of increase of the 38 stock (recruitment f u n c t i o n ) Note that the approach f o l l o w s the Lagrangian technique i n which net p r o f i t i s maximized s u b j e c t t o a r e c r u i t m e n t / h a r v e s t i n g c o n s t r a i n t with i t s attached shadow p r i c e . F o l l o w i n g Neher's r o u t i n e , the f i r s t s tep i s to maximize t h i s flamiltonian f o r a l l p o s s i b l e values of the s t a t e and c o s t a t e v a r i a b l e s d u r i n g the time p e r i o d . The raaximium p r i n c i p l e shows that t h i s i s accomplished when the flamiltonian i s maximized with r e s p e c t to the c o n t r o l . Before t h i s can be accomplished, however, a p r i o r minimum c o s t programming problem must be s o l v e d (Clark, 1976, p.239). Once t h i s i s done, " c " can be assumed t o be the minimum co s t form of h a r v e s t i n g . If we assume that the Hamiltonian i s l i n e a r i n p r o d u c t i o n , the maximum p r i n c i p l e w i l l be u n s u c c e s s f u l i n s o l v i n g f o r the optimal s t a t e and c o s t a t e values by s e t t i n g the p a r t i a l d e r i v a t i v e s of the Hamiltonian with r e s p e c t t o the c o n t r o l equal t o z e r o . The reason i s t h a t the p a r t i a l w i l l not c o n t a i n any form of the c o n t r o l v a r i a b l e . I n s t e a d , the g e n e r a l i z e d Pontryagin v e r s i o n which permits d i s c o n t i n u i t i e s i n the c o n s t r a i n t s i s used. That i s , the c o n t r o l i s e i t h e r on or o f f and operates along a boundary at some point i n time; c a l l e d a s w i t c h i n g f u n c t i o n , i t i s s e t equal to zero. He can then d e r i v e an "upper semi-continuous correspondence" (Neher, 1974b, pp.40-41) f o r the s t a t e and c o s t a t e v a r i a b l e s ; and the c o n t r o l i s e i t h e r switched on or o f f u n t i l an i n t e r i o r s o l u t i o n i s found. The d e c l i n e of the "present value shadow p r i c e " must egual the s t a t e v a r i a b l e ' s c o n t r i b u t i o n to the Hamiltonian i f the 39 second c o n d i t i o n i s t o be f u l f i l l e d (Neher* 1974b, pp.40-41). Neher c a l l s t h i s the zero net p r o f i t c o n d i t i o n because i t r e q u i r e s t h a t the s t a t e v a r i a b l e be a p p l i e d t o the po i n t where the value of the marginal product o f the f i s h stock p l u s i t s c a p i t a l g a i n s equals marginal net c o s t of an uncaught f i s h . The l a s t i s j u s t the foregone market r a t e of r e t u r n on t h a t uncaught f i s h l e s s the value of i t s own growth r a t e o f r e t u r n . That i s , the value of the marginal product of the f i s h stock p l u s the c a p i t a l gains (given by the equation of motion f o r the shadow p r i c e ) equals the f i n a n c i a l c o s t of an uncaught f i s h minus the value o f a p p r e c i a t i o n a t the b i o l o g i c a l own r a t e of i n t e r e s t (Neher, 1974b, p.41). I t t h i s time d e r i v a t i v e of the shadow p r i c e i s equal to zero, as i t w i l l be a t the s i n g u l a r i t y , then there a re no c a p i t a l g a i n s o r l o s s e s i n the i m p l i c i t c o s t of the f i s h s t o c k (Quirk and Smith, 1970, pp.12-13). The f i n a l c o n d i t i o n the s o l u t i o n must f u l f i l l i f i t i s to be o p t i m a l , i s t h a t the f i s h stock obey the b i o t e c h n i c a l c o n s t r a i n t s (Neher, 1974b, pp..41-42) given by F (x) i n ex p r e s s i o n (1-2). T h i s c o n d i t i o n i s f u l f i l l e d when the p a r t i a l d e r i v a t i v e of the Hamiltonian with r e s p e c t t o the shadow p r i c e i s e q u a l t o the n a t u r a l r a t e of i n c r e a s e o f the stock ( F ( x ) ) . Neher c a l l s t h i s the resource c o n s t r a i n t c o n d i t i o n f o r obvious reasons. The s o l u t i o n w i l l be s t a t i o n a r y i f the time d e r i v a t i v e of the s w i t c h i n g f u n c t i o n i s equal to zero (Clark and Munro, 1975, p.104); we have an i n t e r i o r s o l u t i o n when the s w i t c h i n g f u n c t i o n i s e q u a l t o zero; and we can use the c a l c u l u s of v a r i a t i o n s E u l e r eguation which i s none other than the time d e r i v a t i v e of the s w i t c h i n g f u n c t i o n (because of the autonomous nature of the 40 model). F o l l o w i n g C l a r k and Hunro {1975, p.95) t h i s l e a d s to an equation f o r the s i n g u l a r s o l u t i o n : (1-3) (1/s) ((d/dx) <(p-c(x))F(x))) = p-c(x) where x = s i n g u l a r or e q u i l i b r i u m s o l u t i o n as C l a r k and Hunro (1975, p.96) e x p l a i n , the l . h . s . "can be i n t e r p r e t e d as an e x p r e s s i o n of marginal user c o s t , i n t h a t i t shows the present c o s t of c a p t u r i n g the marginal increment of f i s h , a c o s t t h a t has to be weighed a g a i n s t the marginal gain from c u r r e n t capture. 1? S i m p l i f y i n g eguation (1-3) l e a d s to (1-3«) F'(x) - ( C (x)*F (x)/(p-c(x)) ) = s Again, C l a r k and Hunro (1975, p.96) e x p l a i n t h a t the l . h . s . " i s the 'own r a t e of i n t e r e s t ' (which) c o n s i s t s of two components: F ' ( x ) , the instantaneous marginal p h y s i c a l product of c a p i t a l , and - (c ' (x) *F (x) / (p-c (x))) , the marginal s t o c k e f f e c t . " The f i r s t i n d i c a t e s the foregone gain i n f u t u r e growth by h a r v e s t i n g an e x t r a f i s h today and the l a t t e r i n d i c a t e s the i n c r e a s e d f u t u r e c o s t s of h a r v e s t i n g you are imposing on y o u r s e l f by h a r v e s t i n g an e x t r a f i s h today, making f u t u r e stock more d i s p e r s e d and thus, more d i f f i c u l t t o c a t c h than today (both through reduced stock and through reduced growth i n the s t o c k ) . In summary, the method f o r o b t a i n i n g an optimal e q u i l i b r i u m s o l u t i o n depends on generating a s i n g u l a r s o l u t i o n which i s then made s p e c i f i c through s u b s t i t u t i o n of parameters from the recruitment and h a r v e s t i n g c o s t f u n c t i o n s . T h i s s o l u t i o n i s then s o l v e d f o r the optimal escapement given the s o c i a l r a t e of d i s c o u n t . I t should be noted, however, t h a t t h i s approach 41 bypasses a number of the c o m p l i c a t i o n s found i n a more general model {see C l a r k , 1976b, pp.88-108). There are three problems a s s o c i a t e d with the d e r i v a t i o n of the e g u i l i b r i u m equation i n e x p r e s s i o n (1-3) which p r e c l u d e i t s use i n t h i s study. F i r s t , e q u i l i b r i u m e q u a t i o n (1-3) i s a continuous time model which assumes t h a t "the response of the p o p u l a t i o n to e x t e r n a l f o r c e s such as h a r v e s t i n g i s i n s t a n t a n e o u s " ( C l a r k , 1976b, p.210) and yet sockeye salmon are only h a r v e s t e d f o r one twelve week p e r i o d i n t h e i r f o u r year l i f e c y c l e . The model f o r sockeye salmon i s a d i s c r e t e time model and depends on the i n t e r - r e l a t i o n s h i p s amongst escapement (parent f i s h l e f t over from h a r v e s t i n g ) , the subsequent recru i t m e n t f o u r years l a t e r and the c a t c h (the d i f f e r e n c e between recruitment and escapement)., Recruitment f o u r years hence i s a f u n c t i o n of escapement today which, i n t u r n , i s a f u n c t i o n of the recruitment ( d e r i v e d from escapement f o u r years p r e v i o u s l y ) minus the c u r r e n t c a t c h . The second problem i s that the equilibrium equation i n e x p r e s s i o n (1-3) i s derived f o r a l i n e a r h a r v e s t i n g f u n c t i o n , whereas the h a r v e s t i n g production f u n c t i o n s used i n t h i s study are n o n l i n e a r . A l l o w i n g f o r a n o n l i n e a r h a r v e s t i n g f u n c t i o n and the d i s c r e t e nature of the r e c r u i t m e n t f u n c t i o n , C l a r k (1976b, pp.250-253) has developed an a l t e r n a t i v e e q u i l i b r i u m e q u a t i o n of the form: (1-4) G(x)((P(R) • P(C))/P(C)) = ( U s ) S where x = optimal escapement G (x) = d e r i v a t i v e of r e c r u i t m e n t f u n c t i o n with r e s p e c t t c x E (B) = d e r i v a t i v e of p r o f i t with r e s p e c t to 42 recruitment from x P (C) = d e r i v a t i v e of p r o f i t with respect to c a t c h from the recruitment and a l l o w i n g o p t i m a l escapement T h i s i s the e q u i l i b r i u m equation used i n t h i s study to s o l v e f o r optimal escapement, recruitment and c a t c h (the d e r i v a t i o n of (1-4) i s g i v e n i n the Appendix a t t a c h e d t o t h i s c h a p t e r ) . As with e x p r e s s i o n (1-3*), the l . h . s . i s the f i s h e r y r a t e of i n t e r e s t c o n s i s t i n g of the marginal p h y s i c a l product of c a p i t a l , G (x), and the marginal stock e f f e c t , ((P (E) + P(C))/P ( C ) ) . Because of the d i s c r e t e nature of the p r o f i t f u n c t i o n , and the impact on c o s t s and p r o f i t s of the i n t e r r e l a t i o n s h i p amongst escapement, r e c r u i t m e n t and c a t c h , the d e r i v a t i v e with r e s p e c t to optimal escapement i s i n d i r e c t l y o b tained through o b s e r v i n g the impact of changes i n c a t c h and r e c r u i t m e n t (given t h a t o p t i m a l escapement) on p r o f i t s . C l a r k (1976b, pp.243-253) e x p l a i n s how the d i s c r e t e v e r s i o n i n e x p r e s s i o n (1-4) can be converted i n t o the continuous v e r s i o n i n e x p r e s s i o n (1-3*) gi v e n a l i n e a r c o s t f u n c t i o n . The t h i r d problem with the d e r i v a t i o n of both e q u i l i b r i u m equation (1-3) and (1-4) i s the assumption t h a t a p r i o r minimum co s t programming problem has been so l v e d to achieve the minimum c o s t form of h a r v e s t i n g . T h i s may be p o s s i b l e with a l i n e a r h a r v e s t i n g f u n c t i o n as assumed f o r (1-3) but i s not p o s s i b l e with n o n l i n e a r h a r v e s t i n g p r o d u c t i o n f u n c t i o n s as assumed i n (1-4). The reason i s t h a t the e q u i l i b r i u m combination of c a t c h and r e c r u i t m e n t suggested by (1-4) c o u l d l e a d to a d i f f e r e n t l e a s t c o s t c o n f i g u r a t i o n of gear to minimize the c o s t s of t a k i n g that c a t c h . T h i s a l t e r e d c o n f i g u r a t i o n o f gear would change the 43 marginal s t o c k e f f e c t i n (1-4) and l e a d t o a d i f f e r e n t e q u i l i b r i u m combination of c a t c h and r e c r u i t m e n t . T h e r e f o r e , the l e a s t c o s t programming s o l u t i o n and the e g u i l i b r i u m v a l u e s f o r c a t c h and r e c r u i t m e n t must be s i m u l t a n e o u s l y determined. One of the i n n o v a t i o n s i n t h i s study i s the development of a programming a l g o r i t h m which i s capable of such simultaneous d e t e r m i n a t i o n . In a d d i t i o n to the i n t e r t e m p o r a l common property problem which a r i s e s from the c a p i t a l t h e o r e t i c nature of an open access f i s h e r y , t h e r e are s p a t i a l m i s a l l o c a t i o n s of h a r v e s t i n g e f f o r t which can a l s o take place because of h a r v e s t i n g a l l o c a t i o n e x t e r n a l i t i e s p e c u l i a r t o a g a u n t l e t f i s h e r y such as t h a t f o r F r a s e r Hiver sockeye salmon. The c o n g e s t i o n l e a d i n g to i n t r a -s e a s o n a l stock e f f e c t s of t h i n n i n g w i l l d r i v e the fisherman to areas where gear i s l e s s crowded. In these poorer areas gear may be l e s s p r o d u c t i v e i n terms of c a t c h a b i l i t y but o v e r a l l r e c r u i t m e n t a v a i l a b i l i t y i s probably higher than i n crowded areas where gear would normally be more p r o d u c t i v e . I t i s easy to confuse a v a i l a b i l i t y with e f f i c i e n c y of gear types. But what i s i n f a c t going on i s a s p a t i a l m i s a l l o c a t i o n of e f f o r t because fishermen are f o r c e d from areas where c a t c h a b i l i t y i s normally h i g h e r but a v a i l a b i l i t y i s reduced because of crowding to areas of normally poorer c a t c h a b i l i t y but where there i s l e s s t h i n n i n g of the stock due t o reduced crowding. T h i s i s p a r t i c u l a r y true f o r the h a r v e s t i n g model used i n t h i s study. In a d d i t i o n t o i n t r a - g e a r i n t e r f e r e n c e of t h i s type i t i s p o s s i b l e f o r there t o be i n t e r - g e a r i n t e r f e r e n c e as c e r t a i n gear types must wait f o r the f i s h t o come to them, whereas other gear 44 types can a c t i v e l y search f o r b l o c k s of f i s h . 4s well as these i n t r a - s e a s o n a l stock e f f e c t s , there are a l s o i n t e r - s e a s o n a l stock e f f e c t s i n an open access f i s h e r y because to one fisherman the shadow p r i c e on enhanced recru i t m e n t i s always l i k e l y to be below the c u r r e n t landed p r i c e o f f i s h . Fishermen tend to h a r v e s t more h e a v i l y today l e a d i n g to an i n c r e a s e i n f u t u r e c o s t s owing to the i n v e r s e r e l a t i o n s h i p between a v a i l a b i l i t y and c o s t s of h a r v e s t i n g . I f f i s h i n g were h a l t e d , the biomass would grow a t a g r e a t e r r e l a t i v e r a t e (Hannesson, 1975, pp. 161-2). But common prop e r t y fishermen ignore t h i s user c o s t because they have no way of c a p t u r i n g the investment " r e t u r n s " . Open access and stock e f f e c t s combine t o produce a sub-optimal a l l o c a t i o n of e f f o r t on a s p a t i a l b a s i s during a s i n g l e season and on an i n t e r t e m p o r a l b a s i s over s e v e r a l seasons. I f access were c l o s e d by handing the f i s h e r y over to a monopolist, he would i n c r e a s e p r o d u c t i v i t y by spreading e f f o r t more evenly over the whole f i s h i n g g a u n t l e t and over time so as to take advantage of a v a i l a b i l i t y d i f f e r e n c e s , d i f f e r e n c e s i n c a t c h a b i l i t y and d i f f e r e n c e s i n gear e f f i c i e n c y . These cost r e d u c t i o n s would come about not because of a v o i d i n g d i m i n i s h i n g r e t u r n s but because of i n c r e a s e d a v a i l a b i l i t y . I f the monopolist had s o l e r i g h t s through time as w e l l , he c o u l d again reduce c o s t s by i n - k i n d investment through a b s t e n t i o n from f i s h i n g . He would e x p l o i t h i s f i s h e r y u n t i l the f i s h e r y r a t e of r e t u r n was equated with the s o c i a l r a t e of d i s c o u n t g i v i n g the optimal s o l u t i o n (Brown, 1974, pp.167-8; and C l a r k and ifunro, 1975, pp.95-96). 45 There are user costs associated with each of the i n t r a -seasonal s p a t i a l and inter-seasonal temporal misallocations of e f f o r t . Even i f the open access fisherman i s a net present value maximizer, he s i l l ignore these costs because of the overwhelming negative user cost associated with common property; there i s no point i n keeping an inventory or re-stocking one because your neighbour simply takes i t from you (Brown, 1974, p.167). Even in a two person f i s h e r y , i t can be shown that both w i l l work to enhance their "take" of the neighbour's inventory to as close to 100% as possible (Clark, 1978). If the fishery i s turned over to a monopoly inventory manager, from his point of view i t makes very l i t t l e difference how the biomass grows: whether i t be from increased f l e s h on e x i s t i n g f i s h or increased numbers of new f i s h . What does matter to him i s the f a c t that catching one more f i s h today could mean more growth or less growth i n tomorrow's biomass depending on the e f f e c t s operating within the population. For sockeye salmon, the manager i s faced with the fact that growth can only take place through new r e c r u i t s because the parents die after spawning. The stock e f f e c t only comes through recruitment, not through surviving adults. And because of the four year cycle, the restocking of inventory i s delayed f o r four years. Clark (1976b, p.7) has shown that i n the case of non-survival of escaping f i s h , the delay can be collapsed into a single period. This would mean, in the case of sockeye, a "period" consisting of four years. In general, Clark has found that for models i n which v u l n e r a b i l i t y of new r e c r u i t s to the f i s h e r y i s delayed, the most rapid approach to equilibrium values i s not optimal. 46 But i n the case of n o n - s u r v i v a l of escaping parents and constant o p p o r t u n i t y c o s t s (both b a s i c assumptions i n t h i s study) , the optimal approach, a c c o r d i n g t o C l a r k , can be approximated by the most r a p i d approach to e q u i l i b r i u m v a l u e s . For t h i s reason, the F r a s e r B i v e r sockeye f i s h e r y w i l l be assumed to operate with a p e r i o d of f o u r years d u r a t i o n with a p p r o p r i a t e adjustments i n the d i s c o u n t r a t e . T h i s should present no s e r i o u s problems as each of the f o u r year c l a s s runs of sockeye within each p e r i o d appears to be independent both from a recruitment (Chapter Two) and a h a r v e s t i n g (Chapter Three) p o i n t of view. It w i l l a l s o be assumed, given constant o p p o r t u n i t y c o s t s , t h a t the most r a p i d approach to the e q u i l i b r i u m biomass l e v e l i s o p t i m a l . T h i s should not be a problem i n terms of sockeye management because the r e c r u i t m e n t s i n every c y c l e year exceed the optimal escapement l e v e l . T h e r e f o r e , i n c y c l e years where op t i m a l escapement i s s m a l l e r than a c t u a l , c a t c h would be i n c r e a s e d ; and i n c y c l e years where optimal escapement i s l a r g e r than a c t u a l , c a t c h would be decreased. Turning t o the r e g u l a t i o n s to be used by the f i s h e r y manager, i t i s w e l l known that c o n t r o l s play an important r o l e i n determining both the c o s t s of h a r v e s t i n g and the c o s t s of management i t s e l f . T h i s study c o n f i n e s i t s e l f to d i r e c t q u a n t i t a t i v e c o n t r o l s over e f f o r t as other systems of r e g u l a t i o n are e i t h e r not f e a s i b l e or p r o h i b i t i v e l y expensive. D i r e c t g u a n t i t a t i v e c o n t r o l over e f f o r t a b s t e n t i o n has taken place h i s t o r i c a l l y through a r e a , time and season c l o s u r e s . One problem with u s i n g these d i r e c t types of c o n t r o l s to reduce exposure of 47 gear to f i s h i s t h a t i t l e a d s fishermen t o i n v e s t i n c a p i t a l to enhance c a t c h i n g power d u r i n g the b r i e f exposure time. The problem with the stock c f boats i s the i r r e v e r s i b l e nature of t h i s c a p i t a l good. In order to avoid i r r e v e r s i b i l i t y i n t h i s study, c a p i t a l goods w i l l be converted i n t o a flow which w i l l be employed as an i n p u t to h a r v e s t sockeye salmon. Only l e a s t c o s t gear and v e s s e l s w i l l be used and the r e s t w i l l be o f f e r e d lump sum compensation f o r t h e i r c a p i t a l l o s s . , In t h i s way, the problems encountered with i r r e v e r s i b l e c a p i t a l s t o c k s can be avoided (see C l a r k e , C l a r k and Munro, 1979). To avoi d problems such as these, the c h o i c e of regime i s a l s o c r i t i c a l . Furthermore, the r i g h t s assigned under the regime must be such as to avoid d i s s i p a t i o n of economic r e n t through i n e f f i c i e n c y or open access problems. Munro (1979) has developed an approach to a n a l y z i n g the transboundary management of a f i s h e r i e s r e source i n which he a l l o w s c o n f l i c t i n management s t r a t e g i e s t o a r i s e through d i f f e r e n c e s i n s o c i a l r a t e s of d i s c o u n t , h a r v e s t i n g c o s t s and consumer t a s t e s i n the two consuming n a t i o n s . Munro demonstrates that i f there are no d i f f e r e n c e s i n s o c i a l d i s c o u n t r a t e s , consumer t a s t e s or h a r v e s t i n g c o s t s , b a r g a i n i n g devolves i n t o s h a r i n g the harvest or the proceeds. For the purposes of t h i s study i t w i l l be assumed that t h e r e are no d i f f e r e n c e s i n s o c i a l d i s c o u n t r a t e s or consumer t a s t e s between the O.S. and Canada but t h a t t h e r e are d i f f e r e n c e s i n h a r v e s t i n g c o s t s . &nd the regime best s u i t e d to maximizing the present value of the F r a s e r R i v e r sockeye f i s h e r y w i l l be assumed to be a s o l e owner: a h y p o t h e t i c a l monopoly f i r m without ownership r i g h t s but with the r i g h t s to 48 use and d i s p o s i t i o n c f the n a t u r a l resource a s s e t and charged with the task of maximizing the present value of the f i s h e r y . I t w i l l be assumed that the powers o f the IPSFC w i l l be expanded to i n c l u d e such r i g h t s and d u t i e s . I t i s widely r e c o g n i z e d (Kasahara and Burke, 1973) that the most s u c c e s s f u l i n t e r n a t i o n a l f i s h e r i e s commission has j u s t such powers under the North P a c i f i c Fur Seals Convention, The t r e a t y was r a t i f i e d i n 1911 and covers p e l a g i c s e a l i n g i n the B e r i n g Sea. The commission i s charged with maximizing p r o f i t ( a l b e i t on a s t a t i c basis) and has f u l l g u a n t i t a t i v e c o n t r o l over e f f o r t . However, i t i s powerless without the unanimous vote of the members. It w i l l be assumed that n e i t h e r the O.S. nor the Canadian government would be prepared t o a l l o t the h a r v e s t i n g of the c a t c h on anything but an equal d i v i s i o n b a s i s between U.S. and Canadian fishermen, ft p o l i c y of h i r i n g only the l e a s t c o s t gear type areas would r e s u l t i n d i f f e r e n t catch p r o p o r t i o n s . , For these reasons, the optimal e q u i l i b r i u m management s o l u t i o n w i l l be c a l c u l a t e d f o r both an unconstrained case i n which the IPSFC can choose to h a r v e s t where i t wishes, and an a l t e r n a t i v e case i n which the IPSFC w i l l be c o n s t r a i n e d to h a r v e s t i n g on an egual c a t c h share b a s i s . (3) Management Model With t h i s t h e o r e t i c a l base f o r the o p t i m i z i n g model i t i s now p o s s i b l e to c o n s t r u c t the management model. The IPSFC w i l l be assumed t o have the r i g h t s t o h a r v e s t and manage the F r a s e r River sockeye salmon and be charged with the r e s p o n s i b i l i t y to maximize the present value of the f i s h e r y . I t w i l l operate 49 w i t h i n the j u r i s d i c t i o n a l and p h y s i c a l l i m i t s o u t l i n e d above and w i l l h i r e g i l l n e t s and purse s e i n e s i n the f o u r , l e a s t c o s t f i s h i n g zones. The F r a s e r R i v e r sockeye f i s h e r y w i l l be assumed to operate with a p e r i o d of f o u r years d u r a t i o n with a p p r o p r i a t e adjustments i n the d i s c o u n t r a t e . T h i s should present no s e r i o u s problems as each of the f o u r year c l a s s runs o f sockeye w i t h i n each p e r i o d appears to be independent both from a r e c r u i t m e n t (Chapter Two) and a h a r v e s t i n g (Chapter Three) point of view. T h i s i m p l i e s t h a t each c y c l e year w i t h i n t h a t f o u r year p e r i o d w i l l have a d i f f e r e n t optimal c a t c h l e v e l ; and t h a t i t w i l l take f c u r y e a r s to reach those optimal e g u i l i b r i u m biomass l e v e l s . T h i s r a p i d approach to the e q u i l i b r i u m biomass l e v e l s w i l l be assumed to be the optimal approach. The r e c r u i t m e n t f u n c t i o n i s the b a s i s of the b i o l o g i c a l system. The h a r v e s t i n g production f u n c t i o n s are the b a s i s of the l e a s t c o s t s p a t i a l l o c a t i o n model. And the Clark-Munro e g u i l i b r i u m eguation i s the b a s i s f o r the dynamic o p t i m i z i n g system. I f the h a r v e s t i n g c o s t model were l i n e a r , the management p a r t of the study would be f a i r l y s t r a i g h t - f o r w a r d . A s i n g l e l e a s t c o s t h a r v e s t i n g gear type area would be chosen f o r each of the f o u r c y c l e years; and the o p t i m a l escapement, re c r u i t m e n t and c a t c h c o u l d be c a l c u l a t e d by combining the i n f o r m a t i o n on the r e c r u i t m e n t model and the h a r v e s t i n g model. T h i s optimal management s o l u t i o n would take i n t o account the impact of escapement on subsequent r e c r u i t m e n t , and the impact of the l a t t e r on the amount of e f f o r t needed to take the s p e c i f i e d c a t c h ; a f i s h e r y r a t e of r e t u r n would then be c a l c u l a t e d i n terms o f the c u r r e n t p r o f i t s d e r i v e d from e x p l o i t a t i o n , l e s s the 50 c o s t s o r b e n e f i t s a s s o c i a t e d with reduced or enhanced f u t u r e c a t c h and rec r u i t m e n t . And the present value of the f i s h e r y would be maximized when t h a t f i s h e r y r a t e of r e t u r n was e q u a l i z e d with the s o c i a l r a t e of d i s c o u n t . However, the h a r v e s t i n g c o s t model i s not l i n e a r , and, as Flowchart (1-1) i l l u s t r a t e s , t h i s r e g u i r e s a s e l e c t i o n of h a r v e s t i n g gear type areas s i m u l t a n e o u s l y with recruitment. The recrui t m e n t model i s i n i t i a l i z e d with a s p e c i f i c escapement and the model then c a l c u l a t e s subsequent r e c r u i t m e n t . Catch i s simply t h a t recruitment minus the s p e c i f i e d escapement. , The h a r v e s t i n g c o s t model then a l l o c a t e s the given c a t c h and recruitment u n t i l i t f i n d s the l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n . Programming c r i t e r i a w i t h i n the o v e r a l l a l g o r i t h m check t o determine i f t h i s combination of escapement, recruitment and c a t c h together with i t s l e a s t c ost h a r v e s t i n g c o n f i g u r a t i o n i s indeed o p t i m a l . , I f not, the program r e -i n i t i a l i z e s the escapement value and goes through the same r o u t i n e as above. The model keeps on i t e r a t i n g i n t h i s way u n t i l i t a c hieves an o p t i m a l s o l u t i o n . The name of the program package used i n the programming was CONOPT which i s pa r t of a l a r g e r package f o r n o n l i n e a r f u n c t i o n o p t i m i z a t i o n c a l l e d UBC NLP (P a t t e r s o n , 1978). CONOPT uses a penalty f u n c t i o n approach t o remove c o n s t r a i n t s . Whenever a c o n s t r a i n t i s v i o l a t e d , the o b j e c t i v e f u n c t i o n value i s adjusted by a smooth f u n c t i o n (using a modified Lagrange m u l t i p l i e r approach) which p l a c e s bounds on the f e a s i b l e parameter values the program can c o n s i d e r . That i s , a guasi-Newton a l g o r i t h m i s used to minimize the o b j e c t i v e f u n c t i o n ; and i f t h i s s o l u t i o n 51 Flowchart : Optimal Management Program 52 v i o l a t e s one of the c o n s t r a i n t s , a penalty f u n c t i o n i s c a l l e d to a d j u s t the o b j e c t i v e f u n c t i o n to be w i t h i n c e r t a i n parameter bounds. , The o p t i m i z i n g a l g o r i t h m e v a l u a t e s the o b j e c t i v e f u n c t i o n and i t s f i r s t p a r t i a l d e r i v a t i v e s , and the c o n s t r a i n t s and t h e i r f i r s t p a r t i a l d e r i v a t i v e s with r e s p e c t to the v a r i a b l e s at any p o i n t . Subject t o the penalty f u n c t i o n , as soon as the p a r t i a l d e r i v a t i v e s are very c l o s e t o zero (within some s p e c i f i e d d i s t a n c e ) the program s t o p s . Programming f o r the Clark-Munro e g u i l i b r i u m values i n v o l v e s a s e a r c h f o r a zero. Because recru i t m e n t , c a t c h and escapement are a l l f u n c t i o n a l l y r e l a t e d , i t i s p o s s i b l e t o re-arrange e x p r e s s i o n (1-4) to g i v e : (1-4«) G(x) ((P(B(x)) + P(C(x)) )/P(C(x))) - (1 + s f = 0 where x = optimal escapement G(x) = d e r i v a t i v e of r e c r u i t m e n t f u n c t i o n with r e s p e c t to x P(B(x)) = d e r i v a t i v e of p r o f i t with r e s p e c t to recruitment as a f u n c t i o n of x P(C(x)) = d e r i v a t i v e of p r o f i t with r e s p e c t t o c a t c h as a f u n c t i o n of x, a l l o w i n g optimal escapement The key, unknown v a r i a b l e i s the escapement l e v e l and a program i s used to search f o r a zero . As soon as a value of escapement i s found which reduces t h i s i m p l i c i t f u n c t i o n very c l o s e to zero (within seme s p e c i f i e d d i s t a n c e ) , the program st o p s . Because of the simultaneous nature of the optimal s o l u t i o n , the problem becomes cne of t r y i n g to b r i n g both these programs t o g e t h e r . , CONOPT would then g i v e a l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n f o r s p e c i f i e d l e v e l s of escapement, r e c r u i t m e n t and c a t c h and the s e a r c h - f o r - z e r o program would then c a l c u l a t e the optimal l e v e l of escapement given t h a t h a r v e s t i n g c o n f i g u r a t i o n . The i n n o v a t i o n i n t h i s study was to simply make 53 the Clark-Munro e q u i l i b r i u m equation a bi n d i n g c o n s t r a i n t i n CGNOPT. Although the programming was n o n - t r i v i a l , the e s s e n t i a l idea i s very simple: CONOPT maximizes p r o f i t using a l e v e l of escapement which s a t i s f i e s the Clark-Munro e q u i l i b r i u m equation. The p r o f i t f u n c t i o n d e r i v a t i v e s i n the l a t t e r c o n s t r a i n t are ev a l u a t e d f o r each l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n suggested by the r e c r u i t m e n t / c a t c h r a t i o d e r i v e d from the c u r r e n t l e v e l of escapement. , In terms of running the program, a l l v a r i a b l e v a l u e s are i n i t i a l i z e d at a s t a r t i n g p o i n t which does not v i o l a t e the c o n s t r a i n t s ( i n c l u d i n g the the zero value f o r the Clark-Hunro e g u a t i o n ) . The program then a l t e r s the escapement l e v e l , l e a d i n g to changes i n c a t c h and r e c r u i t m e n t which then l e a d to a d i f f e r e n t , l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n s u b j e c t t o a l l the c o n s t r a i n t s . These i t e r a t i o n s continue u n t i l an escapement l e v e l i s chosen which si m u l t a n e o u s l y maximizes p r o f i t s and s a t i s f i e s the Clark-Hunro e q u i l i b r i u m e quation. In t h i s way, both the s p a t i a l and the i n t e r t e m p o r a l dimensions of the problem are s i m u l t a n e o u s l y s a t i s f i e d . I SOU BOY AND CONCLUSIONS Given the t h e o r e t i c a l b a s i s f o r the o p t i m i z i n g model i n the survey of the l i t e r a t u r e and the assumptions needed f o r the management model, i t i s now p o s s i b l e t o summarize the e m p i r i c a l approach used i n t h i s study to analyse o p t i m a l management of the Fr a s e r River sockeye f i s h e r y . The l i f e c y c l e of the sockeye and t h e i r r e t u r n m i g r a t i o n demonstrates the two p a r t nature of the problem: a recru i t m e n t 54 model which i n t e r a c t s with a h a r v e s t i n g model. The presence of a very strong four year c y c l e i n both these models s i m p l i f i e s the task o f developing them and p l a y s a c r i t i c a l r o l e i n the management o p t i m i z i n g process by p e r m i t t i n g the f i s h e r y to be broken up i n t o f o u r separate year c l a s s f i s h e r i e s with f o u r year l a g s between escapement and r e c r u i t m e n t . The e v o l u t i o n of f i s h e r i e s economics has culminated i n a c a p i t a l t h e o r e t i c approach t o choosing a management s o l u t i o n . T h i s s o l u t i o n i s determined by the i n t e r a c t i o n between the own r a t e of growth o f the stock (recruitment model) and the s e n s i t i v i t y of h a r v e s t i n g c o s t s and p r o f i t s to changes i n c a t c h and r e c r u i t m e n t s i z e ( h a r v e s t i n g model) . The stock e f f e c t s and the problems a s s o c i a t e d with open access on the f i s h i n g grounds can be r e s o l v e d by g i v i n g the f i s h e r y manager monopoly powers. The economically f e a s i b l e c o n t r o l would be d i r e c t g u a n t i t a t i v e a l l o c a t i o n of e f f o r t t o achieve the l e a s t c o s t s p a t i a l d i s t r i b u t i o n of h a r v e s t i n g e f f o r t and the o p t i m a l escapement f o r each c y c l e year. It w i l l be assumed t h a t the IPSFC i s granted use and d i s p o s i t i o n r i g h t s over the F r a s e r R i v e r sockeye salmon and that they are charged with maximizing the present value of the resource r e n t . The economic p r o f i t e x t r a c t e d from the resource a s s e t can then be used to compensate v e s s e l owners who are f o r c e d out of the sockeye f i s h e r y . Apart from t h i s "as i f " compensation, d i s t r i b u t i o n a l a s p e c t s w i l l be i g n o r e d . T h i s i s p a r t i a l l y j u s t i f i e d on the b a s i s t h a t a l l f a c t o r i n p u t s w i l l be paid t h e i r o p p o r t u n i t y c o s t s i f they are used i n the h a r v e s t i n g p r o c e s s . However, the IPSFC w i l l use the f a c t o r s i n d i f f e r i n g 55 amounts and periods of time depending on the c y c l e year. , The c o s t s o f t h i s "excess c a p a c i t y " w i l l be assumed to be borne by the v e s s e l owners and no attempt w i l l be made to i n c l u d e i t i n the o p p o r t u n i t y c o s t f a c t o r r e n t a l payments. Turning from these assumptions to the management approach used in t h i s study, the optimal management s o l u t i o n f o r the F r a s e r R iver sockeye f i s h e r y w i l l be obtained i n three s t e p s . In the f i r s t , the h a r v e s t i n g model developed i n Chapter Three w i l l be transformed i n t o a c o s t model. , N onlinear programming w i l l then be used t o so l v e f o r l e a s t c o s t h a r v e s t i n g p r o p o r t i o n s amongst the s i x most important gear type areas, given recruitment and escapement (Chapter Three)., The second step i n v o l v e s t a k i n g the l e a s t c o s t gear type areas f o r B.C. and the U.S. and s u b s t i t u t i n g them p l u s the r e c r u i t m e n t model (Chapter Two) i n t o the Clark-Munro e g u i l i b r i u m e g u a t i o n . The d i s c r e t e form o f t h i s eguation i s used as sockeye are only h a r v e s t e d at d i s c r e t e i n t e r v a l s i n t h e i r l i f e c y c l e . . The most important and u s e f u l technigue at t h i s stage i s the s p l i t t i n g o f the F r a s e r sockeye i n t o f o u r separate year c l a s s e s . Each year c l a s s w i l l be assumed t o be harvested q u i t e s e p a r a t e l y from any other year c l a s s a t d i s c r e t e f o u r year i n t e r v a l s , although s o l v i n g the e q u i l i b r i u m equation f o r the f o u r year c l a s s e s s i m u l t a n e o u s l y with i m p l i c i t i n t e r a c t i o n s i s p o s s i b l e , s p l i t t i n g the stock i n t o f o u r year c l a s s f i s h e r i e s l e a d s t o a gre a t improvement. T h i s permits o p t i m a l escapement to be d i f f e r e n t f o r each year c l a s s , whereas simultaneous d e t e r m i n a t i o n would y i e l d o n l y a s i n g l e optimal escapement f o r a l l f o u r year c l a s s e s . 56 The t h i r d step toward the management s o l u t i o n i n v o l v e s c a l c u l a t i o n of optimal r e c r u i t m e n t , from the r e c r u i t m e n t f u n c t i o n , g i v e n optimal escapement; and c a l c u l a t i o n of optimal c a t c h , given the production f u n c t i o n s f o r the l e a s t c o s t areas p l u s o p t i m a l escapement and re c r u i t m e n t . The r e s u l t s a re then f e d back i n t o the l e a s t c o s t program t o determine the l e a s t cost c a t c h p r o p o r t i o n s given the new l e v e l s of escapement and recru i t m e n t . T h i s i t e r a t i o n between the recr u i t m e n t model and the h a r v e s t i n g model c o n t i n u e s u n t i l the program converges to a s o l u t i o n . I t i s then p o s s i b l e to c a l c u l a t e the net revenue from h a r v e s t i n g of t h a t optimal c a t c h with the l e a s t c o s t gear type areas and to compare t h i s h y p o t h e t i c a l net revenue with the a c t u a l average net revenue experienced under IPSFC management over t h e twenty-nine years from 1947 to 1975. The r e s u l t s would appear to support the hypothesis that a c t u a l management d i d not achieve o p t i m a l a l l o c a t i o n of e f f o r t over t h a t twenty-nine 'year p e r i o d at c o n s i d e r a b l e c o s t i n terms of foregone r e n t which co u l d have been accumulated. I t should be made c l e a r , however, that the IPSFC has never been charged with the r e s p o n s i b i l i t y f o r o p t i m a l a l l o c a t i o n of e f f o r t but has attempted, g i v e n the l e g a l c o n s t r a i n t s of the convention, to a l l o c a t e e f f o r t i n as econ o m i c a l l y e f f i c i e n t a manner as p o s s i b l e . Less than h a l f the h y p o t h e t i c a l gain i n p o t e n t i a l economic rent comes about from a s p a t i a l r e l o c a t i o n of h a r v e s t i n g gear and the r e s t comes from the o p t i m a l e q u i l i b r i u m values f o r escapement, r e c r u i t m e n t and c a t c h . I t appears t h a t a l a r g e p a r t of these gains a r i s e s from a r e d u c t i o n i n h a r v e s t i n g c o s t s r a t h e r than by i n c r e a s i n g revenue 57 through enhanced catches. T h i s emphasizes a need f o r economic c o n s i d e r a t i o n s as both economic theory and the e m p i r i c a l work i n t h i s study i n d i c a t e t h a t p r o f i t s can be s i g n i f i c a n t l y enhanced through r e d u c t i o n s i n c o s t r a t h e r than through i n c r e a s e s i n revenue only as a r e s u l t of l a r g e r s u s t a i n a b l e catches. APPENDIX D e r i v a t i o n Of The Clark-Hunro E q u i l i b r i u m Equation To d e r i v e a formula f o r the optimal e q u i l i b r i u m s o l u t i o n to a d i s c r e t e time f i s h e r y model with a n o n l i n e a r h a r v e s t i n g f u n c t i o n C l a r k (1976b, pp.252-253) employs the d i s c r e t e maximum p r i n c i p l e . The o b j e c t i v e f u n c t i o n a l i s given by (1-5) J = JE a*-*P(R,C) where a = 1/(1 +s) s = s o c i a l r a t e o f d i s c o u n t P(R,C) = p r o f i t f u n c t i o n dependent on recruitment (R) and cat c h (C) The Hamiltonian i s (1-6) H = a*"iP(B,C) .• K ( t ) ( F ( R - Q - R) where K (t) = c o s t a t e v a r i a b l e (shadow p r i c e ) F (R-C) = subsequent r e c r u i t m e n t dependent on escapement (R-C) i n pr e v i o u s time p e r i o d (recruitment f u n c t i o n ) The maximum p r i n c i p l e i m p l i e s (Neher's f i r s t c o n d i t i o n ) : (1-7) H (C) = a*~*P(C) - K(t)G(x) = 0 where H (C) = p a r t i a l d e r i v a t i v e o f the Hamiltonian with r e s p e c t t o c a t c h P(C) = p a r t i a l d e r i v a t i v e of p r o f i t with respect t o c a t c h G (x) = d e r i v a t i v e c f the recruitment f u n c t i o n with r e s p e c t to escapement (x) 58 A f t e r re-arrangement t h i s g i v e s (1-8) K (t) = a*-»P (C)/G (x) and (1 -9) K(t) - K(t-1) = ( a * - i - a*-*)P<C)/G(x) Neher's second c o n d i t i o n i s f u l f i l l e d when (1-10) K(t) - K(t-1) = -H(E) where H(B) = p a r t i a l d e r i v a t i v e of the Hamiltonian with r e s p e c t t o the s t a t e v a r i a b l e (recruitment) and the r i g h t hand s i d e i s (1-11) -H (B) = -a*-ip(B) - a" t-»P(C)(1 - 1/G (x)) where P (8) = p a r t i a l d e r i v a t i v e of the p r o f i t f u n c t i o n with respect to the recr u i t m e n t Eguating e x p r e s s i o n s (1 - 9 ) and (1-11) and s i m p l i f y i n g l e a d s to (1-4) G(x)((P(B) • P(C))/P(C)) = ( U s ) * The reason the d i s c o u n t f a c t o r i s r a i s e d to the f i f t h power i s because of the f o u r year lag between escapement and subsequent r e c r u i t m e n t . Neher's t h i r d c o n d i t i o n i s a u t o m a t i c a l l y met by G (x), t h e n a t u r a l r a t e of growth i n the s t o c k a l l o w i n g f o r h a r v e s t i n g . 59 CHAPTER TWO: THE RECROITMENT FONCTION I STOCK MANAGEMENT The management of the sockeye f i s h e r y r e q u i r e s two d e c i s i o n s : how much of the c u r r e n t r e c r u i t m e n t can be harvested and i n what way. Whatever the manager allows as an escapement bears a c o s t i n terms of foregone c a t c h i n the c u r r e n t p e r i o d but o f f e r s a r e t u r n f o u r years l a t e r (and subseguently) i n terms of some recr u i t m e n t . The l a r g e r the f u t u r e recruitment, the l a r g e r the p o t e n t i a l c a t c h and the s m a l l e r the h a r v e s t i n g c o s t s due to i n c r e a s e d a v a i l a b i l i t y . Thus the manager can " i n v e s t " f o r the f u t u r e by a b s t a i n i n g from h a r v e s t i n g today. T h i s chapter i s concerned with e s t i m a t i n g t h i s b a s i c b i o l o g i c a l investment f u n c t i o n . The model used i s the Ricker form of r e c r u i t m e n t f u n c t i o n which i s the standard c h o i c e i n salmon s t u d i e s . The data used i n the e s t i m a t i o n process i n c l u d e o b s e r v a t i o n s on recruitment of biomass to the f i s h e r y , the h a r v e s t i n g c a t c h made by the fishermen and the count of f i s h e s caping to the spawning grounds. And the nature of the b i o l o g i c a l f u n c t i o n i s determined by which of r e p r o d u c t i o n , growth or reduced n a t u r a l m o r t a l i t y o f f s e t s f i s h i n g m o r t a l i t y . Taking i n t o account the c y c l i c a l nature of environmental i n f l u e n c e s , the r e c r u i t m e n t model i m p l i e s t h a t c o n t r o l over f i s h i n g m o r t a l i t y i s the main determinant of subsequent re c r u i t m e n t . I t i s i n t h i s way t h a t the f i s h e r y manager c o n t r o l s the r e t u r n on h i s i n - k i n d investment. The management d e c i s i o n to c a t c h the c u r r e n t stock or allow i t to escape as an i n d i r e c t form of investment i n a f u t u r e stock 60 r e q u i r e s knowledge of the p r o d u c t i v i t y of that investment. Consequently i t i s important to understand the rudiments of the b i o l o g i c a l processes i n v o l v e d i n the mechanisms which c o n t r o l the stock. Many b i o l o g i s t s b e l i e v e the most important c o n t r o l on the s i z e o f a salmon stock i s achieved through d e n s i t y dependence. That i s , the f a c t o r s enhancing or r e t a r d i n g s u r v i v a b i l i t y and growth are turned on or o f f by the degree of s t o c k d e n s i t y i n a p a r t i c u l a r stage i n the l i f e c y c l e . I t i s u s u a l l y assumed that there i s some form of environmental l i m i t on the stock s i z e . That i s , the a q u a t i c environment can only support a c e r t a i n s i z e of s t o c k which i m p l i e s that there w i l l be some maximum f u t u r e r e c r u i t m e n t , and i n v e s t i n g through g r e a t e r a b s t e n t i o n y i e l d s no r e t u r n . .Often r e f e r r e d to as the e c o l o g i c a l or environmental n i c h e , the u l t i m a t e l i m i t on the stock s i z e i s , t h e r e f o r e , determined by the e x t e r n a l environment. And d e n s i t y dependence w i l l be assumed to c o n t r o l the s i z e of the s t o c k w i t h i n that b a s i c environmental l i m i t . In a d d i t i o n to d e n s i t y dependent i n t r a - s p e c i e s c o n t r o l s t h e r e are i n t e r - s p e c i e s (predator-prey) i n t e r a c t i o n s which are a l s o thought to c o n t r o l the numbers and growth of the stock of i n t e r e s t at d i f f e r e n t stages and o f t e n i n a d e n s i t y dependent way. However, the manager i s i n t e r e s t e d i n the biomass which i s weight times numbers and e i t h e r or both of weight and numbers may be a f f e c t e d , o f t e n i n v e r s e l y , by d e n s i t y . What i s of importance i s the investment f u n c t i o n as i t r e l a t e s escapement to subseguent biomass r e c r u i t m e n t . The use of c a t c h data to e s t imate t h a t f u n c t i o n c o m p l i c a t e s an a l r e a d y complex s i t u a t i o n 61 by i n t r o d u c i n g the new element of f i s h i n g m o r t a l i t y . , I d e a l l y t h i s investment f u n c t i o n should take i n t o account a l l environmental i n f l u e n c e s , a l l i n t e r - s p e c i e s and i n t r a -s p e c i e s i n t e r a c t i o n s plus the e f f e c t s of h a r v e s t i n g but r e l i a b l e data simply do not e x i s t to develop such a s i m u l a t i o n model. Environmental f a c t o r s such as the temperature of the water and i t s chemical composition can c o n t r o l s u r v i v a l (numbers) and the metabolic r a t e (growth) by o p e r a t i n g e i t h e r d i r e c t l y on the s p e c i e s i t s e l f o r i n d i r e c t l y through a f f e c t i n g i t s food supply. Competition f o r that l i m i t e d food supply and f o r the environmental niche are important sources of c o n t r o l as are the s p e c i e s which form the food supply or which prey on the s p e c i e s of i n t e r e s t . These c o m p e t i t i v e and predator-prey r e l a t i o n s h i p s can occur through both i n t e r - s p e c i e s and i n t r a - s p e c i e s i n t e r a c t i o n s . I n t r a - s p e c i e s i n t e r a c t i o n can a l s o a f f e c t stock s i z e i n d i r e c t l y through the impact of d e n s i t y on r e p r o d u c t i o n and growth r a t e s . For sockeye, these d e n s i t y e f f e c t s on r e p r o d u c t i o n appear to be p a r t i c u l a r l y important. In e s t i m a t i n g the investment f u n c t i o n the manager i s faced with the task of d e c i d i n g which environmental, i n t e r - s p e c i e s and i n t r a - s p e c i e s f a c t o r s are to be t r e a t e d as parameters and which can be handled as v a r i a b l e s . He must a l s o decide which f a c t o r s are most important i n d i f f e r e n t s t a g e s of the l i f e - c y c l e . , The c r i t i c a l f a c t o r i n making these d e c i s i o n s i s the q u a l i t y of the data. Commercial c a t c h data, f o r example, merely g i v e c a t c h f i g u r e s on an annual b a s i s , the average s i z e of f i s h caught and some estimate of the e f f o r t i n v o l v e d i n h a r v e s t i n g the c a t c h . And y e t , most commercial f i s h e r i e s , i n c l u d i n g the 62 sockeye f i s h e r y , are based on p r o l i f i c s t o c k s with l a r g e production s u r p l u s e s . , That i s , q u i t e a l a r g e part of the recruitment can be harvested l e a v i n g only a f r a c t i o n of the s t o c k to escape and reproduce. And i t has been observed t h a t l a r g e r production s u r p l u s e s are a s s o c i a t e d with stocks s u b j e c t to the g r e a t e s t v a r i a b i l i t y i n the environment ( L a r k i n , 1972, p.322). Thus the s t o c k s of g r e a t e s t f i s h e r y i n t e r e s t are the very ones f o r which c o n t r o l over abundance cannot be e s t a b l i s h e d by r e g u l a t i n g f i s h i n g m o r t a l i t y o n l y . As a r e s u l t , a c l o s e r approximation to the i d e a l investment f u n c t i o n i s achieved when b i o l o g i s t s allow the environment to e n t e r : o f t e n i n a s t o c h a s t i c way. I n t h i s study, however, environmental f a c t o r s w i l l be permitted t o enter i n a d e t e r m i n i s t i c way. .; D e t e r m i n i s t i c models based on i n t r a - s p e c i e s i n t e r a c t i o n s i n v o l v e a simultaneous s o l u t i o n of the b i o l o g i c a l r e c r u i t m e n t f u n c t i o n and the h a r v e s t i n g p r o d u c t i o n f u n c t i o n (as i l l u s t r a t e d i n F i g u r e (2-1) below) . An i d e n t i f i c a t i o n problem a r i s e s because o f the simultaneous nature of the s o l u t i o n : how does one d i s t i n g u i s h between recruitment e f f e c t s and h a r v e s t i n g e f f e c t s ? O bservations on c a t c h cannot help and t h i s i s why the assumption of steady s t a t e e q u i l i b r i u m i s e s s e n t i a l t o most d e t e r m i n i s t i c models. That i s , h a r v e s t i n g e f f e c t s are e x a c t l y o f f s e t by r e c r u i t m e n t e f f e c t s i n e q u i l i b r i u m and there i s no need to d i s t i n g u i s h between them. Thus, a second h e r o i c assumption i s t h a t h a r v e s t i n g can be t r e a t e d l i k e a parameter and the nature of the recruitment f u n c t i o n i s determined by which of r e p r o d u c t i o n , growth or reduced n a t u r a l m o r t a l i t y o f f s e t s f i s h i n g m o r t a l i t y . The r e s u l t i n g e q u i l i b r i u m , or s u s t a i n a b l e , 63 y i e l d i s t h a t s u r p l u s p r o d u c t i o n which, when harvested, l e a v e s the stock abundance unchanged d u r i n g the period under o b s e r v a t i o n . F i s h i n g m o r t a l i t y can be o f f s e t because of d e n s i t y dependence: f o r example, f i s h i n g a l s o reduces the d e n s i t y of the parents, the spawn i s s m a l l e r , and more l a r v a e s u r v i v e because o f the reduced pressure on the l i m i t e d food supply (environmental c e i l i n g ) . In order to study the i n t e r a c t i o n between c a t c h and f i s h i n g i n t e n s i t y , we develop a p r o d u c t i o n f u n c t i o n to e x p l a i n h a r v e s t i n g i n r e l a t i o n to the biomass. , The most common hyp o t h e s i s f o r the harvest p r o d u c t i o n f u n c t i o n i s (2 - 1 ) C = gfB where C = c a t c h gf = instantaneous r a t e of f i s h i n g m o r t a l i t y B = biomass of f i s h a v a i l a b l e to the gear The y i e l d i n numbers i s p r o p o r t i o n a l to e f f o r t i f we assume q i s c o n s t a n t . T h i s assumption i s based on the b e l i e f t h a t i n e g u i l i b r i u m the stock l e v e l i s constant and, on p r o b a b a l i s t i c grounds, t h a t t h i s constant stock l e v e l i s e g u a l l y d i s p e r s e d throughout the f i s h i n g grounds at a l l times. As b i o l o g i s t s have shown, t h i s equal s p a t i a l and temporal d i s p e r s i o n of s t o c k i s h i g h l y u n l i k e l y because o f such f a c t o r s as s e a s o n a l changes i n f e e d i n g grounds, m i g r a t i o n , the e f f e c t of weight and age on v e r t i c a l a l l o c a t i o n ( a f f e c t i n g a c c e s s to s u r f a c e - s e t nets) and the f a c t t h a t younger f i s h d i s p l a y seasonal s c h o o l i n g behavior (making them e a s i e r to c a t c h with s e i n e n e t s ) . Furthermore, the u n i t of e f f o r t i s u n l i k e l y t o have remained constant over l o n g e r p e r i o d s of time as t e c h n i c a l 6 4 i n n o v a t i o n i n c r e a s e s both the power of boats and the e f f e c t i v e n e s s of gear.,Increases i n the s i z e of v e s s e l decrease the number o f t r i p s r e q u i r e d to unload the c a t c h , thus the a c t u a l time of exposure of the stock to the gear i n c r e a s e s . That i s , the r a t i o o f steaming time to f i s h i n g time i s a l t e r e d , and yet most o b s e r v a t i o n s are f o r the two combined (Hannesson, 1974, chapt.2). For these reasons, the 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 i s u n l i k e l y t o be c o n s i s t e n t and may be i n f l u e n c e d by the biomass i t s e l f or the e l a p s e o f time. Thus not only i s the biomass u n l i k e l y t o be i n e q u i l i b r i u m but even the t o o l s used to i n d i r e c t l y measure biomass are changing. Consequently we must s p e c i f y a model which w i l l allow us to analyse changes i n "q". T h i s model can be d e r i v e d from one of t h r e e types of models used by b i o l o g i s t s t o d e s c r i b e the investment f u n c t i o n f o r commercial f i s h e r i e s . , Both the f i r s t , the l o g i s t i c model, and the second, the a n a l y t i c model, are used f o r s p e c i e s which become v u l n e r a b l e to f i s h i n g m o r t a l i t y at a c e r t a i n age and remain v u l n e r a b l e f o r the r e s t of t h e i r l i v e s . The t h i r d , c a l l e d the r e c r u i t m e n t model, was developed f o r s p e c i e s l i k e salmon which become v u l n e r a b l e t o f i s h i n g gear at one p a r t i c u l a r stage i n the l i f e c y c l e and are not v u l n e r a b l e i n l a t e r s t a g e s . Thus the stock c o n s i s t s of only a s i n g l e c o h o r t or year c l a s s . /Reproduction depends on numbers and weight of those l e f t over from the one v u l n e r a b l e stage. And the biomass depends on the numbers and weight of those s u r v i v i n g from r e p r o d u c t i o n and p a s s i n g through stages p r i o r to the v u l n e r a b l e one. 65 I I I I I RECBOTTMENT MODEL The model to be used f o r the F r a s e r River sockeye f i s h e r y i s the r e c r u i t m e n t model which c o n c e n t r a t e s on the r e l a t i o n s h i p between parents and new r e c r u i t s . Stock c o n t r o l operates through the e f f e c t o f d e n s i t y dependence on r e p r o d u c t i o n and then growth and m o r t a l i t y i n subseguent s t a g e s . The number of mature f i s h which escape h a r v e s t i n g i s the most important v a r i a b l e . Data s e r i e s on escapement and c a t c h are a v a i l a b l e from 1938 to 1975 f o r F r a s e r River sockeye and have been c o l l e c t e d i n a c o n s i s t e n t manner durin g t h a t p e r i o d . T h e r e f o r e , a long , r e l a t i v e l y d i s t o r t i o n - f r e e s e r i e s of data a r e a v a i l a b l e t o permit e s t i m a t i o n of an escapement-recruitment r e l a t i o n s h i p i f i t can be assumed t h a t r e c r u i t m e n t i s completely accounted f o r by the sum of c a t c h and escapement. As w i l l be e x p l a i n e d below, t h i s i s j u s t i f i e d f o r the F r a s e r River sockeye f i s h e r y . E g u i l i b r i u m i s an important assumption f o r the rec r u i t m e n t model., I t i n v o l v e s the simultaneous s o l u t i o n of two r e l a t i o n s h i p s : the recruitment from a given s t o c k , and the parent stock which r e s u l t s from a given recruitment s u b j e c t e d to h a r v e s t i n g . F i g u r e (2-1) i l l u s t r a t e s the i n t e r s e c t i o n of these two r e l a t i o n s h i p s : Figure (2-1): Simultaneous S o l u t i o n of Harvesting and Recruitment £ Where R = recruitment 66 E = escapement y = r e c r u i t m e n t r e l a t i o n s h i p x = e q u i l i b r i u m parent escapement from h a r v e s t i n g Curve y i l l u s t r a t e s the escapement-recruitment r e l a t i o n s h i p and d e r i v e s i t s shape and p o s i t i o n from d e n s i t y dependent e f f e c t s . Recruitment i s d e f i n e d as the biomass which becomes v u l n e r a b l e t o f i s h i n g gear a t a s p e c i f i c s tage i n the l i f e c y c l e . Note how recruitment i n c r e a s e s r a p i d l y with escapement but at a decreasing r a t e . The l i n e x r e p r e s e n t s the e g u i l i b r i u m escapement which r e s u l t s from e g u i l i b r i u m r e c r u i t m e n t s u b j e c t e d to e q u i l i b r i u m h a r v e s t i n g . As f i s h i n g i n t e n s i t y i s i n c r e a s e d the l i n e x r o t a t e s i n an a n t i - c l o c k w i s e f a s h i o n i n d i c a t i n g t h a t e q u i l i b r i u m escapement from a given recruitment d e c l i n e s as the c a t c h r a t e i n c r e a s e s although r e c r u i t m e n t i n c r e a s e s a t f i r s t and then d e c l i n e s as i n t e n s i t y i n c r e a s e s . At p o i n t "B" f o r example, escapement i s E 1 , r e c r u i t m e n t f o r t h a t escapement would be B l , and, e x t e n d i n g back to the 45 degree l i n e , the d i f f e r e n c e between p o i n t A (the e g u i v a l e n t of R1 along the E axis) and E1 i s the c a t c h . Only i n e q u i l i b r i u m w i l l the simultaneous s o l u t i o n be given by the i n t e r s e c t i o n of x and y. The e q u i l i b r i u m i n t e r s e c t i o n s of f i s h i n g e f f o r t i n t e n s i t y with the escapement-recru i t m e n t r e l a t i o n s h i p w i l l then t r a c e out the d i f f e r e n t s u s t a i n a b l e c a t c h l e v e l s a s s o c i a t e d with d i f f e r e n t l e v e l s of escapement. I t i s t o be expected that the h i g h e s t c a t c h l e v e l s would be a s s o c i a t e d with i n t e r m e d i a t e escapement l e v e l s . Note t h a t as the i n t e r s e c t i o n s are along y, while x can be r o t a t e d p a r a m e t r i c a l l y , the l o c u s of e q u i l i b r i u m p o i n t s would be curve y-There are some ge n e r a l o b s e r v a t i o n s which should be 67 f u l f i l l e d by the p a r a b o l i c r e l a t i o n s h i p between r e c r u i t s and escapement (Ricker, 1975, p. 281): i t should be c o n s t r a i n e d to pass through the o r i g i n t o comply with the b a s i c assumption that without a d u l t s , there can be no r e p r o d u c t i o n ; the r a t e of r e c r u i t m e n t , R/E, should move i n v e r s e l y with the number of escaping a d u l t s ; and recruitment must exceed the st o c k of e s c a p i n g a d u l t s a t seme p o i n t , otherwise the stock would become e x t i n c t . The path to equilibrium takes s e v e r a l p e r i o d s to converge. I f f i s h i n g i n t e n s i t y i n c r e a s e s , the r e s u l t i n g escapement w i l l be reduced from a given r e c r u i t m e n t . , T h i s w i l l l e a d to reduced r e c r u i t m e n t i n the next p e r i o d and a s m a l l e r escapement which i n t u r n , w i l l l e a d to a f u r t h e r r e d u c t i o n i n recruitment f o r the f o l l o w i n g p e r i o d {Paulik and Greenough, 1966). One advantage of working with the F r a s e r R i v e r Sockeye s t o c k i s t h a t t h i s e q u i l i b r i u m problem i s not very important. As the parent stock d i e s a f t e r spawning, a change i n escapement l e v e l l e a d s to e q u i l i b r i u m w i t h i n one f o u r year c y c l e . As t h e r e i s only one year c l a s s f o r each c y c l e and a l l f o u r c y c l e s are assumed independent, there are no stock s t r u c t u r e problems a s s o c i a t e d with a l t e r a t i o n s i n the year c l a s s a r r a y within a given s t o c k . One major assumption of the model used i n t h i s study i s t h a t the " f o c u s " i s s t a b l e , although i t i s thought that t h i s i s not s t r i c t l y c o r r e c t f o r sockeye salmon. The convergence to e q u i l i b r i u m a p p l i e s i n the case o f any d i s t u r b a n c e , not j u s t from imposed f i s h i n g m o r t a l i t y ; and the escapement-recruitment r e l a t i o n i s the mechanism f o r p o p u l a t i o n s t a b i l i t y and c o n t r o l . , E x t e r n a l f a c t o r s such as i n t e r - s p e c i e s i n t e r a c t i o n s and 68 environ mental i n f l u e n c e s may have profound e f f e c t s on the escapement-recruitment r e l a t i o n s h i p , a l t e r i n g both the shape and p o s i t i o n of the f u n c t i o n . Most o f t e n there i s no data a v a i l a b l e f o r these e x t e r n a l f a c t o r s and they are u s u a l l y l e f t to enter through the e r r o r term. T h i s i s a s e r i o u s problem i n the case of a migratory s p e c i e s l i k e salmon because d i f f e r e n t stages of the l i f e c y c l e are spent i n t o t a l l y d i f f e r e n t environments and the e f f e c t s of d e n s i t y i n one stage may be u n r e l a t e d to s u r v i v a l i n a subseguent stage. The F r a s e r River sockeye have four d i s t i n c t year c l a s s c y c l e s because the l a g between escapement and recruitment i s f o u r y e a r s . For most c y c l e years three year o l d and f i v e year o l d f i s h r e p r e s e n t l e s s than 5% of the r e c r u i t m e n t . I f one r e g r e s s e s r e c r u i t m e n t on escapement (in the l o g form of the R i c k e r f u n c t i o n , e x p r e s s i o n (2-7) below) f o r the F r a s e r River sockeye the f i t i s poor. A glance at a s c a t t e r diagram of r e c r u i t m e n t a g a i n s t escapement i s enough e x p l a n a t i o n : the dots appear t o be randoaly d i s t r i b u t e d . However, one technigue which does not appear to have been used before i s the i n t r o d u c t i o n of dummy v a r i a b l e s i n slope and i n i n t e r c e p t form to a l l o w f o r d i f f e r e n c e s i n parameters amongst the f o u r year c l a s s c y c l e s . Furthermore, s t a t i s t i c a l t h e o r y , emphasizing the use of as much i n f o r m a t i o n as p o s s i b l e , suggests t a k i n g account of the f a c t t h a t data e x i s t a l l o w i n g escapement to be disaggregated i n t o eleven r a c e s of sockeye a s s o c i a t e d with c e r t a i n l a k e s i n the F r a s e r R i v e r system. Regressing recruitment on disaggregated escapement using c y c l e dummies l e a d s t o a c o n s i d e r a b l e i n c r e a s e i n e x p l a n a t o r y power. T h i s would appear t o i n d i c a t e that 69 environmental i n f l u e n c e s or i n t e r - s p e c i e s i n t e r a c t i o n s or both can be modeled through the r e c o g n i t i o n of c y c l e d i f f e r e n c e s i n the recruitment f u n c t i o n f o r the F r a s e r Biver sockeye s t o c k . Turning to the d e n s i t y dependent i n t r a - s p e c i e s i n t e r a c t i o n s , the v a r i o u s recruitment models are d i s t i n g u i s h e d by t h e i r assumptions about the agent of d e n s i t y dependent c o n t r o l and i t s r e l a t i o n s h i p to the escapement of parent f i s h (Chapman, 1973; Dahlberg, 1973; D o i , 1973; L a r k i n , 1973; P a u l i k , 1973; P a u l i k and Greenough, 1966; and B i c k e r , 1975, chapt. 11). Host o f these assume the f e c u n d i t y of the parent stock remains cons t a n t and that the number of eggs hatched i s p r o p o r t i o n a l t o the parent stock. B i c k e r , however, assumes t h a t the number of p r e d a t o r s i s p r o p o r t i o n a l to the parent stock and that m o r t a l i t y i n the e a r l y stages of l i f e i s p r o p o r t i o n a l to the p r e d a t o r s . Thus, n a t u r a l m o r t a l i t y of the progeny i s p r o p o r t i o n a l to the d e n s i t y of the parent stock. Predators are the agents of d e n s i t y dependent r e g u l a t i o n ( L a r k i n , 1973); m o r t a l i t y operates i n a compensatory f a s h i o n and i s r e l a t e d to s t o c k abundance at a s p e c i f i c time. Thus, numbers i n one stage are a simple f u n c t i o n of numbers s u r v i v i n g from a p r e v i o u s stage where s u r v i v a l i s a f u n c t i o n of parent d e n s i t y : (2-2) N (t+1) = H (t) S (E) where N = numbers a l i v e S(E) = f r a c t i o n s u r v i v i n g i n each period and i s a f u n c t i o n of escapement of mature a d u l t s The compensatory m o r t a l i t y assumption i m p l i e s t h a t the s u r v i v a l r a t i o i s a monotonically d e c r e a s i n g f u n c t i o n of the escapement and B i c k e r assumes the r a t e i s e x p o n e n t i a l : (2 - 3 ) S (E) = a (exp (-bE) ) 70 where E = escapement of a d u l t s t o c k i n numbers a, b = parameters and as the progeny are assumed p r o p o r t i o n a l to the parent stock, r e c r u i t m e n t i s given by (2-4) R = aE(exp(-bE)) where R = r e c r u i t m e n t i n numbers to the f i s h e r y In the B i c k e r approach, only "compensatory" e f f e c t s ( d e n s i t y e f f e c t s on growth and/or m o r t a l i t y i n c r e a s e with d e n s i t y ) are permitted t o work and yet there i s evidence t o show th a t "depensatory" e f f e c t s ( d e n s i t y e f f e c t s on growth and/or m o r t a l i t y decrease with density) operate i n at l e a s t one or two stages f o r most s t o c k s . I t i s thought t h a t depensatory p r e d a t i o n i s one of the main causes of c y c l e s i n r e c r u i t m e n t (Dahlberg, 1S73). P a u l i k and Greenough (1966) suggest a l t e r i n g the R i c k e r s u r v i v a l f u n c t i o n s to allow both compensatory and depensatory f a c t o r s to operate. For stock c o n t r o l t o work i n a d e n s i t y dependent way means t h a t compensatory e f f e c t s must e v e n t u a l l y outweigh the depensatory e f f e c t s , otherwise d e n s i t y dependence as a form of stock c o n t r o l would disappear. For the R i c k e r model the m o d i f i c a t i o n to e x p r e s s i o n (2-4) i m p l i e s : (2-5) R = aE J (exp(-bE)) where d = index of d e n s i t y dependence In g e n e r a l there are as many recruitment models as there are f u n c t i o n a l forms to express the r e l a t i o n s h i p between recruitment and escapement. D i f f e r e n t a p r i o r i assumptions about the density-dependent f a c t o r s o f t e n l e a d to the same f u n c t i o n a l forms. Hot only i s there a l a c k of data on which t o base a p r i o r i judgements but f l u c t u a t i o n s of the environment make i t 71 d i f f i c u l t t o have any c o n f i d e n c e i n f u n c t i o n s which do f i t the o b s e r v a t i o n s cn c a t c h and escapement. Thus, the f u n c t i o n s are u s e f u l " i n t e r p o l a t i o n s " but cannot be s a i d to be e s t i m a t o r s of b i o l o g i c a l parameters f o r the u n d e r l y i n g p o p u l a t i o n (Chapman, 1973, p.328). However, f o r a f i s h e r y manager who i s only i n t e r e s t e d i n e s t i m a t i n g the r e t u r n on h i s a b s t e n t i o n investment, a reasonable i n t e r p o l a t i o n i s a l l t h a t i s needed. Perhaps the most u s e f u l way to d i s t i n g u i s h between a l t e r n a t i v e f u n c t i o n a l forms i s the way i n which the e r r o r terra e n t e r s the r e l a t i o n . The e r r o r most l i k e l y summarises the unknown e x t e r n a l f a c t o r s which cannot be modeled e i t h e r because of l a c k of knowledge of the way they e n t e r or because of l a c k of data f o r use i n e m p i r i c a l f i t t i n g ( L a r k i n , 1973, p.322). But should the environmental e f f e c t s be a d d i t i v e or m u l t i p l i c a t i v e ? From the p o i n t of view of l o g i c "we should expect the e f f e c t of the p h y s i c a l environment normally to be m u l t i p l i c a t i v e . . . i f c o n d i t i o n s are f a v o r a b l e , a l l ( f i s h ) have a chance of b e n e f i t t i n g ; i f u n f a v o r a b l e , a c e r t a i n f r a c t i o n (not a f i x e d number) w i l l be l o s t " ( B icker, 1975, p.274).,The next s t e p i s to determine the freguency d i s t r i b u t i o n of the e r r o r . Both Ric k e r (1973a, pp.337-338) and a l i e n (1973, pp.353-355) have found that the r a t i o R/E i s l o g normally d i s t r i b u t e d f o r many s t o c k s . T h i s skewed d i s t r i b u t i o n of r e c r u i t m e n t f o r g i v e n escapement i s l i k e l y to be generated by m u l t i p l i c a t i v e environmental e f f e c t s . The hypothesis a l s o r e c e i v e s support from the o b s e r v a t i o n d i s c u s s e d e a r l i e r , t h at l a r g e r s u r p l u s p r o d u c t i o n s are a s s o c i a t e d with s t o c k s s u b j e c t to the l a r g e s t environmental v a r i a t i o n . 72 ft m u l t i p l i c a t i v e e r r o r term can be added to the B i c k e r model i n such a way as to r e t a i n the i n t r i n s i c l i n e a r i t y of the f u n c t i o n s : (2-6) B = a E d (exp (-bl)) exp (#) where (#) = e r r o r term T h i s e x p r e s s i o n can be transformed i n t o a l i n e a r e s t i m a t o r by t a k i n g n a t u r a l l o g s : (2-7) l n ( B / ( E d ) ) = l n a - bE + (#) I_I SOCKEYE J_ECJPJTMEST FUNCTION As e x p r e s s i o n (2-4) above i n d i c a t e s , the B i c k e r form of recruitment model needs o b s e r v a t i o n s on B and E to e s t i m a t e the parameters a and b. .  These data have been c o l l e c t e d i n a c o n s i s t e n t manner by the IPSFC s i n c e 1938, Escapement data are based on a c t u a l counts of f i s h swimming i n the spawning streams and r i v e r s . Observations of c a t c h are taken from commercial c a t c h r e p o r t s and DOE estimated c a t c h r e p o r t s and i n c l u d e sockeye taken o f f the west coa s t of Vancouver I s l a n d , those caught by O.S. and Canadian fishermen i n the IPSFC convention a r e a , those caught by Canadian fishermen i n Johnstone S t r a i t , and those caught by n a t i v e I n d i a n s from the sockeye esc a p i n g up the F r a s e r B i v e r . Recruitment w i l l be assumed e q u a l to the sum of the escapement and the c a t c h . As the F r a s e r R i v e r sockeye f i s h e r y has been o p e r a t i n g s i n c e before the t u r n of the century, and t h e r e have been no dramatic changes i n f i s h i n g or i n the environment s i n c e 1938, i t i s probably s a f e to assume t h a t i t has been i n e q u i l i b r i u m s i n c e 1938. , 73 The escapement data d i s p l a y a c y c l i c a l t r e n d f o r number and s i z e of spawners and timing of spawning. I f the only form of sockeye stock c o n t r o l were e x t e r n a l , f l u c t u a t i o n s i n environmental f a c t o r s are so great that changes i n abundance would be g u i t e l a r g e . Sockeye r e c r u i t m e n t does f l u c t u a t e but i n a c y c l i c a l f a s h i o n and only t o a l i m i t e d e x t e n t . The f i r s t and t h i r d y e ars of the c y c l e are about the same s i z e , twice as l a r g e as the r e c r u i t m e n t f o r the second c y c l e year but only h a l f the s i z e of the recruitment f o r the f o u r t h c y c l e year. The data can be used t o account f o r these e f f e c t s by a n a l y s i n g the impact of year c l a s s c y c l e s i n both weight and numbers on the b a s i c escapement-recruitment r e l a t i o n . There are eleven races of sockeye in the F r a s e r B i v e r system which means there i s a m u l t i - s p e c i e s h a r v e s t i n g problem (Bicker, 1973b, p. 1280). However, to determine race p r o p o r t i o n of the subsequent rec r u i t m e n t and the i n f l u e n c e of gear type c a t c h on race r e c r u i t m e n t t o each subsequent f i s h i n g zone would r e q u i r e i n t r a - s e a s o n a l a n a l y s i s and complicate an a l r e a d y complex model., Ho attempt w i l l be made to s o l v e t h i s m u l t i -s p e c i e s h a r v e s t i n g problem., With d i f f e r e n c e s i n f e c u n d i t y , growth and m o r t a l i t y r a t e s , however, i t i s p o s s i b l e t o analyse the s i g n i f i c a n t l y d i f f e r e n t impacts o f these e l e v e n races on the escapement-recruitment r e l a t i o n , owing t o the a v a i l a b i l i t y of data by race escapement s i n c e 1938. Turning t o the e s t i m a t i o n process, the whole o b j e c t i v e has been to achieve the best parameter es t i m a t e s p o s s i b l e f o r p r e d i c t i o n purposes. Subseguent work i n t h i s study i s based on the assumption t h a t confidence can be placed i n p r e d i c t e d l e v e l s 74 of r e c r u i t m e n t given d i f f e r e n t l e v e l s of escapement. Otherwise, the c a l c u l a t i o n s f o r optimal management become meaningless, although use of c y c l e dummies and disaggregated race escapement does p r o v i d e a t e s t a b l e h y p o t h e s i s , the r e c r u i t m e n t model hypothesis i s not being t e s t e d so much as being used t o d e r i v e the best parameter estimates f o r recruitment to the f i s h e r y . On the grounds of s i m p l i c i t y , i t was decided that i n t r i n s i c a l l y l i n e a r f u n c t i o n s would be used; s p e c i f i c a l l y , (2-8) R = aE d (exp <-bE)) exp(#) (2-9) E = aE d (exp (#) ) The a b i l i t y to transform these i n t o l i n e a r models makes o r d i n a r y l e a s t squares e s t i m a t i o n e a s i e r (Kmenta, 1971, p.451). The l i n e a r v e r s i o n s of expr e s s i o n s (2-8) ( f o l l o w i n g Ricker) and (2-9) are: (2-10) ln(R/EV) = i n a - bE + (#) (2-11) InR = i n a + d(lnE) • (#) Ex p r e s s i o n (2-10) presents a problem: the dependent v a r i a b l e , R, i s m o d ified i n some way by the independent v a r i a b l e , E, In a d d i t i o n to the e m p i r i c a l problems which r e s u l t from having l i n e a r t r a n s f o r m a t i c n s of the same v a r i a b l e s appear on both s i d e s of the eguation, there i s a problem t h a t i f the v a r i a n c e of E i s l a r g e r than the varia n c e of R, i t w i l l tend to dominate the r a t i o i n the e s t i m a t i o n . , I t i s p o s s i b l e t o re-arrange (2-10) to give (2-12) InR = l n a - bE + d (InE) In a d d i t i o n t o improving the e s t i m a t i o n procedure, t h i s form has the b e n e f i t of p e r m i t t i n g the data to determine the depensation 75 term, d. However, one consequence of us i n g (2-12) i s that m u l t i c o l l i n e a r i t y i s l i k e l y t o be high between E and i t s monotonic t r a n s f o r m a t i o n , InE. T h i s can be reduced by using i n s t r u m e n t a l v a r i a b l e , BE, i n place o f e i t h e r of the E*s (Kraenta, 1971, pp.309-313). WE i s simply escapement m u l t i p l i e d by the average c y c l e weight of sockeye and i s thus a non-monotonic t r a n s f o r m a t i o n of E. Furthermore, the use of biomass r a t h e r than numbers would take i n t o account the f a c t o r of s i z e of f i s h and i t s impact on s u r v i v a l during escapement upstream. The e s t i m a t i o n process was d i v i d e d i n t o two stages. In the f i r s t , e x p r e s s i o n s (2-10), (2-11) and (2-12) were estimated using disaggregated forms of E and WE; that i s , the e l e v e n race escapements ( i n numbers or biomass form) were used i n pl a c e of the aggregate forms of E and WE. In the second stage, the f u n c t i o n a l forms which gave the best parameter e s t i m a t e s were used with dummy v a r i a b l e s t o mark the year c l a s s c y c l e s . Dummy seguences i n both slope and i n t e r c e p t form with two year (every other year) , f o u r year (once every f o u r years) , two/two year (two good f o l l o w e d by two bad y e a r s ) , and e i g h t year c y c l e s were used. As an a l t e r n a t i v e way of p i c k i n g c y c l e s , a polynomial d i s t r i b u t e d l a g was used with l e n g t h s of l a g varying from two to e i g h t y e a r s . T h i s technique would imply t h a t the c y c l e i s some f u n c t i o n of c o n d i t i o n s i n the environment i n pr e v i o u s years and assumes t h a t the weights on the pr e v i o u s escapements f o l l o w a polynomial o f a s p e c i f i e d degree (Kmenta, 1971, p.492-495). The sockeye rec r u i t m e n t hypothesis does not sup p l y any a p r i o r i s p e c i f i c a t i o n s f o r the degree and l e n g t h of the polynomial and 76 an i t e r a t i v e approach was employed t o all o w the data t o determine the most s t a t i s t i c a l l y s i g n i f i c a n t r e s u l t s u sing the 8-sguared d e l e t e approach. As the r e s u l t s were not as s i g n i f i c a n t using d i s t r i b u t e d l a g s as they were with c y c l e dummies i n s l o p e and i n t e r c e p t form, the l a t t e r were used i n s t e a d o f the former. In terms of goodness of f i t the e s t i m a t i o n proceeded i n the f o l l o w i n g way. Using the b a s i c B i c k e r f u n c t i o n a l form with recruitment i n terms of numbers regressed on escapement i n numbers, the B-sguared was .28, When escapement and recru i t m e n t were put i n biomass form (weight times numbers) the f i t improved to .35, When c y c l e e f f e c t s were permitted to enter through escapement i n dummy form, the f i t improved t o .64. And d i s a g g r e g a t i n g escapement i n t o the eleven r a c e s r a i s e d R-squared to .88. / The r e s u l t s f o r t h i s l a s t e s t i m a t i o n are given i n the Appendix (expression (2-14)). These parameter e s t i m a t e s w i l l not be used i n the management model because of the c o m p l e x i t i e s i n v o l v e d i n modeling a m u l t i - s p e c i e s h a r v e s t i n g system. T h e r e f o r e , the parameter e s t i m a t e s which depend on recruitment i n biomass form regressed on escapement i n biomass form d i s t i n g u i s h e d by year c y c l e w i l l be used i n the management model i n Chapter Four. The b a s i c f u n c t i o n a l form i s t h a t of e x p r e s s i o n (2-8) and the parameter e s t i m a t e s a re reproduced i n e x p r e s s i o n (2-13) below. A t e s t was performed to check on the p o s s i b i l i t y o f changes i n parameter values over the t h i r t y - f o u r years under o b s e r v a t i o n using m o d i f i e d c h i - s g u a r e v a r i a b l e s t o compute the F s t a t i s t i c 77 {Kmenta, 1971, p.373). Both of the best estimated f u n c t i o n s {(2-13) and (2-14)) passed t h i s t e s t with no problem which would i n d i c a t e t h a t b a s i c parameter v a l u e s have not changed s i g n i f i c a n t l y over the whole p e r i o d . , One problem of importance i n the e s t i m a t i o n was the presence of a u t o r e g r e s s i o n . The success of the year c l a s s c y c l e dummies would i n d i c a t e t h a t the major i n f l u e n c e on recruitment i s the escapement fo u r years e a r l i e r . I t i s l i k e l y , however, t h a t t h e e f f e c t of d i s t u r b a n c e s does c a r r y over i n t o o t h e r year c l a s s c y c l e s . The experience with d i s t r i b u t e d l a g s would tend to support t h i s and i n d i c a t e s that the g r e a t e r the number of periods between d i s t u r b a n c e s , the s m a l l e r the c o v a r i a n c e . For t h i s reason, i t i s probably s a f e to s p e c i f y a f i r s t order a u t o r e g r e s s i v e scheme (Kmenta, 1971, pp.271-273). To t e s t the s i g n i f i c a n c e of the c a l c u l a t e d c o e f f i c i e n t of c o r r e l a t i o n of d i s t u r b a n c e s , a standard t w o - t a i l t t e s t was employed. I t i s usual i n economics to use a o n e - t a i l t e s t because i t i s assumed t h a t i f a u t o c o r r e l a t i o n e x i s t s , i t i s p o s i t i v e . But i n b i c l o g y i t i s g u i t e p o s s i b l e t h a t environmental f a c t o r s c o u l d l e a d to n e g a t i v e a u t o c o r r e l a t i o n which i s the reason f o r the t w o - t a i l t e s t . The Cochrane-Orcutt r o u t i n e was used with the best estimated f u n c t i o n t o determine the c o e f f i c i e n t of c o r r e l a t i o n of the d i s t u r b a n c e s . In f a c t , t h i s t e s t d i d i n d i c a t e s i g n i f i c a n t f i r s t order a u t o c o r r e l a t i o n . I t i s p o s s i b l e t o i n c l u d e parameter est i m a t e s a d j u s t e d f o r a c o e f f i c i e n t of c o r r e l a t i o n d i f f e r e n t from zero i n the estimated r e c r u i t m e n t f u n c t i o n . However, to do so would make the optimal escapement problem e x c e e d i n g l y complex. That i s , as 78 e x p l a i n e d i n Chapter Four below, one of the advantages of using dummies to d i s t i n g u i s h between year c l a s s e s i s t h a t the recrui t m e n t f o r each year c l a s s can then be i s o l a t e d from the re c r u i t m e n t f o r the remaining t h r e e year c l a s s e s . The most exhaustive b i o l o g i c a l study of the F r a s e r River sockeye salmon ( F o e r s t e r , 1968) would a l s o tend to support the view that interdependence amongst year c l a s s e s does e x i s t ( p a r t i c u l a r l y i n the freshwater l a k e r e s i d e n c e period) but t h a t i t i s too weak and i n c o n s i s t e n t t o have much impact. For these reasons, the a u t o r e g r e s s i o n problem w i l l be ignored i n subseguent work. The f i n a l r e s u l t s of e s t i m a t i o n gave the f o l l o w i n g parameters with R-squared i n brackets a f t e r the dependent v a r i a b l e and t s t a t i s t i c s i n br a c k e t s a f t e r the r e l e v a n t explanatory v a r i a b l e . Note t h a t the e s t i m a t i o n procedure was not s p l i t i n t o f o u r c y c l e p e r i o d s , but the r e s u l t s below are given i n c y c l e year form t o make e x p l i c i t the manner i n which biomass recruitment (HR) can be p r e d i c t e d from escapement i n numbers (E) and i n biomass (WE) form. (2-13) WR(.64) = exp{-.000000694227E) (-5.4) D41 WE 1« 1 5 (75.4) D4 2WE*,i 3(54.3) D43WE* . i s (74.4) D44 WE t, * 8 (69. 8) Where Wfi = biomass (weight times numbers) r e c r u i t m e n t WE = mass of escapement (weight times numbers) lagged four periods E = escapement i n numbers lagged four p e r i o d s p r e f i x D = i n d i c a t e s t h a t t h i s v a r i a b l e i s i n dummy form 41,2,3,4 = f o u r year c y c l e with the weight i n the f i r s t , second, t h i r d or f o u r t h year of c y c l e The f u n c t i o n a l form used t o d e r i v e these parameter e s t i m a t e s was taken from e x p r e s s i o n (2-12)., Note t h a t the compensatory parameter "b" i n (2-12) i s indeed negative i n (2-13) and that the depensatory parameter "d" i n (2-12) i s p o s i t i v e 79 i n (2-13)., The expression i n (2-13) implies that biomass recruitment i s d i r e c t l y related to cycle year biomass escapement with depensatory c o e f f i c i e n t s distinguished by year class cycle. To o f f s e t t h i s e f f e c t , biomass recruitment i s inversely related (through the exponential operator) with escapement i n numbers but the compensatory c o e f f i c i e n t i s not sensitive to the year class cycle. The purpose of t h i s chapter has been to estimate the recruitment function: the basic investment function which t e l l s the fishery manager what " i n kind" return to expect on his "in kind" investment i n future recruitment through current abstention. External e f f e c t s do not have to be l e f t out and can be included by allowing cycle differences between year classes to be represented by dummy variables and by allowing the e f f e c t s of weight on cycle year differences to be included through the use of biomass forms of the variables rather than just the numbers form. The use of these technigues with a modified form of the Bicker exponential recruitment function leads to parameter estimates which support the hypothesis that recruitment four years hence can be predicted from current escapement. As the period analysis did not indicate any s i g n i f i c a n t change i n parameter values for the t h i r t y four year period under study, these parameter estimates f o r the escapement-recruitment function can be used with confidence in the Clark-Munro eguilibrium eguation to estimate the optimal escapement f o r each year cycle. 80 APPENDIX Compilation Of Catch And Escapement Data Although observations on catch of sockeye salmon date back to 1894, observations on escapement from the fis h e r y are only available a f t e r 1937. There i s a d i s t i n c t truncation i n catch data r e s u l t i n g from the landslide into the Fraser River i n 1913. Thus i t can be assumed that any accumulated recruitment potential was well used up by 1938 and that the sockeye stocks have been i n seme form of eguilibrium ever since. The basic source for the data i s the observations made by s c i e n t i s t s working f o r the IPSFC as published each year i n the IPSFC annual reports. The other major source of data i s the Department of the Environment (DOE) revisions of the IPSFC catch data (Anderson, 1976). Data on escapement by race of sockeye are available f o r the forty-four creeks and streams feeding into the eleven major r i v e r - l a k e systems which belong to the Fraser River watershed. Data on catch are available for sockeye entering the f i s h i n g gauntlet prior to ascending the Fraser River to spawn. I t i s not available by race. Fraser sockeye caught i n non-treaty waters are not included in the IPSFC catch t o t a l s . . T h i s i s the reason for including the DOE revisions. By summing revised catch and escapement, the migration of recruitment figure i s obtained. The best explanation of observations on escapement i s given i n the 1943 IPSFC annual report: "1) The t o t a l run may be i n d i v i d u a l l y counted through an opening i n a fence or weir. 2) The t o t a l number of salmon may be estimated by determining the r a t i o 81 e s t a b l i s h e d by marking a known number i n the run, or by counting a sample which has a known r a t i o to the t o t a l . ,3) A comparison of the magnitude of the runs from year to year may be based on counts which may be assumed to be c o n s t a n t but a c t u a l l y an unknown p r o p o r t i o n of the whole" (IPSFC, 1943, p.38). H e i r s are seldom used because of the c o s t and the methods used most f r e q u e n t l y are v i s u a l counts of l i v e spawners and p h y s i c a l counts of the dead a f t e r spawning. Hap (1-1) i l l u s t r a t e s the extent of the job i n v o l v e d : "each r a c i a l escapement i s enumerated by t a g g i n g , by proven l i n e count i n d i c e s or i n some cases by weirs" (IPSFC, 1953, p.36). In a d d i t i o n to the usual e r r o r s a r i s i n g from changes i n water l e v e l , temperature and weather, the g r e a t e s t source of e r r o r i s t h a t escapement o b s e r v a t i o n s are f o r n e t escapement and not gross escapement. That i s , the count i s not of the numbers escap i n g fishermen's nets but of numbers r e a c h i n g the spawning grounds. M o r t a l i t y i n the upstream m i g r a t i o n can vary from 5% to as much as 60S. Thus true r e c r u i t m e n t i s not g i v e n by summing net escapement and c a t c h . However, counts on the spawning grounds do g i v e a more accurate f i g u r e f o r the number of spawners i n v o l v e d i n the f o r m a t i o n of the the next year c l a s s . A p o s s i b l e j u s t i f i c a t i o n f o r using net r a t h e r than gross i s that "only the peak of the run i s p r o p e r l y s y n c h r o n i z ed with the r e p r o d u c t i v e environment and ...the f i s h from the beginning or end of each run are not normally capable of r e p r o d u c t i o n at a maximum r a t e because o f t h e i r e a r l y or l a t e m i g r a t i o n " (IPSFC, 1955, p.8). T h i s i s h e l p f u l i n terras of escapement but s t i l l does not help i n c a l c u l a t i n g m i g r a t i o n t o t a l s . I t w i l l be assumed, t h e r e f o r e , t h a t escapement m o r t a l i t y i s f i x e d cn a p r o p o r t i o n a t e b a s i s a c c o r d i n g t o year c l a s s c y c l e which i m p l i e s t h a t the measurement 82 e r r o r f o r r e c r u i t m e n t i s n u l t i p l i c a t i v e . The i n d e x system of e n u m e r a t i n g escapement i s most o f t e n based on a v i s u a l c o u n t . T h i s does n o t g i v e t h e a b s o l u t e f i g u r e s but does a l l o w c o m p a r i s o n o f t h e i n d e x year by y e a r . I f the p o p u l a t i o n i s c o u n t e d i n any y e a r , t h e i n d e x i n p a s t y e a r s can be a d j u s t e d t o r e f l e c t the t r u e a b s o l u t e c o u n t r a t h e r t h a n the r e l a t i v e c o u n t . T h i s a c c o u n t s f o r t h e many s u b s e g u e n t c o r r e c t i o n s i n p r e v i o u s y e a r s ' escapement t o t a l s . These a n n u a l r e v i s i o n s meant t h a t escapement t o t a l s f o r t h e f o r t y - f o u r streams and c r e e k s from 1938 to 1975 had to be checked and r e -checked t c make s u r e the l a t e s t c o r r e c t i o n s had been i n c l u d e d . The d i f f e r e n c e between g r o s s and n e t escapement i s a l s o a f f e c t e d by t h e I n d i a n c a t c h which t a k e s p l a c e a f t e r t h e salmon escape t h e c o m m e r c i a l f i s h e r y and b e f o r e t h e y r e a c h the spawning g r o u n d s . The 19 44 a n n u a l r e p o r t i n c l u d e d a r e p o r t on the c a t c h s t a t i s t i c s f o r t h e I n d i a n f i s h e r y ( p p .65-74) . The I n d i a n c a t c h s t a t i s t i c s a r e o b t a i n e d from P r o t e c t i o n O f f i c e r s of C a n a d a , Department o f F i s h e r i e s . These i n s p e c t o r s c o n t r o l the t a k i n g of sockeye f o r f o o d by t h e I n d i a n p o p u l a t i o n r e s i d i n g t h r o u g h o u t the F r a s e r E i v e r w a t e r s h e d . The IPSFC e s t i m a t e s o f I n d i a n c a t c h d a t e back o n l y to 1941. F o e r s t e r ' s e s t i m a t e s of I n d i a n c a t c h are used f o r 1938, 1939 and 1940 ( F o e r s t e r , 1968, p . 5 8 ) . A n o t h e r p o t e n t i a l s o u r c e of e r r o r i s the b l o c k a g e o f the H e l l ' s Gate s e c t i o n o f the F r a s e r . The f i s h w a y s were c o m p l e t e d and used by the spawners i n 1945 ( I P S F C , 1946, p . 5 ) . Subsequent c o u n t s i n d i c a t e d t h a t these f i s h w a y s l e d t o an average 100% i n c r e a s e i n escapement t o spawning grounds above H e l l ' s G a t e . T h i s o n c e - a n d - f o r - a l l change i n escapement s h o u l d not a f f e c t 83 estimation too seriously because the figures on escapement had been " i n s i t u " counts on spawning beds. And as recruitment only included net escapement and not gross escapement, i t meant that recruitment and escapement pr i o r to 1945 were net of the e f f e c t s of the Hell's Gate obstruction. Subsequent figures would include the increased net escapement both i n recruitment t o t a l s and in escapement t o t a l s . , That i s , subsequent counts would simply indicate a larger escapement from the fishery and the observations would be for a d i f f e r e n t range of the escapement-recruitment function. I t was not u n t i l 1952 that the importance of cycles was noted. It was recognized that the "factors c o n t r o l l i n g quadrennial dominance in productivity i n the reproducing areas above Hell's Gate may perhaps be of greater importance than the size of escapement " (IPSFC, 1952, p.26). To avoid changing the c y c l i c character "escapements are controlled to f i t the h i s t o r i c a l pattern of productivity i n each of the four cycle years. & large escapement i s provided in the year of the dominant run, a reduced number of f i s h i n the year of sub-dcminant run and lesser numbers of f i s h are permitted to escape under normal circumstances in the two 'off-years'" (IPSFC, 196 5, p. 3) .. The j u s t i f i c a t i o n f or t h i s policy i s that consistent size of escapement would tend to i n t e r f e r e with the predator, f i s h , food and prey r e l a t i o n s and lead to cycles of a d i f f e r e n t sequence. This means that escapement regulation which was i n i t i a t e d i n 1946 has not o f f s e t the natural cycles. This gives even more support to the use of cycle dummies to act as a proxy for external influences. Catch s t a t i s t i c s are recorded by the IPSFC through the log 84 book system i n s t i t u t e d i n 1941 (IPSFC, 1941 , p. 9 ) . & check of cannery records plus the f i s h t i c k e t s , buyer t a l l i e s and wharf t a l l i e s at each cannery or buyer are used to enumerate daily landings (IPSFC, 1944 , pp.52-53) . The system has not changed r a d i c a l l y as the o r i g i n a l system has worked well. The IPSFC i s charged with the task cf dividing the convention catch equally between the O.S- and Canada. Thus there has always been a great deal of pressure for accuracy. In some years the sockeye become vulnerable to f i s h i n g in non-convention areas such as the west coast of Vancouver Island and i n Johnstone S t r a i t . U n t i l 1 9 5 8 , the IPSFC did not publish data f o r these catches despite the obvious d i s t o r t i o n s to migration t o t a l s . This was not possible after 1958 , the year i n which a t h i r d of the commercial catch of Fraser sockeye was taken i n Johnstone S t r a i t . DOE estimates of non-convention catch are used as a check on the IPSFC published catch reports and the former do agree with the maximum of the two IPSFC estimates for each year. The DOE estimates are l i k e l y to be more accurate because the IPSFC has no j u r i s d i c t i o n over non-convention catch. Lack cf data prior to 1951 i s unlikely to be too serious an omission as few Fraser sockeye were caught i n non-convention areas. 85 Beeruitment F u n c t i o n E s t i m a t i o n The e s t i m a t i o n procedure was d i v i d e d i n t o two stages. The key to the eleven race escapements i s g i v e n below with the freshwater r e s i d e n c e l a k e s i n b r a c k e t s {see Map (1-1) f o r the spawning areas of the Fr a s e r R i v e r watershed): ELF = Lower F r a s e r ( P i t t ) EH = H a r r i s o n (Harrison) ELC = L i l l o o e t , F r a s e r Canyon (Nahatlatch) ESA = Seton-Anderson (Seton and Anderson) EST = South Thompson (Shuswap and Adams) ENT = North Thompson EC = C h i l c o t i n (Chilko) EQ = Quesnel (Quesnel) EN = Nechako (F r a n c o i s and Fras e r ) ES = St u a r t (Takla, Trembleur, Stuart) ENE = Northeast (Bowron) B = t o t a l m i g r a t i o n of F r a s e r sockeye E = t o t a l escapement lagged f o u r p e r i o d s 1 = n a t u r a l l o g a r i t h m i c operator; any v a r i a b l e p r e f i x e d by t h i s symbol i s i n l o g form w" = weight operator; any v a r i a b l e p r e f i x e d by t h i s symbol i s i n biomass r a t h e r than numbers form; i t i s lagged f o u r p e r i o d s when used with escapement SER = standard e r r o r of r e g r e s s i o n C = cons t a n t D = dummy operator; any v a r i a b l e p r e f i x e d by t h i s symbol i s i n dummy form OLSQ = o r d i n a r y l e a s t sguares POL = polynomial d i s t r i b u t e d l a g CORC = Cochrane-Orcutt i t e r a t i v e l e a s t sguares r o u t i n e to estimate rho Rho = c o e f f i c i e n t of c o r r e l a t i o n between s u c c e s s i v e d i s t u r b a n c e s Stage One LB, LHR, and L(R/E) were r e g r e s s e d on C p l u s the ap p r o p r i a t e form of E or i t s transform ( i e : HE or LE) plus e l e v e n race escapements e i t h e r i n biomass or numbers form. A l l the best parameter est i m a t e s were then transformed back i n t o n o n - l i n e a r form and a c t u a l recruitment was r e g r e s s e d on 86 p r e d i c t e d r e c r u i t m e n t t o check the f i t . Stage Two LR, LWR and L(8/E) were r e g r e s s e d on C, the a p p r o p r i a t e form o f E (l o g transform or biomass) and the best of the eleven race escapements i n a p p r o p r i a t e form (numbers o r biomass) e i t h e r i n dummy form or PDL form with the seguences or l a g s as above. In every case the dummy seguences gave b e t t e r r e s u l t s than the PDL and the l a t t e r technigue was dropped i n f a v o r of the former i n subsequent work. That i s , the hypothesis t h a t the c y c l e i s some f u n c t i o n of c o n d i t i o n s i n the environment i n previous years does not appear to be supported. The a l t e r n a t i v e hypothesis that environmental i n f l u e n c e s simply a c t i n a fou r year c y c l e does appear t o be supported. The depensatory parameter was taken from the LWE c o e f f i c i e n t i n the expre s s i o n (2-12) form of the B i c k e r model (the same as (2-13) without the c y c l e dummies, j u s t WE) and used i n c o n j u n c t i o n with the L (B/E) v e r s i o n . The best parameter e s t i m a t e s are given i n eguation (2-14) below. T h i s v e r s i o n had a c o r r e c t e d B-squared of .81 and a c r i t i c a l t s t a t i s t i c of 2.08 a t the 5% l e v e l of s i g n i f i c a n c e . (2-14) LOG (WR/WE*. »*)) (.88) = +. 43(3.2) -.11E(-4.9) -. 13D4lE(-3.0) -. 18D42E(-4. 2) 15WEH{3.0) -. 32D43WEH(-3.5) -. 03BEST (3.4) +. 18EC (3. 1) -. 37D224EC (-4.2) • . 23D43WELC (3.7) -.2lEQ(-2.2) -.32WESA(-2.7) +2. 6D22WES4 (2.7) Where WB = biomass (weight times numbers) WE = mass of escapement (weight times numbers) E = escapement i n numbers lagged four p e r i o d s 87 p r e f i x D = i n d i c a t e s t h a t t h i s v a r i a b l e i s i n dummy form 4 1 , 2 , 3 , 4 - f o u r year c y c l e with the weight i n the f i r s t , second, t h i r d or f o u r t h year o f c y c l e (1 0 0 0) D 2 1 , 2 2 = two year c y c l e with the weight i n the f i r s t or second year o f c y c l e (1 0) D224 = f o u r year c y c l e with the weight i n the second and t h i r d year of c y c l e (0 1 1 0) HEfi = mass escapement t o Ha r r i s o n WEST = mass escapement to South Thompson EC = escapement i n numbers t o C h i l c o t i n HELC = mass escapement t o l i l l o o e t , F r a s e r Canyon EQ = escapement (in numbers) to Quesnel WESa = mass escapement to Seton-anderson The second step i n v o l v e d e s t i m a t i o n i n s p l i t p e r i o d form. The b e s t e s t i m a t o r had a sum of squared r e s i d u a l s (SSR) of . 2 4 3 5 f o r the period 1942 to 1958, an SSB o f . 0986 f o r the p e r i o d 1959 t o 1975 and an SSR of 1.2848 f o r the period 1942 t o 1975. These gave an F s t a t i s t i c of 1.743 which was w e l l below the c r i t i c a l F value of 3 .26 a t the 5$ l e v e l of s i g n i f i c a n c e and meant t h a t the n u l l h y p o t h e s i s of unchanged parameter values c o u l d be accepted. The t h i r d step i n v o l v e d the use of CORC to estimate the value o f rho f o r the best e s t i m a t o r . The H i l d r e t h - L u r o u t i n e was used as a check on CORC to make sure the i t e r a t i o n d i d not stop a t a l o c a l minimum SER r a t h e r than at the g l o b a l . The best e s t i m a t o r had a value f o r rho of .64 with an R-sguared o f . 9 . The f i n a l step involved c o n v e r t i n g the parameter e s t i m a t e s back i n t o n o n - l i n e a r form and r e g r e s s i n g r e c r u i t m e n t a g a i n s t p r e d i c t e d . The best parameter estimates (expression (2 -14) above) had an R-sguared of .9 and an SER of . 7 . 88 CHAPTER THREE: HARVESTING: THE PRODUCTION AND COST FUNCTIONS I INTRODUCTION The r e c r u i t m e n t f u n c t i o n estimated i n Chapter Two w i l l t e l l the manager of the f i s h e r y what recruitment t o expect f o u r years a f t e r a l l o w i n g a c e r t a i n l e v e l o f escapement. But the manager needs t o he a b l e to compare t h i s investment r e t u r n with the r e t u r n he could expect i f he were to h a r v e s t more of that escapement and i n v e s t the net p r o f i t a t the s o c i a l r a t e of r e t u r n . , In making t h i s comparison the next s t e p i s t o estimate the c o s t s o f h a r v e s t i n g a s p e c i f i e d c a t c h from a given recruitment cf F r a s e r River sockeye salmon.,But before going to a d e s c r i p t i o n of the procedure f o r determining these c o s t s , i t i s u s e f u l t o review some of the c h a r a c t e r i s t i c s of the F r a s e r R i v e r sockeye salmon f i s h e r y . T h i s review w i l l h e l p provide some i n s i g h t i n t o the a p p r o p r i a t e s p e c i f i c a t i o n of the salmon production f u n c t i o n from which h a r v e s t i n g c o s t s are e v e n t u a l l y d e r i v e d . As d e s c r i b e d i n Chapter One and Hap (1-2), the f i r s t zone t o r e c e i v e biomass recruitment i s the west c o a s t of Vancouver I s l a n d where t r o l l f i s h i n g f o r sockeye i s s u c c e s s f u l (net f i s h i n g i s i l l e g a l ) but i s not s u b s t a n t i a l i n terms of t o t a l h a r v e s t . The reason i s t h a t sockeye stop f e e d i n g and conc e n t r a t e t h e i r energy i n s t e a d on the spawning m i g r a t i o n when they enter the g a u n t l e t . Thus they are not v u l n e r a b l e t o l u r e s nor to b a i t on hooks. T h i s e x p l a i n s why v i r t u a l l y no sockeye are caught by the s p o r t f i s h e r y ( S i n c l a i r , 1960, p.36). Once the reduced 89 r e c r u i t m e n t passes M i s s i o n on the Fraser B i v e r , commercial f i s h i n g i s i l l e g a l . ,Native r i g h t s have permitted Indian f i s h i n g o f sockeye i n c e r t a i n areas of the F r a s e r B i v e r system. T h i s Indian f i s h e r y i s modest i n comparison t o commercial sockeye h a r v e s t i n g and most of the c a t c h i s made f o r t h e i r own consumption. However, the I n d i a n c a t c h does a f f e c t the m i g r a t i o n up the F r a s e r B i v e r and w i l l be i n c l u d e d on an average c y c l e year b a s i s i n subsequent work with escapement. The gear types used i n the f i s h e r y c o n s i s t of t r o l l s , g i l l n e t s and purse s e i n e s i n Canadian waters and g i l l n e t s , r e e f n e t s and purse seines i n Washington waters. A study made of the important trends i n the B.C. salmon f i s h e r y (Campbell and Roberts, 1967) i n d i c a t e d that the t r o l l c a t c h of sockeye was almost n e g l i g i b l e . The Department of the Environment (Canada, 1951-1975) r e p o r t s on c a t c h by gear support t h i s view, p a r t i c u l a r l y i n i n s i d e waters where the t r o l l c a t c h of sockeye r e p r e s e n t s l e s s than 11, by weight, of the t o t a l t r o l l c a t c h of salmon. The o n l y s i g n i f i c a n t t r o l l c a t c h of sockeye i s o f f the west c o a s t o f Vancouver I s l a n d . The t r a d i t i o n a l type of gear used to c a t c h sockeye has been the g i l l n e t . I t s e f f e c t i v e n e s s depends on the r e l a t i o n s h i p between the mesh s i z e and the weight of the f i s h . I f the mesh i s too s m a l l , the head of the f i s h cannot penetrate deeply enough i n t o the net to enmesh the g i l l s . I f the mesh i s too l a r g e , the f i s h simply swim through without too much t r o u b l e . S t u d i e s conducted by the IPSFC culminated i n 1948 i n an e i g h t i n c h l i n e n mesh and i n 1951 i n an e i g h t and t h r e e - g u a r t e r i n c h nylon mesh. G i l l n e t f i s h i n g i s e s s e n t i a l l y p a s s i v e i n t h a t the nets are 90 set, usually in f a i r l y shallow and confined channels of water, and the fishermen wait for migrating f i s h to become entangled. There i s a law prohibiting the use of the boat to herd the f i s h into the net. But fishermen are permitted to patrol the net to judge when i t i s f u l l enough to p u l l and clear. With the increasing use of purse seines in the early part of the gauntlet, the biomass a v a i l a b i l i t y f o r g i l l n e t fishermen i n the l a t e r parts of the gauntlet has declined steadily. , The a v a i l a b i l i t y to Fraser l i v e r g i l l n e t t e r s dropped from nearly 100% of the potential Canadian catch i n 1944 to 65% i n 1956 (IPSFC, 1956, p.20)., As a r e s u l t , g i l l n e t fishermen started moving into the e a r l i e r part of the gauntlet, although with limited success at f i r s t (IPSFC, 1954, p.27 and 1955, p.25). However, as f i s h i n g methods adjusted to the new technigue of " d r i f t netting" i n unconfined waters, the g i l l n e t catch in the S t r a i t of Juan de Fuca has become very s i g n i f i c a n t , e s p e c i a l l y in years when sockeye d r i f t rather than swim. A hybrid form of g i l l n e t i s the reefnet used by a small proportion of Washington fishermen. As the name implies, the net i s used i n the v i c i n i t y of reefs where waters are very confined and sockeye are forced into heavy concentrations. Fishermen wait u n t i l the concentration builds up and then quickly set the reefnets. Once again, increasing use of gear i n e a r l i e r parts of the gauntlet has caused a steady decline i n biomass a v a i l a b i l i t y to reefnets u n t i l , i n 1963, there were no sockeye available at a l l (IPSFC, 1963, p.16, and 1966, p.17). To counteract t h i s problem, the IPSFC adjusted t h e i r open time of day regulations to permit the reefnet open days to st a r t e a r l i e r than f o r any 91 other type of gear throughout the season (IPSFC, 1967, p.25). A f t e r the g i l l n e t , the purse s e i n e i s the most important type o f gear f o r h a r v e s t i n g salmon. I t i n v o l v e s a c t i v e f i s h i n g i n the sense t h a t the boat must s e a r c h f o r salmon and, when a s i z a b l e block i s s i g h t e d , the set i s made by e n c i r c l i n g the f i s h with a c u r t a i n of net which i s then c l o s e d at the bottom by p u r s i n g the s e i n e . The entrapped f i s h are then b r a i l e d i n t o the hold o f the v e s s e l . Because of the s i z e and complexity of the o p e r a t i o n , the average purse s e i n e c a r r i e s a crew of f i v e as opposed to the one man t r o l l and g i l l n e t o p e r a t i o n s . The investment i n the purse s e i n e and i t s gear i s p r o p o r t i o n a t e l y greater as w e l l . Purse s e i n i n g has been c o n s i d e r a b l y s t r e a m l i n e d s i n c e the i n t r o d u c t i o n of the P u r e t i c power block i n 1953 and the use of a drum r a t h e r than a t a b l e to h a u l the net (IPSFC, 1954, p.22; 1955, p-25; and 1956, p.19). These i n n o v a t i o n s have p e r m i t t e d the average crew s i z e to f a l l from e i g h t cr more to f i v e with c a t c h per man i n c r e a s i n g s i m u l t a n e o u s l y ( S i n c l a i r , 1960, pp.213-214; and Boyce, Bevan, et a l . , 1963, p.2). If a block of sockeye are found m i l l i n g c l o s e to the s u r f a c e , the purse seine i s an extremely e f f e c t i v e type of gear. Once the s e t i s made and the net pursed, the f i s h can be b r a i l e d aboard w i t h i n a s h o r t time. Thus, the c a t c h i n g power of purse s e i n e s i s much higher, on average, than f o r g i l l n e t s or t r o l l s but the v a r i a n c e i s a l s o much g r e a t e r . In some years c e r t a i n gear types are more s u c c e s s f u l than others and i n other years, l e s s s u c c e s s f u l . C a t c h a b i l i t y i s the f a c t o r which can be used to account f o r much of the year-to-year 92 and area-to-area differences i n catch per unit of e f f o r t and appears to be influenced by the c y c l i c a l nature of run sizes, weight of f i s h etc. A v a i l a b i l i t y of recruitment for a l l cycle years, however, appears to follow a f a i r l y standardized path. The analysis of sockeye salmon migration performed by the Royce-Bevan study in order to construct time of entry curves indicated the following average pattern: about 9H% on the Canadian side in the outer S t r a i t of Juan de Fuca; 20% then leave the S t r a i t of Juan de Fuca to spend one day in the Westbeach area in Washington waters; these l a t e r r e j o i n the remaining 30% i n the San Juan Islands area in Washington waters; then about 95% of the remainder swim past and through the Point Roberts area on the Washington side; and f i n a l l y , 100% of the remainder swim into the Fraser River mouth (Royce, Bevan, et a l . , 1963, p.21). There are da i l y e f f o r t data available f o r each of four main areas i n Washington waters and eleven main areas i n Canadian waters. But the Royce-Bevan study had the greatest success using data aggregated on a monthly basis for the following areas only: the west coast of Vancouver Island, S t r a i t of Juan de Fuca, southern Vancouver Island including Maru S t r a i t and Stuart Island, Westbeach, San Juan Islands (Bosario Straits) , Point Roberts, the mouth of the Fraser River and the Fraser River i t s e l f (1963, p.31) . Whereas the Royce-Bevan group looked at only six years of data (1956 to 1961), the present study uses twenty-five years (1951 to 1975) of observations on gear type e f f o r t to estimate the production function. And despite the many changes i n gear location in the gauntlet during that time, there i s a surprising 93 c o n s i s t e n c y i n the gear types used i n each of the above named areas over t h a t p e r i o d of time. These are the same areas of sockeye m i g r a t i o n chosen by S i n c l a i r as well (1960, p.29). For the most p a r t the f a v o r i t e gear type f o r each area i s a r e s u l t of c a t c h a b i l i t y c o n d i t i o n s t h e r e which are p e c u l i a r to the needs of the gear type. I t i s important to note, however, t h a t c e r t a i n gear types have been p r o h i b i t e d from h a r v e s t i n g i n c e r t a i n areas by the IPSFC (1953, p.7). In p a r t i c u l a r , area 19 was c l o s e d to commercial f i s h i n g t o permit s p o r t f i s h i n g ( S i n c l a i r , 1960, p.34) and areas to seaward o f the black l i n e on Hap (1-2) are c l o s e d to net f i s h i n g . , Purse s e i n e s , f o r example, have been t r i e d over the years i n the F r a s e r B i v e r below Miss i o n but with very l i t t l e s u c c e s s . The reason i s t h a t by t h a t l a t e stage i n the g a u n t l e t , sockeye are coming through i n only s m a l l groups.,, There i s very l i t t l e bunching except a t the peak and a block o f f i s h i s r a r e . G i l l n e t s are i d e a l l y s u i t e d f o r t h i s type of f i s h i n g , another example i s r e e f n e t s i n the Westbeach area which have met with very l i t t l e s uccess. Again both purse s e i n e s and g i l l n e t s were t r i e d i n area 21 with some success but they were p r o h i b i t e d a f t e r 1957. And area 19 has long been poor f o r Canadian fishermen because the sockeye have a tendency to shoot by with c u r r e n t s i n t o Washington waters. Thus the f o l l o w i n g h a r v e s t i n g zones have become a s s o c i a t e d , e i t h e r through custom or r e g u l a t i o n , with the f o l l o w i n g areas and gear t y p e s . Note that the o f f i c i a l areas w i t h i n each zone have been aggregated to s i m p l i f y the a n a l y s i s . In every case these a g g r e g a t i o n s were checked with the department of f i s h e r i e s concerned and they conform with o f f i c i a l p r a c t i s e s . W i thin each 94 zone, there may be more than one gear type operating. To avoid confusion over the expression "gear type zone" when these are r e a l l y f i s h i n g zones , they are referred to as "gear type areas" i n this study. This also avoids the confusion which can arise when r e f e r r i n g to sixteen "gear types" when, i n f a c t , there are only four ( t r o l l s , g i l l n e t s , purse seines and reefnets). Therefore, there are sixteen gear type areas operating within the seven f i s h i n g zones. The seven zones with their associated gear type areas are, zone one: west coast of Vancouver Island, consisting of areas 21, 23, half of 24, and International Area "C", harvested by t r o l l ; zone two: the Canadian section of the S t r a i t of Juan de Fuca, consisting of areas 19 and 20, harvested by g i l l n e t s and purse seines; zone three: the Washington section of S t r a i t of Juan de Fuca (after 1957) and Westbeach, harvested by g i l l n e t s and purse seines; zone four: the San Juan Islands i n Washington waters, harvested by g i l l n e t s , purse seines and reefnets; zone f i v e : Point Roberts i n Washington waters, harvested by g i l l n e t s , purse seines and reefnets; zone s i x : the S t r a i t of Georgia in Canadian waters, consisting of areas 17 and 18, harvested by t r o l l s , g i l l n e t s and purse seines; and zone seven: the Fraser River area, consisting of areas 29A, 29B, 29C and 29D, harvested by t r o l l s and g i l l n e t s . These areas and gear types account f o r 95% of the commercial sockeye salmon harvest i n most years i n the f i s h i n g gauntlet. These seven harvesting zones and their sixteen gear type harvesting areas have been aggregated, therefore, from t h i r t y -four d i f f e r e n t gear type areas. I t w i l l be assumed that within 95 each zone recruit-tent a v a i l a b i l i t y i s the same for a l l gear types but that c a t c h a b i l i t y i s d i f f e r e n t . From one harvesting zone to another i t w i l l be assumed that the recruitment a v a i l a b i l i t y has been reduced by the catch taken by a l l the gear types i n the previous zone. Of the sixteen gear type areas which can be distinguished i n the gauntlet, only six have accounted for 90% of the average catch taken over the period 1951 to 1975. Although production functions were estimated for a l l sixteen gear type areas, the ten gear type areas accounting for only 10% of the average catch were a l l more costly i n terms of harvesting than the six gear type areas accounting for 90% of the average catch. For t h i s reason, only the six major gear type areas w i l l be discussed in terms of both production function estimation and programming for cost minimization. These areas are as follows: CZ2GN and CZ2PS, Canadian g i l l n e t s and purse seines i n the outer S t r a i t of Juan de Fuca; 0SZ4GN and 0SZ4PS, O.S. g i l l n e t s and purse seines in the San Juan Islands; 0SZ5PS, O.S. purse seines i n the Point Boberts area; and CZ7GH, Canadian g i l l n e t s i n the mouth of the Fraser Biver., In addition to sockeye, four other species of salmon are caught i n the gauntlet during the Fraser Biver sockeye season. However, the multi-species nature of escapement and harvesting w i l l be ignored for most of t h i s study to concentrate on harvesting and escapement fo r sockeye only. The catch of other salmon w i l l be recognized only to the extent that i t lowers the costs of harvesting for sockeye. The model used w i l l be very basic in that the average cycle catch of other salmon w i l l be allowed to flow into each zone to be harvested by gear operating 96 there f o r purposes o f c a t c h i n g sockeye,,Gear w i l l be permitted to harvest ether salmon e i t h e r t o the extent of remaining c a p a c i t y a f t e r h a r v e s t i n g sockeye or to the h i s t o r i c a l extent a s s o c i a t e d with average c y c l e c a t c h of other salmon per u n i t of e f f o r t , whichever i s s m a l l e r . Thus the model assumes t h a t gear concentrate cn c a t c h i n g sockeye and harvest o t h e r salmon on an i n c i d e n t a l b a s i s only a c c o r d i n g to excess c a p a c i t y . N e v e r t h e l e s s , sockeye may be an i n c i d e n t a l c a t c h f o r some gear i n some years because the fishermen are i n t e n t on h a r v e s t i n g pink salmon. For t h i s reason, c a t c h a b i l i t y of sockeye may vary on a year c y c l e b a s i s simply because of the j o i n t production nature o f h a r v e s t i n g . although data do e x i s t t o permit e s t i m a t i o n of recruitment and p r o d u c t i o n f u n c t i o n s f o r these other salmon, because of the c o m p l e x i t i e s i n v o l v e d i n mode l l i n g a m u l t i - s p e c i e s f i s h e r y , no attempt w i l l be made to estimate e i t h e r r e c r u i t m e n t or production f u n c t i o n s f o r pink, chum, Chinook or coho salmon. T h i s i s p a r t i a l l y j u s t i f i e d on the b a s i s of the f a c t t h a t the c o s t m i n i m i z a t i o n program w i l l choose the l e a s t c o s t method f o r h a r v e s t i n g sockeye and i n so doing w i l l a u t o m a t i c a l l y exclude a p o r t i o n of the c a t c h of "o t h e r " salmon. Sockeye w i l l r e p r e s e n t a very l a r g e component of the c a t c h f o r the l e a s t c o s t areas simply because of the d e n s i t y of sockeye which are permitted through the g a u n t l e t to be caught by those l e a s t c o s t h a r v e s t i n g areas. Both pink and sockeye salmon are managed by the IPSFC and the r e p o r t s i n d i c a t e t h a t the job i s not easy. , although the peaks of the sockeye and pink runs do not c o i n c i d e , there i s 97 enough o v e r l a p to cause problems with s p e c i e s escapement. Pink salmon represent the g r e a t e r weight of "other" salmon caught i n "on" y e a r s , while chum, chinook and coho r e p r e s e n t a s i z e a b l e weight i n the " o f f " years f o r pink. T h i s can l e a d to s e r i o u s problems with investment i n enhancement or even i n i n - k i n d a b s t e n t i o n investment. I f the r e p r o d u c t i o n p o t e n t i a l of sockeye i s r a i s e d without a corresponding i n c r e a s e f o r another s p e c i e s , the i n c r e a s e d h a r v e s t i n g r a t e needed f o r c a t c h i n g the enhanced sockeye c a t c h c o u l d s e r i o u s l y d e p l e t e the other stock ( C r u t c h f i e l d and Pontecorvo, 1969, p. 180). In summary, the t h i r t y f o u r o r i g i n a l gear type areas i n the g a u n t l e t have been aggregated i n t o seven major f i s h i n g zones c o n s i s t i n g of one or more gear type areas, to g i v e s i x t e e n gear type areas i n a l l . Within each of the seven major f i s h i n g zones, i t i s assumed that recruitment a v a i l a b i l i t y i s the same to each gear type o p e r a t i n g w i t h i n t h a t zone. However, recruitment a v a i l a b i l i t y v a r i e s from zone to zone a c c o r d i n g to the s i z e of catches made i n p r e v i o u s zones. Within each zone, gear type c a t c h a b i l i t y i s permitted to vary on a c y c l e year b a s i s , although production f u n c t i o n s f o r a l l s i x t e e n gear type areas were estimated, i n f a c t only s i x gear type areas proved f e a s i b l e i n terms of c o s t . Only these s i x gear type a r e a s o p e r a t i n g i n zones two, f o u r , f i v e and seven w i l l be r e t a i n e d f o r work with the l e a s t c o s t program and the management model. 98 I I PBODOCTIOH FONCJION ESTIHATION (1) P r o d u c t i o n F u n c t i o n S p e c i f i c a t i o n The approach taken i n t h i s study i s to c a l c u l a t e c o s t s by e s t i m a t i n g a production f u n c t i o n f i r s t . The most common hyp o t h e s i s f o r the harvest p r o d u c t i o n f u n c t i o n i s (3-1) C = gBf where C = c a t c h g = c a t c h a b i l i t y of biomass (constant) B = r e c r u i t m e n t a v a i l a b i l i t y ( v a r i a b l e ) f = a composite u n i t o f e f f o r t ( v a r i a b l e ) For a non-migratory s p e c i e s , g i s sometimes assumed t o be c o n s t a n t which i m p l i e s t h a t the stock of f i s h i s e q u a l l y d i s p e r s e d throughout the f i s h i n g grounds at a l l times. For a migratory s p e c i e s l i k e sockeye, the f i r s t change i s to make B s e n s i t i v e t o the movement of the biomass through the g a u n t l e t . However, i n t h i s study there w i l l be no i n t r a - s e a s o n a l a n a l y s i s of a v a i l a b i l i t y d i f f e r e n c e s . In any one season, the r e c r u i t m e n t w i l l be a v a i l a b l e t o the f i r s t f i s h i n g area i n the g a u n t l e t i n unmodified form. I t w i l l be a v a i l a b l e t o the second f i s h i n g area i n a reduced amount because of the f i s h i n g i n the f i r s t a r e a . As the m i g r a t i o n proceeds, r e c r u i t m e n t a v a i l a b i l i t y i s c o n t i n u a l l y reduced by the c a t c h i n the preceding area u n t i l o n l y the escapement i s l e f t a f t e r the l a s t f i s h i n g a r e a . As t o t a l catch r e p r e s e n t s about 70% of r e c r u i t m e n t ( l e a v i n g 30% f o r escapement), the r e d u c t i o n i n a v a i l a b i l i t y can be c o n s i d e r a b l e by the time the seventh f i s h i n g zone has been reached. Given the r e c r u i t m e n t a v a i l a b i l i t y from a previous zone, c e r t a i n f a c t o r s operate which a f f e c t the c a t c h a b i l i t y of the 99 biomass which i s a v a i l a b l e from t h a t p r e v i o u s zone. I f recruitment i s composed o f s m a l l e r f i s h , these w i l l tend to s c h o o l and swim c l o s e r to the s u r f a c e which makes them more v u l n e r a b l e to purse s e i n e s and, because the g i l l c ircumference i s s m a l l e r , l e s s v u l n e r a b l e t o g i l l n e t s . On the other hand, l a r g e r f i s h tend to swim more independently and a t g r e a t e r depths, making them l e s s v u l n e r a b l e t o s u r f a c e nets l i k e purse s e i n e s but more v u l n e r a b l e to g i l l n e t s . Other f a c t o r s a f f e c t i n g c a t c h a b i l i t y are weather, t i d e s , c u r r e n t s and water temperature. However, i n the above, w e l l known, f o r m u l a t i o n f o r h a r v e s t i n g production f u n c t i o n s , i t i s sometimes assumed that c a t c h a b i l i t y cf the a v a i l a b l e biomass i s c o n s t a n t on an i n t e r -seasonal b a s i s . Catch i s r e g r e s s e d on e f f o r t and the biomass re c r u i t m e n t to a r r i v e at parameter estimates. P a r t of the reason f o r t h i s u n r e a l i s t i c c a t c h a b i l i t y assumption has a r i s e n from the data c o n s t r a i n t s which have f o r c e d b i o l o g i s t s to estimate stock abundance u s i n g c a t c h per u n i t of e f f o r t (assuming constant c a t c h a b i l i t y ) i n a n a l y t i c models. T h i s assumption w i l l not be necessary t o the b i o l o g i c a l work i n t h i s study because of the use of escapement to estimate r e c r u i t m e n t independently. Thus, i t w i l l be p o s s i b l e to estimate v a r y i n g c a t c h a b i l i t y of biomass as w e l l . Many s t u d i e s have t r e a t e d f i s h c a t c h a b i l i t y as a constant (see, f o r example, C r u t c h f i e l d and Pontecorvo, 1969, chap t. 2; Gordon, 1954, pp. }39-142; Hannesson, 1975, pp.152-153; IPSFC, 1944, pp.23-26; and S i n c l a i r , 1960, p.67). Hannesson d i s c u s s e s the f a c t t h a t a v a i l a b i l i t y (what i s c a l l e d c a t c h a b i l i t y i n t h i s study) i s a f u n c t i o n of both e f f o r t and biomass f o r cod but 1 0 0 concludes t h a t i n the long-run cod are probably evenly d i s p e r s e d and t h a t a v a i l a b i l i t y ( c a t c h a b i l i t y i n t h i s s t u d y ) , t h e r e f o r e , i s c onstant (1974, chapt.2, p a r t 4 ) . , Although biomass a v a i l a b i l i t y of cod v a r i e s with the s i z e of the stock, even f o r cod the c a t c h a b i l i t y appears to vary with year c l a s s s t r e n g t h , g e o g r a p h i c a l l o c a t i o n and weather. Why e l s e would cod fishermen wait f o r the r i g h t t i d e s and weather and jockey f o r the "best" p o s i t i o n s on the f i s h i n g grounds? The h a r v e s t i n g production f u n c t i o n i n e x p r e s s i o n (3-1) i m p l i e s t hat the more d i s p e r s e d the f i s h , i n a given f i s h e r y , the more e f f o r t must be expended to c a t c h them. ,To c l a r i f y the d i s t i n c t i o n between a v a i l a b i l i t y and c a t c h a b i l i t y of biomass, and t e c h n i c a l change of e f f o r t , i t i s u s e f u l t o modify e x p r e s s i o n (3-1) to g i v e : (3-2) C = gBpf where C = c a t c h g = biomass c a t c h a b i l i t y index B = r e c r u i t m e n t a v a i l a b i l i t y p = e f f o r t e f f i c i e n c y index f = h a r v e s t i n g e f f o r t Bote t h a t c a t c h a b i l i t y as w e l l as a v a i l a b i l i t y can now a f f e c t the s c a l e of o p e r a t i o n . /For most h a r v e s t i n g production f u n c t i o n e s t i m a t i o n , B and f are allowed to vary a c c o r d i n g t o the data. But g i s sometimes assumed t o be c o n s t a n t . /If t h i s i s the case, then a c t u a l f l u c t u a t i o n s i n biomass c a t c h a b i l i t y are absorbed by an e r r o r term. T h i s procedure can r e s u l t i n i n f e r i o r f i t s f o r the p r o d u c t i o n f u n c t i o n . Most r e s e a r c h e r s have then turned to modifying the e f f o r t index to allow f o r changes i n e f f i c i e n c y . The biomass index i s u s u a l l y l e f t alone and f l u c t u a t i o n s i n c a t c h per u n i t of e f f o r t are diagnosed i n terms of changes i n 101 gear e f f i c i e n c y . I t i s t r u e t h a t the u n i t of e f f o r t i s s u b j e c t to c r i t i c i s m because i t i s a composite measure of a v e s s e l and i t s complement of men and gear m u l t i p l i e d by the number of hours or days of f i s h i n g . And i t i s obvious t h a t v i r t u a l l y a l l components of such an index must have changed over time. For example, although i t i s probably s a f e to assume t h a t l a b o u r p r o d u c t i v i t y has changed very l i t t l e i n the B.C. salmon f i s h e r y (Canada, DOE, 1971) , not only has r e l a t i v e e f f i c i e n c y between gear types changed c o n s i d e r a b l y but changes w i t h i n gear types have been very r a p i d s i n c e World War I I . T h e r e f o r e , some account must be taken of the l a r g e i n c r e a s e i n the c a t c h i n g a b i l i t y of the d i f f e r e n t gear types. Any attempt to measure t e c h n i c a l i n n o v a t i o n i n gear or v e s s e l s i s going t o be of l i t t l e use unless some adjustment f o r f l u c t u a t i o n s i n c a t c h a b i l i t y can be made. I f not, the measure of e f f o r t w i l l have absorbed the f l u c t u a t i o n s i n biomass c a t c h a b i l i t y over the years as w e l l . The technique employed by Tomkins and B u t l i n (1975, p.115) to " e l i m i n a t e the e f f e c t s of varying stock abundance between y e a r s " i n v o l v e d s t a n d a r d i z i n g the average c a t c h e s of other gear types " r e l a t i v e to the performance of the d r i f t boat s e c t o r i n each year, s i n c e there has been very l i t t l e change i n the technology of d r i f t net f i s h i n g over the years" , The problem with using t h i s approach i s t h a t i t assumes that biomass c a t c h a b i l i t y was constant wi t h i n each season f o r a l l gear types. While t h i s i s u n l i k e l y even i n a cod f i s h e r y , i t i s impossible i n the sockeye salmon g a u n t l e t . Some means, e x t e r n a l t o the p r o d u c t i o n f u n c t i o n , must be found 1 0 2 to a d j u s t e f f o r t f o r t e c h n i c a l change. I f the e s t i m a t i n g process i t s e l f i s allowed to compensate f o r these e r r o r s i n e f f o r t measurement, f l u c t u a t i o n s i n c a t c h a b i l i t y of biomass and t e c h n i c a l change w i l l become confused. The problem i s that c a t c h per u n i t of e f f o r t v a r i e s from season to season f o r three b a s i c reasons..The f i r s t reason i s t h a t the a v a i l a b i l i t y of f i s h v a r i e s from season to season. Most h a r v e s t i n g p r o d u c t i o n f u n c t i o n work does not s u f f e r from a lac k of o b s e r v a t i o n s on a v a i l a b i l i t y and t h i s i n f l u e n c e can be removed. The second reason i s that gear e f f i c i e n c y changes from season to season because of changes i n v e s s e l and gear p r o d u c t i v i t y . The t h i r d reason f o r f l u c t u a t i o n s i n c a t c h per u n i t o f e f f o r t a r i s e s from the nature o f the stock of f i s h (whether i t s c h o o l s or not), i t s s p a t i a l d i s p e r s i o n ( t i d e s and c u r r e n t s prevent equal d i s p e r s i o n ) and the v e r t i c a l displacement i n the water (how deep the f i s h swim). T h i s a f f e c t s the c a t c h a b i l i t y to v a r i o u s gear types. Observations cn c a t c h a b i l i t y do not e x i s t f o r many f i s h e r i e s . any attempt t o allow the e s t i m a t i n g process to account f o r changes i n p r o d u c t i v i t y i s bound t o pick up the e f f e c t s of c a t c h a b i l i t y . T h i s i s why i t i s so important to account f o r changes i n p r o d u c t i v i t y or c a t c h a b i l i t y o u t s i d e the e s t i m a t i n g process i t s e l f . In the present study, independent measures of t e c h n i c a l change e x t e r n a l to the p r o d u c t i o n f u n c t i o n are a v a i l a b l e both i n IPSFC r e p o r t s and i n measures of gear and v e s s e l value changes. And i t w i l l be p o s s i b l e t o a d j u s t the measure of e f f o r t . The improvement i n f i t i s s i g n i f i c a n t f o r most of the gear type 103 areas i n the F r a s e r River sockeye f i s h i n g g a u n t l e t . However, i t i s not p o s s i b l e to measure f l u c t u a t i o n s i n c a t c h a b i l i t y e x t e r n a l l y to the pro d u c t i o n f u n c t i o n . I n s t e a d , dummy v a r i a b l e s i n s l o p e and i n t e r c e p t form w i l l be used with biomass recru i t m e n t (adjusted f o r a v a i l a b i l i t y i n each f i s h i n g zone i n the gauntlet) t o allow the e s t i m a t i o n process i t s e l f t o estimate the c a t c h a b i l i t y parameters. The very s t r o n g f o u r year c y c l e which r e s u l t s would tend to support the hypothesis that c a t c h a b i l i t y i s mainly i n f l u e n c e d by year c l a s s c y c l e s which may r e s u l t from c y c l i c a l environmental f a c t o r s . As i t i s h i g h l y u n l i k e l y t h a t t e c h n i c a l progress advances and r e t r e a t s i n a f o u r year p a t t e r n , i t i s probably s a f e to assume t h a t t h i s method of adjustment i s not c o n f u s i n g c a t c h a b i l i t y of a g i v e n recruitment of sockeye with enhanced c a t c h i n g power of gear ( e f f i c i e n c y ) . In summary, biomass r e c r u i t m e n t can be adjusted t o r e f l e c t a v a i l a b i l i t y i n each f i s h i n g zone by s u b t r a c t i n g the pr e v i o u s zones' c a t c h from a v a i l a b l e biomass recr u i t m e n t i n t h a t zone. E f f i c i e n c y indexes and gear and v e s s e l value indexes can be used t o a d j u s t e f f o r t f o r t e c h n i c a l change i n a manner e x t e r n a l to the production f u n c t i o n . , And the e s t i m a t i o n process i t s e l f , through the use of dummy v a r i a b l e s which d i s t i n g u i s h biomass a v a i l a b i l i t y a c c o r d i n g to c y c l e year, can be used t o estimate the c a t c h a b i l i t y parameters. The h a r v e s t i n g model used i n t h i s study w i l l be a n o n - l i n e a r homogeneous p r o d u c t i o n f u n c t i o n : (3-3) C - a ( ( p f ) b ) (B^) (exp(#)) where C = gear type catch by area a,b = parameters pf = gear type e f f o r t by area adjusted by an index of t e c h n i c a l progress i n gear type p r o d u c t i v i t y (in d e f l a t o r form) B = biomass a v a i l a b i l i t y by gear type area 1 0 4 q = c a t c h a b i l i t y parameter # = e r r o r term E x p r e s s i o n (3-3) i s i n t r i n s i c a l l y n o n - l i n e a r and i n l o g a r i t h m i c form looks l i k e : ( 3 - 4 ) InC = l n a • b ( l n ( p f ) ) • q ( l n ( B ) ) + # E x p r e s s i o n ( 3 - 4 ) i s the b a s i c equation used i n a l l production f u n c t i o n parameter e s t i m a t i o n . (2) T e c h n i c a l Progress Before e s t i m a t i o n can take p l a c e the e f f o r t data have to be a d j u s t e d f o r t e c h n i c a l p r o g r e s s . Furthermore, the data are f o r d e l i v e r i e s or l a n d i n g s only, not f o r the amount of time spent s e a r c h i n g , c a t c h i n g and p r o c e s s i n g . A d e l i v e r y or l a n d i n g i s assumed to be one day's worth of e f f o r t i n the data r e p o r t s . Thus the e f f o r t v a r i a b l e M f " must be a d j u s t e d t o r e f l e c t the t r u e amount of e f f o r t and the p r o d u c t i v i t y index "p" must be c a l c u l a t e d t o r e f l e c t changes i n t h a t p r o d u c t i v i t y . F i r s t we d e a l with the problems experienced with the r e p o r t e d measure of e f f o r t over time. Some v e s s e l s may d e l i v e r twice a day while others may d e l i v e r every two or t h r e e days; and yet the f i r s t i s o f f i c i a l l y r e p o r t e d to have used two u n i t s o f e f f o r t to c a t c h the harvest while the second i s r e p o r t e d as having used a s i n g l e u n i t of e f f o r t . The f a v o r i t e f i s h i n g p o r t s and f i s h i n g grounds have changed over the p e r i o d 1951 to 1975 to the extent t h a t gear has moved out to the open areas of the g a u n t l e t . Thus running d i s t a n c e s have changed, although with the advent of more powerful engines, i t i s not c l e a r how much of 105 t h i s i n c r e a s e i n d i s t a n c e s has been o f f s e t by g r e a t e r speed. Gear s t i l l f i s h i n g i n t r a d i t i o n a l areas i s u n l i k e l y to have experienced much change i n running times or d i s t a n c e s . T h e r e f o r e , with more powerful engines, e l e c t r o n i c equipment and s h i p - t o - s h i p r a d i o , i t must be assumed t h a t s e a r c h time has been reduced over the years f o r some gear but not f o r other s . The l e n g t h of exposure of the gear t o the biomass has a l s o changed because the hauling and c l e a r i n g of gear has been g r e a t l y speeded up with the a d d i t i o n of power to s k i f f s , r o l l e r s , p u l l e y s and b r a i l e r s . T h i s means t h a t purse s e i n e s can be set twelve times a day i n s t e a d o f n i n e ; g i l l n e t s can be c l e a r e d more o f t e n and more q u i c k l y , r e s u l t i n g i n fewer f i s h l o s t from unmeshing and p r e d a t i o n , and t r o l l s can h a u l , c l e a r and r e - b a i t t h e i r l i n e s more o f t e n and more q u i c k l y . There has a l s o been a tendency to i c e the cat c h i n o r d e r t o f o r e s t a l l d e l i v e r y u n t i l an area i s c l o s e d or t o permit running from a c l o s i n g area to an opening area without the need t o spend time d e l i v e r i n g . D e l i v e r i e s or l a n d i n g s r e c o r d the a c t u a l number o f times the v e s s e l unloaded i t s c a t c h , b e c a u s e of the e f f o r t i n v o l v e d i n t r a v e l l i n g , s e a r c h i n g , f i s h i n g and p r o c e s s i n g during a day, d e l i v e r i e s do gi v e an approximation t o days f i s h i n g f o r some gear. The ex c e p t i o n would be the l a r g e r open water gear types with on-board i c i n g f a c i l i t i e s . For t h i s reason, i t w i l l be assumed t h a t adjustment of the e f f o r t v a r i a b l e can be made through the index of t e c h n i c a l p r o d u c t i v i t y . T h i s p r o d u c t i v i t y index w i l l have to r e f l e c t both changes i n p r o d u c t i v i t y and the need to ad j u s t the e f f o r t v a r i a b l e to r e f l e c t t r u e e f f o r t . . For 106 some areas d e l i v e r i e s are a very c l o s e approximation to true d a i l y e f f o r t but f o r e t h e r s e f f o r t must be a d j u s t e d to g i v e the t r u e d a i l y e f f o r t . To t h i s end, two indexes of p r o d u c t i v i t y f o r each gear type were developed. The f i r s t was an index of gear e f f i c i e n c y which simply r e f l e c t e d the major t e c h n i c a l i n n o v a t i o n s over the p e r i o d under study. The second was an index of gear and v e s s e l values which should r e f l e c t both gear e f f i c i e n c y changes and the changes i n open-water v e s s e l s i z e and f a c i l i t i e s . , For gear o p e r a t i n g i n c o n f i n e d waters, i t was assumed t h a t once d e l i v e r i e s had been converted f o r e f f i c i e n c y changes (the f i r s t i n d e x ) , the 1951 e q u i v a l e n t d e l i v e r i e s would be the same as 1951 e q u i v a l e n t days of f i s h i n g e f f o r t . These g i l l n e t and purse s e i n e e f f i c i e n c y indexes do not handle the problem of changes i n v e s s e l s i z e and power which reduce running and search times and permit c a t c h to be h e l d over s e v e r a l days before the v e s s e l i s f o r c e d to make a d e l i v e r y . However, running time and search time are not of great importance to v e s s e l s l o c a t e d near t r a d i t i o n a l f i s h i n g grounds. Nor i s unloading a problem because c a n n e r i e s have l o c a t e d c l o s e to t r a d i t i o n a l f i s h i n g grounds and o f f e r t e n d e r i n g s e r v i c e s when they cannot l o c a t e c l o s e enough. In many c a s e s , because of the c o n f i n e d working space i n t r a d i t i o n a l f i s h i n g a r e a s , i t i s not a great advantage to have a l a r g e v e s s e l . And, as the v e s s e l s are c l o s e to d e l i v e r y p o i n t s , i t i s e a s i e r t o d e l i v e r once a day than to take the time t o i c e the c a t c h . T h i s i s not the case f o r v e s s e l s engaged i n h a r v e s t i n g i n more open waters i n the g a u n t l e t . These o p e r a t i o n s i n v o l v e a c e r t a i n amount of running 107 and s e a r c h i n g . Both seine and g i l l n e t v e s s e l s i n v o l v e d i n open water o p e r a t i o n s have grown s t e a d i l y l a r g e r and more powerful with more s o p h i s t i c a t e d e l e c t r o n i c gear and i c i n g f a c i l i t i e s . T h i s has i n c r e a s e d the number of e f f e c t i v e hours of f i s h i n g and the number of days per d e l i v e r y . The second index of p r o d u c t i v i t y , t h e r e f o r e , was designed t o p i c k up these open water e f f e c t s through i n d e x i n g the value o f gear and v e s s e l s used i n salmon f i s h i n g . Gear and v e s s e l value indexes were c o n s t r u c t e d f o r purse s e i n e s and g i l l n e t s based on data a v a i l a b l e i n annual r e p o r t s from the f e d e r a l Department of F i s h e r i e s . In b r i e f , an index of average gear and v e s s e l values d e f l a t e d by the a p p r o p r i a t e p r i c e index and a d j u s t e d f o r l i c e n s e c a p i t a l i z a t i o n s was c o n s t r u c t e d f o r g i l l n e t s and f o r purse s e i n e s . The c o n s t r u c t i o n methods are d i s c u s s e d i n g r e a t e r d e t a i l i n the appendix below. These gear and v e s s e l value indexes should pick up the i n f l u e n c e of the switch from t a b l e to drum s e i n e s and from l i n e n to nylon nets, simply because more e f f i c i e n t v e s s e l s and gear are worth more. Thus these gear and v e s s e l value indexes can be used i n p l a c e of e f f i c i e n c y indexes to convert d e l i v e r i e s i n t o 1951 e g u i v a l e n t days of f i s h i n g e f f o r t . I t was assumed, t h e r e f o r e , t h a t t h i s second index of p r o d u c t i v i t y would a d j u s t e f f o r t both f o r changes i n p r o d u c t i v i t y and f o r the a c t u a l number of days of e f f o r t i n v o l v e d i n making a d e l i v e r y . 108 J l l Biomass A v a i l a b i l i t y And C a t c h a b i l i t y E q u a l l y as important as the e f f o r t v a r i a b l e i s the biomass v a r i a b l e . There does not appear to be any pu b l i s h e d data a v a i l a b l e f o r biomass and we are l e f t with the re c r u i t m e n t v a r i a b l e estimated i n connection with the recruitment f u n c t i o n i n Chapter Two. Preparing t h i s v a r i a b l e f o r use i n parameter e s t i m a t i o n f o r the pr o d u c t i o n f u n c t i o n i n v o l v e s two steps. The f i r s t i s to a d j u s t downward the flow of biomass recruitment i n t o each o f the major zones i n the g a u n t l e t f o r catches made i n the previous zone. T h i s e s t a b l i s h e s the biomass a v a i l a b i l i t y to each f i s h i n g zone. The second step i s to estimate the c a t c h a b i l i t y parameter t o r e f l e c t the gear type c a t c h a b i l i t y of sockeye from the a v a i l a b l e biomass i n each zone. , From the s u c c e s s f u l use of c y c l e v a r i a b l e s i n the recruitment f u n c t i o n i n Chapter Two, i t would be l o g i c a l to assume t h e r e i s probably a c y c l i c a l element i n biomass c a t c h a b i l i t y . For example, i t i s known that average weight of f i s h p l a y s a c r i t i c a l r o l e i n terms of both s c h o o l i n g behavior and v e r t i c a l d i s p e r s i o n i n the water. I t w i l l be assumed t h a t the many d i s p e r s i o n e f f e c t s can be summarized by the c h a r a c t e r of each of the f o u r year c y c l e s . Thus each year c l a s s of the fo u r year c y c l e i s assumed t o be d i f f e r e n t from the other three year c l a s s e s i n terms of c a t c h a b i l i t y . Furthermore, the c a t c h a b i l i t y f o r any s i n g l e year c l a s s w i l l be assumed to be c o n s i s t e n t from one fou r year p e r i o d to another. In terms of the production f u n c t i o n t h i s c y c l i c a l v a r i a t i o n w i l l be estimated through the use of four year s l o p e and i n t e r c e p t dummies. The l a t t e r should a l s o p i c k up changes i n gear type g a u n t l e t 109 position due to harvesting for pinks rather than sockeye. Therefore, dummies i n intercept and slope form f o r biomass, numbers and weight were used i n cycles of two years, two/two years (two good followed by two bad), four years and eight years. Cycles of odd numbered years were t r i e d with l i t t l e success. . Returning to the f i r s t step, the adjustment of recruitment for biomass a v a i l a b i l i t y , the following procedure was used. For each year of the twenty-five year period, known biomass recruitment of sockeye was used as the available biomass f o r zone one. The biomass for zone two was simply the biomass for zone one minus the actual h i s t o r i c a l catch of sockeye in zone one. I t was assumed that within each zone there was no serious gear interference between g i l l n e t s and purse seines: the IPSFC opens areas for d i f f e r e n t gear at di f f e r e n t times, The biomass for zone three was the zone two biomass minus the g i l l n e t , purse seine and Sooke trap catches in zone two. (As the f i s h traps in Sooke Harbour only operated for seven years between 1951 and 1958, no production function was estimated for them. However, the i r catch must be subtracted from zone two biomass to give the true biomass in zone three.) This biomass calculation routine was followed for the subsequent zones. According to the IPSFC, the catch i n Johnstone S t r a i t consists of biomass which separates i t s e l f from the main body of sockeye before entering the gauntlet and swims through waters north of Vancouver Island i n order to reach Johnstone S t r a i t . In most recent years the Johnstone S t r a i t catch represents less than 5% of the t o t a l sockeye catch. But i n some years i t has 110 reached as high as 33%. For this reason i t i s important to discover just how the Johnstone S t r a i t sockeye got there. Tests were performed with the production functions to see i f i n c l u s i o n or exclusion of Johnstone S t r a i t catch - i n the biomass of sockeye made s i g n i f i c a n t changes. In each test i t became clear that exclusion of the Johnstone S t r a i t catch from the bicmass from every zone except zone seven led to s i g n i f i c a n t l y poorer r e s u l t s . These tests would support the hypothesis that sockeye caught i n Johnstone S t r a i t do not detach themselves from the main body of sockeye u n t i l they have passed through each zone i n the gauntlet with the exception of zone seven (mouth of the Fraser River)• That i s , they appear to reach Johnstone S t r a i t by swimming through the gauntlet waters south of Vancouver Island. There appears to be some support f o r this hypothesis i n a recent study by Anderson (1976) . For some reason the sockeye appear to swim past the mouth of the Fraser River and into Johnstone S t r a i t . The b i o l o g i c a l reason could be connected with the fact that the years of largest Johnstone S t r a i t catch are also years when the recruitment of sockeye i s exceptionally large. A v a i l a b i l i t y of sockeye must depend primarily on the biomass of sockeye migrating through the gauntlet. However, for gear types l i k e t r o l l , g i l l n e t and reefnet, the actual numbers of f i s h rather than the biomass weight are probably more meaningful.,An estimate for available recruitment i n numbers for each year was obtained by dividing available biomass by the average weight of sockeye f o r that year. The reason numbers may be important i s that g i l l n e t s catch salmon through entanglement 111 of the g i l l s i n the net. For parse s e i n e gear the biomass i s l i k e l y to be of g r e a t e r importance because the f i s h are caught i n b l o c k s r a t h e r than i n d i v i d u a l l y . , What the parameter e s t i m a t i o n showed, however, was t h a t a l l gear o p e r a t i n g l a t e r i n the g a u n t l e t were more s e n s i t i v e to numbers than gear o p e r a t i n g i n e a r l i e r p a r t s of the g a u n t l e t . T h i s was supported by the f a c t t h a t i n years of poor rec r u i t m e n t (fewer blocks) almost a l l gear types were more s e n s i t i v e t o numbers; whereas i n years of heavy r e c r u i t m e n t almost a l l gear types were more s e n s i t i v e to the biomass form. T h e r e f o r e , t h i s f u r t h e r adjustment to the biomass v a r i a b l e appears to be j u s t i f i e d . (4) The H a r v e s t i n g P r o d u c t i o n F u n c t i o n s Given e f f o r t , d e r i v e d from d e l i v e r i e s and adjusted f o r t e c h n i c a l change, and r e c r u i t m e n t , a d j u s t e d f o r catches i n p r e v i o u s zones and i n c y c l e year dummy form to allow f o r the i n f l u e n c e of c a t c h a b i l i t y , the r e s u l t s f o r the s i x major gear type areas are as f o l l o w s (with R-sguared i n brackets a f t e r the dependent v a r i a b l e and t s t a t i s t i c i n b r a c k e t s a f t e r the r e l e v a n t explanatory v a r i a b l e ) : (3-5) lnC22G»S (.82) = -12.95 (2.3) + 1.3lnCZ2GND (7.7) • 1.14lnD4lCZ2N(3. 1) +1. 18lnD42CZ28 (3. 1) + 1. 151nD43CZ2N(3. 1) >1.04lnD44CZ2B (3. 3) (3-6) lnCZ2PSS ( .7)= -20.01 (3 .7) 51lnCZ2PSD (2. 1) + 1.79lnCZ2B(5.2) (3-7) lnUSZ4GNS (.81) = .781n0SZ4GND (6.0) +.42lnD41USZ4B (6. 0) «-.481nD42USZ4N (6. 5) +.421nD4 30SZ4B(6. 2) +.4 11nD44USZ4B (6. 1) (3-8) lnDSZ4PSS{.89) = .681nOSZ4PSD (3. 7) + . 6lnD410SZ4N (5. 9) • . 6 1lnD42USZ4N (6. 6) +. 541nD430SZ4B (6. 1) +. 551nD44USZ4B (6.5) 112 (3-9) 1-USZ5PSS (.88) = .561nUSZ5PSD (2.7) +.6 11nD4lOSZ4B (5. 8) +.681nD42USZ4N (6.2) +.6 1lnD43USZ4B (-5. 9) + .621nD44USZ4B (6. 3) (3-10) lnCZ7GNS (.70) = .791nCZ7GND (3. 5) +. 42D41CZ7B (2.9) + . 48D42CZ7N (2 . 9) +.43D43CZ7B (3.0) +. 43D44CZ7B (3. |) Where GN = g i l l n e t PS = purse s e i n e s u f f i x S = sockeye c a t c h i n weight s u f f i x B = sockeye r e c r u i t m e n t i n weight a f t e r d eduction of previous zones' catches s u f f i x N = sockeye r e c r u i t m e n t i n numbers a f t e r d eduction of previous zones' c a t c h s u f f i x D = e f f o r t i n 1951 e q u i v a l e n t days of f i s h i n g s u f f i x W = average c y c l e weight of sockeye p r e f i x D = v a r i a b l e i s i n s l o p e dummy form 41,2,3,4 = f o u r year c y c l e with the weight i n the f i r s t , second, t h i r d or f o u r t h year of c y c l e p r e f i x CZ = Canadian zone p r e f i x USZ = U.S. .zone These are the parameter e s t i m a t e s which w i l l be used i n the co s t programming to determine the l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n . Note that only the f i r s t two gear type areas have i n t e r c e p t s and th a t none of the gear type areas are s e n s i t i v e to the c y c l e i n f l u e n c e i n dummy i n t e r c e p t form. /However, except f o r zone two purse s e i n e s , a l l gear type areas appear t o be s e n s i t i v e t o the c y c l e i n f l u e n c e i n slope form. I t i s a l s o i n t e r e s t i n g t o note that i n c y c l e year two (D42) almost a l l gear type areas a re more s e n s i t i v e to recruitment i n numbers form, whereas i n c y c l e year f o u r they a re a l l s e n s i t i v e t o rec r u i t m e n t i n biomass form. T h i s i s reasonable as c y c l e year two has a rec r u i t m e n t one quarter the s i z e of t h a t f o r c y c l e year f o u r . That i s , the f i s h tend t o migrate i n blocks i n the f o u r t h c y c l e year because of the l a r g e numbers but tend t o swim i n d i v i d u a l l y i n c y c l e year two. The f i n a l p o i n t of i n t e r e s t i s ,the comparison between 1 1 3 r e c r u i t m e n t e l a s t i c i t i e s . The l o g c o e f f i c i e n t s f o r r e c r u i t m e n t i n both biomass and numbers form i s much l a r g e r f o r the open water gear types i n zone two than f o r the c o n f i n e d water gear types i n zones f o u r , f i v e and seven. W i t h i n these two c a t e g o r i e s , purse s e i n e recruitment e l a s t i c i t i e s are much l a r g e r than those f o r g i l l n e t s . That i s , purse s e i n e s , i n g e n e r a l , are f a r more s e n s i t i v e to recruitment than g i l l n e t s . T h i s i m p l i e s t h a t u n i t e f f o r t c o s t s drop r a p i d l y f o r purse s e i n e s , compared with g i l l n e t s , when l a r g e r e c r u i t m e n t i s a v a i l a b l e . T h i s same c o n c l u s i o n a l s o a p p l i e s to the comparison between open water and c o n f i n e d water gear types. The c y c l e i n f l u e n c e can be seen more c l e a r l y by breaking the h a r v e s t i n g production f u n c t i o n s f o r Canadian zone two g i l l n e t s i n t o f o u r c y c l e years, i n v e r t i n g the f u n c t i o n t o put i t i n terms of d e l i v e r i e s r e g u i r e d to l a n d a c a t c h of sockeye given a c e r t a i n r e c r u i t m e n t i n biomass or numbers term, and t a k i n g the an t i l o g s . (3-5*) CZ2GND = 288310. 9 (CZ2GNS - 9 T) D41Z2G»-*.n D"2Z2GN-».»* D43Z2GN-»,i* D44Z2GB-*>oi Note t h a t e f f o r t i n each c y c l e year i s d i r e c t l y r e l a t e d to the sockeye c a t c h i n the same way, 288310.9(CZ2GNS » 9 7 ) . However, e f f o r t i n each c y c l e year i s i n v e r s e l y r e l a t e d to r e c r u i t m e n t a v a i l a b i l i t y i n a d i f f e r e n t way f o r each c y c l e year. For the f i r s t t h r e e years, recruitment i n the form of numbers i s used, whereas for the f o u r t h c y c l e year i t appears as biomass. Furthermore, the negative exponents are d i f f e r e n t f o r each year of the f o u r year c y c l e . These d i f f e r e n c e s r e f l e c t the c a t c h a b i l i t y v a r y i n g from c y c l e year to c y c l e year. 114 The major c o n c l u s i o n which can be drawn a f t e r working with the p r o d u c t i o n f u n c t i o n s and the c a t c h a b i l i t y exponents i s that the way sockeye swim, t h e i r s c h o o l i n g behavior, the areas i n which they p r e f e r to swim and the routes they take during m i g r a t i o n are c l o s e l y r e l a t e d t o the year c l a s s i n which they were spawned. Thus mature sockeye not only r e t u r n to the same n a t a l stream to spawn, they appear to take the same rou t e as t h e i r p a r e n t s to get t h e r e . I t i s i n t e r e s t i n g t o note that f o r zone two purse s e i n e s , biomass i s a more powerful e x p l a n a t o r y v a r i a b l e than the dummy v e r s i o n s i n g e n e r a l . From 1951 to 1958, there were salmon t r a p s o p e r a t i n g i n Sooke Harbour and the c a t c h and e f f o r t f o r t r a p s and purse s e i n e s were repo r t e d i n aggregated form. As e x p l a i n e d i n the s e c t i o n of the Appendix d e a l i n g with the c o m p i l a t i o n of c a t c h and e f f o r t data, an i n t e r p o l a t i o n r o u t i n e was d e v i s e d t o t r y and separate the two gear types from each other but there may have been s e r i o u s e r r o r s i n v o l v e d l e a d i n g t o d i s t o r t i o n s i n the e f f o r t and c a t c h v a r i a b l e s which may have prevented c a t c h a b i l i t y from being estimated i n a s i g n i f i c a n t way. For f i f t e e n out o f the s i x t e e n gear type areas, c a t c h a b i l i t y of sockeye was a s i g n i f i c a n t improvement over the use of biomass. As the f i t s were so good f o r key areas such as zone f o u r g i l l n e t s and purse s e i n e s , zone f i v e purse s e i n e s and zone seven g i l l n e t s , the hypothesis t h a t biomass a v a i l a b i l i t y should be a d j u s t e d to r e f l e c t c a t c h a b i l i t y would appear to be supported. The use of biomass adj u s t e d f o r c a t c h a b i l i t y appears to be more important than unadjusted biomass i n e x p l a i n i n g c a t c h of sockeye f o r almost a l l of the s i x t e e n gear type a r e a s which 115 have accounted f o r 95% of the sockeye c a t c h over the l a s t t w e n t y - f i v e years. I l l THE HARVESTING COST FONCTIQNS JJLL Production F u n c t i o n -Conversion In order t o form a p r o f i t f u n c t i o n f o r use i n the C l a r k -Munro e q u i l i b r i u m equation, the p r o d u c t i o n f u n c t i o n s must be converted i n t o h a r v e s t i n g c o s t f u n c t i o n s . P r o f i t i s simply t o t a l revenue minus t o t a l c o s t s . And t o t a l revenue i s given by the c a t c h of sockeye times the p r i c e per pound. T o t a l c o s t s are given by the sum of the c o s t s a s s o c i a t e d with t a k i n g a s p e c i f i e d c a t c h o f sockeye i n each gear type area given a c e r t a i n r ecruitment to each f i s h i n g zone. ; As demonstrated i n e x p r e s s i o n (3-5*) above, c o s t f u n c t i o n s are d e r i v e d from the production f u n c t i o n s by i n v e r t i n g the f u n c t i o n so t h a t d e l i v e r i e s are some f u n c t i o n of c a t c h and biomass a v a i l a b i l i t y , t a k i n g the a n t i l o g s and m u l t i p l y i n g by gear type c o s t s . Taking the f i r s t c y c l e years only f o r the p r o d u c t i o n f u n c t i o n s i n e x p r e s s i o n s (3-5) to (3-10) we g e t , (3-11) CZ2GND = 44.83(288310.9) (CZ2GNS« 9 7) (CZ2N - 1 ** 1) (3-12) CZ2PSD = 240.55(1.04) (CZ2PSS 1* 9 6) (CZ2B"*3 * 5) (3-13) USZ4GND = 37. 07 (USZ4GNS1 • z a ) (0SZ48--54) (3-14) 0SZ4PSD = 179.87 (0SZ4PSS 1 • * 6) (0SZ4N-- 8 8) (3-15) 0SZ5PSD = 179.87 (DSZ5PSS 1 » 7 9 ) (USZ5B-*.° 9) (3-16) CZ7GND = 37.07 (CZ7GNS * • 2 7 ) (CZ7B-. S») The f i r s t term on the r.h.s. of each expression i s the u n i t c o s t 116 a s s o c i a t e d with a s i n g l e 1951 e g u i v a l e n t day of e f f o r t f o r t h a t gear type area. Note how much h i g h e r u n i t e f f o r t c o s t s are f o r open water g i l l n e t s and s e i n e s (CZ2GN and CZ2PS) as compared with c o n f i n e d water g i l l n e t s and purse s e i n e s (L7SZ5PS and CZ7GN). GNS and PSS r e f e r t o sockeye catches ( i n pounds) made by g i l l n e t s and purse s e i n e s r e s p e c t i v e l y . N r e f e r s to r e c r u i t m e n t i n numbers while B r e f e r s t o r e c r u i t m e n t i n biomass form. T o t a l c o s t f o r c y c l e year one i s simply the sum of (3-11) through (3-16) as determined by the program. That i s , r i g h t hand s i d e s of a l l s i x equations are summed: while the program a l l o c a t e s catch and r e c r u i t m e n t t o each area f o r the c a t c h and r e c r u i t m e n t values. a s i m i l a r format i s f o l l o w e d f o r the other c y c l e years using d i f f e r e n t e f f o r t c o s t s and year c l a s s c a t c h a b i l i t y exponents. The use of c y c l e dummies permits the f o u r year c l a s s e s of sockeye to be s p l i t i n t o f o u r separate f i s h e r i e s so t h a t each year c l a s s can be t r e a t e d as a s t o c k s u b j e c t e d to h a r v e s t i n g every f o u r years i n the c y c l e . Mathematical programming must be used to i s o l a t e the lowest cost gear type h a r v e s t i n g areas because no other technique can vary h a r v e s t i n g c o n f i g u r a t i o n s as many times as i s necessary t o f i n d the cheapest way to h a r v e s t . The c o s t f u n c t i o n s g i v e n i n e x p r e s s i o n s (3-11) to (3-16) are n o n l i n e a r which means th a t n o n l i n e a r programming techniques are r e g u i r e d t o solve f o r minimum c o s t . The l o g c o e f f i c i e n t s i n e x p r e s s i o n s (3-5) to (3-10) have become the exponents when i n a n t i - l o g form i n e x p r e s s i o n s (3-11) to (3-16). T h e i r s i z e f o r a l l gear type areas imply t h a t these Cobb-Douglas p r o d u c t i o n f u n c t i o n s are not l i n e a r l y homogeneous 117 but d i s p l a y i n c r e a s i n g r e t u r n s to s c a l e , these s c a l e e f f e c t s vary a c c o r d i n g t o c y c l e year because the c a t c h a b i l i t y exponents vary, although the e f f o r t exponents remain the same (expressions (3-5) to (3-10)). The only area f o r which t h i s i s not t r u e i s Canadian zone two purse s e i n e s which i s not s e n s i t i v e t o c y c l e year c a t c h a b i l i t y . , The stock o f f i s h , i e : r e c r u i t m e n t , i s a f r e e i n p u t to the h a r v e s t i n g process, whereas e f f o r t bears a c o s t . The co s t minimization program does not take advantage of these s c a l e economies so much by a l t e r i n g s c a l e ( i n p u t s are changed p r o p o r t i o n a t e l y ) as i t does by a l t e r i n g the flow of r e c r u i t m e n t . That i s , some areas, such as Canadian zone two g i l l n e t s , have very high r e c r u i t m e n t e l a s t i c i t i e s but these are o f f s e t by very high e f f o r t e l a s t i c i t i e s (expression (3-5)). T h e r e f o r e , the program ta k e s advantage of the f r e e i n p u t (recruitment) only t o the extent t h a t i t can o f f s e t the c o s t l y i n p u t ( e f f o r t ) and no s i n g l e area w i l l be chosen as the most c o s t e f f i c i e n t . I nstead, a combination of areas w i l l be used t o e x p l o i t the recru i t m e n t e l a s t i c i t i e s . In other words, d i r e c t i n g more and more of the rec r u i t m e n t i n t o one area i m p l i e s a l a r g e r c a t c h must be taken by t h a t area (otherwise i t w i l l have to be taken by another area r e c e i v i n g l e s s and l e s s of the r e c r u i t m e n t ) . . T h i s , i n t u r n , i m p l i e s t h a t more e f f o r t i s needed and e f f o r t c o s t s money. One problem a s s o c i a t e d with the c o s t programming f o r h a r v e s t i n g o t h e r salmon as w e l l as sockeye i s t h e r e c o g n i t i o n of the o p p o r t u n i t y c o s t of the other salmon c a t c h foregone f o r the i n c r e a s e d sockeye c a t c h . . I t w i l l be assumed t h a t the only h a r v e s t i n g of concern i s of sockeye salmon, ftny other salmon 118 caught at the same time s i l l reduce the costs of harvesting sockeye but no attempt w i l l be made to deliberately harvest other salmon. Catch of other salmon w i l l be permitted up to the amount of excess capacity l e f t after harvesting sockeye. This excess capacity w i l l be the average (1951 to 1975) cycle weight per unit of gear type e f f o r t (for a l l salmon) minus the weight of sockeye harvested per unit of e f f o r t . Given excess capacity per delivery, other salmon w i l l be permitted to be caught up t o the average (1951 to 1975) cycle weight of other salmon per delivery. One or the other of these capacity constraints w i l l be binding or the maximum permissable cycle year catch of other salmon for the fishery w i l l be binding, whichever of the three i s smaller. One problem which arises out of t h i s method of harvesting other salmon i s that no d i s t i n c t i o n i s made between the four species of other salmon. Each species of salmon has a d i f f e r e n t price per pound with the highest price twice as large as the smallest, ft compromise adjustment was made; of the f i v e races of salmon: sockeye, chinook, chum, coho and pink, sockeye has t r a d i t i o n a l l y received the higher price per pound. Chinook and coho are worth about 80% to 90% of the sockeye price and chum and pink about 45% to 67%. The d i f f e r e n t i a l s have narrowed over the period 1951 to 1975 and are s l i g h t l y larger for Washington than for B.C. The p r o p o r t i o n of the races of other salmon caught depends on the gear type area. Just as i t i s impossible to do other than guess with regard to the displacement e f f e c t , i t i s impossible to know how catch proportions would change with the altered patterns of harvesting sockeye. However, the e f f e c t of 119 the pink harvest i n a l t e r n a t i n g years i s so overwhelming t h a t i t i s p o s s i b l e t c at l e a s t d i s t i n g u i s h between the cat c h value of other salmon i n a l t e r n a t i n g y e a r s . The f o l l o w i n g compromise was used t o r e f l e c t both the narrowing of d i f f e r e n t i a l s over the years and the a l t e r e d p a t t e r n s of c a t c h i n g other salmon. In " o f f " years f o r the pink, other salmon are valued a t 67% of the sockeye p r i c e per pound and i n "on" years f o r the pink, other salmon are valued at 50% of the sockeye p r i c e per pound. These r a t h e r c o n s e r v a t i v e estimates w i l l tend t o understate the impact of other salmon h a r v e s t i n g i n the net p r o f i t comparisons below. As e x p l a i n e d i n Chapter One, a l l f i x e d c o s t s are converted i n t o v a r i a b l e c o s t s . Thus f i x e d c o s t s appear i n an amortized way egual to the d e p r e c i a t i o n p l u s the opportunity c o s t s o c i a l r a t e of r e t u r n on the money i n v e s t e d i n the c a p i t a l eguipment. There are only a few c o s t o b s e r v a t i o n s a v a i l a b l e f o r the f u l l twenty-f i v e year p e r i o d under study and much b r i d g i n g and i n t e r p o l a t i o n was r e q u i r e d t o b u i l d c o n s i s t e n t indexes o f gear type c o s t s . The only c o n s o l a t i o n i s t h a t d e s p i t e r a d i c a l l y d i f f e r e n t parameters f o r the pr o d u c t i o n f u n c t i o n s , the c o s t programming i n d i c a t e d t h a t the s i x major gear type areas are very c o m p e t i t i v e i n terms of c o s t s of h a r v e s t i n g . No s i n g l e area stands out as being very high c o s t or very low c o s t . 120 J i l l L e j s t Cost Programming Flowchart ( 1 - 1 ) i n Chapter One introduced the i d e a of the programming needed to o b t a i n o p t i m a l v a l u e s f o r escapement, recruitment and c a t c h and the accompanying l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n to take t h a t c a t c h given the re c r u i t m e n t . The programming i n t h i s chapter r e l a t e s s t r i c t l y t o the choosing of the l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n given s p e c i f i e d l e v e l s of escapement, r e c r u i t m e n t and c a t c h . No attempt i s being made to choose optimal values. In f a c t , the s p e c i f i e d l e v e l s are simply the average year c y c l e h i s t o r i c a l t o t a l s f o r escapement, re c r u i t m e n t and c a t c h which are to be a l l o c a t e d to each gear type h a r v e s t i n g area by the program. The purpose of p r e s e n t i n g the l e a s t c o s t program at t h i s p o i n t i s simply to make e x p l i c i t the nature of the problem i n v o l v e d i n choosing l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n s . Furthermore, the r e s u l t s are a very u s e f u l guide t o the gains t o be achieved from o p t i m a l s p a t i a l a l l o c a t i o n of h a r v e s t i n g gear. The l e a s t c o s t programming r e s u l t s give us some idea of the rent l o s s i n c u r r e d by the s p a t i a l m i s a l l o c a t i o n a r i s i n g from the common property problem i n an open access f i s h e r y . That i s , fishermen do not go to the area where c a t c h a b i l i t y i s g r e a t e s t f o r t h e i r gear type, given a v a i l a b i l i t y ; i n s t e a d , they migrate to e a r l i e r p o r t i o n s of the g a u n t l e t t o har v e s t i n areas where rec r u i t m e n t a v a i l a b i l i t y i s g r e a t e r because c a t c h has not reduced a v a i l a b i l i t y so much i n previous zones but where c a t c h a b i l i t y may be poor f o r t h e i r gear type. By e l i m i n a t i n g t h i s common property problem, the l e a s t c ost program can then a l l o c a t e catch and recru i t m e n t so as to take advantage of the 121 technical e f f i c i e n c y of d i f f e r e n t gear types and th e i r s e n s i t i v i t y to recruitment and area c a t c h a b i l i t y . The format used f o r laying out the programming problem was taken from Beale (1968) and Pfaffenberger and Walker (1976, pp. 3-13 3). Bather than program f o r each of the twenty-five years from 1951 to 1975, a representative average cycle year f o r each of the four cycle years was used instead. The study i s concerned with finding the lowest cost gear type area on average f o r each cycle year. As some gear type areas were not exploited u n t i l the 1960's and as there was a process of migration to e a r l i e r areas of the gauntlet throughout the twenty-five years, the average cycle year should give a better r e f l e c t i o n of costs of harvesting. Flowchart (3-1) i l l u s t r a t e s the rather complex nature of the l e a s t cost program. The program takes the average recruitment for the cycle year and allows i t to flow through to zone two. The recruitment observations are i n biomass form (weight times numbers) but as some areas are more sensitive to numbers, a routine i s included which transforms biomass into numbers using the average cycle year weight of sockeye. The program then subtracts the gear type sockeye catches (i f there are any) i n zone two from the recruitment and allows t h i s reduced recruitment to flow through to zone four. The same procedure as above i s followed for zones four and five and the remaining recruitment flows through to zone seven a f t e r the average cycle year Johnstone S t r a i t catch i s subtracted. After recruitment i s further reduced by gear type catch in zone seven, the reduced recruitment i s permitted to flow through as average ( j i u . N e . - n T w o C a r K H e s RCOtUtT—.-»T ' STRAIT £»«««»/ tout se^ e~ C—TCM«f -,o>oe TUJO / / , / 1 SoCK-se CATCH es Foue / / , HeDuct* / eft foe {t\lV\Jl T0N& I Flit A Cwt-H Flowchart (3-1): Least Cost Program to Harvest Sockeye Assuming Average Historical Cycle Recruitment, Catch and Escapement. _ o w _ T w o T l « t CO.T4 zowe. * Dt % e k t » c P. ICS CO»TS N P T W I C (tibia*) T > » - i v - - l « < T t M * » C*a« T s p t C o v r t 123 c y c l e y e a r Indian c a t c h and escapement up the F r a s e r B i v e r . With the escapement, r e c r u i t m e n t and c a t c h c o n s t r a i n t s s a t i s f i e d , the va r i o u s unknowns i n the t o t a l c o s t f u n c t i o n w i l l then be e s t a b l i s h e d {see e x p r e s s i o n s (3-11) to (3-17)). The program then checks the p a r t i a l d e r i v a t i v e s o f the c o s t f u n c t i o n to determine i f , i n f a c t , a minimum has been achieved. I f not, the program r e - s h u f f l e s the c a t c h and re c r u i t m e n t a l l o t m e n t s a c c o r d i n g to a penalty f u n c t i o n a l g o r i t h m based on the g r a d i e n t s of those p a r t i a l d e r i v a t i v e s , a f t e r a number o f i t e r a t i o n s , the c o n f i g u a t i o n w i l l indeed give the minimum c o s t , a few t e s t s were performed to ensure that a g l o b a l minimum r e s u l t e d but as the catch p r o p o r t i o n s changed so r a d i c a l l y from one i t e r a t i o n to another, i t i s probably safe to assume the s o l u t i o n s are indeed g l o b a l . Turning t o Flowchart (3-2) i n which both sockeye and other salmon are caught, the same average c y c l e year v a l u e s were used as above with the a d d i t i o n of the average c y c l e year c a t c h of other salmon, gear type c a p a c i t y f o r t o t a l c a t c h and gear type c a p a c i t y f o r other salmon. These c a p a c i t i e s are based on a l i n e a r t r a n s f o r m a t i o n of the average c y c l e year c a t c h , by weight, of a l l salmon per u n i t of e f f o r t f o r the f i r s t and other salmon only per u n i t of e f f o r t f o r the second. The program takes the r e g u i r e d gear type d e l i v e r i e s t o land the proposed c a t c h of sockeye, determines the a v a i l a b l e c a p a c i t y and compares i t with the average c y c l e year gear type c a p a c i t y f o r c a t c h i n g a l l salmon. I f the proposed c a t c h of sockeye exceeds the c a p a c i t y , i n terms of weight of salmon per d e l i v e r y , no other salmon are permitted t o be caught. I f the proposed 124 Flowchart (3-2): Least Cost Program to Harvest Sockeye and Other Salmon Assuming Average Historical Cycle Recruitment, Catch and Escapement. K u C TUlo f i M i i Sen*. Ti« f*VTl MtW«» OTH«< i C FOUR C O B T t ffl AMI t©Twe«. ea*™ M"*« «•"<** COT-CM tC«H*wtT£% COST1 CPrre H CcCTS ^ I n r f t C^Mtt J 1 c STOP 125 c a t c h of sockeye i s l e s s than the c a p a c i t y i n terms of weight of salmon per d e l i v e r y , other salmon are permitted to be taken. The second c a p a c i t y c o n s t r a i n t then comes i n t o e f f e c t . Other salmon are permitted t o be caught up to e i t h e r : a) the t o t a l salmon gear type catch c a p a c i t y (the f i r s t c a p a c i t y c o n s t r a i n t ) or b) the other salmon gear type c a t c h c a p a c i t y (the second c a p a c i t y c o n s t r a i n t ) . In the event t h a t both these c a p a c i t i e s are very l a r g e , a t h i r d c o n s t r a i n t equal to the maximum average c y c l e year c a t c h of other salmon l i m i t s the t o t a l catch of other salmon landed by any s i n g l e gear type and by a l l combined. Thus f o r each gear type area, the c a t c h of other salmon i s l i m i t e d by one or other of the c a p a c i t y c o n s t r a i n t s ; and a l l gear type catches of o t h e r salmon combined are l i m i t e d by the t h i r d c o n s t r a i n t . Once again, the program always e v a l u a t e s the p a r t i a l d e r i v a t i v e s of the c o s t f u n c t i o n a f t e r each i t e r a t i o n to determine i f a minimum has been reached. Without a j o i n t production f u n c t i o n f o r c a t c h i n g both sockeye and other salmon, the approximating technique o u t l i n e d above cannot be used with c o n f i d e n c e . For t h i s reason, the main c r i t e r i o n used f o r choosing the l e a s t cost/most p r o f i t a b l e gear type areas w i l l depend only on the r e s u l t s f o r c a t c h i n g sockeye. However, an example showing the economic r e n t earned by h a r v e s t i n g f o r both sockeye and other salmon w i l l be i n c l u d e d f o r comparison purposes. .. I f the t o t a l c o s t f u n c t i o n s were l i n e a r with given or f i x e d r e c r u i t m e n t , escapement and c a t c h , the c o s t program would choose a s i n g l e area t o harvest the t o t a l c a t c h . With constant marginal o p p o r t u n i t y c o s t s , one area would always be l e a s t c o s t . , T h i s 126 l e a s t c o s t f u n c t i o n could then be converted i n t o a p r o f i t f u n c t i o n by s u b t r a c t i n g c o s t s from the value of the c a t c h . T h i s p r o f i t f u n c t i o n c o u l d then be used with the recruitment f u n c t i o n i n the l i n e a r c o s t v e r s i o n of C l a r k and Munro's optimal equation (Clark, 1976, pp.243-245). The l a t t e r would give the o p t i m a l escapement which c o u l d then be used with the r e c r u i t m e n t f u n c t i o n to g i v e optimal r e c r u i t m e n t . The former would then be s u b t r a c t e d from the l a t t e r t o g i v e o p t imal c a t c h . But note that the c a t c h and the biomass a v a i l a b i l i t y o b t a i n e d from the o p t i m i z i n g process would be d i f f e r e n t from those used to choose the l e a s t c o s t area. From e x p r e s s i o n s (3-11) to (3-16) i t i s obvious that t h i s c o u l d have an important impact on which area was the l e a s t c o s t because some areas are very s e n s i t i v e to biomass a v a i l a b i l i t y while others are not. T h e r e f o r e , a s m a l l e r or l a r g e r c a t c h (reduced or i n c r e a s e d cost) a s s o c i a t e d with a s m a l l e r or l a r g e r recruitment ( i n c r e a s e d or reduced cost) c o u l d cause changes i n the r e l a t i v e c o s t s of h a r v e s t i n g . I t i s probable t h a t the changes f o r l i n e a r cost f u n c t i o n s would not be so r a d i c a l as to cause the program t o choose a d i f f e r e n t area as t h e new l e a s t c o s t h a r v e s t i n g area. However, with n o n l i n e a r c o s t f u n c t i o n s recruitment e l a s t i c i t i e s can play a l a r g e r o l e i n determining the l e a s t cost area. C e r t a i n gear type areas w i l l be cheap to use f o r h a r v e s t i n g up to a c e r t a i n p o i n t a f t e r which i t may be cheaper to h a r v e s t the next increment i n another gear type area which has not been used b e f o r e . One would expect, t h e r e f o r e , t h a t the l e a s t c o s t program wculd choose t o h a r v e s t any g i v e n c a t c h using most o f the s i x gear type areas s i m u l t a n e o u s l y . With r a r e 127 e x c e p t i o n , t h i s i s e x a c t l y what happened when t h e c o s t program was a p p l i e d using average c y c l e r e c r u i t m e n t , c a t c h and escapement. T h i s approach means t h a t the op t i m a l s o l u t i o n depends c r i t i c a l l y on what areas are being used and i n what p r o p o r t i o n s . Using the n o n l i n e a r v e r s i o n of C l a r k and Kunro's eguation (Clark, 1976, pp.252-253) the optimal escapement, and subseguently the optimal r e c r u i t m e n t and c a t c h , can be c a l c u l a t e d . But the changes i n recruitment and c a t c h which r e s u l t from o b t a i n i n g the e g u i l i b r i u m values f o r the o p t i m a l s o l u t i o n may have a profound e f f e c t on the l e a s t c o s t way of h a r v e s t i n g . C o s t s move d i r e c t l y with catch and i n v e r s e l y with recruitment; and as the c a t c h / r e c r u i t m e n t p r o p o r t i o n s are changed, the s o l u t i o n i s bound to be d i f f e r e n t . For t h i s reason i t i s important t h a t l e a s t c o s t gear type areas and the optimal escapement, recruitment and c a t c h be determined s i m u l t a n e o u s l y . JJ1 J e t P r o f i t Comparisons The f i r s t net p r o f i t comparison permits the cost program t o choose the l e a s t c o s t h a r v e s t i n g p r o p o r t i o n s given average c y c l e year c a t c h of sockeye (no other salmon), r e c r u i t m e n t , escapement, Johnstone S t r a i t c a t c h and Indian c a t c h . 128 TABLE J3j__l__ UNCONSTBAINED AVEBAGE HISTOBICAL CY.CLE CATCH OF SOCKEYE ONLY IN OOP'S OF POUNDS, PBOPOBTIONS OF GEAR TYPE ABEA CATCHES AND RESULTING COSTS AND PROFITS IN 00 0J.S OF 1951 DOLLARS CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 CATCH 18,359 11,717 19,380 34,930 CZ2G 60% 0% 201 55% CZ2P 8% 2% 6% 30% USZ4G 5% 48% 15% 0% US Z4P 7% 17% 9% 5% USZ5P 7% 7% , 10% 3% CZ7G 13% 26% 40% 7% VALUE $6,294 $4,173 $6,993 $11,916 COST $5,536 $3,161 $4,800 $3,965 NET PROFIT $758 $1,012 $2,193 $7,952 ACTUAL » $246 $466 $1,145 $5,748 GAIN IN " $512 $546 $1,047 $2,203 Note t h a t "Net P r o f i t " i s economic r e n t a c c o r d i n g to the program's c a l c u l a t i o n s given average h i s t o r i c a l catch and a new a l l o c a t i o n o f h a r v e s t i n g e f f o r t . " A c t u a l " net p r o f i t i s what can be c a l c u l a t e d from given average h i s t o r i c a l c a t c h and gear type e f f o r t . According to these c o s t c a l c u l a t i o n s , fishermen are r e c e i v i n g an economic rent, p a r t i c u l a r l y i n the f o u r t h c y c l e year when economic " r e n t " exceeds the " c o s t " of l a n d i n g the average sockeye c a t c h . The "Gain i n " net p r o f i t i n d i c a t e s how much e x t r a economic r e n t c o u l d have been earned i f h a r v e s t i n g had been l e a s t c o s t . Even i f o p p o r t u n i t y c o s t e s t i m a t e s are too c o n s e r v a t i v e and what i s recorded as a c t u a l net p r o f i t i s r e a l l y p a r t of o p p o r t u n i t y c o s t , f o l l o w i n g a l e a s t c o s t program could have y i e l d e d a f a i r l y s u b s t a n t i a l economic r e n t as recorded i n the gain i n net p r o f i t . T h i s g i v e s some id e a of the r e n t l o s s i n c u r r e d from the common property t r a d e - o f f of a v a i l a b i l i t y f o r c a t c h a b i l i t y . 129 Turning to the changes i n p r o p o r t i o n s of gear type catches, i t i s i n t e r e s t i n g to note how recruitment e l a s t i c i t i e s are so important to open water gear (Canadian zone two) i n the f o u r t h year o f very l a r g e runs, a l t e r n a t i v e l y , the c o n f i n e d water g i l l n e t s (USZ4G,CZ7G) do well i n the second c y c l e year with low r e c r u i t m e n t when there are very few b l o c k s moving through the g a u n t l e t . In a d d i t i o n to wide swings amongst d i f f e r e n t gear t y p e s , i t i s obvious that catches are anything but e q u a l l y d i v i d e d between the O.S and Canada. I t i s h i g h l y u n l i k e l y t h a t the IPSFC would be permitted to h a r v e s t the sockeye catch without d i v i d i n g the harvest e q u a l l y between both c o u n t r i e s . Table (3-2) i l l u s t r a t e s the case f o r an equal d i v i s i o n of c a t c h between O.S. and Canadian fishermen. Comparing the r e s u l t s i n T a b l e s (3-1) and (3-2) i t i s obvious that Munro 1s (1979) plan f o r using the lowest c o s t country f o r h a r v e s t i n g the t o t a l c a t c h and p e n s i o n i n g o f f the highest c o s t country would indeed i n c r e a s e economic r e n t . But what i s s u r p r i s i n g i s t h a t with the e x c e p t i o n of the f o u r t h c y c l e year, the d i f f e r e n c e s i n r e n t are not l a r g e r . Despite l a r g e d i f f e r e n c e s i n a c t u a l c a t c h of sockeye and other salmon and the d i f f e r e n c e s i n c o s t between gear t y p e s , the economic r e n t s earned are very c l o s e . T h i s would c o n f i r m the hypothesis t h a t fishermen are indeed mobile between ar e a s (and p o s s i b l y c o u n t r i e s ) and very p r o f i t c o n s c i o u s . ..Or perhaps, what i s even more important, i s that d e s p i t e severe c o n s t r a i n t s on t h e i r attempts to l i m i t p o t e n t i a l l y d i s a s t r o u s o v e r f i s h i n g , the IPSFC has been able t o a p p o r t i o n gear type catches i n such as way as t o preserve the economic v i a b i l i t y of the f i s h e r y . 130 The second net p r o f i t comparison i s s i m i l a r t o the f i r s t except that the IPSFC i s now c o n s t r a i n e d to an equal d i v i s i o n of the c a t c h . TABLE ( 3 - 2 ) : CONSTRAINED AVERAGE HISTORICAL CYCLE CATCH OF SOCKEYE ONLY IN OOP'S OF POUNDS, PROPORTIONS OF GEAR TYPE AREA CATCHES AND RESULTING COSTS AND PROFITS IN OOP'S OF 1951 DOLLARS CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 CATCH 18,359 11,717 19,380 34,930 CZ2G 27% 0% 13% 17% CZ2P 7% 2% 7% 23% 0SZ4G 26% 29% 28% 6% 0SZ4P 14% 13% 15% 29% USZ5P 10% 8% 6% 15% CZ7G 16% 48% 30% 10% VALUE $6,294 $4,173 $6,993 $11,916 COST $5,692 $3,167 $4,86P $5,131 NET PROFIT $602 $1,006 $2,133 $6,785 ACTUAL " $246 $466 $1,145 $5,748 GAIN IN " $356 $54 0 $988 $1,037 Except f o r c y c l e year two, i t appears t h a t moving i n t o e a r l i e r p a r t s of the g a u n t l e t has made good economic sense f o r Canadian g i l l n e t t e r s moving from zone seven t o zone two. In two out of f o u r c y c l e years, the S t r a i t of Juan de Fuca g i l l n e t t e r s harvest a l a r g e r p r o p o r t i o n of the catch than the g i l l n e t t e r s l e f t i n the mouth of the F r a s e r R i v e r , Once a g a i n , recruitment e l a s t i c i t i e s are very important f o r purse s e i n e s which f a r e very p o o r l y i n c y c l e year two (23% of catch) but are rewarded with l a r g e c a t c h e s i n c y c l e year f o u r (67% of c a t c h ) . I g n o r i n g the catch o f other salmon, t h i s i s the most f e a s i b l e case f o r minimizing the c o s t s of h a r v e s t i n g F r a s e r R i v e r sockeye. I t appears from these h y p o t h e t i c a l examples that open access has r e s u l t e d i n a s i g n i f i c a n t l o s s o f economic rent 131 i n terms of s p a t i a l m i s a l l o c a t i o n of h a r v e s t i n g e f f o r t . The t h i r d and f o u r t h net p r o f i t comparisons permit the cost program to h a r v e s t other salmon as w e l l as sockeye given average c y c l e year c a t c h e s of sockeye and other salmon, r e c r u i t m e n t , escapement, Johnstone S t r a i t c a t c h and Indian c a t c h . Table (3-3) g i v e s the comparisons assuming the IPSFC i s unconstrained by any d i v i s i o n of the c a t c h and Table (3-4) assumes the IPSFC i s c o n s t r a i n e d by an egual catch d i v i s i o n agreement. TABLE J3__3JL_: 0NCQNSTBAINED AVEHAGE HISTOBICAL CYCLE CATCH OF SOCKEYE AND OTHEB SALMON IN 000J.S POUNDS_ PBOPOBTIONS OF GEAR TYPE AREA CATCHES OF SOCKEYE AND BESOTTING PJ_OFITS IN OOO^S OF 1951 DOLLARS CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 SOCKEYE 18,359 11,717 19,380 34,930 ALL OTHEB 42,006 8,540 29,532 10,943 % CATCH " 100% 100% 100% 100% C22G 55% 34% 52% 50% CZ2P 8% 3% 10% 27% 0SZ4G 6% 24% 2% 1% 0SZ4P 7% 11% 4% 9% DSZ5P 9% 6% 3% 10% CZ7G 15% 22% 29% 3% NET PROFIT $7,933 $2,817 $7,385 $10,421 ACTUAL » $6,688 $2,302 $5,733 $7,844 GAIN IN » $1,246 $515 $1,652 $2,578 132 TABLE (3-4): CONSTRAINED AVERAGE HISTORICAL CYCLE CATCH OF SOCKEYE AND OTHER SALMON IN 000'S OF POUNDS^ PROPORTIONS OF GEAR TYPE AREA CATCHES OF SOCKEYE JIND RESULTING PROFITS IN 000J.S OF 1951 DOLLARS CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 SOCKEYE 18,359 11,717 19,380 34,930 ALL OTHER 33,605 8,540 22,740 9,739 % CATCH " 80% 100% 77% 89% CZ2G 36% 37% 20% 37% CZ2P 6% 2% 10% 12% USZ4G 36% 32% 22% 14% USZ4P 3% 13% 18% 24% USZ5P 11% 6% 10% 12% CZ7G 8% 10% 20% 1% NET PROFIT $6,073 $2,802 $6,332 $9,405 ACTUAL " $6,688 $2,302 $5,733 $7,844 GAIN IN $-615 $500 $600 $1,561 Comparing the unconstrained catches of sockeye i n Tables (3-1) and (3-3) r e v e a l s j u s t how important the catch of other salmon i s i n terms of red u c i n g gear type e f f o r t c o s t s . Sockeye c a t c h p r o p o r t i o n s are not too d i f f e r e n t f o r c y c l e years one and f o u r . But note the r a d i c a l s h i f t i n c a t c h p r o p o r t i o n s f o r c y c l e year two. In p a r t i c u l a r , Canadian zone two g i l l n e t s which were not used when c a t c h i n g sockeye only are now used t o the extent o f t a k i n g t h i r t y - s e v e n percent of the sockeye c a t c h when other salmon are permitted t o be caught. A s i m i l a r r a d i c a l change can be detected f o r c y c l e year t h r e e . Comparing the unconst r a i n e d and c o n s t r a i n e d cases of c a t c h i n g both sockeye and other salmon (Tables (3-3) and (3-4)) i t i s c l e a r t h a t there i s a case to be made f o r i n t e r n a t i o n a l b a r g a i n i n g over the cat c h d i v i s i o n s . The most obvious e f f e c t of c o n s t r a i n i n g the sockeye ca t c h i s the r e d u c t i o n i n c a t c h of 133 other salmon i t would be p o t e n t i a l l y p o s s i b l e t o take, The c o n s t r a i n e d r e s u l t s i n d i c a t e that t h e r e i s not enough c a p a c i t y l e f t over from h a r v e s t i n g the s p e c i f i e d sockeye ca t c h to c a t c h the other salmon, whereas the unconstrained program i s a b l e to allow enough c a p a c i t y to land one hundred percent of the p o t e n t i a l c a t c h of other salmon f o r each c y c l e year. I t i s important to remember that the programming system used t o h a r v e s t other salmon i s only an approximation to the r e a l h a r v e s t i n g process. T h e r e f o r e , the p r o f i t comparisons are not very u s e f u l . What i s i n t e r e s t i n g , however, i s that economic r e n t , whether a c t u a l or h y p o t h e t i c a l , i s r e a l l y r a t h e r l a r g e . And one can presume that i n a proper m u l t i - s p e c i e s h a r v e s t i n g program, l e a s t c o s t a l l o c a t i o n of gear would l e a d to even l a r g e r economic r e n t (see f o r example the g a i n i n net p r o f i t i n Table (3-3) where 1005? of the p o t e n t i a l catch of other salmon i s taken). F i n a l l y a comparison of the c o n s t r a i n e d c a t c h p r o p o r t i o n s i n T a bles (3-2) and (3-4) r e v e a l s the importance of Canadian zone two g i l l n e t s i n l a n d i n g other salmon as compared with g i l l n e t s o p e r a t i n g i n the mouth of the Fraser River: o n l y pink salmon out o f the other four s p e c i e s , spawn i n the F r a s e r R i v e r system e x c l u s i v e l y . Table (3-2) i n d i c a t e s t h a t Canadian zone two g i l l n e t s are important i n terms of l e a s t c o s t h a r v e s t i n g of sockeye but Table (3-3) demonstrates t h a t t h i s open water gear type i s v i t a l t c l e a s t c o s t h a r v e s t i n g of o t h e r salmon. In a sense the other salmon case i s always unconstrained because there i s no egual d i v i s i o n of the c a t c h of o t h e r salmon between 0,S. and Canadian fishermen. T h i s i s not e x a c t l y c o r r e c t i n r e a l 134 l i f e as the IPSFC dees d i v i d e the pink salmon c a t c h e q u a l l y . . Rs e x p l a i n e d e a r l i e r , however, no attempt i s made t o de a l with t h i s m u l t i - s p e c i e s h a r v e s t i n g problem except to reduce the c o s t s of c a t c h i n g sockeye salmon. Thus, no p r o v i s i o n was made i n the program to d i v i d e the c a t c h of other salmon e g u a l l y . One c o n c l u s i o n which emerges from a l l the comparisons i s th a t l e a s t c o s t s p a t i a l a l l o c a t i o n of the gear can be p r o f i t a b l e d e s p i t e the c l o s e n e s s i n p r o f i t a b i l i t y of many of the areas. What i s of great i n t e r e s t i s t h a t so l i t t l e e x t r a p r o f i t can be obtained through optimal a l l o c a t i o n o f h a r v e s t i n g e f f o r t . T h i s would support the hypothesis t h a t fishermen have a shrewd i d e a of the lowest c o s t areas to harvest i n terms of biomass a v a i l a b i l i t y and the c a t c h a b i l i t y p e c u l i a r to t h e i r gear types. However, as the IPSFC r e g u l a t e s gear type e f f o r t w i t h i n the c o n s t r a i n t s p r o s c r i b e d by the con v e n t i o n , i t i s more c o r r e c t to say t h a t the IPSFC has been extremely a d r o i t at ta k i n g many of these r e l e v a n t b i o l o g i c a l and economic parameters i n t o account i n l i m i t i n g gear type area h a r v e s t i n g . The l o s s i n p r o f i t i n d i c a t e d by p r o f i t comparisons a r i s e s because of the s p a t i a l m i s a l l o c a t i o n o f e f f o r t r e s u l t i n g from the open access nature of the f i s h e r y . T h i s chapter has taken the a v a i l a b l e e f f o r t data and adj u s t e d them t o r e f l e c t t r u e e f f o r t p l u s changes i n p r o d u c t i v i t y . The a v a i l a b l e r e c r u i t m e n t data have been adusted t o r e f l e c t a v a i l a b i l i t y i n each f i s h i n g zone. The adjusted e f f o r t and the adjusted biomass a v a i l a b i l i t y have then been used t o estimate parameters f o r the s i x major gear type area production f u n c t i o n s a l l o w i n g c a t c h a b i l i t y t o vary by c y c l e year 135 through the use of dummies. These production f u n c t i o n s have been converted i n t o c o s t f u n c t i o n s which i l l u s t r a t e that t o t a l c o s t v a r i e s d i r e c t l y with s i z e of c a t c h but i n v e r s e l y with biomass a v a i l a b i l i t y (depending on the c y c l e c a t c h a b i l i t y parameter) . Because the c o s t f u n c t i o n s are n o n l i n e a r i t was e x p l a i n e d t h a t the c o s t program would choose to a l l o t the a v a i l a b l e c a t c h on a p r o p o r t i o n a t e b a s i s amongst the gear type areas. With s e v e r a l gear type areas h a r v e s t i n g s i m u l t a n e o u s l y i t f o l l o w s t h a t the o p t i m a l escapement, recr u i t m e n t and c a t c h d e r i v e d from C l a r k and Munro's optimal eguation w i l l depend on the c a t c h p r o p o r t i o n s , which i n t u r n depend ( d i r e c t l y on c a t c h and i n v e r s e l y on recruitment) on the o p t i m a l s o l u t i o n . For t h i s reason the optimal program must i n v o l v e a simultaneous s o l u t i o n f o r the l e a s t c o s t h a r v e s t i n g p r o p o r t i o n s and the optimal escapement, r e c r u i t m e n t and c a t c h . However, i t has been p o s s i b l e , using average c y c l e year t o t a l c a t c h and r e c r u i t m e n t , t o compare net p r o f i t o b tained through l e a s t c o s t h a r v e s t i n g with a c t u a l h i s t o r i c a l net p r o f i t : without using optimal values f o r r e c r u i t m e n t and c a t c h . And f i n a l l y , the c o s t program i s permitted t o harvest other salmon to demonstrate how ca t c h p r o p o r t i o n s change when oth e r salmon are harvested as w e l l as sockeye. 136 APPENDIX Compilation Of Catch And E f f o r t Data Data on c a t c h of sockeye, c a t c h of a l l o t h e r salmon s p e c i e s (during the sockeye harvest p e r i o d ) , d e l i v e r i e s and days f i s h i n g are a v a i l a b l e by gear type on a d a i l y b a s i s s i n c e 1951 f o r a l l areas i n the g a u n t l e t . None of these data are a v a i l a b l e i n machine rea d a b l e form and a l l of i t had to be compiled by hand from books a v a i l a b l e a t the Departments of F i s h e r i e s headquarters i n B.C. and Olympia Washington. If the s t a t i s t i c s had been compiled on a d a i l y b a s i s , i t would have i n v o l v e d h a n d l i n g over 200,000 sep a r a t e o b s e r v a t i o n s f o r the f i v e s p e c i e s of salmon plu s d e l i v e r i e s f o r t h i r t y gear type areas f i s h e d f o r an average of three and a h a l f days a week f o r an average of t h i r t e e n weeks a year f o r t w e n t y - f i v e years. The d a i l y data had been aggregated i n t o monthly form and, although t h i s i s l e s s a c c u r a t e , i t reduced the number of o b s e r v a t i o n s to 14,000 f o r the f i v e s p e c i e s of salmon p i n s d e l i v e r i e s f o r the t h i r t y gear type areas f i s h e d f o r three months a year f o r t w e n t y - f i v e years. The Boyce-Bevan p r o j e c t found the monthly aggregates as a c c u r a t e as the d a i l y b a s i s t o t a l s f o r t h e i r study of h a r v e s t i n g . However, the problem with monthly aggregates i s that the t i m i n g of sockeye runs does not conform p r e c i s e l y with the J u l y , August, September schedule. In some ye a r s the runs s t a r t e a r l i e r or run l a t e r ; but f o r the most p a r t these account f o r l e s s than 5% of the t o t a l c a t c h f o r the whole g a u n t l e t . The reason f o r not i n c l u d i n g June and October f o r a l l p a r t s of the g a u n t l e t i s t h a t fishermen are out c a t c h i n g salmon o t h e r than 1 3 7 sockeye; and to include June and October totals for d e l i v e r i e s , sockeye catch and other salmon catch would d i s t o r t the picture for sockeye d e l i v e r i e s . That i s , to include June and October d e l i v e r i e s for a l l parts of the gauntlet would overstate the d e l i v e r i e s necessary to take 95% of the sockeye run i n July, August and September only and overstate the proportion of other salmon to sockeye.. A study was made of the guantlet patterns over the years and the following months were chosen as the best o v e r a l l aggregates to approximate the d e l i v e r i e s needed to take at le a s t 95% of the sockeye catch and the other salmon associated with the sockeye harvest. For the outer gauntlet: June, July, August and September; for the middle of the gauntlet: July, August, and September; for the end of the gauntlet: July, August, September, and October. The data for B.C.,waters were in terms of weight for the Canadian sector of the guantlet. The Washington data were i n terms of numbers of f i s h for the U.S. sector. The average seasonal weight for each species was obtained by dividing the Puget Sound catch i n weight by the Puget Sound numbers for each year as reported i n the State of Washington, Department of F i s h e r i e s , F i s h e r i e s S t a t i s t i c a l Beport for each year. These yearly average weights were then used with numbers of salmon to convert them into salmon catch i n terms of weight for the U.S. sector. , For 1974 and 1975 the Washington data were not available in aggregate form by months f o r major areas and the daily t o t a l s f o r disaggregated areas were aggregated by hand following the 138 J u l y , august, September schedule ( a l l U.S. f i s h i n g takes place i n the middle of the gauntlet) . From 1951 to 1955 i n c l u s i v e , the B.C. .data d i d not d i s t i n g u i s h between f i s h t r a p s i n Sooke harbour and purse s e i n e s o u t s i d e the harbour. An i n t e r p o l a t i o n r o u t i n e was devised based on IPSFC r e p o r t s of the catch f o r the Sooke t r a p s , which s p l i t the t o t a l s i n t o those f o r t r a p s and those f o r s e i n e s . A l l t o t a l s f o r both B.C and Washington were c a l c u l a t e d twice t o check f o r e r r o r s . The IPSFC r e p o r t s of y e a r l y gear type catches f o r the whole g a u n t l e t by numbers were m u l t i p l i e d by y e a r l y average weights of sockeye as reported by the IPSFC to get y e a r l y gear type catches by weight for the Canadian s e c t o r and the U.S. s e c t o r . The Canadian sum of y e a r l y gear type catches by area and the U.S. sum of y e a r l y gear type c a t c h e s by area were compared with the IPSFC t o t a l s as a f i n a l check on sockeye c a t c h . There was l e s s than a 5% d i s c r e p a n c y e i t h e r way i n most years. Where the d i s c r e p a n c i e s d i d exceed 5% i n a year, the t o t a l s f o r that year were completely r e c a l c u l a t e d . as the IPSFC t o t a l s are c a l c u l a t e d on the b a s i s o f weight s l i p s a t the c a n n e r i e s and not from the data as reported by Washington or B.C., i t i s a c r e d i t t o a l l concerned t h a t there are only two or t h r e e years out of t w e n t y - f i v e f o r each gear type where the d i s c r e p a n c i e s exceed 5%. The o n l y e x c e p t i o n i s the Canadian t r o l l c a t c h , most of which i s taken i n areas o u t s i d e the j u r i s d i c t i o n of the IPSFC, as the Canadian t r o l l c a t c h i s i n s i g n i f i c a n t i n most ye a r s , no attempt was made to d i s c o v e r the cause of the d i s c r e p a n c i e s . , 139 T e c h n i c a l Change The IPSFC has kept a r e c o r d of major t e c h n i c a l i n n o v a t i o n s and performed t e s t s to measure t h e i r impact on gear e f f i c i e n c y . As a check on the IPSFC c a l c u l a t i o n s , other sources such as monthly i s s u e s of Western F i s h e r i e s s i n c e 1951, C r u t c h f i e l d and Pontecorvo (1969), the Royce-Bevan p r o j e c t (1963), the S i n c l a i r r e p o r t (1960) and the Washington S t a t e , Department of F i s h e r i e s , F i s h e r i e s Research Papers (1954, pp.48-51) were used as w e l l . There was s u r p r i s i n g agreement amongst the sources with r e s p e c t t o the dates of the advent of new technology and only minor d i f f e r e n c e s on the estimates of the impact on gear e f f i c i e n c y . The major changes occurred i n the e a r l y 1950*s with the replacement of l i n e n f i l a m e n t with nylon f o r the n e t s , the a p p l i c a t i o n of power to p u l l e y b l o c k s , the i n t r o d u c t i o n of the powered drum f o r s e i n e s , the replacement of l i n e n l i n e s with wire l i n e s , the a p p l i c a t i o n of power t o gurdies ( r e e l s f o r t r o l l l i n e s ) and the s u b s t i t u t i o n of h y d r a u l i c power t r a n s m i s s i o n f o r mechanical gear t r a n s m i s s i o n . Even though nylon nets d i d save time f o r purse s e i n e s i n terms of r e p a i r s , the b i g g e s t impact from n y l c n nets was f e l t by the g i l l n e t fishermen. The power drum and P u r e t i c power block were very important i n improving purse s e i n e e f f i c i e n c y by r e p l a c i n g the cumbersome r o l l e r t a b l e . Wire l i n e s and powered g u r d i e s meant fewer broken l i n e s and f a s t e r h a u l i n g and c l e a r i n g f o r t r o l l s . I t must be remembered, however, t h a t c o s t s aving t e c h n i c a l i n n o v a t i o n i s g u i t e d i f f e r e n t from technology which i n c r e a s e s the a c t u a l e f f i c i e n c y of gear h a r v e s t i n g , I t i s the l a t t e r which must be measured to o b t a i n an e f f i c i e n c y index. With t h i s i n 140 mind and using the sources d i s c u s s e d above, e f f i c i e n c y indexes were c o n s t r u c t e d f o r g i l l n e t s and purse s e i n e s . C a l c u l a t i o n Of Costs and P r i c e s In c a l c u l a t i n g c o s t s two a l t e r n a t i v e methods can be used. The f i r s t c o n s i s t s of using the lowest d a i l y c a t c h per u n i t of gear type e f f o r t f o r each season. T h i s approximates the o p p o r t u n i t y c o s t f o r one day's e f f o r t by t h a t gear type (Hannesson, 1974, p.72-74)..Daily f i g u r e s are a v a i l a b l e f o r the U.S. but only weekly t o t a l s are a v a i l a b l e f o r Canada. They can be d i v i d e d by the known l e g a l number of f i s h i n g days per week to get the average d a i l y minimum f o r t h a t week. However, t h i s would have r e g u i r e d s e a r c h i n g and comparing 200,000 d a i l y t o t a l s by hand to a r r i v e at a f i g u r e which i s probably a very poor approximation t o o p p o r t u n i t y c o s t . Instead a second method of c a l c u l a t i n g c o s t s was used. This i n v o l v e d s p l i t t i n g e f f o r t i n t o t h r e e c o s t components: l a b o u r , o p e r a t i n g and f i x e d c o s t s . Only one o b s e r v a t i o n on c o s t was a v a i l a b l e f o r Washington f i s h e r i e s . I t was obtained by s u r v e y i n g v e s s e l owners i n 1962 as p a r t o f the Royce-Bevan study (1963). The r e s u l t s of the survey are d i s c u s s e d i n the study but the c o s t data are not l i s t e d anywhere and i s given i n only h i g h l i g h t form. For example, the average gear and v e s s e l value f o r g i l l n e t s i n Washington i n 1962 was estimated by Royce-Bevan (1963, pp.. 121-22) as $6,000 t o $7,000 which compares with my 1962 c o s t c a l c u l a t i o n of $6,908 f o r average gear and v e s s e l value f o r g i l l n e t s i n B.C. Dr.. C r u t c h f i e l d , a member of the Royce-Bevan study group, e x p l a i n e d t o me t h a t there are no c o s t f i g u r e s a v a i l a b l e f o r 141 Washington except for t h e i r 1962 survey, and that the survey gave only a rough approximation as the reporting system was not complete and the sampling procedures were non-random. No figures were available f o r reefnets as the fishermen surveyed refused to respond. The above comparisons indicate that g i l l n e t values were quite s i m i l a r between Washington and B.C. but that purse seine values differed by about 25%. Errors of measurement could account for the discrepancy p a r t i c u l a r l y as the bias i n the sample came from a low response amongst smaller vessel owners. In the l i g h t of these r e s u l t s and noting that data are not available f o r any other years i n Washington, i t was decided that B.C. cost data would be used f o r both B.C. and Washington fishermen. The other serious omission i s the absence of cost data for reefnets. As t h i s gear does not operate in B. C« waters, there are no cost estimates for them i n B.C.; nor did the Boyce-Bevan study produce any reef.net estimates. I t i s known that reefnets are highly competitive with g i l l n e t s i n Washington waters and that the crew from one gear type are not adverse to switching to the other. Thus productivity differences between the gear types i n Washington waters are probably clo s e l y approximated by cost differences. And the former can be used as an estimator for the la t t e r . In a study conducted by Junge (1959) estimates were made of comparative productivity differences between purse seines, g i l l n e t s and reefnets. The r e s u l t s indicated that reefnets were 2.83 times as productive as g i l l n e t s on average. The increase i n net e f f i c i e n c y which resulted from replacement of linen by nylon 142 nets i n the e a r l y 1950's i s thought to have had more a p p l i c a t i o n to g i l l n e t s than to r e e f n e t s because of the more h i g h l y s p e c i a l i z e d nature of r e e f n e t f i s h i n g . Thus r e e f n e t c o s t s w i l l be assumed to be 2.5 times g i l l n e t c o s t s on average f o r the p e r i o d 1951 to 1975 i n t h i s study. Turning t o the c a l c u l a t i o n of labour c o s t s , from d i s c u s s i o n s with f i s h e r i e s o f f i c i a l s i n both departments of f i s h e r i e s i n B.C. and Washington and from p u b l i s h e d r e p o r t s on fishermen's incomes (Canada, "Some Economic Aspects.,,," 1971; and Hunter, 1971), a g r i c u l t u r e and the f o r e s t i n d u s t r y appear to be the most popular forms of a l t e r n a t i v e employment f o r fishermen. As the a g r i c u l t u r a l wage index i s of d o u b t f u l accuracy due t o non-monetized a s p e c t s , the f o r e s t i n d u s t r y average weekly wage was chosen as the best a l t e r n a t i v e . , In the 1950»s fishermen used t o f i s h f o r approximately e i g h t hours a day, f i v e days a week f o r a f o r t y hour work week. By the 1970»s t h i s had f a l l e n to three and a h a l f days a week because of area c l o s u r e s . However, during those days most fishermen worked a twelve hour day f o r approximately a f o r t y - t w o hour week. Thus the e f f o r t i n days o f f i s h i n g , once c o r r e c t e d by e f f i c i e n c y or gear and v e s s e l value indexes, would probably c l o s e l y approximate the number of hours worked i n 1951., The days of f i s h i n g e f f o r t given by the p r o d u c t i o n f u n c t i o n s i s i n terms of 1951 e f f o r t . Thus d i v i d i n g the average weekly wage i n the f o r e s t i n d u s t r y by f i v e s hould give the d a i l y o p p o r t u n i t y c o s t f o r labour i n the salmon f i s h i n g i n d u s t r y . I t w i l l be assumed t h a t 1.2 men work on g i l l n e t s and t r o l l s on average. For purse s e i n e s the r e c e n t average has been 5.5 men 143 (see for example, Fraser, 1975, p.66) but i n the early 1950's i t was as high as 7 men because of the large table seines. Thus an index i s used to correct for the decline i n numbers working purse seines. The average weekly wage for the forest industry in B.C. was obtained from S t a t i s t i c s Canada (catalogue:72-202) f o r the years 1951 to 1961 and from Cansim ( S t a t i s t i c s Canada, Cansim 1493. 1. 1, code=d704260) for the years 1961 to 1975. The average was taken from the weekly wages for July, August and September each year. One f i n a l note on labour costs re l a t e s to the use of l a t e r years' higher average wages with l a t e r years' d e l i v e r i e s converted into 1951 eguivalent d e l i v e r i e s . Obviously i f e f f o r t i s i n 1951 delivery terms, then no more should be charged than 1951 eguivalent days of labour. Thus the index of labour costs per unit of e f f o r t (one day's fishing) must be deflated by the e f f i c i e n c y or gear and vessel value index used to i n f l a t e the units of e f f o r t . Thus i f three and a half days of f i s h i n g i n 1975 costs $57 per day in labour (twelve hour days) for a t o t a l of $20 0, then f i v e days of 1951 f i s h i n g e f f o r t incurred in 1975 should cost $39.90 per day (eight hour days) for a t o t a l of $200. That i s , the costs have been deflated by the same index used to i n f l a t e e f f o r t . For fishermen the re n t a l cost of boat c a p i t a l consists of depreciation on gear and vessel value plus the market rate of return foregone on the c a p i t a l value of the vessel plus the value of the license attached to the vessel. This l a s t figure was obtained only for the years 1973, 1974, and 1975. The reason 144 i s that p r i o r to 1973, l i c e n s e s bore very l i t t l e value as new v e s s e l s were simply l a r g e r than the o l d . I t was not u n t i l the t c n - f o r - t o n r e g u l a t i o n s came i n t o e f f e c t t h a t fishermen were f o r c e d t o buy more l i c e n s e s to b u i l d a l a r g e r v e s s e l . The c a p i t a l value of l i c e n s e s was c a l c u l a t e d by s u b t r a c t i n g adjusted gear and v e s s e l value from a c t u a l gear and v e s s e l v a l u e . and f i n a l l y , the a p p r o p r i a t e i n f l a t i o n - f r e e market r a t e o f r e t u r n on c a p i t a l was assumed to be 3.5% (Campbell, 1973). From the p o i n t of view of s o c i e t y the op p o r t u n i t y c o s t of boat c a p i t a l (market r e n t a l s ) should not i n c l u d e the market r a t e o f r e t u r n foregone on the c a p i t a l value of the l i c e n s e s attached t o the v e s s e l . T h i s i s an example of a s i t u a t i o n where p r i v a t e o p p o r t u n i t y c o s t s exceed s o c i a l o p p o r t u n i t y c o s t s . . U n f o r t u n a t e l y , t h i s p r i v a t e o p p o r t u n i t y c o s t was not excluded i n the work with the c o s t f u n c t i o n s i n both Chapters Three and Four. L i c e n s e values were used only f o r 1973, 1974 and 1975, t h e r e f o r e , the d i s c r e p a n c i e s were not too great and only occured i n c y c l e s one, three and fo u r . The d i s c r e p a n c i e s are as f o l l o w s with the higher (wrong) d a i l y c o s t f o l l o w e d by the lower ( c o r r e c t ) d a i l y c o s t . For c o n f i n e d water g i l l n e t s : c y c l e one, 37.07/36.99; c y c l e t h r e e , 35.60/35.59; c y c l e f o u r , 35.05/34.86. For open water g i l l n e t s : c y c l e one, 44.83/44.77; c y c l e t h r e e , 45.44/45.44; c y c l e f o u r , 46.10/45.96. For c o n f i n e d water purse s e i n e s ; c y c l e one, 179.87/178.89; c y c l e t h r e e , 172.33/172.05; c y c l e f o u r , 171.11/169.44. For open water purse s e i n e s : c y c l e one, 240.55/239.49; c y c l e t h r e e , 247.67/247.32; c y c l e f o u r , 246. 17/244.29. 145 As the d i s c r e p a n c i e s were so s m a l l , i t was decided not to re-run a l l the optimal and c o s t programming as the expense i n computer time would have been very high and the r e s u l t s are not s e n s i t i v e t o changes i n co s t (see the r e s u l t s i n Tables (4-4) and (4-5) i n Chapter Pour). F o l l o w i n g the l e a d of s e v e r a l s t u d i e s o f salmon f i s h i n g c o s t s i n B.C. (Buchanen and Campbell, 1957; S i n c l a i r , 1960, pp. 163-238; Royce-Bevan, 1963, pp. 33-45, 121-122; Campbell, 1969, pp.48-57; Canada, "Some Economic Aspects...," 1971; Canada, "An A n a l y s i s cf Gross Returns..,," 1971; Grauer, 1973; and Canada, "You Are Operating... ,*' 1971) d e p r e c i a t i o n on v e s s e l values was estimated at 7,5% f o r a l l gear types a n n u a l l y . For gear values the a p p r o p r i a t e annual d e p r e c i a t i o n appears t o be 33% f o r s e i n e v e s s e l s and 10 0% f o r g i l l n e t s and t r o l l s . As the main gear expense i s the net or t r o l l l i n e s which l a s t approximately three years f o r s e i n e s and one year f o r g i l l n e t s and t r o l l s , these appear to fee reasonable. Foregone market r a t e of r e t u r n on gear and v e s s e l value was assumed equal t o an i n f l a t i o n - f r e e 3.5% P^r year. The t h r e e components of the r e n t a l c o s t f o r boat c a p i t a l were d i v i d e d by s e v e n t y - f i v e t o g i v e the average expected r e n t a l per day of f i s h i n g . Fishermen are a c t u a l l y engaged i n f i s h i n g f o r only about s e v e n t y - f i v e days out o f every year on average but f r e g u e n t l y spend another s e v e n t y - f i v e days a year aboard the boat e i t h e r r e p a i r i n g i t or preparing i t f o r f i s h i n g . V e s s e l s i n the sockeye f i s h e r y are only used t h r e e and a h a l f days a week now i n s t e a d of f i v e but i t i s thought by most fishermen t h a t the a c t u a l wear and t e a r on a weekly b a s i s i s about the same f o r the 146 reasons d i s c u s s e d under labour c o s t s . Thus, f o r s i m i l a r reason, f i x e d c o s t r e n t a l s must a l s o be d e f l a t e d by the same index used to i n f l a t e e f f o r t to the 195 1 e q u i v a l e n t . Operating c o s t s are the most d i f f i c u l t t o c a l c u l a t e as there are no indexes and the few o b s e r v a t i o n s a v a i l a b l e are s c a t t e r e d over the years. The c o s t r e f e r e n c e s used f o r c a l c u l a t i n g d e p r e c i a t i o n were used to t r y and r e l a t e o p e r a t i n g c o s t s , i f p o s s i b l e , to the t h r e e c o n s i s t e n t and continuous indexes a v a i l a b l e : gear and v e s s e l v a l u e , wages i n a l t e r n a t i v e employment, and e f f o r t i n 1951 days of f i s h i n g . For the years 1S51 to 1956, the monthly magazine, Western F i s h e r i e s , was used as a source f o r v e s s e l insurance r a t e s , gear d e p r e c i a t i o n , v e s s e l d e p r e c i a t i o n , food and f u e l c o s t s ; f o r the years 1953 to 1954, Buchanen and Campbell (1957) were used f o r c o s t breakdowns on f u e l , b a i t , gear r e p a i r s and purchases, v e s s e l r e p a i r s and engine purchases, wages, l i c e n s e f e e s , i n t e r e s t c o s t s and equipment r e n t a l s ; f o r 1958 the S i n c l a i r r e p o r t (1960, pp. 163-238) was used f o r broad breakdowns on the r e l a t i o n s h i p between f i x e d , o p e r a t i n g and labour c o s t s ; f o r the years 1959 to 1962, the Eoyce-Bevan study (1963, pp.121-122) gave the rough v a l u e s f o r v e s s e l s and gear; f o r the years 1965 t o 1967, Campbell (1969, pp.48-57) was used f o r e s t i m a t e s f o r f u e l , food, v e s s e l value, gear value, maintenance, i n s u r a n c e , wages, wharfage and d e p r e c i a t i o n c o s t s ; f o r the years 1966 to 1970, s t u d i e s by the Canadian department of f i s h e r i e s ("Some Economic Aspects...," 1971; and " A n a l y s i s of Gross Returns...," 1971) were used f o r f i g u r e s on v e s s e l values, income earned, a l t e r n a t i v e employment earnings and average number of days f i s h i n g per year; and f o r 147 1971, the study by Grauer (1973) was used f o r f i g u r e s on gear r e p a i r s and purchases, v e s s e l and engine r e p a i r s and purchases, food, f u e l , eguipment l e a s i n g , i n s u r a n c e , d e p r e c i a t i o n , wages and l i c e n s e f e e s . Food c o s t s were d i r e c t l y r e l a t e d to the number of days i n v o l v e d i n a f i s h i n g day and four meals per day per man were assumed. 1951 p r i c e s f o r the average meal were taken from S t a t i s t i c s Canada (catalogue: 62-002) as were food c o s t indexes. These were used to d e f l a t e p r i c e s f o r the years 1951 to 1975 to put food c o s t s i n 1951 d o l l a r s . With the advent of the F i s h i n g V e s s e l Insurance Plan sponsored by the f e d e r a l government and taking i n t o account the refunds on p r i v a t e insurance premiums on plans o p e r a t i n g before the f e d e r a l scheme, i t was assumed t h a t b a s i c v e s s e l f i r e i n s u r a n c e amounted to 2.5% of the v e s s e l value each year. , F u e l c o s t s were estimated on the f o l l o w i n g b a s i s : purse s e i n e s are mainly d i e s e l and use twelve g a l l o n s per s e t and make an average of s i x s e t s per day f o r a t o t a l of seventy-two g a l l o n s per day; t r o l l s are mainly g a s o l i n e and use 2.5 g a l l o n s per hour of f i s h i n g f o r twelve hours per day f o r a t o t a l of t h i r t y g a l l o n s per day; and g i l l n e t s use two g a l l o n s of g a s o l i n e per hour and f i s h f o r twelve hours a day f o r a t o t a l of twenty-f o u r g a l l o n s per day. As these are a l l maximum number of f i s h i n g hours and 1951 e f f o r t assumes fewer f i s h i n g hours, t h i s should allow f o r an a d d i t i o n a l two to f o u r hours of steaming time per day. 19 51 wholesale (commercial) p r i c e s f o r g a s o l i n e and d i e s e l were taken from S t a t i s t i c s Canada (catalogue: 62-002) from which 143 f u e l c o s t indexes were a l s o obtained. These were used t o d e f l a t e f u e l c o s t s f o r the years 1951 to 1975 to put f u e l c o s t s i n 1951 d o l l a r s . Eguipment l e a s i n g and r e n t a l s are extremely d i f f i c u l t to estimate as shore f a c i l i t i e s d i f f e r , e l e c t r o n i c eguipment has re p l a c e d o l d e r types of eguipment and wharfage and s l i p charges are simply not a v a i l a b l e . Judging from the s c a t t e r e d o b s e r v a t i o n s , i t would appear that they average about 2.5% of v e s s e l value each year. The same problem a p p l i e s with regard to b a i t and i c e , taxes, l i c e n s e s and f e e s . These were lumped together as miscellaneous and were valued at 2% of gear and v e s s e l value each year. The r e s u l t s appeared t o conform reasonably well with the few o b s e r v a t i o n s a v a i l a b l e . F i n a l l y , f o r the same reasons as f o r labour and f i x e d c o s t r e n t a l s , o p e r a t i n g c o s t s were d e f l a t e d by the same index used to i n f l a t e days of f i s h i n g i n t o 1951 e g u i v a l e n t s . Thus a l l c o s t s were put i n 1951 constant d o l l a r s f i r s t through using t h e i r a p p r o p r i a t e p r i c e indexes and were then f u r t h e r d e f l a t e d by e f f i c i e n c y or gear and v e s s e l value indexes t o put them i n 1951 e g u i v a l e n t days o f e f f o r t c o s t s . P r i c e s f o r the v a r i o u s r a c e s o f salmon caught i n B.C. were taken from two sou r c e s : B r i t i s h Columbia Catch S t a t i s t i c s (Canada, Department c f the Environment, F i s h e r i e s S e r v i c e , 1951-1S75) and the Annual S t a t i s t i c a l Beview of Canadian F i s h e r i e s (Canada, Department of the Environment, V o l s . 1 - 9 ) . From 1951 to 1971 the sockeye p r i c e s were used as given but f o r the p e r i o d 1972 to 1975 adjustments were r e q u i r e d . According t o v a r i o u s sources i n the Department of F i s h e r i e s i n B.C. and amongst 149 fishermen, a f t e r 1971 the p r i c e s paid at the p r o c e s s o r s no long e r r e f l e c t e d true landed p r i c e . The union had agreed to allow landed p r i c e s to be l e s s than t r u e landed p r i c e s i n order to allow boat owners t o use lower p r i c e s f o r c a l c u l a t i n g shares t o pay t h e i r workers, Apparently many boat owners were i n s e r i o u s f i n a n c i a l d i f f i c u l t y because of the r a p i d r i s e i n v e s s e l b u i l d i n g and r e p a i r c o s t s . The boat owner (or skipper) was paid a bonus at the end of the season based on h i s landed c a t c h which brought the landed p r i c e of f i s h up t o the a c t u a l landed p r i c e . What i s pu b l i s h e d i s the announced p r i c e and not the a c t u a l p r i c e . To convert the reported p r i c e i n t o a c t u a l p r i c e f o r the years 1972 to 1975 the f o l l o w i n g procedure was used. Landings and Landed Value of sockeye were taken from the Annual S t a t i s t i c a l fieview of Canadian F i s h e r i e s f o r the years 1953 to 1975, Wholesale Product and Wholesale Product Value f o r sockeye f o r the same p e r i o d and from the same source were a l s o compiled. The landed p r i c e was c a l c u l a t e d by d i v i d i n g landed value by l a n d i n g s i n weight (metric tons converted t o pounds). The wholesale p r i c e was c a l c u l a t e d by d i v i d i n g wholesale product value by wholesale product weight a d j u s t e d because the l a t t e r was reported i n thousands of 48 pound cases. The r a t i o of landed t o wholesale p r i c e was then c a l c u l a t e d and i t v a r i e d from a high of 36.4% i n 1968 to a low of 26.9% i n 1973. .The p e r i o d 1972 t o 1975 was c o n s i s t e n t l y lower, c o n f i r m i n g the r e p o r t s of the d i f f e r e n c e between a c t u a l and repo r t e d p r i c e s of landed sockeye. The average r a t i o f o r the p e r i o d 1953 to 1971 was c a l c u l a t e d as 33.7%. T h i s average r a t i o was a p p l i e d to the period 1972 to 1975 150 t o i n f l a t e r e p o r t e d p r i c e s up to a c t u a l p r i c e s f o r landed sockeye. P r i c e s f o r the f i v e races of salmon caught i n Washington waters were taken from two s o u r c e s : unreported sockeye p r i c e s obtained by the Department of F i s h e r i e s i n Washington and the r a t i o o f Catch Value to T o t a l Pounds Landed as r e p o r t e d i n the annual i s s u e s of the F i s h e r i e s S t a t i s t i c a l Report (State of Washington, 1951, 1957-1974). The two sockeye p r i c e s were e x a c t l y e q u i v a l e n t and i t i s to be assumed t h a t the unreported p r i c e s are simply c a l c u l a t i o n s made of the r a t i o of Catch Value to T o t a l Pounds Landed. There were no unreported p r i c e s f o r the p e r i o d 1952 t o 1956 and there were no annual i s s u e s o f the F i s h e r i e s S t a t i s t i c a l Report p u b l i s h e d i n those years. T h i s would tend to c o n f i r m the f a c t t h a t the unreported p r i c e s were based on those S t a t i s t i c a l Reports. Breakdowns by gear type f o r sockeye catches were not a v a i l a b l e f o r B.C. They were a v a i l a b l e f o r the peri o d 1965 to 1974 f o r Washington i n the form of unreported p r i c e s . The d i f f e r e n t i a l s were not great as a l l three gear types use nets and i t i s undamaged f r o z e n f i s h which command a premium. Hor were the d i f f e r e n t i a l s c o n s i s t e n t : some gear types earned more i n some years and l e s s i n o t h e r s . I t i s true t h a t t r o l l gear i n B.C. are reported t o r e c e i v e about 5% more per pound because the f i s h are undamaged by nets and are packed i n d i v i d u a l l y i n i c e . However, because of the premium f o r undamaged f r o z e n f i s h , both purse s e i n e s and g i l l n e t s have taken great care over the years t o i c e t h e i r c a t c h where p o s s i b l e and to minimize the net damage. For t h i s reason, no attempt was made to c a l c u l a t e 151 d i f f e r e n t i a l s i n sockeye p r i c e by gear type. Canadian p r i c e s were used f o r the c a t c h e s made by both B.C. and Washington fishermen. The d i f f e r e n t i a l between the two p r i c e s has remained f a i r l y s t a b l e , i n c l u d i n g the exchange r a t e d i f f e r e n t i a l , a t about 155? t o 17% higher f o r sockeye s o l d i n t o Washington. T h i s e f f e c t w i l l be picked up i n Chapter Four below where t h e U.S. c o s t s are permitted t o be 15% l e s s than f o r Canadian fishermen. . The p r i c e s of other salmon were then taken as a percent of sockeye p r i c e s and the c a t c h of other salmon were then taken as a percent of sockeye catches over the years. Although c a t c h p r o p o r t i o n s v a r i e d over the years and the d i f f e r e n t i a l s f o r race p r i c e s a l s o changed over the years, the most dramatic change was i n the c a t c h of pink salmon every second year. For t h i s reason and f o r the reasons d i s c u s s e d i n the programming s e c t i o n above, the value of other salmon caught was estimated as 50% of sockeye p r i c e s f o r Canada and the U.S. i n "on" years f o r the pink and as 67% of sockeye p r i c e s i n " o f f " years f o r the pink. Gear And V e s s e l Value Indexes The gear and v e s s e l value indexes used i n Chapter Three to c o r r e c t r e p o r t e d e f f o r t were c a l c u l a t e d as f o l l o w s . Observations on average value f o r g i l l n e t , purse s e i n e and t r o l l gas and d i e s e l v e s s e l s i n B.C. were a v a i l a b l e from 1953 to 1975. Observations on average gear value f o r g i l l n e t , purse s e i n e and t r o l l v e s s e l s i n B.C. were a v a i l a b l e from 1953 t o 1971. Observations on average y e a r l y expenditure on gear f o r g i l l n e t , purse s e i n e and t r o l l v e s s e l s i n B.C. were a v a i l a b l e from 1967 152 t c 1975, Observations on average y e a r l y new engine expenditures f o r the same three gear types f o r the same p e r i o d were a l s o a v a i l a b l e . a l l these o b s e r v a t i o n s were taken from F i s h e r y S t a t i s t i c s o f B.C. (1951-1975) . The same s e r i e s were a l s o a v a i l a b l e i n the anuual May i s s u e o f the magazine, Western F i s h e r i e s (1951-1956)./ The f i r s t s tep was the c o n s t r u c t i o n of a complete s e r i e s on average v e s s e l value. T h i s i n v o l v e d m u l t i p l y i n g average v e s s e l values by the number of r e p o r t i n g v e s s e l s t o a r r i v e at t o t a l v e s s e l v a l u e s f o r each gear type f o r gas and d i e s e l . The t o t a l gas v e s s e l values and t o t a l d i e s e l v e s s e l v a l u e s were summed to get the t o t a l f l e e t value by gear type. These t o t a l f l e e t values were then d i v i d e d by t o t a l v e s s e l numbers (both gas and d i e s e l ) i n each gear type to a r r i v e a t the r e s p e c t i v e average v e s s e l values f o r each f l e e t . The second step i n v o l v e d the c o n s t r u c t i o n of a Boat B u i l d i n g and Repair Index (BBBPX). A BBBPX was a v a i l a b l e from S t a t i s t i c s Canada (catalogue: 62-515) from 1956 to 1959. ft BBBPX was a v a i l a b l e from Cansim ( S t a t i s t i c s Canada, Cansim 429,1, code=d587101) from 1961 to 1971. ft BBBPX was a v a i l a b l e from S t a t i s t i c s Canada (catalogue: 62-002) from 1971 to 1975. These indexes were a l l c o n s i s t e n t i n terms of the sources used by S t a t i s t i c s Canada to generate BBBPX. However, there was no continuous index f o r the period 1951 to 1975., To i n t e r p o l a t e f o r the missing years a Transport index was used. T h i s was d e r i v e d from sub-indexes such as BBBPX and was continuous f o r the whole p e r i o d ( S t a t i s t i c s Canada, c a t a l o g u e : 62-002). For the years 1951-1955 and 1960, the BBBPX 153 interpolated as equal to 1.05 the Transport index for those years f o r the same base. This p a r t i a l l y interpolated BBRPX for 1951 to 1960 was then s p l i c e d onto the BBBPX which was availa b l e from 1961 to 1971. And f i n a l l y , the l a t t e r was spliced onto the BBRPX from 1971 to 1975. The re s u l t i n g BBRPX with a base of 1951=10 0 was then compared with the overal Transport index, the Consumer Price index and the General Wholesale Price index with sim i l a r bases. The BBRPX conformed favorably with the c h a r a c t e r i s t i c that i t accelerated at a faster rate than any of the others. This i s consistent with S t a t i s t i c s Canada reports on the more rapid i n f l a t i o n in the boat building and repair industry (various issues of catalogue; 62-515 and 62-002). The t h i r d step removed the influence of the license l i m i t a t i o n program on the value of vessels. This involved defla t i n g average vessel values for g i l l n e t s , purse seines and t r o l l s by the BBRPX from 1953 to 1975, These deflated values were then regressed on the years 1962 to 1972 to i s o l a t e the 10 year trends p r i o r to license value increases. The f i t for the g i l l n e t , purse seine and t r o l l vessel value trends were a l l between an R-sguared of .85 and .95. These trends were then extended, using the parameter estimates, for the years 1973 to 1975 to derive adjusted vessel values for those years. These adjusted vessel values were then i n f l a t e d with BBRPX and substituted for the average vessel value f o r the years 1973 to 1975. The adjusted vessel value indexes were then subtracted from the average vessel values indexes (unadjusted) to a r r i v e at estimates of the influence of license values on vessel c a p i t a l values. The results are rather s i m i l a r to newspaper reports on 154 per ton l i c e n s e values. The f o u r t h step i s s i m i l a r to the f i r s t s t e p i n the c o n s t r u c t i o n of a complete s e r i e s on average gear v a l u e . I t i n v o l v e d m u l t i p l y i n g average gear value by the number of r e p o r t i n g v e s s e l s t c d e r i v e t o t a l gear values f o r each gear type f o r gas and d i e s e l . T o t a l f l e e t v a l u e s were then c a l c u l a t e d by summing gas and d i e s e l f i g u r e s and then transformed i n t o average gear type values by d i v i d i n g by f l e e t s i z e . A s i m i l a r procedure was i n v o l v e d i n d e r i v i n g average y e a r l y gear expenditure by gear type. F i r s t d i f f e r e n c e s on a y e a r l y b a s i s were taken of average gear value and compared with average gear expenditure f o r the years 1966 to 1971., An average m u l t i p l i e r was c a l c u l a t e d which gave a reasonably c o n s i s t e n t r e l a t i o n between these f i r s t d i f f e r e n c e s and the y e a r l y expenditures f o r each gear type. These m u l t i p l i e r s were then used with the average gear expenditure f i g u r e s f o r the years 1972 t o 1975 to estimate average gear values by gear type f o r the missing years. The r e s u l t a n t index was c a l l e d a d j u s t e d gear value. The f i f t h s t e p i n v o l v e d summing the a d j u s t e d average v e s s e l value and a d j u s t e d average gear value f o r each gear type to give a gear and v e s s e l value index. T h i s was d e f l a t e d by BBBPX. F i n a l l y , i t was assumed that gear and v e s s e l values d i d not change f o r the years 1951 to 1953. The reason t h i s was done was t h a t t h e r e were no data a v a i l a b l e anywhere t o i n d i c a t e how to i n t e r p o l a t e f o r the missing y e a r s , 1951 and 1952. The only j u s t i f i c a t i o n f o r assuming t h a t d e f l a t e d gear and v e s s e l value indexes were unchanged f o r those three e a r l y years i s t h a t the 155 e f f i c i e n c y indexes as compiled i n Chapter Three i n d i c a t e d no s e r i o u s change i n the p r o d u c t i v i t y or e f f i c i e n c y of c a t c h i n g f i s h f o r each gear type i n the years 1951 to 1953. Furthermore, v e s s e l values are l i k e l y to r e a c t with a l a g to any p r o d u c t i v i t y changes. Comparing the e f f i c i e n c y indexes with gear and v e s s e l value indexes i n d i c a t e s that the l a t t e r are more l i k e moving averages of the former. That i s , i t takes time f o r t e c h n o l o g i c a l change to show up i n gear and v e s s e l values. 156 CHAPTER FOUR: THE OPTIMAL MANAGEMENT SOLUTION I INTRODUCTION Using the rec r u i t m e n t f u n c t i o n from Chapter Two and the c o s t f u n c t i o n s from Chapter Three, the management model presented i n t h i s chapter w i l l d e r i v e the e q u i l i b r i u m values needed f o r optimal management of the F r a s e r River sockeye f i s h e r y . The f i r s t p a r t of t h i s chapter b r i n g s together a l l the assumptions b a s i c t o the management model which have been d i s c u s s e d i n the previous three c h a p t e r s . The management model w i l l then be used i n f o u r h y p o t h e t i c a l case s t u d i e s and the r e s u l t s compared with the a c t u a l h i s t o r i c a l management experience. In the f i r s t case sockeye o n l y w i l l be caught u s i n g a c o n s t r a i n e d example i n which the IPSFC w i l l be f o r c e d to use both U. S. and Canadian fishermen t o land 50% of the c a t c h each. An unco n s t r a i n e d example w i l l a l s o be used i n which the IPSFC can a p p o r t i o n the c a t c h as i t sees f i t amongst the lowest c o s t gear type areas. In the second case, the c o n s t r a i n e d example of h a r v e s t i n g sockeye only w i l l be used to examine the impact of a l t e r i n g d i s c o u n t r a t e s on op t i m a l escapement, re c r u i t m e n t and c a t c h and the net p r o f i t . In the t h i r d case, c o s t s w i l l be va r i e d , using both c o n s t r a i n e d and unconstrained examples, to examine the impact of a l t e r i n g c o s t s on o p t i m a l escapement, rec r u i t m e n t and c a t c h and on the l e a s t c o s t gear type h a r v e s t i n g p r o p o r t i o n s . t In the f o u r t h case, other salmon, as we l l as sockeye, w i l l be permitted t o be caught up t o the c a p a c i t y l i m i t s d e s c r i b e d i n Chapter Three. Both an unconstrained and a 157 c o n s t r a i n e d example s i l l re used. 41.1 the p r e v i o u s cases w i l l be based on annual net p r o f i t once e g u i l i b r i u m has been achieved. A f u l l comparison between the h i s t o r i c a l net p r o f i t and the net p r o f i t d e r i v e d from the optimal management model should i n c l u d e the p e r i o d s d u r i n g which the management model a d j u s t s to the e g u i l i b r i u m values. The best way to do t h i s appears to be to study the p e r i o d 1951 to 1975 p l u s the f o u r years from 1947 to 1950 d u r i n g which escapement l e v e l s would h y p o t h e t i c a l l y be a d j u s t e d to the optimal values. Compounding the annual net p r o f i t s forward from 1947 to 1975 w i l l then permit comparison of present values f o r both the h i s t o r i c a l experience and the o p t i m a l management case. The f i n a l s e c t i o n of t h i s chapter presents a b r i e f summary of the whole study and the major c o n c l u s i o n s to emerge from working with the management model. The f i r s t b a s i c assumption of the management model i s that the IPSFC has been given the monopoly powers of a s o l e owner and charged with the task of maximizing the present worth of the F r a s e r B i v e r sockeye f i s h e r y . I t w i l l be assumed t h a t l e a s t c o s t gear are p a i d t h e i r o pportunity c o s t s and t h a t net revenue i s d i v i d e d e q u a l l y between the O.S..and Canadian governments. The methods used by the IPSFC to achieve i t s management o b j e c t i v e s w i l l i n c l u d e q u a n t i t a t i v e c o n t r o l of gear type e f f o r t i n the s i x t e e n d i f f e r e n t gear type areas. T h i s i s s i m i l a r to i t s present method of c o n t r o l l i n g escapement except t h a t i t w i l l now be permitted to c l o s e ten gear type areas completely and to c o n t r o l the gear type numbers i n v o l v e d i n h a r v e s t i n g the remaining open areas. 158 It w i l l be assumed that the most r a p i d approach t o the e g u i l i b r i u m l e v e l of escapement i s the op t i m a l h a r v e s t p o l i c y (see C l a r k , 1976b, pp.245-248). And i t w i l l be assumed, f o r the purposes o f t h i s study, t h a t those o p t i m a l escapement l e v e l s c o u l d have been achieved i n the f o u r years b e f o r e 1951. In some cases the r e s u l t s suggest t h a t h i s t o r i c a l l y the IPSFC has allowed too l a r g e an escapement and i n o t h e r s , too s m a l l an escapement., For c y c l e years where there has been u n d e r f i s h i n g , i n c r e a s i n g e x p l o i t a t i o n i s no problem. But f o r o v e r f i s h e d c y c l e y e a r s , a l l o w i n g the optimal escapement w i l l e n t a i l a s a c r i f i c e the f i r s t y ears. A c h i e v i n g these o p t i m a l l e v e l s o f escapement i n each c y c l e year i s very simple as the sub-optimal r e c r u i t m e n t s i n a l l cases exceed the optimal escapements which are r e q u i r e d . T h e r e f o r e , the IPSFC simply reduces or i n c r e a s e s e f f o r t i n each c y c l e year to a l l o w the op t i m a l escapement. Except f o r t h i s adjustment period, however, the op t i m a l s o l u t i o n s are e q u i l i b r i u m values which i m p l i e s they are the same f o r each point i n time t h e r e a f t e r . In order t o harvest the r e c r u i t m e n t r e s u l t i n g from t h i s o p t i m a l escapement, the IPSFC needs some guide t o choosing the l e a s t c o s t methods. These are d e r i v e d i n Chapter Three from production f u n c t i o n s using 1951 e g u i v a l e n t days of f i s h i n g e f f o r t and a v a i l a b l e recruitment v a r y i n g by year c l a s s c y c l e c a t c h a b i l i t y . Although there are s i x t e e n t r a d i t i o n a l gear type areas, only s i x w i l l be used. For the three t r o l l gear a r e a s , sockeye i s s t i l l o nly an i n c i d e n t a l c a t c h d e s p i t e g r e a t improvement i n c a t c h i n g e f f i c i e n c y . Both r e e f n e t areas appear to 159 be marginal even at the best of times. Recruitment to the 0.3. side of the outer S t r a i t of Juan de Fuca and the Westbeach area i s highly r e s t r i c t e d making zone three g i l l n e t s and purse seines i n e f f i c i e n t . And the recruitment to the Canadian area in the S t r a i t of Georgia outside the mouth of the Fraser Siver i s highly variable. Except f o r years of exceptionally large recruitment when sockeye migrate past the mouth of the Fraser into Johnstone S t r a i t * t h i s S t r a i t of Georgia f i s h i n g area could never land a substantial portion of the catch. For these reasons a l l these ten areas experienced higher costs on average than the six most productive and consistently less costly areas. And i t w i l l be assumed that the IPSFC w i l l close them to a l l f i s h i n g during the sockeye season. This leads to the problem of redundant gear i n certain areas (Clarke, Clark and Munro, 1 9 7 9 ) . This redundant gear can be accommodated by either forcing i t into other salmon f i s h e r i e s (outside the six week sockeye fishery) or by relocating the gear i n the six most productive gear type areas. The redundant gear types which cannot be relocated, such as reefnets, can be compensated out of the larger rent which accrues through the optimal management of the fi s h e r y . For the six remaining gear type areas, some cycle years are good and some are bad i n terms of excess capacity. In pa r t i c u l a r , cycle four makes very heavy use of purse seines and open water g i l l n e t s because of the larger recruitment e l a s t i c i t i e s which make them sensitive to a v a i l a b i l i t y i n years of large migrations. But other years make much heavier use of g i l l n e t s . That i s , the exponents on the recruitment variables 160 f o r purse s e i n e s and open water g i l l n e t s are l a r g e r than f o r g i l l n e t s i n c o n f i n e d waters and t h i s reduces u n i t e f f o r t c o s t s c o n s i d e r a b l y i n years of l a r g e r e c r u i t m e n t a v a i l a b i l i t y such as f o r c y c l e f o u r . A c l o s e r study of h a r v e s t i n g p a t t e r n s r e v e a l s the f o l l o w i n g : c y c l e year two i s poor f o r a l l gear type areas as i t i s an o f f year f o r pink salmon and the s m a l l e s t c y c l e year re c r u i t m e n t f o r sockeye. C y c l e years one and three are good f o r g i l l n e t s i n terms of sockeye and they are a l s o "on" years f o r pink. Thus a l l gear types appear to b e n e f i t from the good sockeye catches and heavy pink c a t c h e s . C y c l e year f o u r i s an o f f year f o r pink which i s more than o f f s e t by the enormous recruitment f o r sockeye. On the s u r f a c e , c y c l e year four appears to be e x c e l l e n t f o r purse s e i n e s and poor f o r g i l l n e t s . But on c l o s e r examination i t t u r n s out t h a t the harvest f o r g i l l n e t s i s only poor i n comparison t o the l a r g e purse s e i n e c a t c h e s . In f a c t the g i l l n e t c atches are almost the same a b s o l u t e l y as those f o r c y c l e years one and t h r e e and i t i s o n l y i n c y c l e year two t h a t they are poor. The v a r i a n c e o f c a t c h i s g r e a t e s t f o r purse s e i n e s and i t i s very convenient t h a t they are the most mobile of the gear types. The m o b i l i t y cf purse s e i n e s i n t o other f i s h e r i e s plus the very l a r g e purse s e i n e c a t c h e s of sockeye i n c y c l e year f o u r w i l l be assumed to adequately compensate purse s e i n e s f o r t h e i r excess c a p a c i t y i n poor years. The d i s c u s s i o n of pink salmon i n t r o d u c e s the problem of the j o i n t production nature of the f i s h e r y , , Because of the c o m p l e x i t i e s i n v o l v e d i n modelling a m u l t i - s p e c i e s h a r v e s t i n g 161 system, t h i s study has focused on the sockeye almost exclusively., This w i l l be continued in searching f o r an optimal management so l u t i o n . The chief objective for the IPSFC w i l l be to maximize the present value of the sockeye. However, an alternative model w i l l be presented which allows other salmon to be harvested on a capacity basis (as i n Chapter Three) i n order to demonstrate the impact of the j o i n t production on the optimal management of sockeye. The choice of the least cost gear type harvesting areas given nonlinear cost functions i s determined by the size of catch, the size of biomass recruitment, the c a t c h a b i l i t y of that biomass and the e f f i c i e n c y of the gear type areas. The gear type area e f f o r t has been standardized into 1951 equivalent days' of e f f o r t (see the Appendix to Chapter Three for details) and c a t c h a b i l i t y for each gear type has been standardized on a year cycle basis. But catch and recruitment depend on the optimal solution which i n turn depends on the proportion of catch taken by each area. As discussed in Chapter One, the optimal solution must determine least cost gear type harvesting proportions and optimal catch and recruitment simultaneously. The system of least cost harvesting uses the cost functions i n Chapter Three and optimal escapement i s obtained by using Clark and Hunro's equilibrium equation. The algorithm was developed by Clark (1976b, pp.252-253) s p e c i f i c a l l y for discrete-time metered models such as the Fraser l i v e r sockeye which uses a modified Ricker form of recruitment function. The s p e c i f i c form of the algorithm used i n t h i s study i s : (4-1) G(x) ((P(R(x)) • P(C(x)))/P(C(x))) - ( H s ) S = 0 162 where x = optimal escapement G (x) = d e r i v a t i v e of r e c r u i t m e n t f u n c t i o n with r e s p e c t to x P(B(x)) = d e r i v a t i v e of p r o f i t with r e s p e c t to r e c r u i t m e n t as a f u n c t i o n of x P (C (x)) = d e r i v a t i v e of p r o f i t with r e s p e c t to c a t c h as a f u n c t i o n of x, a l l o w i n g optimal escapement D = d i s c o u n t f a c t o r egual to one p l u s the s o c i a l r a t e of discount In the model i n t h i s study there i s a f o u r year l a g between escapement and recruitment, t h e r e f o r e the d i s c o u n t f a c t o r i s r a i s e d to the f i f t h power to r e f l e c t t h i s compounding e f f e c t . The f i s h e r y w i l l be broken up i n t o f o u r s e p a r a t e c y c l e year f i s h e r i e s which w i l l a llow f o r the d i f f e r e n t p e r i o d i c i n f l u e n c e s f o r each c y c l e year. T h e r e f o r e , the C l a r k a l g o r i t h m i n e x p r e s s i o n (4-1) w i l l be used t o o b t a i n the optimal v a l u e s assuming each o f the f o u r f i s h e r i e s i s harvested only once every f o u r years. Flowchart (4-1) i l l u s t r a t e s the a l t e r a t i o n s needed to transform the l e a s t c o s t h a r v e s t i n g program (Flowchart (3-1)) i n t o a program which s i m u l t a n e o u s l y determines the optimal escapement and the l e a s t c o s t gear type h a r v e s t i n g p r o p o r t i o n s to l a n d the subsequent c a t c h . As e x p l a i n e d i n Chapter One, the program c l o s e s the loop between the r e s i d u a l (escapement) and the i n i t i a l v a l u e s (recruitment and c a t c h ) . Given i n i t i a l values f o r c a t c h and recruitment, the model a p p o r t i o n s the gear type c a t c h so as to minimize c o s t of h a r v e s t i n g , c a l c u l a t e s the own r a t e of r e t u r n on the f i s h e r y and s u b t r a c t s the (compounded) s o c i a l r a t e of r e t u r n . I f the r e s u l t i n g value i s d i f f e r e n t from z e r o , the program s p e c i f i e s a new l e v e l of escapement, uses the r e c r u i t m e n t f u n c t i o n to e s t i m a t e recruitment and subsequent c a t c h , and goes through the same steps as above. These 1 6 3 Flowchart ( 4 - 1 ) : Program to Maximize Present Worth o f Sockeye F i shery Assuming Sockeye Harves t ing Only. | W \ T i A L AL.60RVTH* 164 i t e r a t i o n s w i l l continue u n t i l a l e v e l o f escapement i s reached where t h e own r a t e of r e t u r n on the f i s h e r y equals the f o u r year compounded s o c i a l r a t e of d i s c o u n t . , Note t h a t i n a l l subsequent work i t i s assumed t h a t the Johnstone S t r a i t c a t c h and the Indian c a t c h w i l l remain at the average c y c l e year l e v e l . As i t i s i m p o s s i b l e t o guess how these c a t c h p r o p o r t i o n s would change and as they r e p r e s e n t o n l y a s m a l l p r o p o r t i o n of t o t a l c a t c h , no attempt w i l l be made to modify them. 11 CASE ONE: HAHVESTING SOCKEYE ONLY Table (4-1) give s the comparison between a c t u a l h i s t o r i c a l average c y c l e year p r o f i t and the annual net p r o f i t obtained from the op t i m a l p o l i c y assuming: only sockeye are caught, an i n f l a t i o n f r e e s o c i a l d i s c o u n t r a t e o f three and a h a l f percent, and the IPSFC c o n s t r a i n e d to an egual d i v i s i o n of the cat c h between O.S. and Canadian fishermen. 165 TABLE (4 -1) : CONSTRAINED OPJ-IMAL CftJCH OF SOCKETE ONLY*. HISTORIC AND OPTIMAL CYCLE YEAR ESCAPEME NT X RECRUITMENT AND CATCH IN OOP'S OF POUNDS; PROPORTION OF GEAR TYPE AREA CATCHES^ AND RESULTING ANNUAL NET PROFIT IN OOP^S OF 1951 DOLLARS USING A DISCOUNT BATE OF THREE AND A HALF PERCENT CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 HISTORIC ESCAPE 5,809 4,274 7,033 15,482 RECRUIT 27,602 18,217 29,992 59,275 CATCH 18,359 11,717 19,380 34,930 OPTIMAL ESCAPE 7,996 8,085 9,437 10,946 RECRUIT 34,887 23,713 32,043 64,504 CATCH 23,456 13,402 19,026 44,695 CZ2G .5% 11S 9% 21% CZ2P 12.5% 4% 9% 24% USZ4G 22% 28% 27% 10% USZ4P 16% 13% 14% 28% USZ5P 12% 9% 9% 12% CZ7G 37% 35% 32% 5% NET PROFIT $1,535 $1,754 $2,507 $8,302 ACTUAL " $246 $466 $1,145 $5,748 GAIN IN " $1,289 $1,288 $1,362 $2,554 As t h i s i s the case c l o s e s t to the h i s t o r i c a l e x p e r i e n c e i t w i l l be used as the r e f e r e n c e case i n subsequent work. The f i r s t p o i n t t o note i s t h a t the op t i m a l program i s assuming e q u i l i b r i u m has been achieved i n the past. Thus "Net P r o f i t " i s the annual c y c l e year p r o f i t assuming o p t i m a l e g u i l i b r i u m escapement, recruitment and c a t c h . The " A c t u a l " net p r o f i t i s the average annual c y c l e year p r o f i t between 1951 and 1975. And the "Gain I n " net p r o f i t i s the d i f f e r e n c e between h y p o t h e t i c a l net p r o f i t and h i s t o r i c a l a c t u a l net p r o f i t : i t g i v e s an i n d i c a t i o n of the gains to be expected from o p t i m a l management of the F r a s e r R i v e r sockeye f i s h e r y . Note t h a t what i s r e f e r r e d 166 to here i s the net p r o f i t on an annual b a s i s once e q u i l i b r i u m has been achieved. The s m a l l e r net p r o f i t experienced i n the adjustment year i s ignored f o r now but w i l l be i n c l u d e d i n the net worth comparisons below., For the f i r s t three c y c l e y e a r s , o p t i m a l escapement i s much l a r g e r than the average h i s t o r i c a l escapement permitted by the IPSFC while o p t i m a l c a t c h i s not much g r e a t e r than h i s t o r i c a l c a t c h f o r the f i r s t two c y c l e years and i s a c t u a l l y s l i g h t l y l e s s f o r c y c l e year t h r e e . In f a c t , the g r e a t e s t i n c r e a s e on a percentage b a s i s i s f o r optimal recruitment. T h i s i l l u s t r a t e s the important r o l e f o r op t i m a l management i n redu c i n g the c o s t s imposed by the common property problem. IPSFC management appears t o have been p r i m a r i l y concerned with an e g u a l i n t e r n a t i o n a l d i v i s i o n o f the cat c h and the prevention of p o t e n t i a l l y d i s a s t r o u s o v e r f i s h i n g r e s u l t i n g from t h i s common pr o p e r t y problem. Apart from c y c l e year f o u r , i t i s apparent that h i s t o r i c a l l e v e l s o f escapement have been lower than optimal which would suggest t h a t the IPSFC has been under c o n s i d e r a b l e pressure from a f i s h e r y dominated by a d r i v e to avoid the common property a b s t e n t i o n c o s t s . Because of these c o s t s , the p r i v a t e d i s c o u n t r a t e s are very high and i t would appear that the IPSFC has been f o r c e d t o use unreasonably high d i s c o u n t r a t e s i n determining escapement l e v e l s (see, f o r example, Table (4-3) below). Optimal management, however, i s concerned with maximizing p r o f i t and t h i s r e g u i r e s b a l a n c i n g the increment i n p r o f i t a s s o c i a t e d with l a r g e r c a t c h e s a g a i n s t the increment i n c o s t i n h a r v e s t i n g t h a t c a t c h . Enhanced r e c r u i t m e n t i s a very important way to reduce f i s h i n g c o s t s and the program has made 167 f u l l use of t h i s f a c t . Once the ccamon property problem has been e l i m i n a t e d , the IPSFC w i l l be a b l e to make f u l l use of these i n t e r t e m p o r a l gains. C y c l e year f o u r i s d i f f e r e n t from the f i r s t three i n that optimal escapement i s s m a l l e r , o p t i m a l recruitment s l i g h t l y l a r g e r but o p t i m a l c a t c h very much l a r g e r than the h i s t o r i c a l e q u i v a l e n t s . T h i s i l l u s t r a t e s very c l e a r l y the r a t h e r unique way i n which r e c r u i t m e n t e l a s t i c i t y e f f e c t s can work i n a f i s h e r y . The e f f i c i e n c y of gear does not a l t e r from c y c l e year to c y c l e year and working with l a r g e r and l a r g e r r e c r u i t m e n t s q u i c k l y drops the average c o s t s of t a k i n g any c a t c h . But by the time very l a r g e r e c r u i t m e n t s , such as those of c y c l e f o u r , are experienced, net p r o f i t can be i n c r e a s e d more r a p i d l y by i n c r e a s i n g c a t c h . The program's e f f o r t s to enhance r e c r u i t m e n t r a t h e r than c a t c h f o r years with poor recruitment while enhancing c a t c h r a t h e r than r e c r u i t m e n t f o r the one c y c l e year where r e c r u i t m e n t i s a l r e a d y high i m p l i e s that the t r a d e o f f between the recruitment e l a s t i c i t y c o s t e f f e c t s and the enhanced revenue from g r e a t e r c a t c h has t i p p e d i n f a v o r of the l a t t e r . The reason i s the r e l a t i o n s h i p amongst escapement, recr u i t m e n t and c a t c h . That i s , recruitment cannot be enhanced much more by s a c r i f i c i n g c a t c h but catch can be enhanced c o n s i d e r a b l y by reducing escapement without a f f e c t i n g r e c r u i t m e n t too s e r i o u s l y . Comparing Tables (4-1) and (3-2) r e v e a l s important changes i n gear type c a t c h p r o p o r t i o n s . T h i s makes sense i n a program t h a t now has the c a p a c i t y to j u g g l e c a t c h and recruitment i n such a way as to take f u l l advantage of areas which are very s e n s i t i v e t o one or the other i n terms of c o s t s a v i n g . T h i s i s 168 the "stock e f f e c t " r e f e r r e d to by C l a r k and Munro (1975)., What i s most i n t e r e s t i n g i s t h a t the s h i f t s amongst gear types from year to year of the c y c l e are not n e a r l y as r a d i c a l i n t h i s optimal program. Some areas are s t i l l favored i n c e r t a i n years but now the c a t c h i s spread more evenly over most areas. However, the s h i f t toward purse s e i n e s and open-water g i l l n e t s i s s t i l l very pronounced f o r the f o u r t h c y c l e year. The f i n a l point worth n o t i n g i s t h a t net p r o f i t i s not always d i r e c t l y r e l a t e d t o s i z e o f c a t c h . Cycle years two, three and f o u r conform t o the c o n v e n t i o n a l n o t i o n t h a t as c a t c h s i z e i n c r e a s e s , so does net p r o f i t . But c y c l e year one has a l a r g e r c a t c h and recruitment than both c y c l e years two and t h r e e and yet e x p e r i e n c e s lower net p r o f i t . The c a t c h a b i l i t y parameter p l a y s a key r c l e because not only i s c y c l e year one an "on" year f o r p i n k s but the run of pinks i s over t h i r t y percent higher than f o r c y c l e year three (the other "on" y e a r ) . I t i s l i k e l y t h a t t h e pink c a t c h i s so va l u a b l e to fishermen t h a t they are r e l o c a t i n g i n the g a u n t l e t a t the expense of c a t c h a b i l i t y of sockeye. A comparison of h i s t o r i c c a t c h , recruitment and net p r o f i t i n d i c a t e s the same tren d . In f a c t , once the c a t c h of other salmon i s i n c l u d e d (Tables (4-6) and (4-7) below) the usual r e l a t i o n s h i p between c a t c h and net p r o f i t i s r e -e s t a b l i s h e d . Table (4-2) g i v e s much the same comparisons as those i n Table (4-1) with the e x c e p t i o n t h a t the IPSFC i s no longer c o n s t r a i n e d t o an egual d i v i s i o n of the c a t c h . 169 TABLE (4-2) : UNCONSTRAINED OPTIMAL CJTCH OP SOCKEYE ONLIx OPTIMAL CYCLE YEAR ESCAPEMENT. RECRUITMENT AND CATCH IN 000* S OF POUNDS; PROPORTION OF GEAR TYPE AREA CATCHES; AND RESULTING NET PROFIT IN OOO'S OF 1951 DOLLARS USING A DISCOUNT RATE OF THREE AND A HALF PJRCENT CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 OPTIMAL ESCAPE 9,584 8,211 9,382 10,922 RECRUIT 35,519 23,776 32,042 64,503 CATCH 22,501 13,339 19,080 44,719 CZ2G 54% 10% 20% 29% CZ2P 12% 5% 8% 24% USZ4G 5% 33% 13% 5% USZ4P 3% 18% 9% 13% USZ5P 8% 7% 8% 10% CZ7G 18% 27% 42% 19% NET PROFIT $2,406 $1,759 $2,538 $8,724 ACTUAL " $246 $466 $1,145 $5,748 GAIN IN " $2,160 $1,303 $1,393 $2,976 The f i r s t d i f f e r e n c e from the previous example i s the i n c r e a s e i n both escapement and re c r u i t m e n t f o r c y c l e years one and two accompanied by a r e d u c t i o n i n c a t c h . U n f e t t e r e d by the egual c a t c h d i v i s i o n c o n s t r a i n t , the program i s , once agai n , attempting to take advantage of s e n s i t i v i t y to g r e a t e r r e c r u i t m e n t s . The pronounced changes i n c a t c h p r o p o r t i o n s would i n d i c a t e how important r e c r u i t m e n t e l a s t i c i t i e s are f o r the open-water g i l l n e t s (CZ2G). Except f o r c y c l e year two, there i s a dramatic i n c r e a s e i n usage o f t h i s p a r t i c u l a r gear. I t i s obvious that once recr u i t m e n t passes a c r i t i c a l l e v e l ( l a r g e r than t h a t i n c y c l e year two) Canadian g i l l n e t s i n the f i r s t and l a s t p a r t s o f the g a u n t l e t are the fa v o r e d gear. , Note the dramatic i n c r e a s e i n net p r o f i t f o r c y c l e year one as a r e s u l t of the r e a l l o c a t i o n of h a r v e s t i n g e f f o r t . The c o s t 170 o f c o n s t r a i n i n g the IPSFC to an equal d i v i s i o n of the c a t c h i n c y c l e year one amounts to n e a r l y $900,000 i n 1951 d o l l a r s . I n c l u d i n g the c a t c h of other salmon does not reduce t h i s s h a r i n g c o s t i f one compares Tables (4-6) and (4-7) below, < F i n a l l y , comparing T a b l e s (4-2) and (3-1) l e a d s to much the same c o n c l u s i o n s as f o r the p r e v i o u s example: s h i f t s i n c a t c h p r o p o r t i o n s between c y c l e years are not n e a r l y as dramatic and the c a t c h appears t o be spread more evenly over most areas. I l l CASE TWO: ALTESIJG DI SCJHJNT SATES Table (4-3) giv e s the c y c l e year changes i n optimal escapement, recr u i t m e n t and ca t c h and net p r o f i t f o r d i f f e r e n t d i s c o u n t r a t e s assuming: only sockeye are caught and t h a t the IPSFC i s c o n s t r a i n e d to d i v i d i n g the ca t c h e g u a l l y between the U.S. and Canada. 171 TABLE (4-3): CONSTRAINED OPTIMAL CATCH OP SOCKEYE ONLY,. OPTIMAL ESCAPEMENT, RECRUITMENT AND CATCH IN 000J.S OF POONDSj,. AND RESULTING NET PROFIT IN 000 «S OF 1951 DOLLARS IX DISCOUNT ESCAPE RECRUIT CATCH PROF IT CYCLE 1 8,089 34,962 23,438 $1,553 3.5% DISCOUNT ESCAPE 7,996 RECRUIT 34,887 CATCH 23,456 PROFIT $1,535 10% DISCOUNT ESCAPE 7,739 RECRUIT 34,651 CATCH 23,478 PROFIT $1,480 CYCLE 2 8,174 23,758 13,35 8 $1,736 8,085 23,713 13,402 $1,754 7,8 37 23,571 13,507 $1,734 CYCLE 3 9,444 32,043 19,019 $2,513 9,437 32,043 19,026 $2,507 9,436 32,043 19,027 $2,511 CYCLE 4 10,947 64,504 44,694 $8,291 10,946 64,504 44,695 $8,302 10,933 64,503 44,707 $8,292 20% DISCOUNT ESCAPE 7,300 7,392 9,432 10,896 RECRUIT 34,151 23,250 32,043 64,502 CATCH 23,416 13,632 19,031 44,743 PROFIT $1,374 $1,684 $2,506 $8,231 30% DISCOUNT ESCAPE 6,811 6,885 9,429 10,869 RECRUIT 33,435 22,776 32,043 64,500 CATCH 23,190 13,665 19,035 44,768 PROFIT $1,235 $1,596 $2,507 $8,317 C l a r k (1976b, pp.47-51) has g i v e n a s i m i l a r comparison f o r the P a c i f i c H a l i b u t and the A n t a r c t i c f i n whale, the former with a high p o p u l a t i o n growth r a t e and the l a t t e r with a low growth r a t e . , Although the example f o r sockeye i s not s t r i c t l y comparable because the program si m u l t a n e o u s l y determines optimal stock v a l u e s and l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n s while C l a r k ' s g i v e s o p t i m a l stock v a l u e s given a l e a s t c o s t h a r v e s t i n g system, the r e s u l t s are r a t h e r s i m i l a r . That i s , l i k e the P a c i f i c H a l i b u t , F r a s e r R i v e r sockeye salmon have a high 172 i n t r i n s i c growth r a t e ( a l b e i t over f o u r years) and the optimal c a t c h l e v e l s are r a t h e r i n s e n s i t i v e to the d i s c o u n t r a t e . B i o l o g i c a l o v e r f i s h i n g does not become op t i m a l u n l e s s the d i s c o u n t r a t e exceeds 10% f o r c y c l e year one, and 30% f o r the other t h r e e c y c l e years. T h i s i l l u s t r a t e s the b a s i c c h o i c e c o n f r o n t i n g the f i s h e r y manager: should he c a t c h more now to enable him to i n v e s t a t the s o c i a l r a t e o f r e t u r n o r should he a b s t a i n i n order to earn at a r a t e of r e t u r n egual to the " f i s h e r y r a t e of r e t u r n " on the stoc k ? High i n t r i n s i c growth r a t e s imply that f u t u r e net p r o f i t s are enhanced not only by i n c r e a s e d harvest but a l s o by the r e d u c t i o n i n h a r v e s t i n g c o s t s r e s u l t i n g from l a r g e r subsequent r e c r u i t m e n t s . The r e s u l t s i n Table (1-3) f o l l o w the t r a d i t i o n a l p a t t e r n o f s a c r i f i c i n g escapement to enhance c u r r e n t c a t c h as di s c o u n t r a t e s r i s e . However, the impact on re c r u i t m e n t i s very s l i g h t f o r the t h i r d and f o u r t h c y c l e y e a r s . For the f i r s t two c y c l e years r e c r u i t m e n t f a l l s s t e a d i l y as d i s c o u n t r a t e s (and catches) r i s e which i n d i c a t e s that i n t r i n s i c growth r a t e s are not as high. F o r the t h i r d and f o u r t h c y c l e years the " f i s h e r y r a t e s of r e t u r n " are so high that i t takes a higher o p p o r t u n i t y c o s t r a t e of r e t u r n t o induce the manager t o s a c r i f i c e f u t u r e catches and rec r u i t m e n t s by c a t c h i n g more today. I t i s important to note t h a t these are the e g u i l i b r i u m values f o r c a t c h and i t i s i m p l i c i t t h a t i n the adjustment year catches would have been higher because of reduced escapements. 173 IV CASE TABEE: VARYING HARVEST COSTS Table (4-4) g i v e s the comparison between a c t u a l h i s t o r i c a l average c y c l e year p r o f i t s and the annual net p r o f i t obtained from the op t i m a l p o l i c y assuming: sockeye o n l y are ha r v e s t e d , an op p o r t u n i t y c o s t s o c i a l r a t e o f di s c o u n t of three and h a l f percent, and the IPSFC c o n s t r a i n e d t o an equal d i v i s i o n of the ca t c h between O.S. and Canadian fishermen. I t i s a l s o assumed that u n i t e f f o r t c o s t s are r a i s e d by 15%. The d i s c o u n t r a t e used i n t r a n s f o r m i n g boat c a p i t a l i n t o a flow i s not r a i s e d as i t i s assumed to be exogenously determined, TABLE (4-4) : CONSTRAINED OPTIHAL CATCH OF SOCKEYE ONM_ OPTIMAL CYCLE YEAR ESCAPEMENT, RECRUITMENT AND CATCH IN OOO^S OF POUNDS; PROPORTION OF GEAR TYPE AREA CATCHES AND RESULTING NET PROFIT IN OOO'S OF 1951 DOLLARS OSIJG A DISCOUNT RATE OF THREE AND A HALF reBCEfiT AND ASSUMING OPPORTUNITY COSTS ARE 15% HIGHER FOR ALL GEAR TY_?ES CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 OPTIMAL ESCAPE 8,943 8,639 9,773 11,000 RECROIT 35,422 23,945 32,021 64,504 CATCH 23,044 13,079 18,669 44,642 CZ2G .5% 11% 9% 21% CZ2P 12.5% 4% 9% 18% USZ4G 22% 30% 27% 16% USZ4P 16% , 1 1 % 13% 23* USZ5P 12% 9% 10% 11% CZ7G 37% 35% 32% 11% NET PROFIT $764 $1,341 $1,865 $7,036 ACTUAL « $-661 $-90 $268 $4,823 GAIN IN « $1,425 $1,431 $1,597 $2,213 The f i r s t p o i n t to note i s t h a t the a c t u a l h i s t o r i c a l net p r o f i t has a l s o been adjusted to r e f l e c t a f i f t e e n percent i n c r e a s e i n c o s t s f o r a l l gear types used to land the average 174 annual h i s t o r i c a l c y c l e year c a t c h . Comparing the gains i n net p r o f i t with those of the r e f e r e n c e case i n Table (4-1) i t i s immediately obvious t h a t the e x t r a degree of freedom g i v e n by c o n t r o l over escapement, re c r u i t m e n t and c a t c h i n the h y p o t h e t i c a l example permits the program to reduce the c a t c h i n a l l f o u r c y c l e years t o compensate f o r i n c r e a s e d h a r v e s t i n g c o s t s . T h i s i s why, d e s p i t e lower net p r o f i t s , the gains i n net p r o f i t a c t u a l l y r i s e f o r the f i r s t three c y c l e years. , I t i s i n t e r e s t i n g to note how i n s e n s i t i v e the model i s to changes i n c o s t . Once a g a i n , the reason would appear t o be the high i n t r i n s i c growth r a t e s : an i n c r e a s e i n c o s t s i s somewhat s i m i l a r to a decrease i n the d i s c o u n t r a t e . Thus cat c h e s are reduced and escapements i n c r e a s e d , as t r a d i t i o n a l theory would suggest, but not by a g r e a t d e a l . F i n a l l y , as one would expect with r e l a t i v e c o s t s unchanged, there i s very l i t t l e change i n the catch p r o p o r t i o n s when compared with those of the r e f e r e n c e case i n Table (4-1) Table (4-5) g i v e s the comparison between a c t u a l h i s t o r i c a l average c y c l e year p r o f i t s and the net p r o f i t s obtained from the optimal p o l i c y assuming: o p p o r t u n i t y c o s t s (except the d i s c o u n t rate) a r e lower by 15% f o r U.S. gear o n l y , sockeye only are h a r v e s t e d , an o p p o r t u n i t y c o s t s o c i a l d i s c o unt r a t e of t h r e e and a h a l f percent and the IPSFC unconstrained as to s h a r i n g the c a t c h . 175 TABLE J4-5j_i UNCONSTRAINED OPTIMAL CATCH OF SOCKEYE ONLY. OPTIMAL CYCLE YEAH ESCAPEMENT. RECRUITMENT AND CATCH IN OOP'S OF POUNDS; PROPORTION OF GEAR TYPE ABEA CATCHES: AND RESULTING NET PBOFIT IN OPP'S OF 1951 DOLL ABS USING A DISCOUNT RATE OF THjgEE AND A HALF PERCENT AND ASSUMING OPPORTUNITY COSTS ARE LOWERED BY 15% FOR U.S. GEAR ONLY CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 OPTIMAL ESCAPE 9,570 7,923 9,357 10,919 RECRUIT 35,519 23,623 32,041 64,503 CATCH 22,515 13,474 19,105 44,721 CZ2G 54% 10% 20% 30% CZ2P 11% 5% 8% 23% USZ4G 5% 33% 14% 6% 0SZ4P 4% 18% 9% 12% USZ5P 9% 7% 6% 14% CZ7G 17% 27% 43% 15% NET PROFIT $2,540 $1,984 $2,683 $9,071 ACTUAL " $700 $744 $1,584 $6,211 GAIN IN " $1,840 $1,240 $1,099 $2,860 A c t u a l h i s t o r i c a l c o s t s have been adjusted t o r e f l e c t a f i f t e e n percent r e d u c t i o n i n c o s t s f o r U.S.,gear only. Although net p r o f i t r i s e s i n every c y c l e year, as one would expect, the gai n s i n net p r o f i t f a l l f o r a l l f o u r c y c l e years when compared with the unconstrained case i n Table (4-2). One p o s s i b l e reason i s t h a t d e c r e a s i n g c o s t s l e a d to an i n c r e a s e i n c a t c h f o r a l l f o u r c y c l e y e a r s with v i r t u a l l y no i n c r e a s e i n re c r u i t m e n t . , E a r l i e r d i s c u s s i o n has r e v e a l e d how s e n s i t i v e the program i s to r e c r u i t m e n t e l a s t i c i t i e s and that open water Canadian gear are favored i n t h i s r e gard. What i s most i n t e r e s t i n g i s to compare the c a t c h p r o p o r t i o n s with those i n Table (4-2). U.S. c a t c h r i s e s from s i x t e e n t o eighteen percent i n c y c l e one, s t a y s the same i n c y c l e two, f a l l s from t h i r t y t o twenty-nine 176 percent i n c y c l e three and r i s e s from twenty-eight to t h i r t y - t w o percent of the catc h i n c y c l e f o u r . As i n the pr e v i o u s example where the impact of r a i s i n g c o s t s was only s l i g h t , the high i n t r i n s i c p o p u l a t i o n growth r a t e s appear t o reduce s e n s i t i v i t y to decreases i n cost as w e l l . Reducing c o s t s i s roughly comparable t o an i n c r e a s e i n the di s c o u n t r a t e and appears to have very l i t t l e i n f l u e n c e i n the face of a high " f i s h e r y r a t e of r e t u r n " . , The perverse r e s u l t i n c y c l e three c o u l d be e x p l a i n e d by the s e n s i t i v i t y o f the program to the c r i t i c a l i n t e r e l a t i o n s h i p between escapement, recr u i t m e n t and c a t c h . That i s , i f the program were only concerned with the s p a t i a l c o n f i g u r a t i o n i t would probably r e a l l o c a t e more of the ca t c h to O.S. zones, but the o p t i m a l i n t e r t e m p o r a l p a r t of the program c a l l s f o r a change i n escapement, recruitment and ca t c h . T h i s l a s t e f f e c t , to which the program i s so s e n s i t i v e i n a l l o c a t i n g c a t c h , presumably outweighs the c o s t e f f e c t and r e s u l t s i n a s l i g h t l y l a r g e r a l l o c a t i o n t o Canadian gear. T h i s t r a d e o f f may, i n f a c t , account f o r the absence of any change i n c y c l e two and the r a t h e r s m a l l changes i n c y c l e years one and f o u r . I CASE JOORj. HARVESTING OTHER SALMON The use of the program i l l u s t r a t e d by Flowchart (3-^2) i n place o f t h a t i n Flowchart (3-1) i s a l l t h a t i s needed to change the o p t i m a l program i l l u s t r a t e d by Flowchart (4-1) i n t o a program t o harvest both sockeye and other salmon by sim u l t a n e o u s l y determining o p t i m a l escapement and l e a s t c o s t gear type h a r v e s t i n g p r o p o r t i o n s t o l a n d the subseguent sockeye 177 catch and the average c y c l e year c a t c h of ot h e r salmon. Once again, the program i l l u s t r a t e d i n Flowchart (4-1) simply c l o s e s the loop between escapement, recr u i t m e n t and c a t c h through the recru i t m e n t f u n c t i o n s u b j e c t t o the optimal c o n t r o l c o n s t r a i n t . Table {4-6) g i v e s the comparison between a c t u a l h i s t o r i c a l average c y c l e year p r o f i t and the net p r o f i t obtained from the optimal p o l i c y assuming: both sockeye and the average c y c l e year harvest of other salmon are caught, an i n f l a t i o n f r e e o p p o r t u n i t y c o s t s o c i a l r a t e of d i s c o u n t of three and a h a l f percent and no c o n s t r a i n t on IPSFC a l l o t m e n t o f c a t c h p r o p o r t i o n s . TABLE (4-6) : UNCONSTRAINED OPTIMAL CATCH OF SOCKEYE AND A V E J M I HISTORICAL CYCLE CATCH OF OTHER SALMON IN OOOJ^ S OF POUNDS^ GEAR TYPE AREA PROPORTIONS: AND RESULTING NET PROFIT IN 00(VS OF 1951 DOLLARS USING A DISCOUNT RATE OF THREE AND A HALF PERCENT CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 SOCKEYE 22,491 13,471 19,111 44,786 ALL OTHER 42,005 8,540 29,532 10,943 CZ2G 54% 34% 52% 50% CZ2P 10% 3% 10% 27% USZ4G 6% 24% 2% 1% DSZ4P 6% 11% 4% 9% 0SZ5P 9% 6% 3% 10% CZ7G 15% 22% 29% 3% NET PROFIT $9,625 $3,768 $7,771 $12,404 ACTUAL « $6,688 $2, 302 $5,733 $7,844 GAIN IN " $2,937 $1,466 $2,038 $4,560 Note that a c t u a l h i s t o r i c a l net p r o f i t has been a d j u s t e d to i n c l u d e the i n c r e a s e d revenue from c a t c h i n g other salmon. The a l l o w a b l e c a t c h o f other salmon i n the h y p o t h e t i c a l example i s the same s i z e as that f o r the a c t u a l h i s t o r i c a l case. Comparing the gains i n net p r o f i t with those i n the unconstrained example 178 i n T a b l e (4-2) i n d i c a t e s t h a t t h e a b i l i t y t o j u g g l e c a t c h o f o t h e r salmon as s e l l as sockeye escapement, r e c r u i t m e n t and c a t c h g i v e s t h e program enough freedom t o r a i s e net p r o f i t even more, p a r t i c u l a r l y i n the f o u r t h c y c l e y e a r . Comparing o p t i m a l c a t c h w i t h t h a t i n T a b l e (4-2) shows c a t c h i n c r e a s e d i n the l a s t t h r e e c y c l e y e a r s b u t r e d u c e d i n the f i r s t . What i s more r e m a r k a b l e , however, i s t h a t c a t c h p r o p o r t i o n s i n c y c l e y e a r one a r e h a r d l y a l t e r e d w h i l e t h o s e i n the r e m a i n i n g c y c l e y e a r s have been d r a m a t i c a l l y a l t e r e d . In e v e r y c y c l e y e a r , e x c e p t two, the o p e n - w a t e r g i l l n e t s (CZ2G) l a n d f i f t y p e r c e n t or more o f the s o c k e y e c a t c h . A l t h o u g h the c a p a c i t y c o n s t r a i n t a p p r o a c h t o h a r v e s t i n g o t h e r salmon does not i n c l u d e a " r e c r u i t m e n t o f o t h e r salmon" v a r i a b l e , i t i s o b v i o u s t h a t t h e c a t c h of " A l l O t h e r " salmon t o g e t h e r w i t h the s o c k e y e r e c r u i t m e n t t o which t h i s g e a r i s so s e n s i t i v e makes t h e use o f t h i s gear type so p r o f i t a b l e . However, c o m p a r i n g t h e s e r e s u l t s w i t h t h o s e f o r t h e s t a t i c c a s e i n T a b l e (3-3) r e v e a l s very l i t t l e change i n c a t c h p r o p o r t i o n s . I t i s o b v i o u s t h a t t h e i m p a c t o f the c a t c h o f o t h e r salmon has an overwhelming i n f l u e n c e i n terms o f p r o f i t on the sockeye c a t c h p r o p o r t i o n s as t h e l a t t e r d e t e r m i n e the c a p a c i t y f o r c a t c h i n g o t h e r s a l m o n . T h a t i s , d e s p i t e the l a r g e changes i n sockeye c a t c h e s , the c o s t d i f f e r e n c e s between g e a r type a r e a s i n t e r m s o f c a t c h i n g s c c k e y e a r e n o t enough t o overcome t h e need f o r gear type c a p a c i t y t o l a n d the l a r g e c a t c h o f o t h e r salmon as c h e a p l y as p o s s i b l e . T a b l e (4-7) g i v e s the same c o m p a r i s o n as t h a t i n T a b l e ( 4 -6 ) w i t h t h e a d d i t i o n a l c o n s t r a i n t t h a t the IPSFC d i v i d e the 179 c a t c h e q u a l l y between the U.S. and Canada.. TABLE J^zlLL CONSTRAINED OPTIMAL CATCH OF SOCKEYE A_ND AVERAGE HISTORICAL CYCLE CATCH OF OTHER SALMON IN 00OJ.S OF PODNDSj, GEAR TYPE AREA PROPORTIONS; AND RESULTING NET PROFIT IN 000J_S OF -1951 DOLLARS USING A DISCOUNT RATE OF THREE AND A HALF PERCENT CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 SOCKEYE 23,155 13,036 18,948 44,730 ALL OTHER 42,005 8,540 29,532 10,943 CZ2G 23% 42% 18% 32% CZ2P 14% 2% 12% 16% USZ4G 13% 18% 22% 16% DSZ4P 21% 17% 18% 26% USZ5P 16% 14% 10% 8% CZ7G 13% 7% 20% 2% NET PROFIT $8,720 $3,843 $6,922 $10,964 ACTUAL " $6,688 $2,302 $5,733 $7,844 GAIN IN " $2,032 $1,541 $1,189 $3,120 Comparing these r e s u l t s with those of the r e f e r e n c e case i n Table (4-1) shows a f a l l i n optimal c a t c h of sockeye i n the f i r s t three c y c l e years and an i n c r e a s e i n c a t c h f o r the f o u r t h c y c l e year. Once a g a i n , the a l l o w a b l e c a t c h of " A l l Other" salmon i s the same as the annual average h i s t o r i c a l c a t c h . Note how the gains i n net p r o f i t i n c r e a s e f o r every c y c l e year except the t h i r d . As i n the p r e v i o u s example, t h i s can be e x p l a i n e d i n terms of the e x t r a degree of freedom i n the program's j u g g l i n g of other salmon c a t c h as well as o p t i m a l sockeye escapement, re c r u i t m e n t and c a t c h . , Once again , note the dramatic change i n c a t c h p r o p o r t i o n s . Despite the egual c a t c h d i v i s i o n c o n s t r a i n t , the program f a v o r s the use of open-water g i l l n e t s to take the Canadian h a r v e s t , p a r t i c u l a r l y i n c y c l e two. T h i s heavy use of open-water g i l l n e t s i n c y c l e year two i s a departure from the usual c a t c h 1 8 0 p r o p o r t i o n s i n other c o n s t r a i n e d examples and can be e x p l a i n e d by the e q u a l l y dramatic r e d u c t i o n i n c a t c h taken by the O.S. zone f o u r g i l l n e t s . The l a t t e r * s s e n s i t i v i t y t o r e d u c t i o n s i n recruitment r e s u l t i n g from l a r g e catches i n zone two, appears t o be overcome by the p r o f i t a b i l i t y of l a n d i n g both sockeye and other salmon i n the previous zone. When these r e s u l t s are compared with the s t a t i c ones i n Table (3-4), i t i s immediately obvious t h a t t h i s p a t t e r n of f a v o r i n g the open-water g i l l n e t s i n c y c l e year two i s important. Note, however, t h a t the c a t c h f o r O.S. zone f o u r g i l l n e t s i s s t i l l very high i n the s t a t i c example. Apart from c y c l e year two, the c a t c h i s new spread more evenly over most areas as i s the case i n s e v e r a l previous examples when the optimal case i s compared with the s t a t i c one. The gains i n net p r o f i t , however, are not s t r i c t l y comparable with those i n Table (3-4) because the s t a t i c program could not c r e a t e enough c a p a c i t y i n c y c l e y e ars one, t h r e e and f o u r to take the a l l o w a b l e catch of other salmon. Once again, even with the equal c a t c h d i v i s i o n c o n s t r a i n t , the program has enough freedom t o j u g g l e escapement, r e c r u i t m e n t and c a t c h to take the f u l l a l l o w a b l e c a t c h o f other salmon i n the o p t i m a l case. VI MAXIMIZING PRESENT WORTH Assuming the r e f e r e n c e case of h a r v e s t i n g sockeye only (Table (4 - 1 ) ) i s the most f e a s i b l e v e r s i o n of the optimal management p o l i c y , i t i s d e s i r a b l e to compare the net worth derived from t h i s case with the a c t u a l h i s t o r i c a l net worth. The adjustment p e r i o d i n v o l v e d i n r e a c h i n g the e g u i l i b r i u m v a l u e s 181 should a l s o be i n c l u d e d t o present a f a i r comparison. I t w i l l be assumed that the f a s t e s t approach to e q u i l i b r i u m i s optimal and, i n the present case, i t w i l l be assumed t h a t o p t i m a l escapement l e v e l s w i l l be permitted f o r the f o u r c y c l e years from 1947 to 1950. The net p r o f i t s from these adjustment years p l u s those from the e q u i l i b r i u m years, 1951 to 1975, w i l l then be compounded forward t o 1975. Except f o r c y c l e year f o u r , the r e s u l t s i n Ta b l e (4-1) show t h a t h i s t o r i c a l management has r e s u l t e d i n o v e r f i s h i n g . T h i s w i l l e n t a i l s a c r i f i c e s i n the adjustment y e a r s i n terms of foregone catches i n 1947, 194 8 and 1949. TABLE (4-8) : Cp.HPO0.HDED JET PR0FI1S OJ LOSSES IN 00__i_ OF 1975 DOLLARS PORING THE PERIOD J947 TQ ,1975, ASSUHIHG ALL HAN AGE MENT ADJUSTMENTS CARRIED OUT IN THE FIRST FOUR YEARS OF THE CYCLE AND CO HP PONDING JflTH A SOCIAL RATE OF RETURN OF __HREE AND A HALF PERCENT CYCLE 1 CYCLE 2 CYCLE 3 CYCLE 4 IPSFC 6,143 11,236 26,659 129,229 OPTIHAL 33,803 36,873 52,328 183,669 DIFFERENCE 27,660 25,637 25,669 54,440 TOTAL FOR ALL FOUR CYCLE YEARS = $133,406 The f i r s t l i n e e n t i t l e d "IPSFC" g i v e s the net p r o f i t s f o r each of the c y c l e years compounded from the p e r i o d when each occurred u n t i l 1975 and summed f o r each c y c l e year. The next l i n e g i v e s the r e s u l t s when the same procedure i s f o l l o w e d f o r the management model»s net p r o f i t s . The t h i r d l i n e g i v e s the d i f f e r e n c e , on a c y c l e year b a s i s , between the p r e v i o u s two t o t a l s . The f i n a l l i n e g i v e s the t o t a l of a l l these d i f f e r e n c e s f o r the twenty-nine year peri o d from 1947 to 1975. Note that 182 these f i g u r e s are a l l i n 197 5 d o l l a r s . The f i r s t c o n c l u s i o n t o emerge i s that present worth under h i s t o r i c a l management (IPSFC) i m p l i e s that fishermen may have been e a r n i n g s u b s t a n t i a l economic ren t . T h i s i s a remarkable f e a t on the pa r t of the IPSFC gi v e n the severe c o n s t r a i n t s of the convention and the c o n s i d e r a b l e pressure a r i s i n g from the common property problem. Secondly, i f we assume t h a t h i s t o r i c a l present worth i s a c t u a l l y o p p o r t u n i t y c o s t s unaccounted f o r , then the gain i n present worth r e s u l t i n g from o p t i m a l management (Difference) i n d i c a t e s the p o t e n t i a l economic r e n t which could have been earned i n the p e r i o d 1947 to 1975 i f the IPSFC had been given the freedom t o manage o p t i m a l l y . And f i n a l l y , the sum of t h i s h y p o t h e t i c a l r e n t i n c r e a s e over h i s t o r i c a l management (Total) i n d i c a t e s that t h e r e would have been more than enough revenue f o r the IPSFC t o compensate redundant v e s s e l owners f o r c e d to r e t i r e from uneconomic gear type h a r v e s t i n g areas. VII SOMAS* AND CONCLOSIONS T h i s t h e s i s has presented an e m p i r i c a l study of the optimal management of the F r a s e r R i v e r sockeye salmon f i s h e r y . Chapter One i n c l u d e d a survey of the r e l e v a n t economics l i t e r a t u r e to provide the t h e o r e t i c a l base f o r the o p t i m i z i n g model and a d i s c u s s i o n o f the b a s i c assumptions and the approach used to g u a n t i f y t h a t model. Chapter Two discus s e d the f a c t o r s i n v o l v e d i n f i s h e r i e s stock management from a b i o l o g i c a l p o i n t of view. A recru i t m e n t f u n c t i o n was d e r i v e d based on the R i c k e r model but r e - s p e c i f i e d to allow f o r the i n f l u e n c e o f the f o u r year c y c l e on parameter 183 values., T h i s r e c r u i t m e n t model e s t a b l i s h e d the e s s e n t i a l l i n k between escapement and subsequent rec r u i t m e n t . Chapter Three presented a r e - s p e c i f i e d v e r s i o n of the t r a d i t i o n a l p r o d u c t i o n f u n c t i o n . An e f f o r t e f f i c i e n c y index was i n c l u d e d to remove exogenous changes i n gear e f f i c i e n c y . And a biomass c a t c h a b i l i t y index was i n t r o d u c e d t o permit r e c r u i t m e n t a v a i l a b i l i t y t o be g u a l i f i e d as t o the degree of c a t c h a b i l i t y . No e x t e r n a l data was a v a i l a b l e to permit c o n s t r u c t i o n of a biomass c a t c h a b i l i t y index and the e s t i m a t i o n process i t s e l f was permitted to d e r i v e values f o r the c a t c h a b i l i t y parameters a l l o w i n g f o r the i n f l u e n c e of the f o u r year c y c l e . Chapter Three a l s o presented a c o s t programming model which a l l o c a t e d gear type e f f o r t a c c o r d i n g to l e a s t c o s t h a r v e s t i n g c o n f i g u r a t i o n s i n the g a u n t l e t f o r each c y c l e year. The net p r o f i t s from t h i s model were then compared with those earned h i s t o r i c a l l y on an average annual b a s i s between 1951 and 1975. The comparisons i n d i c a t e d t h a t even i f the egual c a t c h d i v i s i o n c o n s t r a i n t were i n e f f e c t , the g a i n i n net p r o f i t from optimal s p a t i a l a l l o c a t i o n would be w e l l worthwhile. A f i n a l c o s t model using a c a p a c i t y c o n s t r a i n t approach to l a n d i n g the h i s t o r i c a l c a t c h e s of other salmon as w e l l as sockeye was a l s o i n c l u d e d , although the c a p a c i t y approach makes the r e s u l t s u n r e l i a b l e , the same pa t t e r n appears to emerge i n terms of h a r v e s t i n g : c e r t a i n gear are b e t t e r or worse than ot h e r s i n d i f f e r e n t c y c l e years and optimal s p a t i a l management can make a worthwhile c o n t r i b u t i o n to net p r o f i t s . Combining the r e c r u i t m e n t model i n Chapter Two with the h a r v e s t i n g c o s t s model i n Chapter Three pro v i d e s the management 184 model used i n t h i s chapter. I t was assumed t h a t the IPSFC would be transformed i n t o a monopoly manager charged with maximizing the present worth of the F r a s e r R i v e r sockeye f i s h e r y . S e v e r a l case s t u d i e s were presented l e a d i n g to the f o l l o w i n g g e n e r a l c o n c l u s i o n s . F i r s t , o p t i m a l management of t h i s f i s h e r y could l e a d to a s i g n i f i c a n t i n c r e a s e i n net p r o f i t . , The i n c r e a s e appears to be l a r g e enough to adeguately compensate redundant v e s s e l owners i n the f i r s t few years. Subseguent net p r o f i t could e i t h e r be d i v i d e d e g u a l l y between the O.S. and Canadian governments or used to enhance the spawning systems i n the F r a s e r R i v e r . The second c o n c l u s i o n to emerge i s that f o r c y c l e years where r e c r u i t m e n t i s low, the program attempts to i n c r e a s e recruitment much more than c a t c h , whereas f o r the f o u r t h c y c l e year where recruitment i s a l r e a d y l a r g e , the program i n c r e a s e s c a t c h much more than r e c r u i t m e n t . T h i s i n d i c a t e s t h a t the " s t o c k - e f f e c t " i s o l a t e d by C l a r k and Munro (1975) plays a very important p a r t i n i n c r e a s i n g p r o f i t s . The c y c l e year c a t c h a b i l i t y c o e f f i c i e n t s e s t a b l i s h the s e n s i t i v i t y of the gear type a r e a s to the s i z e of r e c r u i t m e n t . And the t r a d e o f f between those u n i t c o s t e f f e c t s r e s u l t i n g from recruitment s e n s i t i v i t y and the revenue e f f e c t s from l a r g e r catches a l l o w s the program, i n c e r t a i n year c y c l e s , t o i n c r e a s e net p r o f i t s more by reducing u n i t e f f o r t c o s t s than by i n c r e a s i n g revenue through enhanced c a t c h e s . These r e s u l t s h i g h l i g h t two p o i n t s . The f i r s t i s that economic theory and the e m p i r i c a l work i n t h i s study i n d i c a t e t h a t p r o f i t s can b e n e f i t j u s t as much, i f not more i n some 185 cases, from r e d u c t i o n s i n c o s t as from i n c r e a s e s i n revenue as a r e s u l t of g r e a t e r c a t c h e s . The second p o i n t i s t h a t the s p e c i f i c a t i o n of the p r o d u c t i o n f u n c t i o n i s c r i t i c a l i n a l l o w i n g t h i s e f f e c t t o emerge. That i s , as much care must be taken i n s p e c i f y i n g c a t c h a b i l i t y of the a v a i l a b l e biomass as has been taken with s p e c i f y i n g c a t c h i n g power ( e f f i c i e n c y ) of gear. Perhaps, however, other economic s p e c i f i c a t i o n s would l e a d to even more i n t e r e s t i n g r e s u l t s . A t h i r d c o n c l u s i o n about the r e s u l t s i s t h a t the equal c a t c h d i v i s i o n c o n s t r a i n t can impose a n o t i c e a b l e c o s t i n c e r t a i n c y c l e years., T h i s i s t r u e even when other salmon are p e r m i t t e d t o be caught i n a d d i t i o n to sockeye. T h i s i s a deadweight l o s s as n e i t h e r p a r t y gains from the i n e f f i c i e n t a l l o c a t i o n of e f f o r t r e s u l t i n g from an egual d i v i s i o n of c a t c h . A f o u r t h c o n c l u s i o n i s t h a t high i n t r i n s i c p o p u l a t i o n growth r a t e s make the model r a t h e r i n s e n s i t i v e to i n t e r e s t r a t e s . The same i n s e n s i t i v i t y t o changes i n c o s t s would imply t h a t p r i c e and c o s t e f f e c t s are not as important i n t h i s model as the u n d e r l y i n g b i o l o g i c a l l i n k between escapement, recru i t m e n t and c a t c h and the r e c r u i t m e n t e l a s t i c i t i e s i n the h a r v e s t i n g p r o d u c t i o n f u n c t i o n s . The l a s t point emerges even more c l e a r l y when other salmon, as well as sockeye, are harvested. The r a d i c a l s h i f t s i n gear type area c a t c h p r o p o r t i o n s i n d i c a t e the c r i t i c a l nature of the r e l a t i o n s h i p between c a t c h and r e c r u i t m e n t a v a i l a b i l i t y and c a t c h a b i l i t y i n terms of the s e n s i t i v i t y to recruitment. A f i f t h c o n c l u s i o n which emerges from the study i s that even i f i t i s not f e a s i b l e t o a l t e r escapement and c a t c h l e v e l s 186 to achieve the o p t i m a l v a l u e s , the r e s u l t s i n Chapter Three i n d i c a t e t h a t t h e r e are worthwhile gains to a l l o c a t i n g gear i n a a s p a t i a l l y l e a s t c o s t manner. However, the present worth comparisons between the h i s t o r i c a l experience and the h y p o t h e t i c a l o p t i m a l management over the p e r i o d 1947-1975 i n d i c a t e the c o s t s of not pursuing an o p t i m a l management p o l i c y i n terms of the economic r e n t which c o u l d be earned. Comparing the l e a s t c o s t programming r e s u l t s i n Chapter Three with the r e s u l t s from h y p o t h e t i c a l o p t i m a l management i n t h i s c h a p t e r l e a d s to the f i n a l c o n c l u s i o n : the gains c o n t r i b u t e d by the o p t i m a l v a l u e s f o r escapement, r e c r u i t m e n t and c a t c h outweigh the gains from l e a s t c ost s p a t i a l a l l o c a t i o n of h a r v e s t i n g gear. T h i s i m p l i e s t h a t fishermen are p r o f i t c o n s c i o u s and, except f o r the s p a t i a l m i s a l l o c a t i o n a r i s i n g from the open access nature of the g a u n t l e t , are very c l o s e to a c h i e v i n g the r i g h t t r a d e - o f f between c a t c h , recruitment a v a i l a b i l i t y and biomass c a t c h a b i l i t y . Of g r e a t e r importance i s the c r i t i c a l r o l e played by the management of the IPSFC, not o n l y i n d i v i d i n g the i n t e r n a t i o n a l catch and preventing c a t a s t r o p h i c o v e r f i s h i n g but i n a s s i g n i n g gear type area catches i n a manner very c l o s e to the o p t i m a l s p a t i a l c o n f i g u r a t i o n . T h i s was done d e s p i t e the severe l e g a l c o n s t r a i n t s o f the convention and the c o n s i d e r a b l e pressure a r i s i n g from the common property problem., &s the comparison between h i s t o r i c a l and optimal c a t c h l e v e l s i n Table (4-1) appears t o i n d i c a t e , o p t i m a l management by the IPSFC c o u l d l e a d to i n c r e a s e s i n both s u s t a i n a b l e c a t c h and r e c r u i t m e n t with subsequent gains i n net economic p r o f i t . 187 There are many i m p l i c a t i o n s which a r i s e from t h i s study of the F r a s e r E i v e r sockeye f i s h e r y . Perhaps the most important i s that the e n l a r g i n g of the economic as w e l l as b i o l o g i c a l base has been shown f c a i d i n t r a n s n a t i o n a l b a r g a i n i n g over the f i s h e r y . T h i s b a r g a i n i n g i s l i k e l y to occur q u i t e soon because of the need to re-assess a l l t r a n s n a t i o n a l f i s h e r i e s i n the l i g h t o f the new two hundred mile economic zone. Furthermore, the r e c e n t enhancement program w i l l soon s t a r t to take e f f e c t with, h o p e f u l l y , s u b s t a n t i a l l y i n c r e a s e d catches of sockeye. As Canada, r a t h e r than the U.S., has made t h i s investment, i t i s obvious t h a t some very hard b a r g a i n i n g on c a t c h s h a r i n g w i l l soon fake p l a c e . With regard to the enhancement program i t s e l f , i t i s i n t e r e s t i n g t o note that i t s avowed purpose i s t o double y i e l d s but at c o n s i d e r a b l e expense i n terms o f enhancing spawning and m i g r a t i o n r o u t e s i n the F r a s e r system; whereas the management model i n t h i s study i n d i c a t e s how net p r o f i t y i e l d s can a p p a r e n t l y be more than doubled a t v i r t u a l l y no e x t r a expense. He are at a c r i t i c a l j uncture i n terms of t h i s f i s h e r y because there i s c u r r e n t o v e r - c a p a c i t y i n terms of e f f o r t and yet the promise of s u b s t a n t i a l i n c r e a s e s i n f u t u r e r e c r u i t m e n t s and c a t c h e s . I f e f f o r t i s simply allowed to expand as i n the past, the p o t e n t i a l i n c r e a s e i n economic re n t w i l l be reduced and d i s s i p a t e d . I n s t e a d , i t i s the hope of t h i s study t h a t the management model presented here can be used t o take advantage of t h i s p o t e n t i a l i n c r e a s e i n economic r e n t by o p t i m a l l y a l l o c a t i n g e f f o r t on both a s p a t i a l and an i n t e r t e m p o r a l b a s i s . The i n c r e a s e s i n economic r e n t which would r e s u l t c o u l d s t a r t paying 1 8 8 back some of the heavy investment expenses of the enhancement program. I t would appear t h a t a l l the p a r t i e s i n v o l v e d i n the Fras e r B i v e r Sockeye f i s h e r y c o u l d gain by r e - n e g o t i a t i n g the terms of the con v e n t i o n , expanding the powers of the IPSFC and a l l o w i n g the IPSFC t o manage the f i s h e r y i n an optimal manner. 189 BIBLIOGRAPHY A g n e l l o , R.J. and L.P. Donnelly (1975), " P r i c e s and Property Ri g h t s i n the F i s h e r i e s , " Southern Economic J o u r n a l , pp. 253-262. 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